\r\n\tTopics covered include but are not limited to: Hydrologic Cycle (Precipitation, Runoff, Infiltration and their Measurement, Land surface interaction); Hydrologic Analysis (Hydrograph, Wave routing, Hydrologic statistics, Frequency Analysis); Applied Hydrology (Applications in Engineering, Sciences and Agriculture, Design storms, Risk analysis, Case studies); Computational Hydrology (Numerical modeling, Hydrologic modeling and forecasting, Flow visualization, Model validation, Parameter estimation); Interdisciplinary Hydrology (Hydrometeorology, Impact of Climate Change, Precipitation data analysis, Mathematical concepts, Natural hazards); Radar Hydrology (Precipitation estimation techniques, Promise and Challenges in Radar technology, Uncertainty in radar precipitation estimates).
\r\n
\r\n\tThe contents covered in this book will serve as a valuable reference guide to students, researchers, government agencies and practicing engineers who work in hydrology and related areas. We hope that this book will open new directions in basic and applied research in hydrological science.
",isbn:"978-1-83962-330-1",printIsbn:"978-1-83962-329-5",pdfIsbn:"978-1-83962-331-8",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"02925c63436d12e839008c793a253310",bookSignature:"Dr. Theodore Hromadka and Dr. Prasada Rao",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9864.jpg",keywords:"Runoff, Land Surface Interaction, Hydrograph, Wave Routing, Design Storms, Risk Analysis, Numerical Modeling, Hydrologic Modeling and Forecasting, Flow Visualization, Hydrometeorology, Precipitation Data Analysis, Radar Hydrology",numberOfDownloads:293,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"June 8th 2020",dateEndSecondStepPublish:"September 11th 2020",dateEndThirdStepPublish:"November 10th 2020",dateEndFourthStepPublish:"January 29th 2021",dateEndFifthStepPublish:"March 30th 2021",remainingDaysToSecondStep:"4 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Principal and founder of Hromadka & Associates, professor United States Military Academy, Professor Emeritus at the California State University, and a Member of Board of Directors and an Adjunct Professor at Wessex Institute of Technology.",coeditorOneBiosketch:"Prasada Rao, Ph.D. is a professor in the Civil and Environmental Engineering Department at California State University, Fullerton. His current research areas relate to Climate Change, Surface and Subsurface flow modeling, and Computational Mathematics. He is also the Associate Director for International Institute for Computational Engineering Mathematics.\r\nCo1 - Biosketch",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"181008",title:"Dr.",name:"Theodore",middleName:null,surname:"Hromadka",slug:"theodore-hromadka",fullName:"Theodore Hromadka",profilePictureURL:"https://mts.intechopen.com/storage/users/181008/images/system/181008.jpg",biography:"Hromadka & Associates’ Principal and Founder, Theodore Hromadka II, PhD, PhD, PhD, PH, PE, has extensive scientific, engineering, expert witness, and litigation support experience. 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1. Introduction
A considerable number of modern high-fidelity Computational Fluid Dynamics (CFD) solvers and codes still adopt either one-dimensional physical models based on the Riemann problem using higher order shape functions, such as higher order Finite Volume (FV) and Discontinuous Galerkin Finite Element (FE) methods for the discrete data representation, or truly multi-dimensional physical models using linear shape functions, like Fluctuation Splitting (FS) schemes. Both of these approaches require fast convergence acceleration techniques in order to compete with conventional solvers based on artificial dissipation or upwind schemes in terms of CPU time. Implicit methods based on the Newton’s rootfinding algorithm are receiving an increasing attention in this context for the solution of complex real-world CFD applications, for example in the analyses of turbulent flows past three-dimensional wings, due to their potential to converge in a very small number of iterations [1, 2]. In this chapter, we consider convergence acceleration strategies for the implicit solution of the Reynolds-averaged Navier-Stokes (RANS) equations based on the FS space discretization using a preconditioned Newton-Krylov algorithm for the integration. The use of a Newton solver requires the inversion of a large nonsymmetric system of equations at each step of the non-linear solution process. Choice of linear solver and preconditioner is crucial for efficiency especially when the mean flow and the turbulence transport equation are solved in fully coupled form. In this study, we use the restarted Generalized Minimal Residuals (GMRES) [3] algorithm for the inner linear solver, preconditioned by a block multilevel incomplete lower-upper (LU) factorization. We present the development lines of the multilevel preconditioning strategy that is efficient to reduce the number of iterations of Krylov subspace methods at moderate memory cost, and shows good parallel performance on three-dimensional turbulent flow simulations.
The chapter is structured as follows. The governing conservation equations for both compressible and incompressible flows are reviewed in Section 2. Section 3 briefly describes the fluctuation splitting space discretization, the time discretization and the Newton-Krylov method used to solve the space- and time-discretized set of governing partial differential equations (PDEs). In Section 4, we present the development of the multilevel preconditioning strategies for the inner linear solver. We illustrate the numerical and parallel performance of the preconditioner for the analysis of turbulent incompressible flows past a three-dimensional wing in Section 5. Some concluding remarks arising from the study are presented in Section 6.
2. Governing equations
In the case of inviscid and laminar flows, given a control volume Ci, fixed in space and bounded by the control surface ∂Ci with inward normal n, the governing equations of fluid dynamics are obtained by considering the conservation of mass, momentum and energy. In the case of viscous turbulent flows, one approach to consider the effects of turbulence is to average the unsteady Navier-Stokes (NS) equations on the turbulence time scale. Such averaging procedure results in a new set of steady equations (the RANS equations) that differ from the steady NS equations for the presence of the Reynolds’ stress tensor, representing the effects of turbulence on the averaged flow field. The appearance of this tensor yields a closure problem, which is often solved by adopting an algebraic or a differential turbulence model. In the present work, we use the Spalart-Allmaras [4] one-equation model for the turbulent viscosity. Thus the integral form of the conservation law of mass, momentum, energy and turbulence transport equations has the form
∫Ci∂Ui∂tdV=∮∂Cin⋅FdS−∮∂Cin⋅GdS+∫CiSdVE1
where U is the vector of conserved variables. For compressible flows, we have U=ρρe0ρuν˜T, and for incompressible, constant density flows, U=puν˜T. The operators F and G represent the inviscid and viscous fluxes, respectively; for compressible flows, we have
F=ρuρuh0ρuu+pIν˜u,G=1Re∞0u⋅τ¯¯+qτ¯¯1PrTν+ν˜∇ν˜,E2
and for incompressible, constant density flows,
F=a2uuu+pIν˜u,G=1Re∞0τ¯¯1PrTν+ν˜∇ν˜.E3
Finally, the source term S has a non-zero entry only in the row corresponding to the turbulence transport equation; its expression is not reported here for brevity, but can be found in [4]. Note that the standard NS equations are retrieved from (1) by removing the source term S and the differential equation associated with the turbulence variable, and setting the effective viscosity and thermal conductivity to their laminar values. The Euler equations are instead recovered by additionally removing the flux vector G.
3. Solution techniques
The model used in this study for the discrete data representation is based on the coupling of an hybrid class of methods for the space discretization, called Fluctuation Splitting (or residual distribution) schemes [5], and a fully coupled Newton algorithm. By “fully coupled” we mean that the mass, momentum and energy conservation equations on one hand, and the turbulent equation on the other, are solved simultaneously rather than in a decoupled or staggered fashion. We discuss in the following subsections, separately, the space and time discretization, the numerical integration of the set of equations resulting from the discretization, and the solution of the large linear system at each Newton’s step.
3.1. Space discretisation
The Fluctuation Splitting approach has features common to both Finite Element (FE) and Finite Volume (FV) methods. Like in standard FE methods, the dependent variables are stored at the vertices of the computational mesh made up of triangles in the two-dimensional (2D) space, and tetrahedra in three-dimensional (2D), and are assumed to vary linearly and continuously in space. Denoting Zi as the nodal value of the dependent variable at the grid point i and Ni as the FE linear shape function, this dependence can be written as
Zxt=∑iZitNix.E4
Note that, although the summation in Eq. (4) extends over all grid nodes, the computational molecule of each node is actually limited only to the set of its nearest neighbors due to the compact support of the linear shape functions. In the compressible case, Roe’s parameter vector
Z=ρρh0ρuν˜TE5
is chosen as the dependent variable to ensure discrete conservation [6]. In the incompressible case, discrete conservation is obtained by simply setting the dependent variable Z equal to the vector of conserved variables U. In our code, we group the dependent variables per gridpoint. The first m entries of the array Z are filled with the m flow variables of gridpoint 1, and these are followed by those of gridpoint 2, and so on. Blocking the flow variables in this way, also referred to as “field interlacing” in the literature, is acknowledged [7, 8, 9] to result in better performances than grouping variables per aerodynamic quantity.
The integral Eq. (1) is discretized over each control volume Ci using a FV-type approach. In two dimensions, the control volumes Ci are drawn around each gridpoint by joining the centroids of gravity of the surrounding cells with the midpoints of all the edges that connect that gridpoint with its nearest neighbors. An example of polygonal-shaped control volumes (so-called median dual cells) is shown by green lines in Figure 1(a). With FS schemes, rather than calculating the inviscid fluxes by numerical quadrature along the boundary ∂Ci of the median dual cell, as would be done with conventional FV schemes, the net inviscid flux Φe,inv over each triangular/tetrahedral element
Φe,inv=∮∂Ten⋅FdSE6
is evaluated by means of a conservative linearization based on the parameter vector [6], and scattered to the element vertices using elemental distribution matrices Bie [5]. The inviscid contribution to the nodal residual RΦi is then assembled by collecting fractions Φie,inv of the net inviscid fluxes Φe,inv associated with all the elements by which the node i is surrounded. This is schematically shown in Figure 1(b). Concerning the viscous terms, the corresponding flux balance is evaluated by surface integration along the boundaries of the median dual cell: node i receives a contribution Φie,vis from cell e which accounts for the viscous flux through the portion of ∂Ci that belongs to that cell. This approach can be shown to be equivalent to a Galerkin FE discretization.
Figure 1.
Residual distribution concept. (a) The flux balance of cell T is scattered among its vertices. (b) Gridpoint i gathers the fractions of cell residuals from the surrounding cells.
Summing up the inviscid and viscous contributions to the nodal residual of gridpoint i one obtains
RΦi=∑e∍iΦie,inv+Φie,vis=∑e∍iBieΦe,inv+Φie,vis.E7
In Eq. (7), the summation ranges over all the elements e that meet in meshpoint i, as shown in Figure 1(b). The construction of the distribution matrices Bie involves the solution of d+1 small (of order m) dense linear systems for each triangular/tetrahedral element of the mesh thus making FS schemes somewhat more expensive than state-of-the-art FV schemes based upon either central differencing with artificial dissipation or upwind discretizations. The relatively high computational cost of FS discretizations has to be accounted for when deciding whether the Jacobian matrix should be stored in memory or a Jacobian-free method be used instead.
3.2. Time discretisation
One route to achieve second-order time accuracy with FS schemes is to use mass matrices that couple the time derivatives of neighboring grid points. This leads to implicit schemes, even if the spatial residual were treated explicitly. Although a more general framework for the derivation of the mass matrices can be devised [10], the approach adopted in our study consists of formulating the FS scheme as a Petrov-Galerkin FE method with elemental weighting function given by
Ωie=Ni−1d+1Im×m−Bie.E8
The contribution of element e to the weighted residual equation for grid point i reads
∫TeΩie∂U∂tdV=∫TeΩie∂U∂Z∂Z∂tdV,E9
where the chain rule is used to make the dependent variable Z appear. Since the conservative variables U are quadratic functions of the parameter vector Z, the transformation matrix ∂U/∂Z is linear in Z and can thus be expanded using the linear shape functions Nj, just as in Eq. (4). A similar expansion applies to the time-derivative ∂Z/∂t. Replacing both expansions in the RHS of Eq. (9), the discrete counterpart of the time derivative of Eq. (9) is given by the contribution of all elements sharing the node i:
In Eq. (10) the index j spans the vertices of the element e and the nodal values ∂Z/∂tj are approximated by the three-level Finite Difference (FD) formula
∂Z∂tj=3Zjn+1−4Zjn+Zjn−12Δt.E11
The matrix Mije in Eq. (10) is the contribution of element e to the entry in the ith row and jth column of the global mass matrix M. Similarly to what is done in the assembly of the inviscid and viscous flux balance, Eq. (7), the discretization of the unsteady term, Eq. (10), is obtained by collecting elemental contributions from all the elements that surround the node i. Second-order space and time accuracy of the scheme described above has been demonstrated for inviscid flow problems by Campobasso et al. [11] using an exact solution of the Euler equations.
3.3. Numerical integration
Writing down the space- and time-discretized form of Eq. (1) for all gridpoints of the mesh, one obtains the following large, sparse system of non-linear algebraic equations
RgUZ=RΦUZ−M1Δt32Zn+1−2Zn+12Zn−1=0E12
to be solved at time level n+1 to obtain the unknown solution vector Un+1. The solution of Eq. (12) is obtained by means of an implicit approach based on the use of a fictitious time-derivative (Jameson’s dual time-stepping [12]) that amounts to solve the following evolutionary problem
dUdτVM=RgUE13
in pseudo-time τ until steady state is reached. Since accuracy in pseudo-time is obviously irrelevant, the mass matrix has been lumped into the diagonal matrix VM and a first-order accurate, two-time levels FD formula
dUdτ≈Un+1,k+1−Un+1,kΔτE14
is used to approximate the pseudo-time derivative in the LHS of Eq. (13). The outer iterations counter k has been introduced in Eq. (14) to label the pseudo-time levels.
Upon replacing Eq. (14) in Eq. (13), an implicit scheme is obtained if the residual Rg is evaluated at the unknown pseudo-time level k+1. Taylor expanding Rg about time level k, one obtains the following sparse system of linear equations
1ΔτkVM−JΔU=RgUn+1,kJ=∂Rg∂UE15
to be solved at each outer iteration until the required convergence of Rg is obtained. Steady RANS simulations are accommodated within the presented integration scheme by dropping the physical time-derivative term in Eq. (12). In the limit Δτk→∞, Eq. (15) recovers Newton’s rootfinding algorithm, which is known to yield quadratic convergence when the initial guess Un+1,0=Un is sufficiently close to the sought solution Un+1. This is likely to occur when dealing with unsteady flow problems because the solution of the flow field at a given physical time starts from the converged solution at the preceding time, and this latter constitutes a very convenient initial state. In fact, it is sufficiently close to the sought new solution to allow the use of the exact Newton’s method (i.e. Δτk=∞ in Eq. (15)) since the first solution step.
The situation is different when dealing with steady flow problems. Newton’s method is only locally convergent, meaning that it is guaranteed to converge to a solution when the initial approximation is already close enough to the sought solution. This is generally not the case when dealing with steady flows, and a “globalization strategy” needs to be used in order to avoid stall or divergence of the outer iterations. The choice commonly adopted by various authors [13, 14], and in this study as well, is a pseudo-transient continuation, which amounts to retain the pseudo-transient term in Eq. (15). At the early stages of the iterative process, the pseudo-time step length Δτk in Eq. (15) is kept small. The advantage is twofold: on one hand, it helps preventing stall or divergence of the outer iterations; on the other hand, it makes the linear system (15) easier to solve by means of an iterative solver since for moderate values of Δτk the term Vm/Δτk increases the diagonal dominance of the matrix. Once the solution has come close to the steady state, which can be monitored by looking at the norm of the nodal residual RΦ, we let Δτk grow unboundedly so that Newton’s method is eventually recovered during the last steps of the iterative process. The time step length Δτk is selected according to the Switched Evolution Relaxation (SER) strategy proposed by Mulder and van Leer [15], as follows:
Δτk=ΔτminCmaxC0RgUn+1,02RgUn+1,k2,E16
where Δτ is the pseudo-time step based upon the stability criterion of the explicit time integration scheme, and C0 and Cmax are user-defined constants controlling the initial and maximum pseudo-time steps used in the actual calculations.
In the early stages of the iterative process, the turbulent transport equation and the mean flow equations are solved in tandem (or in a loosely coupled manner, following the nomenclature used by Zingg et al. [16]): the mean flow solution is advanced over a single pseudo-time step using an analytically computed, but approximate Jacobian while keeping turbulent viscosity frozen, then the turbulent variable is advanced over one or more pseudo-time steps using a FD Jacobian with frozen mean flow variables. Due to the uncoupling between the mean flow and turbulent transport equations, this procedure will eventually converge to steady state, but never yields quadratic convergence. Close to steady state, when a true Newton strategy preceded by a “short” pseudo-transient continuation phase can be adopted, the mean flow and the turbulence transport equation are solved in fully coupled form, and the Jacobian is computed by FD. For the sake of completeness, we give further details of each of these two steps in the following two paragraphs.
3.3.1. Tandem solution strategy with (approximate) Picard linearization
Consider re-writing the steady nodal residual RΦ, see [17] for full details, as
RΦU=C−DU,E17
where C and D are (sparse) matrices that account for the convective and diffusive contributions to the nodal residual vector RΦ. Matrix D is constant for isothermal, incompressible flows whereas it depends upon the flow variables through molecular viscosity in the case of compressible flows. Matrix C depends upon U for both compressible and incompressible flows. Both matrices can be computed analytically as described in [17]. What we refer to as a Picard linearization consists in the following approximation
J≈C−D,E18
which amounts to neglect the dependence of matrices C and D upon U when differentiating the residual, written as in Eq. (17).
Once the mean flow solution has been advanced over a single pseudo-time step using the approximate Picard linearization, keeping the turbulent viscosity frozen, the turbulent variable is advanced over one or more (typically ten) pseudo-time steps using a FD Jacobian approximation (described in Section 3.3.2) with frozen mean flow variables. Blanco and Zingg [18] adopt a similar strategy, but keep iterating the turbulence transport equation until its residual has become lower than that of the mean flow equations. The loosely coupled solution strategy is a choice often made as it “allows for the easy interchange of new turbulence models” [19] and also reduces the storage [18], compared to a fully coupled approach. However, due to the uncoupling between the mean flow and the turbulent transport equations, the tandem solution strategy never yields quadratic convergence nor it is always able to drive the nodal residual to machine zero. The last statement cannot be generalized, since Blanco and Zingg [16, 20] report convergence to machine zero for their loosely coupled approach on two-dimensional unstructured grids. However, even if convergence to machine zero is difficult to achieve, the nodal residual is always sufficiently converged for any practical “engineering” purpose and close enough to “true” steady-state solution to be a good initial guess for Newton’s method.
3.3.2. Fully coupled solution strategy with FD Newton linearization
Once the tandem solution strategy has provided a good approximation to the steady flow, or when dealing with unsteady flows, in which case the solution at a given time level is generally a good approximation to the one sought at the next time level, it becomes very attractive to take advantage of the quadratic convergence of Newton’s method. In order to do so, however, the mean flow and the turbulence transport equations must be solved fully coupled and the Jacobian matrix J must be accurate. We take advantage of the compactness of the computational stencil required by FS schemes to compute a close approximation to the true Jacobian matrix even for second-order accurate discretizations. The analytical evaluation of the Jacobian matrix, though not impossible [5, 21], is rather cumbersome and thus this approach is not pursued here.
When the equations are fully coupled, the structure of the Jacobian matrix J is naturally organized into small dense blocks of order m. This has implications both in terms of storage, since it is possible to reduce the length of the integer pointers that define the Compressed Sparse Row (CSR) data structure of the sparse matrix, and also in the design of the preconditioner for solving the large linear system at each Newton step, where division operations can be efficiently replaced by block matrix factorizations. We will address these issues in detail in the next section. Two neighboring gridpoints, i and j, in the mesh will contribute two block entries, Jij and Jji, to the global Jacobian matrix J. Each of these two block entries (say Jij, for instance) will be computed by assembling elemental contributions coming from all the cells that share vertex i, as follows
Jij=∑e∍iJije.E19
Eq. (19) follows by applying the sum rule of differentiation and by observing that the nodal residual itself is a sum of contributions from the elements that share vertex i, see Figure 1(b). Specifically, element Jije accounts for the contribution of cell e to the residual change at gridpoint i, due to a change in the state vector of a neighboring gridpoint j that belongs to the same element e. The contribution of cell e to the element pq of the block Jij is computed from the following one-sided FD formula
Ji,jep,q=RgeipUiÛjq…−RgeUiUj…ε1≤p,q≤m;i,j∈e,E20
where Rgeip is the pth component of the contribution of cell e to the nodal residual of gridpoint i. In Eq. (20) we have emphasized that Rgei only depends upon the flow state of the d+1 vertices of cell e, which include both i and j. The first partial derivative, RgeipUiÛjq… is computed by perturbing the qth component of the conserved variables vector at gridpoint j as follows
Ûjq=uj1uj2…ujq+εujq…,ujm,E21
where ε is a “small” quantity. Due to the use of a one-sided FD formula, the FD approximation (20) of the Jacobian entry is affected by a truncation error which is proportional to the first power of ε. Small values of ε keep the truncation error small, but too small values may lead to round-off errors. Following [21], ε is computed as
εx=εmcmaxx1sgnx.E22
From a coding viewpoint, the same loop over all cells used to build the nodal residual Rg is also used to assemble matrix J. The operations to be performed within each cell are the following: i) perturb each of the m components of the conserved variables vector of the d+1 vertices of cell e; ii) evaluate the residual contribution to each of the vertices; iii) calculate the Jacobian entries according to Eq. (20). While looping over cell e, this contributes d+12 block entries to the global matrix J. Moreover, it follows that the cost of a Jacobian evaluation is equal to m×d+1 residual evaluations, which can be quite a large number. For instance, for a 3D compressible RANS calculation using a one-equation turbulence model, m×d+1=24. In this study, it was decided to store the Jacobian matrix in memory rather than using a Jacobian-free (JFNK), as the Jacobian matrix is relatively sparse even for a second-order accurate discretization due to the compactness of the stencil. The JFNK approach avoids assembling and, more important, storing the Jacobian matrix. However, the matrix-vector product is replaced with the Jacobian matrix by FD formulae which requires extra costly FS residual evaluations. Note that the JFNK method still requires the construction of a preconditioner to be used by the iterative linear solver, often an Incomplete Lower Upper factorization, which is typically constructed using a lower order approximation of the residual vector. Matrix-free preconditioners might also be used [22], saving a huge storage at the expense of extra CPU cost. These latter are referred to as MFNK methods. Although JFNK or MFNK approaches should certainly be favored from the viewpoint of memory occupation, it cannot always be “assumed that the Jacobian-free matrix-vector products are inherently advantageous in terms of computing time” [13].
The compactness of the FS stencil, which never extends beyond the set of distance-1 neighbors even for a second-order-accurate space-time discretization, offers two advantages. On one hand, apart from the truncation and round-off errors involved in the FD derivatives, the numerical Jacobian matrix is a close approximation of the analytical Jacobian, even for a second-order-accurate discretization. This feature is crucial for retaining the quadratic convergence properties of Newton’s algorithm. On the other hand, it is considerably sparser than that obtained using more traditional FV discretizations, which typically extend up to distance-2 [1] or even distance-3 neighbors [8]. In these latter cases, contributions from the outermost gridpoints in the stencil have to be neglected [1], or at least lumped [8], when constructing the Jacobian approximation upon which the ILU(ℓ) preconditioner is built. These approximations are a potential source of performance degradation as reported in [8]. The memory occupation required to store the Jacobian matrix still remains remarkable. Moreover, not only the Jacobian matrix, but also its preconditioner needs to be stored in the computer memory. It is therefore clear that a key ingredient that would help reducing memory occupation is an effective preconditioner having a relatively small number of non-zero entries, as close as possible to that of the Jacobian matrix. This demanding problem is addressed in the next section.
4. Linear solve and preconditioning
The previous discussions have pointed that the solution of the large nonsymmetric sparse linear system (15) at each pseudo-time step is a major computational task of the whole flow simulation, especially when the mean flow and the turbulence transport equations are solved in fully coupled form, the Jacobian is computed exactly by means of FD, and the size of the time-step is rapidly increased to recover Newton’s algorithm. For convenience, we write system (15) in compact form as
Ax=b,E23
where A=aij is the large and sparse coefficient matrix of, say, size n, and b is the right-hand side vector. It is well established that, when A is highly nonsymmetric and/or indefinite, iterative methods need the assistance of preconditioning to transform system (23) into an equivalent one that is more amenable to an iterative solver. The transformed preconditioned system writes in the form M−1Ax=M−1b when preconditioning is applied from the left, or AM−1y=b with x=M−1y when preconditioning is applied from the right. The matrix M is a nonsingular approximation to A called the preconditioner matrix. In the coming sections, we describe the development of an effective algebraic preconditioner for the RANS model.
4.1. Multi-elimination ILU factorization preconditioner
Incomplete LU factorization methods (ILUs) are an effective, yet simple, class of preconditioning techniques for solving large linear systems. They write in the form M=L¯U¯, where L¯ and U¯ are approximations of the L and U factors of the standard triangular LU decomposition of A. The incomplete factorization may be computed directly from the Gaussian Elimination (GE) algorithm, by discarding some entries in the L and U factors according to various strategies, see [3]. A stable ILU factorization is proved to exist for arbitrary choices of the sparsity pattern of L¯ and U¯ only for particular classes of matrices, such as M-matrices [23] and H-matrices with positive diagonal entries [24]. However, many techniques can help improve the quality of the preconditioner on more general problems, such as reordering, scaling, diagonal shifting, pivoting and condition estimators [25, 26, 27, 28]. As a result of this recent development, in the past decade successful experience have been reported using ILU preconditioners in areas that were of exclusive domain of direct solution methods like, in circuits simulation, power system networks, chemical engineering plants modeling, graphs and other problems not governed by PDEs, or in areas where direct methods have been traditionally preferred, such as structural analysis, semiconductor device modeling, computational fluid dynamics (see [29, 30, 31, 32, 33]).
Multi-elimination ILU factorization is a powerful class of ILU preconditioners, which combines the simplicity of ILU techniques with the robustness and high degree of parallelism of domain decomposition methods [34]. It is developed on the idea that, due to sparsity, many unknowns of a linear system are not coupled by an equation (i.e. they are independent) and thus they can be eliminated simultaneously at a given stage of GE. If the, say m, independent unknowns are numbered first, and the other n−m unknowns last, the coefficient matrix of the system is permuted in a 2×2 block structure of the form
PAPT=DFEC,E24
where D is a diagonal matrix of dimension m and C is a square matrix of dimension n−m. In multi-elimination methods, a reduced system is recursively constructed from (24) by computing a block LU factorization of PAPT of the form
DFEC=L0GIn−m×UW0A1,E25
where L and U are the triangular factors of the LU factorization of D, A1=C−ED−1F is the Schur complement with respect to C, In−m is the identity matrix of dimension n−m, and we denote G=EU−1 and W=L−1F. The reduction process can be applied another time to the reduced system with A1, and recursively to each consecutively reduced system until the Schur complement is small enough to be solved with a standard method such as a dense LAPACK solver [35]. Multi-elimination ILU factorization preconditioners may be obtained from the decomposition (25) by performing the reduction process inexactly, by dropping small entries in the Schur complement matrix and/or factorizing D approximately at each reduction step. These preconditioners exhibit better parallelism than conventional ILU algorithms, due to the recursive factorization. Additionally, for comparable memory usage, they may be significantly more robust especially for solving large problems as the reduced system is typically small and better conditioned compared to the full system.
The factorization (25) defines a general framework which may accommodate for many different methods. An important distinction between various methods is rooted in the choice of the algorithm used to discover sets of independent unknowns. Many of these algorithms are borrowed from graph theory, where such sets are referred to as independent sets. Denoting as G=VE the adjacency graph of A, where V=v1v2…vn is the set of vertices and E the set of edges, a vertex independent set S is defined as a subset of V such that
∀vi∈S,∀vj∈S:vivj∉E.E26
The set S is maximal if there is no other independent set containing S strictly [36]. Independent sets in a graph may be computed by simple greedy algorithms which traverse the vertices in the natural order 1,2,…,n, mark each visited vertex v and all of its nearest neighbors connected to v by an edge, and add v and each visited node that is not already marked to the independent set [37]. As an alternative to the greedy algorithm, the nested dissection ordering [38], mesh partitioning, or further information from the set of nested finite element grids of the underlying problem can be used [39, 40, 41].
The multilevel preconditioner considered in our study is the Algebraic Recursive Multilevel Solvers (ARMS) introduced in [25], which uses block independent sets computed by the simple greedy algorithm. Block independent sets are characterized by the property that unknowns of two different sets have no coupling, while unknowns within the same set may be coupled. In this case, the matrix D appearing in (24) is block diagonal, and may typically consist of large-sized diagonal blocks that are factorized by an ILU factorization with threshold (ILUT [42]) for memory efficiency. In the ARMS implementation described in [25], first the incomplete triangular factors L¯, U¯ of D are computed by one sweep of ILUT, and an approximation W¯ to L¯−1F is also computed. In a second loop, an approximation G¯ to EU¯−1 and an approximate Schur complement matrix A¯1 are derived. This holds at each reduction level. At the last level, another sweep of ILUT is applied to the (last) reduced system. The blocks W¯ and G¯ are stored temporarily, and then discarded from the data structure after the Schur complement matrix is computed. Only the incomplete factors of D at each level, those of the last level Schur matrix, and the permutation arrays are needed for the solving phase. By this implementation, dropping can be performed separately in the matrices L¯, U¯, W¯, G¯, A¯1. This in turns allows to factor D accurately without incurring additional costs in G¯ and W¯, achieving high computational and memory efficiency. Implementation details and careful selection of the parameters are always critical aspects to consider in the design of sparse matrix algorithms. Next, we show how to combine the ARMS method with matrix compression techniques to exploit the block structure of A for better efficiency.
4.2. The variable-block ARMS factorization
The discretization of the Navier-Stokes equations for turbulent compressible flows assigns five distinct variables to each grid point (density, scaled energy, two components of the scaled velocity, and turbulence transport variable); these reduce to four for incompressible, constant density flows, and to three if additionally the flow is laminar. If the, say ℓ, distinct variables associated with the same node are numbered consecutively, the permuted matrix has a sparse block structure with non-zero blocks of size ℓ×ℓ. The blocks are usually fully dense, as variables at the same node are mutually coupled. Exploiting any available block structure in the preconditioner design may bring several benefits [43], some of them are explained below:
Memory. A clear advantage is to store the matrix as a collection of blocks using the variable-block compressed sparse row (VBCSR) format, saving column indices and pointers for the block entries.
Stability. On indefinite problems, computing with blocks instead of single elements enables a better control of pivot breakdowns, near singularities, and other possible sources of numerical instabilities. Block ILU solvers may be used instead of pointwise ILU methods.
Complexity. Grouping variables in clusters, the Schur complement is smaller and hopefully the last reduced system is better conditioned and easier to solve.
Efficiency. A full block implementation, based on higher level optimized BLAS as computational kernels, may be designed leading to better flops to memory ratios on modern cache-based computer architectures.
Cache effects. Better cache reuse is possible for block algorithms.
It has been demonstrated that block iterative methods often exhibit faster convergence rate than their pointwise analogues for the solution of many classes of two- and three-dimensional partial differential equations (PDEs) [44, 45, 46]. For this reason, in the case of the simple Poisson’s equation with Dirichlet boundary conditions on a rectangle 0ℓ1×0ℓ2 discretized uniformly by using n1+2 points in the interval 0ℓ1 and n2+2 points in 0ℓ2, it is often convenient to number the interior points by lines from the bottom up in the natural ordering, so that one obtains a n2×n2 block tridiagonal matrix with square blocks of size n1×n1; the diagonal blocks are tridiagonal matrices and the off-diagonal blocks are diagonal matrices. For large finite element discretizations, it is common to use substructuring, where each substructure of the physical mesh corresponds to one sparse block of the system. If the domain is highly irregular or the matrix does not correspond to a differential equation, finding the best block partitioning is much less obvious. In this case, graph reordering techniques are worth considering.
The PArameterized BLock Ordering (PABLO) method proposed by O’Neil and Szyld is one of the first block reordering algorithms for sparse matrices [47]. The algorithm selects groups of nodes in the adjacency graph of the coefficient matrix such that the corresponding diagonal blocks are either full or very dense. It has been shown that classical block stationary iterative methods such as block Gauss-Seidel and SOR methods combined with the PABLO ordering require fewer operations than their point analogues for the finite element discretization of a Dirichlet problem on a graded L-shaped region, as well as on the 9-point discretization of the Laplacian operator on a square grid. The complexity of the PABLO algorithm is proportional to the number of nodes and edges in both time and space.
Another useful approach to compute dense blocks in the sparsity pattern of a matrix A is the method proposed by Ashcraft in [48]. The algorithm searches for sets of rows or columns having the exact same pattern. From a graph viewpoint, it looks for vertices of the adjacency graph VE of A having the same adjacency list. These are also called indistinguishable nodes or cliques. The algorithm assigns a checksum quantity to each vertex, using the function
chku=∑uw∈Ew,E27
and then sorts the vertices by their checksums. This operation takes ∣E∣+∣V∣log∣V∣ time. If u and v are indistinguishable, then chku=chkv. Therefore, the algorithm examines nodes having the same checksum to see if they are indistinguishable. The ideal checksum function would assign a different value for each different row pattern that occurs but it is not practical because it may quickly lead to huge numbers that may not even be machine-representable. Since the time cost required by Ashcraft’s method is generally negligible relative to the time it takes to solve the system, simple checksum functions such as (27) are used in practice [48].
On the other hand, sparse unstructured matrices may sometimes exhibit approximate dense blocks consisting mostly of non-zero entries, except for a few zeros inside the blocks. By treating these few zeros as non-zero elements, with a little sacrifice of memory, a block ordering may be generated for an iterative solver. Approximate dense blocks in a matrix may be computed by numbering consecutively rows and columns having a similar non-zero structure. However, this would require a new checksum function that preserves the proximity of patterns, in the sense that close patterns would result in close checksum values. Unfortunately, this property does not hold true for Ashcraft’s algorithm in its original form. In [49], Saad proposed to compare angles of rows (or columns) to compute approximate dense structures in a matrix A. Let C be the pattern matrix of A, which by definition has the same pattern as A and the non-zero values are equal to 1. The method proposed by Saad computes the upper triangular part of CCT. Entry ij is the inner product (the cosine value) between row i and row j of C for j>i. A parameter τ is used to gauge the proximity of row patterns. If the cosine of the angle between rows i and j is smaller than τ, row j is added to the group of row i. For τ=1 the method will compute perfectly dense blocks, while for τ<1 it may compute larger blocks where some zero entries are padded in the pattern. To speed up the search, it may be convenient to run a first pass with the checksum algorithm to detect rows having an identical pattern, and group them together; then, in a second pass, each non-assigned row is scanned again to determine whether it can be added to an existing group. Two important performance measures to gauge the quality of the block ordering computed are the average block density (av_bd) value, defined as the amount of non-zeros in the matrix divided by the amount of elements in the non-zero blocks, and the average block size (av_bs) value, which is the ratio between the sum of dimensions of the square diagonal blocks divided by the number of diagonal blocks. The cost of Saad’s method is closer to that of checksum-based methods for cases in which a good blocking already exists, and in most cases it remains inferior to the cost of the least expensive block LU factorization, i.e. block ILU(0).
Our recently developed variable-block variant of the ARMS method (VBARMS) incorporates an angle-based compression technique during the factorization to detect fine-grained dense structures in the linear system automatically, without any users knowledge of the underlying problem, and exploits them to improve the overall robustness and throughput of the basic multilevel algorithm [50]. It is simpler to describe VBARMS from a graph point of view. Suppose to permute A in block form as
where the diagonal blocks A˜ii, i=1,…,p are ni×ni and the off-diagonal blocks A˜ij are ni×nj. We use upper case letters to denote matrix sub-blocks and lower case letters for individual matrix entries. We may represent the adjacency graph of A˜ by the quotient graph of A+AT [36]. Calling B the partition into blocks given by (28), we denote as G/B=VBEB the quotient graph obtained by coalescing the vertices assigned to the block A˜ii (for i=1,…,p) into a supervertex Yi. In other words, the entry in position ij of A˜ is a block of dimension ∣Yi∣×∣Yj∣, where ∣X∣ is the cardinality of the set X. With this notation, the quotient graph G/B=VBEB is defined as
VB=Y1…Yp,EB=YiYj∃v∈Yiw∈Yjs.t.vw∈E.E29
An edge connects two supervertices Yi and Yj if there exists an edge from a vertex in Aii to a vertex in Ajj in the graph VE of A+AT.
The complete pre-processing and factorization process of VBARMS consists of the following steps.
Step 1. Find the block ordering PB of A such that, upon permutation, the matrix PBAPBT has fairly dense non-zero blocks. We use the angle-based graph compression algorithm proposed by Saad and described earlier to compute exact or approximate block structures in A.
Step 2. Scale the matrix at Step 1 in the form S1PBAPBTS2 using two diagonal matrices S1 and S2, so that the 1-norm of the largest entry in each row and column is smaller or equal than 1.
Step 3. Find the block independent sets ordering PI of the quotient graph G/B=VBEB. Apply the permutation to the matrix obtained at Step 2 as
PIS1PBAPBTS2PIT=DFEC.E30
We use a simple form of weighted greedy algorithm for computing the ordering PI. The algorithm is the same as the one used in ARMS, and described in [25]. It consists of traversing the vertices G/B in the natural order 1,2,…,n, marking each visited vertex v and all of its nearest neighbors connected to v by an edge and adding v and each visited node that is not already marked to the independent set. We assign the weight ∥Y∥F to each supervertex Y.
In the 2×2 partitioning (30), the upper left-most matrix D is block diagonal like in ARMS. However, due to the block permutation, the diagonal blocks of D are additionally block sparse matrices, as opposed to simply sparse matrices in ARMS and in other forms of multilevel incomplete LU factorizations, see [51, 52]. The matrices F, E, C are also block sparse because of the same reason.
the unknown solution vector and the right-hand side vector of system (35), the solution process with the above multilevel VBARMS factorization consists of level-by-level forward elimination followed by an exact solution on the last reduced system and suitable inverse permutation. The solving phase is sketched in Algorithm 1.
In VBARMS, we perform the factorization approximately, for memory efficiency. We use block ILU factorization with threshold to invert inexactly both the upper leftmost matrix Dℓ≈L¯ℓU¯ℓ at each level ℓ, and the last level Schur complement matrix Aℓmax≈L¯SU¯S. The block ILU method used in VBARMS is a straightforward block variant of the one-level pointwise ILUT algorithm. We drop small blocks B∈RmB×nB in L¯ℓ, U¯ℓ, L¯S, U¯S whenever ∥B∥FmB⋅nB<t, for a given user-defined threshold t. The block pivots in block ILU are inverted exactly by using GE with partial pivoting. In assembling the Schur complement matrix Aℓ+1 at level ℓ, we take advantage of the finest block structure of Dℓ, Fℓ, Eℓ, Cℓ, imposed by the block ordering PB on the small (usually dense) blocks in the diagonal blocks of Dℓ and the corresponding small off-diagonal blocks in Eℓ and Fℓ; we call optimized level-3 BLAS routines [53] for computing Aℓ+1 in Eq. (36). We do not drop entries in the Schur complement, except at the last level. The same threshold is applied in all these operations.
The VBARMS code is developed in the C language and is adapted from the existing ARMS code available in the ITSOL package [54]. The compressed sparse storage format of ARMS is modified to store block vectors and block matrices of variable size as a collection of contiguous non-zero dense blocks (we refer to this data storage format as VBCSR). First, we compute the factors L¯ℓ, U¯ℓ and L¯ℓ−1Fℓ by performing a variant of the IKJ version of the Gaussian Elimination algorithm, where index I runs from 2 to mℓ, index K from 1 to I−1 and index J from K+1 to nℓ. This loop applies implicitly L¯ℓ−1 to the block row DℓFℓ to produce UℓL¯ℓ−1Fℓ. In the second loop, Gaussian Elimination is performed on the block row EℓCℓ using the multipliers computed in the first loop to give EℓU¯ℓ−1 and an approximation of the Schur complement Aℓ+1. Then, after Step 1, we permute explicitly the matrix at the first level as well as the matrices involved in the factorization at each new reordering step. For extensive performance assessment results of the VBARMS method, we point the reader to [50].
Algorithm 1VBARMS_Solve(Aℓ+1,bℓ). The solving phase with the VBARMS method.
Require:ℓ∈N∗, ℓmax∈N∗, bℓ=fℓgℓT
1: Solve Lℓy=fℓ
2: Compute gℓ′=gℓ−EℓUℓ−1y
3: ifℓ=ℓmaxthen
4: Solve Aℓ+1zℓ=gℓ′
5: else
6: Call VBARMS_Solve(Aℓ+1,gℓ′)
7: end if
8: Solve Uℓyℓ=y−Lℓ−1Fℓzℓ
5. Numerical experiments
In this section, we illustrate the performance of the VBARMS method for solving a suite of block structured linear systems arising from an implicit Newton-Krylov formulation of the RANS equations in the turbulent incompressible flow analysis past a three-dimensional wing. On multicore machines, the quotient graph G/B is split into distinct subdomains, and each of them is assigned to a different core. Following the parallel framework described in [55], we separate the nodes assigned to the ith subdomain into interior nodes, that are those coupled by the equations only with the local variables, and interface nodes, those that may be coupled with the local variables stored on processor i as well as with remote variables stored on other processors (see Figure below).
The vector of the local unknowns xi and the local right-hand side bi are split accordingly in two separate components: the subvector corresponding to the internal nodes followed by the subvector of the local interface variables
xi=uiyi,bi=figi.E38
The rows of A indexed by the nodes of the ith subdomain are assigned to the ith processor. These are naturally separated into a local matrix Ai acting on the local variables xi=uiyiT, and an interface matrix Ui acting on the remotely stored subvectors of the external interface variables yi,ext. Hence, we can write the local equations on processor i as
Aixi+Ui,extyi,ext=biE39
or, in expanded form, as
BiFiEiCiuiyi+0∑j∈NiEijyj=figi,E40
where Ni is the set of subdomains that are neighbors to subdomain i and the submatrix Eijyj accounts for the contribution to the local equation from the jth neighboring subdomain. Note that matrices Bi, Ci, Ei, and Fi still preserve the fine block structure imposed by the block ordering PB. From a code viewpoint, the quotient graph is initially distributed amongst the available processors; then, the built-in parallel hypergraph partitioner available in the Zoltan package [56] is applied on the distributed data structure to compute an optimal partitioning of the quotient graph that can minimize the amount of communications.
At this stage, the VBARMS method described in Section 4.2 can be used as a local solver for different types of global preconditioners. In the simplest parallel implementation, the so-called block-Jacobi preconditioner, the sequential VBARMS method can be applied to invert approximately each local matrix Ai. The standard Jacobi iteration for solving Ax=b is defined as
xn+1=xn+D−1b−Axn=D−1Nxn+b,E41
where D is the diagonal of A, N=D−A and x0 is some initial approximation. In cases we have a graph partitioned matrix, the matrix D is block diagonal and the diagonal blocks of D are the local matrices Ai. The interest to consider the block Jacobi preconditioner is its inherent parallelism, since the solves with the matrices Ai are performed independently on all the processors and no communication is required.
If the diagonal blocks of the matrix D are enlarged in the block-Jacobi method so that they overlap slightly, the resulting preconditioner is called Schwarz preconditioner. Consider again a graph partitioned matrix with N nonoverlapping sets Wi0, i=1,…,N and W0=∪i=1NW0i. We define a δ-overlap partition
Wδ=⋃i=1NWiδE42
where Wiδ=adjWiδ−1 and δ>0 is the level of overlap with the neighboring domains. For each subdomain, we define a restriction operator Riδ, which is an n×n matrix with the jjth element equal to 1 if j∈Wiδ, and zero elsewhere. We then denote
Ai=RiδARiδ.E43
The global preconditioning matrix MRAS is defined as
MRAS−1=∑i=1sRiTAi−1RiE44
and named as the Restricted Additive Schwarz (RAS) preconditioner [3, 57]. Note that the preconditioning step still offers a good scope par parallelism, as the different components of the error update are formed independently. However, due to overlapping some communication is required in the final update, as the components are added up from each subdomain. In our experiments, the overlap used for RAS was the level 1 neighbors of the local nodes in the quotient graph.
A third global preconditioner that we consider in this study is based on the Schur complement approach. In Eq. (40), we can eliminate the vector of interior unknowns ui from the first equations to compute the local Schur complement system
Siyi+∑j∈NiEijyj=gi−EiBi−1fi≡gi′,E45
where Si denotes the local Schur complement matrix
Si=Ci−EiBi−1Fi.E46
The local Schur complement equations considered altogether write as the global Schur complement system
where the off-diagonal matrices Eij are available from the parallel distribution of the linear system. One preconditioning step with the Schur complement preconditioner consists in solving approximately the global system (47), and then recovering the ui variables from the local equations as
ui=Bi−1fi−FiyiE48
at the cost of one local solve. We solve the global system (47) by running a few steps of the GMRES method preconditioned by a block diagonal matrix, where the diagonal blocks are the local Schur complements Si. The factorization
Si=LSiUSiE49
is obtained as by-product of the LU factorization of the local matrix Ai,
Ai=LBi0EiUBi−1LSiUBiLBi−1Fi0USiE50
which is by the way required to compute the ui variables in Eq. (48).
5.1. Results
The parallel experiments were run on the large-memory nodes (32 cores/node and 1 TB of memory) of the TACC Stampede system located at the University of Texas at Austin. TACC Stampede is a 10 PFLOPS (PF) Dell Linux Cluster based on 6400+ Dell PowerEdge server nodes, each outfitted with 2 Intel Xeon E5 (Sandy Bridge) processors and an Intel Xeon Phi Coprocessor (MIC Architecture). We linked the default vendor BLAS library, which is MKL. Although MKL is multi-threaded by default, in our runs we used it in a single-thread mode since our MPI-based parallelisation employed one MPI process per core (communicating via the shared memory for the same-node cores). We used the Flexible GMRES (FGMRES) method [58] as Krylov subspace method, a tolerance of 1.0e−6 in the stopping criterion and a maximum number of iteration equal to 1000. Memory costs were calculated as the ratio between the sum of the number of non-zeros in the local preconditioners and the sum of the number of non-zeros in the local matrices Ai.
In our experiments, we analyzed the turbulent incompressible flow past a three-dimensional wing illustrated in Figure 2 using the EulFS code developed by the second author [59]. The geometry, called DPW3 Wing-1, was proposed in the 3rd AIAA Drag Prediction Workshop [35]. Flow conditions are 0.5∘ angle of attack and Reynolds number based on the reference chord equal to 5⋅106. The freestream turbulent viscosity is set to 10% of its laminar value. In Tables 1 and 2 we show experiments with the parallel VBARMS solver on the five meshes of the DPW3 Wing-1 problem. On the largest mesh we report on only one experiment, in Table 2, as this is a resource demanding problem. In Table 3 we report on a strong scalability study on the problem denoted as RANS2 by increasing the number of processors. Finally, in Table 4 we show comparative results with parallel VBARMS against other popular solvers; the method denoted as pARMS is the solver described in [55] using default parameters while the method VBILUT is a variable-block incomplete lower-upper factorization with threshold from the ITSOL package [54]. The results of our experiments show that the proposed preconditioner is effective to reduce the number of iterations especially in combination with the Restricted Additive Schwarz method, and exhibits good parallel scalability. A truly parallel implementation of the VBARMS method that may offer better numerical scalability will be considered as the next step of this research.
Figure 2.
Geometry and mesh characteristics of the DPW3 Wing-1 problem proposed in the 3rd AIAA drag prediction workshop. Note that problems RANS1 and RANS2 correspond to the same mesh, and are generated at two different Newton steps.
Matrix
Method
Graph time (s)
Factorization time (s)
Solving time (s)
Total time (s)
Its
Mem
BJ + VBARMS
17.3
8.58
41.54
50.13
34
2.98
RANS1
RAS + VBARMS
17.4
10.08
42.28
52.37
19
3.06
SCHUR + VBARMS
17.6
11.94
55.99
67.93
35
2.57
BJ + VBARMS
17.0
16.72
70.14
86.86
47
4.35
RANS2
RAS + VBARMS
16.8
21.65
80.24
101.89
39
4.49
SCHUR + VBARMS
17.5
168.85
173.54
342.39
24
6.47
BJ + VBARMS
27.2
99.41
187.95
287.36
154
4.40
RANS3
RAS + VBARMS
25.2
119.32
90.47
209.79
71
4.48
SCHUR + VBARMS
22.0
52.65
721.67
774.31
140
4.39
Table 1.
Experiments on the DPW3 Wing-1 problem. The RANS1, RANS2 and RANS3 test cases are solved on 32 processors. We ran one MPI process per core, so in these experiments we used shared memory on a single node.
Matrix
Method
Graph time (s)
Factorization time (s)
Solving time (s)
Total time (s)
Its
Mem
BJ + VBARMS
51.5
12.05
105.89
117.94
223
3.91
RANS4
RAS + VBARMS
43.9
14.05
91.53
105.58
143
4.12
SCHUR + VBARMS
39.3
15.14
289.89
305.03
179
3.76
RANS5
RAS + VBARMS
1203.94(1)
16.80
274.62
291.42
235
4.05
Table 2.
Experiments on the DPW3 Wing-1 problem. The RANS4 and RANS5 test cases are solved on 128 processors. Note (1): due to a persistent problem with the Zoltan library on this run, we report on the result of our experiment with the metis (sequential) graph partitioner [60].
Solver
Number of processors
Graph time (s)
Total time (s)
Its
Mem
8
38.9
388.37
27
5.70
16
28.0
219.48
35
5.22
RAS + VBARMS
32
17.0
101.49
39
4.49
64
16.0
54.19
47
3.91
128
18.2
28.59
55
3.39
Table 3.
Strong scalability study on the RANS2 problem using parallel graph partitioning.
Matrix
Method
Factorization time (s)
Solving time (s)
Total time (s)
Its
Mem
pARMS
—
—
—
—
6.63
RANS3
BJ + VBARMS
99.41
187.95
287.36
154
4.40
BJ + VBILUT
20.45
8997.82
9018.27
979
13.81
pARMS
—
—
—
—
5.38
RANS4
BJ + VBARMS
12.05
105.89
117.94
223
3.91
BJ + VBILUT
1.16
295.20
296.35
472
5.26
Table 4.
Experiments on the DPW3 Wing-1 problem. The RANS3 test case is solved on 32 processors and the RANS4 problem on 128 processors. The dash symbol − in the table means that in the GMRES iteration the residual norm is very large and the program is aborted.
6. Conclusions
The applicability of Newton’s method in steady flow simulations is often limited by the difficulty to compute a good initial solution, namely, one lying in a reasonably small neighborhood of the sought solution. This problem can now be overcome by introducing some approximations in the first stages of the solution procedure. In the case of unsteady flow problems, on the other hand, the use of Newton’s method in conjunction with a dual-time stepping procedure is even more effective since the flow field computed at the preceding physical time level is likely to be sufficiently close to the sought solution at the next time level to allow the use of Newton’s algorithm right from the beginning of the sub-iterations in pseudo-time. On the downside of Newton-Krylov methods is the need for efficiently preconditioned iterative algorithms to solve the sparse linear system arising at each inner iteration (Newton step). The stiffness of the linear systems to be solved increases when the Jacobian is computed “exactly” and the turbulence transport equations are solved fully coupled with the mean flow equations.
In this chapter, we have presented a block multilevel incomplete factorization preconditioner for solving sparse systems of linear equations arising from the implicit RANS formulation. The method detects automatically any existing block structure in the matrix, without any user’s prior knowledge of the underlying problem, and exploits it to maximize computational efficiency. The results of this chapter show that, by taking advantage of this block structure, the solver can be more robust and efficient. Other recent studies on block ILU preconditioners have drawn similar conclusions on the importance of exposing dense blocks during the construction of the incomplete LU factorization for better performance, in the design of incomplete multifrontal LU-factorization preconditioners [61] and adaptive blocking approaches for blocked incomplete Cholesky factorization [62]. We believe that the proposed VBARMS method can be useful for solving linear systems also in other areas, such as in Electromagnetics applications [63, 64, 65].
Acknowledgments
The authors acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported in this chapter. URL: http://www.tacc.utexas.edu. The authors are grateful to Prof. Masha Sosonkina at the Department of Modeling Simulation & Visualization Engineering, Old Dominion University for help and insightful comments.
Nomenclature
a
artificial sound speed
d
dimension of the space, d=2,3
e0
specific total energy
h0
specific total enthalpy
ℓ
level of fill in incomplete lower upper factorizations
m
number of degrees of freedom within a gridpoint
n
order of a matrix
nnz
number of non-zero entries in a sparse matrix
n
unit inward normal to the control surface
p
static pressure
q
flux vector due to heat conduction
t
time
u
velocity vector
x
vector of the d Cartesian coordinates
Ci
median dual cell (control volume)
∂Ci
boundary of the median dual cell (control surface)
E
edges of a graph
GA
graph of matrix A
M
global mass matrix
ML
left preconditioning matrix
Ma
mach number
I
identity matrix
Ni
shape function
P
permutation matrix
PrT
turbulent Prandtl number
RΦ
spatial residual vector
Re
Reynolds’ number
T
triangle or tetrahedron
U
conserved variables vector
V
vertices of a graph
VM
lumped mass matrix
z
parameter vector
α
angle of attack
Δt
physical time step
Δτ
pseudo-time step
ΔU
=Un+1,k+1−Un+1,k
εmc
machine zero
ρ
density
ν
kinematic viscosity
ν˜
working variable in the turbulence transport equation
τ
pseudo-time variable
τ¯¯
Newtonian stress tensor
Φ
flux balance
Ωie
Petrov Galerkin weighting function
i
nodal index or row index of a matrix
j
nodal index or column index of a matrix
e
cell index
∞
Free-stream condition
inv
inviscid
k
inner iterations counter
n
physical time step counter
T
transpose
vis
viscous
\n',keywords:"computational fluid dynamics, Reynolds-averaged Navier-Stokes equations, Newton-Krylov methods, linear systems, sparse matrices, algebraic preconditioners, incomplete LU factorization, multilevel methods",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/59067.pdf",chapterXML:"https://mts.intechopen.com/source/xml/59067.xml",downloadPdfUrl:"/chapter/pdf-download/59067",previewPdfUrl:"/chapter/pdf-preview/59067",totalDownloads:727,totalViews:191,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"April 25th 2017",dateReviewed:"October 27th 2017",datePrePublished:null,datePublished:"February 14th 2018",dateFinished:null,readingETA:"0",abstract:"Implicit methods based on the Newton’s rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynolds-averaged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of three-dimensional test cases.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/59067",risUrl:"/chapter/ris/59067",book:{slug:"computational-fluid-dynamics-basic-instruments-and-applications-in-science"},signatures:"Bruno Carpentieri and Aldo Bonfiglioli",authors:[{id:"92921",title:"Dr.",name:"Bruno",middleName:null,surname:"Carpentieri",fullName:"Bruno Carpentieri",slug:"bruno-carpentieri",email:"bcarpentieri@gmail.com",position:null,institution:{name:"Free University of Bozen-Bolzano",institutionURL:null,country:{name:"Italy"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Governing equations",level:"1"},{id:"sec_3",title:"3. Solution techniques",level:"1"},{id:"sec_3_2",title:"3.1. Space discretisation",level:"2"},{id:"sec_4_2",title:"3.2. Time discretisation",level:"2"},{id:"sec_5_2",title:"3.3. Numerical integration",level:"2"},{id:"sec_5_3",title:"3.3.1. Tandem solution strategy with (approximate) Picard linearization",level:"3"},{id:"sec_6_3",title:"3.3.2. Fully coupled solution strategy with FD Newton linearization",level:"3"},{id:"sec_9",title:"4. Linear solve and preconditioning",level:"1"},{id:"sec_9_2",title:"4.1. Multi-elimination ILU factorization preconditioner",level:"2"},{id:"sec_10_2",title:"4.2. The variable-block ARMS factorization",level:"2"},{id:"sec_12",title:"5. Numerical experiments",level:"1"},{id:"sec_12_2",title:"5.1. Results",level:"2"},{id:"sec_14",title:"6. Conclusions",level:"1"},{id:"sec_15",title:"Acknowledgments",level:"1"},{id:"sec_17",title:"Nomenclature",level:"1"}],chapterReferences:[{id:"B1",body:'Wong P, Zingg DW. Three-dimensional aerodynamic computations on unstructured grids using a Newton-Krylov approach. 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Krakow: Springer-Verlag. 2014. pp. 520-530'},{id:"B64",body:'Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra JJ, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D. LAPACK Users’ Guide. third ed. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1999'},{id:"B65",body:'Li X. An overview of SuperLU: Algorithms, implementation, and user interface. ACM Transactions on Mathematical Software. September 2005;31(3):302-325'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Bruno Carpentieri",address:"bcarpentieri@gmail.com",affiliation:'
Free University of Bolzano-Bozen, Faculty of Computer Science, Italy
Scuola di Ingegneria, University of Basilicata, Italy
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Trohidou and M. 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Sahajwalla",authors:[{id:"19010",title:"Associate Prof.",name:"Rita",middleName:null,surname:"Khanna",fullName:"Rita Khanna",slug:"rita-khanna"}]},{id:"14029",title:"GCMC Simulations of Gas Adsorption in Carbon Pore Structures",slug:"gcmc-simulations-of-gas-adsorption-in-carbon-pore-structures",signatures:"Maria Konstantakou, Anastasios Gotzias, Michael Kainourgiakis, Athanasios K. Stubos and Theodore A. Steriotis",authors:[{id:"22788",title:"Dr.",name:"Theodore A.",middleName:null,surname:"Steriotis",fullName:"Theodore A. 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Behar, Ch. Kabakchiev, I. Garvanov and H. Rohling",authors:[{id:"2768",title:"Dr.",name:"Christo",middleName:null,surname:"Kabakchiev",fullName:"Christo Kabakchiev",slug:"christo-kabakchiev"},{id:"18669",title:"Dr.",name:"Vera",middleName:null,surname:"Behar",fullName:"Vera Behar",slug:"vera-behar"},{id:"18671",title:"Dr.",name:"Ivan",middleName:null,surname:"Garvanov",fullName:"Ivan Garvanov",slug:"ivan-garvanov"},{id:"18672",title:"Prof.",name:"Hermann",middleName:null,surname:"Rohling",fullName:"Hermann Rohling",slug:"hermann-rohling"}]},{id:"14034",title:"Practical Monte Carlo Based Reliability Analysis and Design Methods for Geotechnical Problems",slug:"practical-monte-carlo-based-reliability-analysis-and-design-methods-for-geotechnical-problems",signatures:"Jianye Ching",authors:[{id:"19783",title:"Prof.",name:"Jianye",middleName:null,surname:"Ching",fullName:"Jianye Ching",slug:"jianye-ching"}]},{id:"14035",title:"A Monte Carlo Framework to Simulate Multicomponent Droplet Growth by Stochastic Coalescence",slug:"a-monte-carlo-framework-to-simulate-multicomponent-droplet-growth-by-stochastic-coalescence",signatures:"Lester Alfonso, Graciela Raga and Darrel Baumgardner",authors:[{id:"18046",title:"Prof.",name:"Lester",middleName:null,surname:"Alfonso",fullName:"Lester Alfonso",slug:"lester-alfonso"},{id:"22501",title:"Dr.",name:"Graciela",middleName:null,surname:"Raga",fullName:"Graciela Raga",slug:"graciela-raga"},{id:"22502",title:"Dr.",name:"Darrel",middleName:null,surname:"Baumgardner",fullName:"Darrel Baumgardner",slug:"darrel-baumgardner"}]},{id:"14036",title:"Monte Carlo Simulation of Room Temperature Ballistic Nanodevices",slug:"monte-carlo-simulation-of-room-temperature-ballistic-nanodevices",signatures:"Ignacio Íñiguez-de-la-Torre, Tomás González, Helena Rodilla, Beatriz G. Vasallo and Javier Mateos",authors:[{id:"22253",title:"Dr.",name:"Tomas",middleName:null,surname:"Gonzalez",fullName:"Tomas Gonzalez",slug:"tomas-gonzalez"},{id:"22450",title:"Dr.",name:"Javier",middleName:null,surname:"Mateos",fullName:"Javier Mateos",slug:"javier-mateos"},{id:"22451",title:"Dr.",name:"Ignacio",middleName:null,surname:"Iñiguez-de-la-Torre",fullName:"Ignacio Iñiguez-de-la-Torre",slug:"ignacio-iniguez-de-la-torre"},{id:"22452",title:"Prof.",name:"Helena",middleName:null,surname:"Rodilla",fullName:"Helena Rodilla",slug:"helena-rodilla"},{id:"22453",title:"Dr.",name:"Beatriz",middleName:null,surname:"Garcia Vasallo",fullName:"Beatriz Garcia Vasallo",slug:"beatriz-garcia-vasallo"}]},{id:"14037",title:"Estimation of Optical Properties in Postharvest and Processing Technology",slug:"estimation-of-optical-properties-in-postharvest-and-processing-technology",signatures:"László Baranyai",authors:[{id:"19663",title:"Dr.",name:"Laszlo",middleName:null,surname:"Baranyai",fullName:"Laszlo Baranyai",slug:"laszlo-baranyai"}]},{id:"14038",title:"MATLAB Programming of Polymerization Processes using Monte Carlo Techniques",slug:"matlab-programming-of-polymerization-processes-using-monte-carlo-techniques",signatures:"Mamdouh A. 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Arif",authors:[{id:"21687",title:"Dr.",name:"Sayyad Zahid",middleName:null,surname:"Qamar",fullName:"Sayyad Zahid Qamar",slug:"sayyad-zahid-qamar"},{id:"21688",title:"Prof.",name:"Anwar Khalil",middleName:null,surname:"Sheikh",fullName:"Anwar Khalil Sheikh",slug:"anwar-khalil-sheikh"},{id:"21689",title:"Prof.",name:"Abul Fazal M.",middleName:null,surname:"Arif",fullName:"Abul Fazal M. Arif",slug:"abul-fazal-m.-arif"},{id:"21690",title:"Prof.",name:"Tasneem",middleName:null,surname:"Pervez",fullName:"Tasneem Pervez",slug:"tasneem-pervez"}]},{id:"14041",title:"Loss of Load Expectation Assessment in Electricity Markets using Monte Carlo Simulation and Neuro-Fuzzy Systems",slug:"loss-of-load-expectation-assessment-in-electricity-markets-using-monte-carlo-simulation-and-neuro-fu",signatures:"H. 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Geisler",authors:[{id:"19631",title:"Dr.",name:"Benjamin",middleName:"Peter",surname:"Geisler",fullName:"Benjamin Geisler",slug:"benjamin-geisler"}]},{id:"14043",title:"Monte Carlo Simulations of Adsorbed Molecules on Ionic Surfaces",slug:"monte-carlo-simulations-of-adsorbed-molecules-on-ionic-surfaces",signatures:"Abdulwahab Khalil Sallabi",authors:[{id:"22986",title:"Prof.",name:"A",middleName:null,surname:"Slabi",fullName:"A Slabi",slug:"a-slabi"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"74803",title:"Neuroendocrinology of Pregnancy: Participation of Sex Hormones",doi:"10.5772/intechopen.95774",slug:"neuroendocrinology-of-pregnancy-participation-of-sex-hormones",body:'
1. Introduction
During pregnancy, the neuroendocrine system undergoes significant hormonal fluctuations determined by stimulatory and inhibitory inputs from the mother and fetus to maintain the internal environment (milieu). This process is regulated mainly by both the maternal brain and the placenta, acting through the maternal-placental-fetal unit (MPFU). It also serves as a protection system against stress and immune responses [1, 2].
Interestingly, the neuroendocrine responses generate a feedback circuit regulated by the placenta. This organ begins its development in days six-seven after conception. It has been considered a passive organ for many years, acting as a barrier between the mother and the fetus, provide nourishing and eliminate metabolism products such as urea, uric acid, and creatinine. However, the placenta is a neuroendocrine organ that can synthesize and release hormones, neuroactive factors, and other mediators, allowing the proper development of the fetus’s maternal tissues to ensure an optimal pregnancy, allowing the fetus to adapt and survive under conditions of stress, infections, hypoxia, and malnutrition [3, 4]. This neuroendocrine mechanism involves at least three different endocrine axes; the hypothalamus-pituitary-gonads axis (HPG), the hypothalamus-pituitary–adrenal gland axis (HPA), and the hypothalamus-pituitary-thyroid axis (HPT), to ensure optimal maternal-fetal development [1].
Specifically, the HPG axis, which is the central axis involved in regulating the reproductive function in vertebrates by a releasing pulsing of GnRH at the hypothalamus and placenta, has a decisive role in the different stages of pregnancy. In this sense, it plays a central role in regulating MPFU development through positive and negative regulation of sex hormones [1].
2. Gonadotropin-releasing hormone (GnRH), FSH, and LH, primary mediators of sex hormones releasing
The GnRH is a hormone synthesized by the hypothalamic neurons. It travels through the portal-pituitary-system to bind to its receptors (GnRHR-I) in pituitary cells (gonadotrophs), activating the synthesis of FSH (Follicle-stimulating hormone) and LH (Luteinizing hormone). These hormones are released into the systemic circulation to act on sex organs regulating both oogenesis and spermatogenesis. Interestingly, GnRH isoforms (GnRH-I and GnRH-II) have also been identified in other tissues, including the testicles, prostate, mammary gland, endometrium, and placenta. In these organs, it has been shown that GnRH-II acts by binding to GnRHR-II receptors [5].
The functions associated with these isoforms are the production of the β-human chorionic gonadotropin (β-hCG) by the syncytiotrophoblast in the early stages of pregnancy. Here, β-hCG intervenes in at least two vital functions, avoiding luteolysis and ensuring Progesterone’s production (P4) until the placenta is implanted. Thus, specific conditions that interfere with this endocrine axis before weeks seven to nine of gestation would culminate in pregnancy loss [5].
Moreover, recent evidence indicates that GnRH is involved in the maternal-fetal environment’s remodeling (milieu) that allows the fetus’s correct implantation. This process is accompanied by increased proliferation of trophoblasts, which invade the decidua and form the outer and inner layers of syncytiotrophoblast, directly contacting maternal tissue. In this condition, the expression of specific metalloproteinases (MMP) is affected. Preclinical studies have shown that both isoforms (GnRH-I and GnRH-II) modify cellular matrix metalloproteinases’ expression. Two of them, MMP-2 and MMP-9, are the most directly involved in the migration and invasion of trophoblasts [5].
In addition to the above, it has been shown that both isoforms can produce proangiogenic cytokines, playing a central role in the rerouting of immune system cells involved in the restructuring of the arteries in the maternal-fetal interface [5]. Therefore, GnRH’s direct participation is vital for all physiological, hormonal, and structural changes that will culminate in the fetus’s correct implantation.
On the other hand, GnRH causes the stimulation of the pituitary hormone’s LH and FSH to regulate the sexual function. However, preclinical studies showed that both hormones are inhibited because of Progesterone and Estrogen increased production during pregnancy. Moreover, FSH and LH level rises on day ten after birth, which correlates to the decrease in sex hormones. In this sense, it has been established that the reduction of sex hormones after delivery performs negative feedback, which can achieve the complete reestablishment of reproductive function two months later after birth [6].
Interestingly, these data provide information valuable in understanding the positive and negative feedback mechanisms that play the sexual hormones during the pregnancy to maintain the MPFU.
3. Progesterone, the “pregnancy hormone”
Progesterone (P4) is considered the “pregnancy hormone” because it is critical for gestational maintenance [3]. During this stage, P4 is produced mainly by the ovary’s luteal body until the twelfth week of pregnancy. After that, its release is principally maintained by the placenta, reaching levels of up to 3 μg/g, while blood concentrations range from 100 to 500 nM, being four to six times its basal levels [7, 8]. These values rise significantly as gestational age progresses. It is involved in both the maintenance and development of the endometrium and inhibiting the uterus’ smooth muscle from preventing premature contractions (spontaneous abortion) [8].
Interestingly, the increase in this hormone’s levels seems to be regulated by an independent mechanism that generally restricts the synthesis of this hormone, being produced by the placental trophoblast cells in response to the stimuli produced in the uterine-fetal microenvironment [9]. At this level, its synthesis is carried out by converting the maternal cholesterol to the pro-hormone pregnenolone into the mitochondrial cytochrome P450. After that, by the action of 3-β-hydroxysteroid dehydrogenases (HSD), it is metabolized to Progesterone. Of the total synthesized Progesterone, 90% go into the maternal circulation, and the remaining 10% goes into fetal circulation [10].
Placental P4 plays an essential role in establishing a pregnancy, as it is responsible for inhibiting uterine contractions that occur at the myometrium’s smooth muscle. In this context, the deficiency of this hormone during the luteal phase has been widely related to infertility and loss of the pregnancy, presenting abortion, a situation that can occur at any stage of pregnancy [8, 9]. Besides, it is involved in the formation of decidua (a layer that coats the endometrium). In this sense, P4 is involved in the structural changes that the uterus undergoes during this period by increasing blood vessels’ permeability and endometrial density. Moreover, it has been suggested that the increase in decidual density is related to a lower likelihood of miscarriage. Also, P4 ensures the integrity of the fetus-maternal interface during the process of trophoblastic invasion and placenta formation [8, 11, 12]. What is more, P4 blocks the early production of T-cell lymphopoiesis protective role intrauterine environment’s immune system (milieu). For that reason, it has been suggested that Progesterone acts as an immunosteroid since a satisfactory pregnancy depends on maternal tolerance to the fetal ‘semi-allograft’ [8].
Similarly, it has been suggested that the increase in P4 levels induces changes in gene expression in the uterine endometrium, indirectly favoring embryo growth [12]. Furthermore, it is a crucial factor between the endocrine and immune system since it has been shown that this hormone is involved in the implantation of tissue, preventing it from being rejected by the mother, a mechanism that appears to be mediated by Th cells (helper T cells), as well as by the interleukins (IL) IL-3, IL-4, IL-5, and IL-10, in such a way, it has been suggested that through inhibition of Th 1 cells and increased production of interleukins, Progesterone is involved in the implantation of the fetus and its maintenance [7, 8].
On the other hand, P4 is involved in regulating the expression of uterine dendritic cells. These are known as antigen-presenting cells (APCs) involved in innate immune response and tolerance maintenance. However, in immature stages, these cells have a tolerogenic phenotype characterized by the low expression of co-stimulating molecules and pro-inflammatory cytokines. Thereby, it has been shown that in the early stages of pregnancy, Progesterone prevents dendritic cells’ maturation. All these previous actions contributed to the maintenance of pregnancy [8].
More interesting, it has been shown that P4 is also involved in reducing gestational stress. In this sense, it has been shown that it can over-express the mPRα gene, which encodes for a membrane receptor present in Cytolytic T lymphocytes CD8 + T cells, and whose increase has been linked to a protective effect against stress-induced abortion [7].
Finally, it is known that P4 levels decrease at the end of pregnancy, a phenomenon that is related to the onset of labor. Hence, an excellent regulatory mechanism of P4 (both at the endocrine and immunological level) from the beginning to the culmination of pregnancy, it is necessary to the implantation, maintenance, and completion of this [12, 13].
4. The modulatory hormones in pregnancy; testosterone (T), androstenedione (A4), and dehydroepiandrosterone (DHEA)
The androgenic hormones T, A4, and DHEA, plays a central role in regulating reproductive processes in many mammalian species. Besides, the presence of androgen receptors has also been demonstrated in different tissues such as the ovary, the myometrium, and placenta, where they are known to participate in implanting the fetus and placentation. In this sense, it has been shown that, once pregnancy occurs, androgen synthesis takes place in the small luteal cells (SLC) of the corpus luteum by stimulation of the human chorionic gonadotrophin (hCG) [3; 14]. In addition to the above, once the placenta has been established, it becomes an independent androgen production source [14]. In this aspect, placental syncytiotrophoblast uses the circulating DHEA, provided by the maternal and fetal adrenal glands, turning it into A4 and T. Which, in turn, as will be discussed later, can be converted to estrogens by different routes to regulate embryonic development [3, 15]. Interestingly, it has been suggested that myometrium could be another important source of androgens during pregnancy; a recent in vitro study showed that this tissue could also produce T and A4 [15].
Suppressively, these hormones are coordinated synthesized during pregnancy. Specifically, it has been shown that T levels increase in the first trimester of pregnancy, reaching a plateau in the second trimester, to later decrease slightly, rising considerably in the last month of pregnancy [14; 15]. Concerning A4, the study carried out by Satué et al. (2018) in mares shows that this hormone rises during gestation, from the second month of pregnancy, reaching a peak maximum in the first stage of pregnancy, and, in the second state, it reduces significantly, reaching its lowest levels in the last month of gestation. However, a clinical study conducted by Makieva et al. (2014) showed that A4 remains stable throughout pregnancy without significant fluctuations. About DHEA, it increases progressively from the first to the fifth month of pregnancy, reaches its highest levels, then begins to decrease between months 6 and 7, reaching its lowest levels in the last month of pregnancy in mares, which is agree with the observed in pregnant women, with levels up to 50% lower than those observed in non-pregnant women, an effect associated with negative E2 feedback to the maternal adrenal glands [14, 15].
The fluctuations in these hormones have specific functions during pregnancy. The significant increase observed in the first months of gestation is associated with the function of the corpus luteum, which uses T for estrogens’ production (analyzed in the next topic), regulating the implantation and decidualization. Later, the decrease observed in the middle of the gestation is related significantly to the development of the fetal gonads, providing the necessary substrates for the synthesis of placental estrogens. So, the primary site of estrogen synthesis at this stage could be the fetus. Finally, T’s elevation in the last stage of pregnancy, but not of A4 and DHEA, could be associated with the restructuring that the cervix must undergo to be prepared for the moment of delivery. At this stage, it has been shown that the cervix can convert T into another metabolite, Dihydrotestosterone (DHT), through the action of 5-alpha-reductase. This androgen is involved in restructuring the cervix’s extracellular matrix tissue, including the structural changes that allow the myometrium’s contractility [14, 15].
Therefore, these interesting data confirm the surprising interrelation and interdependence between estrogens and androgens produced by MPFU to protect and ensure pregnancy’s proper development.
5. Estrogens in pregnancy, an orchestral regulatory mechanism
Estrogens are a group of four different steroid hormones: Oestrone (E1), 17β-Oestradiol (E2), Oestriol (E3), and Oesterol (E4), cyclically synthesized in response to changes during the ovarian cycle, specifically during the pre-ovulation phase, favoring folliculogenesis. However, estrogens also play a central role in the growth of the uterus and mammary gland. It increases the blood flow indispensable for transporting nutrients between the uterus and the fetus [16]. During pregnancy and up to the time of delivery, significant amounts of estrogens are released by the maternal-fetus-placental unit, formed by the luteal body, placenta, and the fetal adrenal cortex [3], suffering significant adjustments between the weeks seven to nine of pregnancy, reaching its highest levels at the time of delivery [17, 18].
17β-Oestradiol (E2) is the most abundant hormone synthesized during pregnancy. In connection with this, until the third month of pregnancy, significant levels of E2 are released by the luteal body, a period from which the primary site of estrogen production is the placenta [3]. It should be noted that the placenta has no autonomic innervation, so these increases occur in response to close communication between the mother and the fetus, where the hormone acts in an autocrine-paracrine form in the development of the mammary gland and uterus, as well as in the development of sexual characteristics in the fetus. This connection allows the placenta and fetus to exchange and share steroid precursors, thus achieving their hormonal self-regulation [18].
Several studies have shown that the increase in estrogen levels is the result of a mutual exchange between the mother and placenta, in which the placenta uses the circulating androgen DHEA produced by the adrenal glands of the MPFU, where it is converted to Testosterone and Androstenedione and then metabolized to E1 and E2 with the help of the cytochrome cyp450 aromatase enzyme [3, 7]. In such a way, both the mother and the fetus contribute to the increase in estrogen synthesis, regulating their production. In addition to this, and due to the high maintenance of this hormone throughout pregnancy, there is sufficient evidence to suggest that regulation in levels of this hormone could also be at the neural level, where E2 could act as a trigger factor of the HPA gland axis. So, the adaptive changes that occur in the mother-fetus are regulated by a positive feedback mechanism, in which the binding of E2 to their receptors at the brain could be sending signals to the adrenal glands for producing a more significant amount of DHEA, thus maintaining their constant levels [19]. Therefore, it seems clear that estrogen levels regulation during pregnancy occurs both locally, by an interaction of the placental-fetal unit and in an autonomic way, with the direct participation of the Central Nervous System.
Concerning its functions, estrogen has been shown to act through binding to nuclear receptors, participating in multiple processes to ensure the maintenance of pregnancy having different roles: in human endometrial explant cultures, they are involved in uterine vascular restructuring by binding to their nuclear receptors present in epithelial and stromal cells of the cervix and endometrium, acting regulating the expression of different genes that control intrauterine growth, maturation of vital organs such as mammary glands for breastfeeding and childbirth [16, 17, 18, 20]. Besides, it promotes the processes of angiogenesis and vasodilation that allow the transfer and exchange of nutrients and oxygen between the placenta and the fetus through uterine and fetal circulation, a process associated with an increase in endothelial production of nitric oxide [3, 21].
On the other hand, in the primary culture of endometrial-epithelial cells (ESC), it has been found that E2 plays an essential role at the beginning of pregnancy by acting in processes such as differentiation and cell proliferation through the secretion of insulin growth factor type 1 (IGF-1) [22]. Also, it increases the rate at which the fertilized egg travels through the fallopian tube, so low estrogen levels promote ectopic pregnancies because the egg stays longer in the fallopian tube [23].
In addition to the above, estrogens E1, E3, and E4, also, play a central role in pregnancy. E1 is the most abundant conjugated estrogen (estrone sulfate) during pregnancy; it increases from the first trimester of pregnancy, reaching its maximum peak in the 35th week of gestation; among its functions, the decrease of estrogenicity has been indicated in the time of delivery [24]. E3 (Oestriol) is also considered a derivative of estradiol, whose primary role during pregnancy is increased uteroplacental blood flow during pregnancy. However, a specific function has also been suggested in the induction of myometrial cells’ contractions through the increase of connexin-4, allowing the restructuring of the myometrium that will trigger the initiation of labor [3]. On the other hand, Oesterol (E4) has an uncertain function during pregnancy since it is produced exclusively by the fetal liver starting up from the ninth week of pregnancy, reaching its significantly elevated levels after week 30, with a peak at week 40. Although its function is unclear, preclinical studies have shown that it can bind to estrogen and progesterone receptors at the uterus, producing histological structural changes and biochemical fluctuations, essential during the differentiation of endometrial cells in pregnancy and delivery [25].
Therefore, during pregnancy, the hyperestrogenic state plays a significant role in maternal-fetal development, being a key piece in fetal growth. Hence, all these actions make the estrogen pleiotropic essential hormones in pregnancy.
6. Prolactin in pregnancy, more than a lactation hormone
Prolactin (PRL) is a protein hormone synthesized by the lactotroph cells of the anterior pituitary gland. Unlike other pituitary hormones, its release is inhibited by Dopamine (DA), a hypothalamic factor produced by dopaminergic neurons located in the arcuate nucleus, which has not only been shown to be able to regulate the release of PRL but can act at the lactotrophic cells, regulating their proliferation [1]. In addition, PRL can control its release, directly stimulating dopaminergic neurons and through direct and indirect mechanisms regulated by E2 [23].
In this sense, a preclinical rat model study showed that in the different reproductive stages, PRL intervenes in a coordinated manner with E2 and Dopamine in regulating the proliferative activity of lactotrophic cells. In this collaborative process, these cells’ activity is elevated in the estrus and delivery stages. But it is decreased during the early stages of pregnancy and lactation, even though PRL levels are increased in all these reproductive stages. In this context, it has been suggested that E2 participates in stimulating the release of PRL during the early stages of pregnancy and lactation by acting at the hypothalamic level regulating both the increase in prolactin levels and the activity of the lactotroph cells when DA is not present, play a dual role in the release of this hormone [23, 26].
Evermore, during pregnancy, essential adaptations occur to allow the release of significant amounts of this hormone by the stimulation caused by the mammary gland and the luteal body [27], with substantial elevations from the twentieth week of pregnancy, until after childbirth [26]. Specifically, PRL has been shown to play a vital role in regulating IL-10 and IL-12 interleukins (essential regulators of immune responses during inflammatory processes). On the one hand, IL-12 interleukin has a pro-inflammatory function, activating itself in response to situations such as stress. On the other hand, IL-10 is an anti-inflammatory cytokine, which intervenes in the regulation of the expression of IL-12. In this sense, it has been shown that, during pregnancy, PRL increases the concentration of IL-10, an effect suggested is associated with the proper maintenance of this [28].
At the clinical level, this hormone has been shown to provide luteotropic support to the luteal body by intervening in the biosynthesis of P4 for its maintenance in the first three months of pregnancy, having an indirect function in the implantation of fetus in the uterus, as well as in the induction of vascular factors necessary for the increase in the volume of the luteal body [1, 29]. On the other hand, it acts directly on the mammary gland, determining the growth and development of alveoli, promoting the expression of genes related to milk synthesis and lactopoiesis. It also helps maintain the luteal body’s integrity and decidual cell survival [30]. Moreover, it is involved in the synthesis of relaxin, a hormone responsible for dilating the cervix during labor, thus facilitating the fetus’s expulsion [27].
PRL, it has been shown to play an essential role in regulating leptins expression in the gestational stage [31]. Leptins are hormones produced mostly by adipocytes, whose central role is related to the regulation of body weight, appetite, and energy homeostasis. The increase in their plasma levels is associated with the rise in the amount of body fat. However, during the gestational stage, vast quantities of leptins are released by the ovary and placenta, remaining constant throughout pregnancy, intervening in the regulation of fetal weight and growth, and with the development of gestational diabetes [32]. In this sense, the increase in PRL levels has been suggested to inhibit the receptor to leptins (LepR), thus blocking the signaling pathways that regulate the development of gestational diabetes [31].
Finally, it has a central role in mother–child recognition by increasing the generation of neurons at the olfactory bulb level, which is essential for such recognition [29]. For that reason, PRL recognizes like a multifaceted hormone, with dual actions during and after delivery.
7. Human placental lactogen, an exclusive metabolic hormone in pregnancy
The human placental lactogen hormone (hPL), known as human Chorionic somatomammotropin, is a polypeptide hormone elevated during pregnancy and is produced exclusively by the placenta [3]. hPL levels are detected between the first and second weeks of placenta gestation. However, it is released into the maternal circulation between the third to sixth week of pregnancy, being possible its detection, which increases until reaching its constant levels with a significant increase at the end of pregnancy with substantial effects after delivery [3].
Although there is controversy regarding its participation during pregnancy, it has been suggested that its primary function is the regulation of maternal metabolism of lipids and carbohydrates, being crucial to maintaining energy homeostasis between mother and fetus. In this sense, at the preclinical level, it has been shown that it stimulates the production of the IGF-1 factor in maternal hepatocytes. It modulates intermediate metabolism by increasing food intake (orexigenic drive), which favors the increase in glucose available for transfer to the fetus and prevents the development of gestational diabetes caused by peripheral resistance, typical at this stage [1, 3].
Moreover, it has been suggested that it has a central role in intrauterine growth because more than 50% of neonates with stunted growth have shown a deficiency in hPL levels. It is also believed that, by stimulating the uptake of glucose, glycerol, and free fatty acids, it could significantly participate in fat deposits, serving as an energy-saving mechanism for the fetus [1, 3].
Furthermore, it is well documented that in a normal state of pregnancy, insulin sensitivity decreases with the advance of the gestational state, which allows the fetus to maintain energy, an effect caused by a joint inhibitory action of peptide hormones (C-reactive protein, leptins, and hPL) on insulin levels causing dysfunction of pancreatic β-cells, named “diabetogenic condition.” However, a clinical study conducted by Ngala et al. (2017) showed that, throughout pregnancy, important maternal factors could predict the development of gestational diabetes mellitus (GDM) in addition to the already known factors of obesity and family history. In this sense, the levels of glucose, insulin, glycosylated hemoglobin (GHb), and hPL, among others, are increased in pregestational pregnant women, an effect not observed in non-diabetic pregnant women. Interestingly, under this condition, E2 and P4 levels decreased in pre-diabetic women, while in healthy women, the levels of both hormones are increased. On the other hand, between weeks 24–28 of gestation, an increase in Progesterone, Estradiol, Leptins, GHb, and Fasting blood glucose (FBG) was observed in developing GMD, an effect associated with the increased insulin resistance. Controversially, although there is little information linking hPL with the development of GDM, it is believed that the decrease in the levels of this hormone after delivery is associated with the reduction in glucose resistance and with the increased risk of diabetes-prediabetes in nursing mothers [33, 34].
Interestingly, hPL participates in lactation by stimulating the breast epithelium, facilitating breast development during the gestational stage. In this process, both hormones (hPL and PRL) act in maternal behavior, suppressing stress responses in the last stage of pregnancy and lactation [1]. In this sense, it has been shown that dopaminergic neurons’ activity can be maintained by hPL [23].
All these results confirm the metabolic action of hPL in pregnancy and lactation, alone or together to other placental and maternal hormones.
8. Human chorionic gonadotropic (hCG), the placental essential hormone
HCG is considered one of the essential hormones during gestational development, having similarities with other members of the same family of glycoproteic proteins such as LH and pituitary FSH. Its synthesis is regulated by the luteal body and placenta, exercising a pleiotropic role during gestation by autocrine and paracrine mechanisms [3]. It is possible to detect significant levels from day eight after fertilization, reaching its maximum levels around the tenth week of development. After which, it maintains at constant levels when the placenta is fully developed. At this point, the luteal body’s secretions are no longer necessary [3].
It participates in the process of steroidogenesis and in the restoration-maintenance of the luteal body, where it acts as a relay system, whose purpose is to prevent menstruation by increasing the synthesis of P4, allowing that the embryo can be implanted in the uterine endometrium, ensuring pregnancy until placental production of Progesterone is well established [3].
The hCG has also been shown to have a structure like Thyroid stimulating hormone (TSH) to bind to the same receptors, having implications for regulating thyrotropic activity. Preclinical studies have shown that maternal TSH decreases at the end of the third trimester of pregnancy. This decrease correlates with increased placental hCG and fetal thyroxine-binding globulin (TGB) [1]. In vitro studies have shown that it can have angiogenic effects; it increases vascular-endothelial growth factor (VEGF) and placental microvascular endothelial cells [3].
Moreover, clinical studies have shown that it is also involved in the differentiation of cytotrophoblast in syncytiotrophoblast, constituting an essential factor in the secretion of relaxing decidual production of PRL. On the other hand, it has androgenic properties. It can promote the synthesis of DHEA by the fetal adrenal cortex, regulating both testicular function and fetal male differentiation during the first weeks of gestation [35]. In addition, it has been suggested that it has participated in other functions; in the immune system, stimulates the production of the anti-inflammatory interleukins IL-8 and IL-10, and inhibits lymphocyte response, preventing rejection of the fetus, suggesting an immunosuppressive role of hCG during pregnancy; it stimulates testicular Leydig cells for testosterone production and provides nutrients and hormones for optimal maintenance of intrauterine microenvironment [3, 35]. So, the metabolic implications of this hormone suggested it like a metabolic hormone in pregnancy.
9. Cortisol and glucocorticoids; is it just stress in pregnancy?
The secretion of cortisol levels during pregnancy is regulated by the placenta, which, by secreting the corticotropin-releasing hormone (CRH), produces an exponential increase in cortisol from the eighth week of gestation up to three times above systemic values [5, 36]. It is present in both the maternal and fetal phases but at different levels; under normal conditions, cortisol levels reach 200 ng/ml at the end of pregnancy, while fetal levels range from around 20 ng/ml [37]. These differences are due to the presence of a natural barrier that prevents maternal cortisol, whose molecular composition can cross the placenta, quickly reaches fetal space [38, 39].
This barrier corresponds to the uterus/fetus interface and is mainly composed of maternal decidua and fetal placenta chorion. Here the regulation of cortisol is carried out through placental glycoprotein P, as well as the enzyme 11-β-hydroxysteroid-dehydrogenase (11-β-HSD) type 2 of trophoblastic and fetal cells, which inactivates cortisol by converting it into cortisone to avoid exposure of the fetus to high levels of cortisol [37, 40]. However, because of its role in organ maturation and labor, fetal cortisol increases towards the end of pregnancy by several mechanisms: a) decrease of 11-β-HSD type 2 in fetal tissues, b) increased synthesis of cortisol by the fetal adrenal gland, and c) increased 11-β-HSD type 1 in fetal tissues, which converts cortisone, into active cortisol [41].
As for the functions of cortisol during pregnancy, glucocorticoids (GC) have been described as participating in the processes of implantation and formation of decidua, as well as in fetal development and maturation, and initiation of childbirth [17, 36, 42]. Elevated levels of GC present during pregnancy are involved in the suppression of inflammation of the uterus, placenta, and fetal membranes, which contributes to maintaining the homeostasis necessary for the maintenance of pregnancy [42]. Moreover, recent evidence suggests that significant increases in cortisol levels play a critical role in the baby’s growth in the postnatal stage [43]. In this sense, studies have shown that high concentrations of cortisol during the fetal phase positively correlated with weight gain within the first five years of postnatal growth, indicating that the higher increase in placental cortisol levels, the more significant weight gain can be observed in children during this stage, suggesting that hormonal changes within the maternal-fetal environment have repercussions in post-birth stages, a highly relevant endocrinological aspect [43].
Conversely, cortisol is also involved in developing pregnancy complications, being responsible for the so-called “Hypothalamic Stress Amenorrhea,” whose consequence is the generation of miscarriages [8, 44]. On the one hand, it has been shown that low maternal cortisol levels compromise the placenta’s structure. In contrast, elevated levels can lead to miscarriages, uterine contractions from placental CRH deregulation, the elevation of fetal cortisol levels, and obstetric alterations by activation of the HPA gland axis [14, 36, 38, 45]. In this sense, two main axes, the HPA, and the sympathetic nervous system-adrenal medulla exerts a negative effect on the reproductive system when activated in stressful situations. In this feedback mechanism, the CRH that is produced at the pituitary can act, in a short negative feedback mechanism, directly inhibiting GnRH at the hypothalamus.
Even more, cortisol act at the pituitary to inhibit the release of LH and FSH, and, consequently, inhibits steroidal ovarian hormones, Estrogen, and P4, resulting in abortion. It has been confirmed in preclinical and clinical studies, where exposure to stressors, such as noise, has been verified to induce miscarriages, with a significant decrease in P4 levels [8, 44]. More interesting, stress increases the excitability of the sympathetic nervous system, resulting in a decrease in blood flow supply to the placenta caused placental hypoxia and increased generation of reactive oxygen species, causing damage to trophoblasts; the outer layer of the blastocyst, responsible for providing nourishing to the embryo [44].
Finally, it has also been suggested that high cortisol levels could mediate a disbalance in T helper cells Th 1 and Th 2, with a specific impact in the decrease of adaptative immune system responses that allow the fetus’s maintenance. However, more studies are needed to confirm this [44]. So, it is evident that cortisol is not just a “stress hormone”; it has several functions supporting the MPFU.
10. Thyroid hormones in pregnancy; regulation by sex hormones?
During pregnancy, high estrogens and corticosteroids induce an increase in TGB levels in the liver, which is significant from week twenty of gestation, reaching its maximum level from week twenty to twenty-four. The rise in TGB during the first half of pregnancy is related to further deiodination of the inner ring of the hormones T4 (Thyroxine) and T3 (Triiodothyronine) at the placenta, which is responsible for the physiological effect attributed to them [46, 47].
As far as the fetus is concerned, it has been shown that there are at least two mechanisms for it to contribute to thyroid hormones: the development of the fetal thyroid gland and the maternal thyroid gland. More interesting, the increase in concentrations of T4 in the first half of pregnancy and the expression of receptors to thyroid hormones in the brain, suggesting its participation in the development of brain structures of the fetus. Moreover, from weeks twelve-fourteen, in which the fetal thyroid begins to synthesize T4, its levels increase progressively, until reaching its maximum levels between week thirty-four to thirty-six, remaining elevated until the delivery time [46].
About iodine levels begin to be detected from ten to eleven weeks of gestation, a stage in which the fetal thyroid can concentrate. Around the twelfth week, the pituitary starts to produce and synthesize TSH and TRH (Thyrotropin-releasing hormone) by the hypothalamic neurons [46].
Before the fetal thyroid develops, the placenta has a particular involvement in maternal-fetal thyroid regulation. It is responsible for exchanging thyroid hormones to the fetus, suggesting an essential role in early fetal growth. Among the functions attributed to thyroid hormones are the brain’s development and the acceleration of fetal pulmonary maturation. The effect has been demonstrated in preclinical and clinical models in which fetal pulmonary growth has been shown to increase after intraamniotic injection of T3 or T4. On the other hand, the effect at the brain level has been demonstrated in intrauterine hypothyroidism conditions. It is related to irreversible damage to the brain and mental disability in children born under these conditions [46, 47].
Interestingly, clinical studies conducted in children whose mothers suffered from hypothyroidism, a condition that occurs in 0.05–0.02% during pregnancy, have shown that these irreversible changes specifically affect neurodevelopment. In this sense, it has been demonstrated that any situation that leads to the development of clinical hypothyroidism (generally associated with Graves’ disease) and hypothyroxinemia (associated with overtreatment of antithyroid drugs) that can occur during the first trimester of pregnancy can lead to a cognitive delay in children, learning disorders, maturational delay, encephalopathy, and seizures among other conditions [48].
Significantly, the increase in TSH, T3, and T4 during pregnancy could have protective effects against fetal anemia because it has been suggested that they may have cardiotonic effects by direct activation of the sympathetic-adrenal nervous system, in addition to being shown to stimulate the production of erythropoietin, which is involved in the production of red blood cells and therefore in the release of oxygen to tissues [47].
In this sense, it is fascinating to understand that sex hormones regulate the release of thyroid hormones and the vital functions involved, like brain development, being crucial during pregnancy and childhood.
11. Conclusions
Pregnancy is a physiological state characterized by critical hormonal changes. Collective participation of the endocrine system is necessary to carry out adequate development and maintenance of both the mother and the fetus. This system is responsible for generating an optimal environment that provides an adequate microenvironment of communication between the maternal-placental-fetal unit, facilitating the exchange of nutrients, hormones, and oxygen, essential throughout the gestational period. These neuroendocrine processes are produced thanks to the synchronous and fluctuating production of sex hormones regulated by endocrine, paracrine, and autocrine mechanisms. Their function is essential before, during, and after the gestational period to ensure the fetus’s correct development and growth.
Acknowledgments
The author thanks the National Council of Science and Technology (CONACyT) for the supporting founding.
Conflict of interest
The author declares no conflicts of interest.
Nomenclature
MPFU
Maternal-placental-fetal unit
HPG
Hypothalamus-pituitary-gonads axis
HPA
Hypothalamus-pituitary–adrenal gland axis
HPT
Hypothalamus-pituitary-thyroid axis
GnRH
Gonadotropin-releasing hormone
GnRHR-I and GnRHR-II
Gonadotropin-releasing hormone receptors I and II
FSH
Follicle-stimulating hormone
LH
Luteinizing hormone
β-hCG
β-Human chorionic gonadotropin
P4
Progesterone
MMP
Metalloproteinases
3β-HSD
3-β-hydroxysteroid-dehydrogenases
11β-HSD
11-β-hydroxysteroid-dehydrogenase
Th cells
Helper T cells
IL
Interleukins
APCs
Antigen-presenting cells
T
Testosterone
A4
Androstenedione
DHEA
Dehydroepiandrosterone
E1
Oestrone
E2
17β-Oestradiol
E3
Oestriol
E4
Oesterol
ESC
Endometrial-epithelial cells
IGF-1
Insulin growth factor type 1
PRL
Prolactin
DA
Dopamine
LepR
Leptin receptors
hPL
Human placental lactogen hormone
TSH
Thyroid Stimulating Hormone
TGB
Thyroxine-binding globulin
VEGF
Vascular-endothelial growth factor
CRH
Corticotropin-releasing hormone
GC
Glucocorticoids
T4
Thyroxine
T3
Triiodothyronine
TRH
Thyrotropin-releasing hormone
SLC
Small luteal cells
GDM
Gestational diabetes mellitus
GHb
Glycosylated Hemoglobin
FBG
Fasting blood glucose
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These changes are regulated by fluctuations in hormones such as Progesterone, Testosterone, Androstenedione, Dehydroepiandrosterone, Estradiol, Prolactin, human Placental Lactogen, human Chorionic Gonadotropin, and Thyroid hormones, which promote the mother’s development and the fetus (maternal-fetal development). Therefore, given the great importance of these hormones during pregnancy, this chapter will explain the preclinical and clinical participation of sex hormones in maternal-fetal development.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/74803",risUrl:"/chapter/ris/74803",signatures:"Luz Irene Pascual Mathey",book:{id:"10313",title:"Sex Hormones",subtitle:null,fullTitle:"Sex Hormones",slug:null,publishedDate:null,bookSignature:"Dr. Courtney Marsh",coverURL:"https://cdn.intechopen.com/books/images_new/10313.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"255491",title:"Dr.",name:"Courtney",middleName:null,surname:"Marsh",slug:"courtney-marsh",fullName:"Courtney Marsh"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Gonadotropin-releasing hormone (GnRH), FSH, and LH, primary mediators of sex hormones releasing",level:"1"},{id:"sec_3",title:"3. Progesterone, the “pregnancy hormone”",level:"1"},{id:"sec_4",title:"4. The modulatory hormones in pregnancy; testosterone (T), androstenedione (A4), and dehydroepiandrosterone (DHEA)",level:"1"},{id:"sec_5",title:"5. Estrogens in pregnancy, an orchestral regulatory mechanism",level:"1"},{id:"sec_6",title:"6. Prolactin in pregnancy, more than a lactation hormone",level:"1"},{id:"sec_7",title:"7. Human placental lactogen, an exclusive metabolic hormone in pregnancy",level:"1"},{id:"sec_8",title:"8. Human chorionic gonadotropic (hCG), the placental essential hormone",level:"1"},{id:"sec_9",title:"9. Cortisol and glucocorticoids; is it just stress in pregnancy?",level:"1"},{id:"sec_10",title:"10. Thyroid hormones in pregnancy; regulation by sex hormones?",level:"1"},{id:"sec_11",title:"11. Conclusions",level:"1"},{id:"sec_12",title:"Acknowledgments",level:"1"},{id:"sec_15",title:"Conflict of interest",level:"1"},{id:"sec_12",title:"Nomenclature",level:"1"}],chapterReferences:[{id:"B1",body:'Voltolini C, Petraglia F. Neuroendocrinology of pregnancy and parturition. Handb Clin Neurol. 2014;124:17-36. doi: 10.1016/B978-0-444-59602-4.00002-2. PMID: 25248577.'},{id:"B2",body:'Lahita RG. The effects of sex hormones on the immune system in pregnancy. Am J Reprod Immunol. 1992 Oct-Dec;28(3-4):136-7. doi: 10.1111/j.1600-0897.1992.tb00775.x. PMID: 1285863.'},{id:"B3",body:'Costa MA. The endocrine function of human placenta: an overview. Reprod Biomed Online. 2016 Jan;32(1):14-43. doi: 10.1016/j.rbmo.2015.10.005. Epub 2015 Oct 27. PMID: 26615903.'},{id:"B4",body:'Behura SK, Dhakal P, Kelleher AM, Balboula A, Patterson A, Spencer TE. The brain-placental axis: Therapeutic and pharmacological relevancy to pregnancy. Pharmacol Res. 2019 Nov;149:104468. doi: 10.1016/j.phrs.2019.104468. Epub 2019 Oct 7. PMID: 31600597; PMCID: PMC6944055.'},{id:"B5",body:'Sasaki K, Norwitz ER. Gonadotropin-releasing hormone/gonadotropin-releasing hormone receptor signaling in the placenta. Curr Opin Endocrinol Diabetes Obes. 2011 Dec;18(6):401-8. doi: 10.1097/MED.0b013e32834cd3b0. PMID: 22024993.'},{id:"B6",body:'Hirano M, Igarashi A, Suzuki M. Dynamic changes of serum LH and FSH during pregnancy and puerperium. Tohoku J Exp Med. 1976 Mar;118(3):275-82. doi: 10.1620/tjem.118.275. PMID: 772883.'},{id:"B7",body:'Zen M, Ghirardello A, Iaccarino L, Tonon M, Campana C, Arienti S, Rampudda M, Canova M, Doria A. Hormones, immune response, and pregnancy in healthy women and SLE patients. Swiss Med Wkly. 2010 Apr 3;140(13-14):187-201. PMID: 20175004.'},{id:"B8",body:'Arck P, Hansen PJ, Mulac Jericevic B, Piccinni MP, Szekeres-Bartho J. Progesterone during pregnancy: endocrine-immune cross talk in mammalian species and the role of stress. Am J Reprod Immunol. 2007 Sep;58(3):268-79. doi: 10.1111/j.1600-0897.2007.00512.x. PMID: 17681043.'},{id:"B9",body:'Yang R, You X, Tang X, Gao L, Ni X. Corticotropin-releasing hormone inhibits progesterone production in cultured human placental trophoblasts. J Mol Endocrinol. 2006 Dec;37(3):533-40. doi: 10.1677/jme.1.02119. PMID: 17170093.'},{id:"B10",body:'Tuckey RC. Progesterone synthesis by the human placenta. Placenta. 2005 Apr;26(4):273-81. doi: 10.1016/j.placenta.2004.06.012. PMID: 15823613.'},{id:"B11",body:'Hızlı D, Köşüş N, Köşüş A, Kasap B, Kafali H, Turhan NÖ. First-trimester reference ranges for decidual thickness and its relation to progesterone levels. J Perinat Med. 2012 Sep;40(5):521-5. doi: 10.1515/jpm-2012-0035. PMID: 23104794.'},{id:"B12",body:'Leitao B, Jones MC, Fusi L, Higham J, Lee Y, Takano M, Goto T, Christian M, Lam EW, Brosens JJ. Silencing of the JNK pathway maintains progesterone receptor activity in decidualizing human endometrial stromal cells exposed to oxidative stress signals. FASEB J. 2010 May;24(5):1541-51. doi: 10.1096/fj.09-149153. Epub 2009 Dec 21. PMID: 20026682; PMCID: PMC2857868.'},{id:"B13",body:'Jeschke U, Mylonas I, Richter DU, Höcker I, Briese V, Makrigiannakis A, Friese K. Regulation of progesterone production in human term trophoblasts in vitro by CRH, ACTH and cortisol (prednisolone). Arch Gynecol Obstet. 2005 Jun;272(1):7-12. doi: 10.1007/s00404-005-0728-0. Epub 2005 Apr 16. PMID: 15834733.'},{id:"B14",body:'Makieva S, Saunders PT, Norman JE. Androgens in pregnancy: roles in parturition. Hum Reprod Update. 2014 Jul-Aug;20(4):542-59. doi: 10.1093/humupd/dmu008. Epub 2014 Mar 18. PMID: 24643344; PMCID: PMC4063701.'},{id:"B15",body:'Satué K, Marcilla M, Medica P, Ferlazzo A, Fazio E. Testosterone, androstenedione and dehydroepiandrosterone concentrations in pregnant Spanish Purebred mare. Theriogenology. 2019 Jan 1;123:62-67. doi: 10.1016/j.theriogenology.2018.09.025. Epub 2018 Sep 26. PMID: 30292857.'},{id:"B16",body:'Corcoran JJ, Nicholson C, Sweeney M, Charnock JC, Robson SC, Westwood M, Taggart MJ. Human uterine and placental arteries exhibit tissue-specific acute responses to 17β-estradiol and estrogen-receptor-specific agonists. Mol Hum Reprod. 2014 May;20(5):433-41. doi: 10.1093/molehr/gat095. Epub 2013 Dec 19. PMID: 24356876; PMCID: PMC4004081.'},{id:"B17",body:'Chang K, Lubo Zhang. Review article: steroid hormones and uterine vascular adaptation to pregnancy. Reprod Sci. 2008 Apr;15(4):336-48. doi: 10.1177/1933719108317975. PMID: 18497342; PMCID: PMC2408771.'},{id:"B18",body:'Gambino YP, Maymó JL, Pérez-Pérez A, Dueñas JL, Sánchez-Margalet V, Calvo JC, Varone CL. 17Beta-estradiol enhances leptin expression in human placental cells through genomic and nongenomic actions. Biol Reprod. 2010 Jul;83(1):42-51. doi: 10.1095/biolreprod.110.083535. Epub 2010 Mar 17. PMID: 20237333.'},{id:"B19",body:'Brunton PJ, Russell JA. The expectant brain: adapting for motherhood. Nat Rev Neurosci. 2008 Jan;9(1):11-25. doi: 10.1038/nrn2280. PMID: 18073776.'},{id:"B20",body:'King AE, Collins F, Klonisch T, Sallenave JM, Critchley HO, Saunders PT. An additive interaction between the NFkappaB and estrogen receptor signalling pathways in human endometrial epithelial cells. Hum Reprod. 2010 Feb;25(2):510-8. doi: 10.1093/humrep/dep421. Epub 2009 Dec 2. PMID: 19955102; PMCID: PMC2806182.'},{id:"B21",body:'Bai J, Qi QR, Li Y, Day R, Makhoul J, Magness RR, Chen DB. Estrogen Receptors and Estrogen-Induced Uterine Vasodilation in Pregnancy. Int J Mol Sci. 2020 Jun 18;21(12):4349. doi: 10.3390/ijms21124349. PMID: 32570961; PMCID: PMC7352873.'},{id:"B22",body:'Shao R. Understanding the mechanisms of human tubal ectopic pregnancies: new evidence from knockout mouse models. Hum Reprod. 2010 Mar;25(3):584-7. doi: 10.1093/humrep/dep438. Epub 2009 Dec 19. PMID: 20023297; PMCID: PMC2817566.'},{id:"B23",body:'Grattan DR. 60 YEARS OF NEUROENDOCRINOLOGY: The hypothalamo-prolactin axis. J Endocrinol. 2015 Aug;226(2):T101-22. doi: 10.1530/JOE-15-0213. Epub 2015 Jun 22. PMID: 26101377; PMCID: PMC4515538.'},{id:"B24",body:'Honjo H, Kitawaki J, Itoh M, Yasuda J, Yamamoto T, Yamamoto T, Okada H, Ohkubo T, Nambara T. Serum and urinary oestrone sulphate in pregnancy and delivery measured by a direct radioimmunoassay. Acta Endocrinol (Copenh). 1986 Jul;112(3):423-30. doi: 10.1530/acta.0.1120423. PMID: 3751457.'},{id:"B25",body:'Holinka CF, Diczfalusy E, Coelingh Bennink HJ. Estetrol: a unique steroid in human pregnancy. J Steroid Biochem Mol Biol. 2008 May;110(1-2):138-43. doi: 10.1016/j.jsbmb.2008.03.027. Epub 2008 Mar 29. PMID: 18462934.'},{id:"B26",body:'Yin P, Arita J. Differential regulation of prolactin release and lactotrope proliferation during pregnancy, lactation and the estrous cycle. Neuroendocrinology. 2000 Aug;72(2):72-9. doi: 10.1159/000054574. PMID: 10971142.'},{id:"B27",body:'Soares MJ. The prolactin and growth hormone families: pregnancy-specific hormones/cytokines at the maternal-fetal interface. Reprod Biol Endocrinol. 2004 Jul 5;2:51. doi: 10.1186/1477-7827-2-51. PMID: 15236651; PMCID: PMC471570.'},{id:"B28",body:'Matalka KZ, Ali DA. Stress-induced versus preovulatory and pregnancy hormonal levels in modulating cytokine production following whole blood stimulation. Neuroimmunomodulation. 2005;12(6):366-74. doi: 10.1159/000091130. PMID: 16557037.'},{id:"B29",body:'Bachelot A, Binart N. Reproductive role of prolactin. Reproduction. 2007 Feb;133(2):361-9. doi: 10.1530/REP-06-0299. PMID: 17307904.'},{id:"B30",body:'Tessier C, Prigent-Tessier A, Ferguson-Gottschall S, Gu Y, Gibori G. PRL antiapoptotic effect in the rat decidua involves the PI3K/protein kinase B-mediated inhibition of caspase-3 activity. Endocrinology. 2001 Sep;142(9):4086-94. doi: 10.1210/endo.142.9.8381. PMID: 11517188.'},{id:"B31",body:'Nagaishi VS, Cardinali LI, Zampieri TT, Furigo IC, Metzger M, Donato J Jr. Possible crosstalk between leptin and prolactin during pregnancy. Neuroscience. 2014 Feb 14;259:71-83. doi: 10.1016/j.neuroscience.2013.11.050. Epub 2013 Dec 4. PMID: 24316468.'},{id:"B32",body:'Butte NF, Hopkinson JM, Nicolson MA. Leptin in human reproduction: serum leptin levels in pregnant and lactating women. J Clin Endocrinol Metab. 1997 Feb;82(2):585-9. doi: 10.1210/jcem.82.2.3731. PMID: 9024259.'},{id:"B33",body:'Ngala RA, Fondjo LA, Gmagna P, Ghartey FN, Awe MA. Placental peptides metabolism and maternal factors as predictors of risk of gestational diabetes in pregnant women. A case-control study. PLoS One. 2017 Jul 21;12(7):e0181613. doi: 10.1371/journal.pone.0181613. PMID: 28732072; PMCID: PMC5521813.'},{id:"B34",body:'Simpson S, Smith L, Bowe J. Placental peptides regulating islet adaptation to pregnancy: clinical potential in gestational diabetes mellitus. Curr Opin Pharmacol. 2018 Dec;43:59-65. doi: 10.1016/j.coph.2018.08.004. Epub 2018 Sep 7. PMID: 30199758.'},{id:"B35",body:'Licht P, Russu V, Wildt L. On the role of human chorionic gonadotropin (hCG) in the embryo-endometrial microenvironment: implications for differentiation and implantation. Semin Reprod Med. 2001;19(1):37-47. doi: 10.1055/s-2001-13909. PMID: 11394202.'},{id:"B36",body:'Field T, Diego M. Cortisol: the culprit prenatal stress variable. Int J Neurosci. 2008 Aug;118(8):1181. doi: 10.1080/00207450701820944. PMID: 18589921.'},{id:"B37",body:'Shams M, Kilby MD, Somerset DA, Howie AJ, Gupta A, Wood PJ, Afnan M, Stewart PM. 11Beta-hydroxysteroid dehydrogenase type 2 in human pregnancy and reduced expression in intrauterine growth restriction. Hum Reprod. 1998 Apr;13(4):799-804. doi: 10.1093/humrep/13.4.799. PMID: 9619527.'},{id:"B38",body:'Diego MA, Jones NA, Field T, Hernandez-Reif M, Schanberg S, Kuhn C, Gonzalez-Garcia A. Maternal psychological distress, prenatal cortisol, and fetal weight. Psychosom Med. 2006 Sep-Oct;68(5):747-53. doi: 10.1097/01.psy.0000238212.21598.7b. PMID: 17012528.'},{id:"B39",body:'Chan J, Rabbitt EH, Innes BA, Bulmer JN, Stewart PM, Kilby MD, Hewison M. Glucocorticoid-induced apoptosis in human decidua: a novel role for 11beta-hydroxysteroid dehydrogenase in late gestation. J Endocrinol. 2007 Oct;195(1):7-15. doi: 10.1677/JOE-07-0289. PMID: 17911392.'},{id:"B40",body:'Myatt L. Placental adaptive responses and fetal programming. J Physiol. 2006 Apr 1;572(Pt 1):25-30. doi: 10.1113/jphysiol.2006.104968. Epub 2006 Feb 9. PMID: 16469781; PMCID: PMC1779654.'},{id:"B41",body:'Myatt L, Sun K. Role of fetal membranes in signaling of fetal maturation and parturition. Int J Dev Biol. 2010;54(2-3):545-53. doi: 10.1387/ijdb.082771lm. PMID: 19924634.'},{id:"B42",body:'Rosen T, Krikun G, Ma Y, Wang EY, Lockwood CJ, Guller S. Chronic antagonism of nuclear factor-kappaB activity in cytotrophoblasts by dexamethasone: a potential mechanism for antiinflammatory action of glucocorticoids in human placenta. J Clin Endocrinol Metab. 1998 Oct;83(10):3647-52. doi: 10.1210/jcem.83.10.5151. PMID: 9768679.'},{id:"B43",body:'Street ME, Smerieri A, Petraroli A, Cesari S, Viani I, Garrubba M, Rossi M, Bernasconi S. Placental cortisol and cord serum IGFBP-2 concentrations are important determinants of postnatal weight gain. J Biol Regul Homeost Agents. 2012 Oct-Dec;26(4):721-31. PMID: 23241122.'},{id:"B44",body:'Tian CF, Kang MH. Common stress and serum cortisol and IL-12 levels in missed abortion. J Obstet Gynaecol. 2014 Jan;34(1):33-5. doi: 10.3109/01443615.2013.830089. PMID: 24359046.'},{id:"B45",body:'Marsman R, Rosmalen JG, Oldehinkel AJ, Ormel J, Buitelaar JK. Does HPA-axis activity mediate the relationship between obstetric complications and externalizing behavior problems? The TRAILS study. Eur Child Adolesc Psychiatry. 2009 Sep;18(9):565-73. doi: 10.1007/s00787-009-0014-y. Epub 2009 Apr 8. PMID: 19353232; PMCID: PMC2721131.'},{id:"B46",body:'Springer D, Jiskra J, Limanova Z, Zima T, Potlukova E. Thyroid in pregnancy: From physiology to screening. Crit Rev Clin Lab Sci. 2017 Mar;54(2):102-116. doi: 10.1080/10408363.2016.1269309. Epub 2017 Jan 19. PMID: 28102101.'},{id:"B47",body:'Thorpe-Beeston JG, Nicolaides KH. Fetal thyroid function. Fetal Diagn Ther. 1993 Jan-Feb;8( 1):60-72. doi: 10.1159/000263749. PMID: 8452651.'},{id:"B48",body:'Temboury Molina MC, Rivero Martín MJ, de Juan Ruiz J, Ares Segura S. Enfermedad tiroidea autoinmunitaria materna: repercusión en el recién nacido [Maternal autoimmune thyroid disease: relevance for the newborn]. Med Clin (Barc). 2015 Apr 8;144(7):297-303. Spanish. doi: 10.1016/j.medcli.2013.10.024. Epub 2014 Jan 30. PMID: 24486115.'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Luz Irene Pascual Mathey",address:"lupascual@uv.mx",affiliation:'
Faculty of Biological Pharmaceutical Chemistry, Veracruzana, University, Gonzalo Aguirre Beltrán s/n, Zona Universitaria, Xalapa, Veracruz, 91000, Mexico
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Angiogenesis is an important process involved in the growth of primary tumors and metastasis. New approaches for controlling the cancer progression and invasiveness can be addressed by limiting the angiogenesis process. An increasingly large number of natural compounds are evaluated as angiogenesis inhibitors. The chorioallantoic membrane (CAM) assay represents an in vivo attractive experimental model for cancer and angiogenesis studies as prescreening to the murine models. Since the discovery of tumor angiogenesis, the CAM has been intensively used in cancer research. The advantages of this in vivo technique are in terms of low time-consuming, costs, and a lower number of sacrificed animals. Currently, a great number of natural compounds are being investigated for their effectiveness in controlling tumor angiogenesis. Potential reducing of angiogenesis has been investigated by our group for pentacyclic triterpenes, in various formulations, and differences in their mechanism were registered. This chapter aims to give an overview on a number of phytocompounds investigated using in vitro, murine models and the chorioallantoic membrane assay as well as to emphasize the use of CAM assay in the study of natural compounds with potential effects in malignancies.",signatures:"Stefana Avram, Roxana Ghiulai, Ioana Zinuca Pavel, Marius Mioc,\nRoxana Babuta, Mirela Voicu, Dorina Coricovac, Corina Danciu,\nCristina Dehelean and Codruta Soica",authors:[{id:"141027",title:"Dr.",name:"Cristina",surname:"Dehelean",fullName:"Cristina Dehelean",slug:"cristina-dehelean",email:"cadehelean@umft.ro"},{id:"173283",title:"Dr.",name:"Dorina",surname:"Coricovac",fullName:"Dorina Coricovac",slug:"dorina-coricovac",email:"dorinacoricovac@umft.ro"},{id:"186372",title:"Prof.",name:"Corina",surname:"Danciu",fullName:"Corina Danciu",slug:"corina-danciu",email:"corina.danciu@umft.ro"},{id:"186680",title:"Dr.",name:"Roxana",surname:"Ghiulai",fullName:"Roxana Ghiulai",slug:"roxana-ghiulai",email:"roxanaghiulai@yahoo.com"},{id:"197894",title:"Prof.",name:"Codruta",surname:"Soica",fullName:"Codruta Soica",slug:"codruta-soica",email:"roxana.ghiulai@umft.ro"},{id:"197929",title:"Dr.",name:"Stefana",surname:"Avram",fullName:"Stefana Avram",slug:"stefana-avram",email:"stefana.feflea@gmail.com"},{id:"202529",title:"Dr.",name:"Ioana Zinuca",surname:"Pavel",fullName:"Ioana Zinuca Pavel",slug:"ioana-zinuca-pavel",email:"ioanaz.pavel@yahoo.com"},{id:"205584",title:"Mr.",name:"Marius",surname:"Mioc",fullName:"Marius Mioc",slug:"marius-mioc",email:"marius.mioc@umft.ro"},{id:"205585",title:"Dr.",name:"Roxana",surname:"Racoviceanu (Babuta)",fullName:"Roxana Racoviceanu (Babuta)",slug:"roxana-racoviceanu-(babuta)",email:"babuta.roxana@yahoo.com"},{id:"205586",title:"Ms.",name:"Mirela",surname:"Voicu",fullName:"Mirela Voicu",slug:"mirela-voicu",email:"mavoicu@yahoo.com"}],book:{title:"Natural Products and Cancer Drug Discovery",slug:"natural-products-and-cancer-drug-discovery",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"141027",title:"Dr.",name:"Cristina",surname:"Dehelean",slug:"cristina-dehelean",fullName:"Cristina Dehelean",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"186372",title:"Prof.",name:"Corina",surname:"Danciu",slug:"corina-danciu",fullName:"Corina Danciu",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/186372/images/system/186372.jpg",biography:"DANCIU CORINA -researcher, pharmacist, Ph.D. (Medicine, 2013)- Doctorate conducted in co-between 'Victor Babeş” University of Medicine and Pharmacy, Timisoara and the Goethe University Clinic of Frankfurt, Germany; Head of Department of Pharmacognosy; 2018 -Habilitation\r\nPresent: Head of Department of Pharmacognosy, Victor Babeş” University of Medicine and Pharmacy, Timisoara, Faculty of Pharmacy\r\n Competencies: pharmacognostical and physicochemical analysis of pure active compounds or total extracts. In vitro assays on cell culture (expertise in proliferation and apoptosis) and in vivo animal models (melanoma and inflammation). \r\nHigh expertise regarding flavonoids and triterpenes. The total cumulated impact factor is over 80, h-index: 10- ISI Web of Science, 13-Google Scholar,: over 60 publications, national and international prices for the presented research, peer-reviewer and guest editor for some specific magazines in the field, member in the editorial board of 6 international journals ,co-author of a patent.\r\nResearch projects: member in 10 research projects \r\nResearch interest: quantitative and qualitative characterization of vegetal extracts, in vivo and in vitro evaluation of pure natural compounds or total extracts for different types of cancer, anti-inflammatory, antioxidant and antibacterial effect",institutionString:"Victor Babeş University of Medicine and Pharmacy",institution:null},{id:"195746",title:"Prof.",name:"Rosa",surname:"Guzmán-Gerónimo",slug:"rosa-guzman-geronimo",fullName:"Rosa Guzmán-Gerónimo",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidad Veracruzana",institutionURL:null,country:{name:"Mexico"}}},{id:"195750",title:"Dr.",name:"Edna",surname:"Alarcón-Aparicio",slug:"edna-alarcon-aparicio",fullName:"Edna Alarcón-Aparicio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"195751",title:"Dr.",name:"Oscar",surname:"García-Barradas",slug:"oscar-garcia-barradas",fullName:"Oscar García-Barradas",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"195752",title:"Dr.",name:"Jose",surname:"Chavez-Servia",slug:"jose-chavez-servia",fullName:"Jose Chavez-Servia",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"196901",title:"Prof.",name:"Wim",surname:"Quax",slug:"wim-quax",fullName:"Wim Quax",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Groningen",institutionURL:null,country:{name:"Netherlands"}}},{id:"197867",title:"MSc.",name:"Christel L.C.",surname:"Seegers",slug:"christel-l.c.-seegers",fullName:"Christel L.C. Seegers",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"197868",title:"Ms.",name:"Rita",surname:"Setroikromo",slug:"rita-setroikromo",fullName:"Rita Setroikromo",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"204660",title:"MSc.",name:"Tania",surname:"Alarcón-Zavala",slug:"tania-alarcon-zavala",fullName:"Tania Alarcón-Zavala",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}]},generic:{page:{slug:"our-story",title:"Our story",intro:"
The company was founded in Vienna in 2004 by Alex Lazinica and Vedran Kordic, two PhD students researching robotics. While completing our PhDs, we found it difficult to access the research we needed. So, we decided to create a new Open Access publisher. A better one, where researchers like us could find the information they needed easily. The result is IntechOpen, an Open Access publisher that puts the academic needs of the researchers before the business interests of publishers.
",metaTitle:"Our story",metaDescription:"The company was founded in Vienna in 2004 by Alex Lazinica and Vedran Kordic, two PhD students researching robotics. While completing our PhDs, we found it difficult to access the research we needed. So, we decided to create a new Open Access publisher. A better one, where researchers like us could find the information they needed easily. The result is IntechOpen, an Open Access publisher that puts the academic needs of the researchers before the business interests of publishers.",metaKeywords:null,canonicalURL:"/page/our-story",contentRaw:'[{"type":"htmlEditorComponent","content":"
We started by publishing journals and books from the fields of science we were most familiar with - AI, robotics, manufacturing and operations research. Through our growing network of institutions and authors, we soon expanded into related fields like environmental engineering, nanotechnology, computer science, renewable energy and electrical engineering, Today, we are the world’s largest Open Access publisher of scientific research, with over 4,200 books and 54,000 scientific works including peer-reviewed content from more than 116,000 scientists spanning 161 countries. Our authors range from globally-renowned Nobel Prize winners to up-and-coming researchers at the cutting edge of scientific discovery.
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In the same year that IntechOpen was founded, we launched what was at the time the first ever Open Access, peer-reviewed journal in its field: the International Journal of Advanced Robotic Systems (IJARS).
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The IntechOpen timeline
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2004
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Intech Open is founded in Vienna, Austria, by Alex Lazinica and Vedran Kordic, two PhD students, and their first Open Access journals and books are published.
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Alex and Vedran launch the first Open Access, peer-reviewed robotics journal and IntechOpen’s flagship publication, the International Journal of Advanced Robotic Systems (IJARS).
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2005
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IntechOpen publishes its first Open Access book: Cutting Edge Robotics.
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2006
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IntechOpen publishes a special issue of IJARS, featuring contributions from NASA scientists regarding the Mars Exploration Rover missions.
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2008
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Downloads milestone: 200,000 downloads reached
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2009
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Publishing milestone: the first 100 Open Access STM books are published
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2010
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Downloads milestone: one million downloads reached
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IntechOpen expands its book publishing into a new field: medicine.
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2011
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Publishing milestone: More than five million downloads reached
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IntechOpen publishes 1996 Nobel Prize in Chemistry winner Harold W. Kroto’s “Strategies to Successfully Cross-Link Carbon Nanotubes”. Find it here.
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IntechOpen and TBI collaborate on a project to explore the changing needs of researchers and the evolving ways that they discover, publish and exchange information. The result is the survey “Author Attitudes Towards Open Access Publishing: A Market Research Program”.
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IntechOpen hosts SHOW - Share Open Access Worldwide; a series of lectures, debates, round-tables and events to bring people together in discussion of open source principles, intellectual property, content licensing innovations, remixed and shared culture and free knowledge.
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2012
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Publishing milestone: 10 million downloads reached
\\n\\t
IntechOpen holds Interact2012, a free series of workshops held by figureheads of the scientific community including Professor Hiroshi Ishiguro, director of the Intelligent Robotics Laboratory, who took the audience through some of the most impressive human-robot interactions observed in his lab.
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\\n\\n
2013
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IntechOpen joins the Committee on Publication Ethics (COPE) as part of a commitment to guaranteeing the highest standards of publishing.
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\\n\\n
2014
\\n\\n
\\n\\t
IntechOpen turns 10, with more than 30 million downloads to date.
\\n\\t
IntechOpen appoints its first Regional Representatives - members of the team situated around the world dedicated to increasing the visibility of our authors’ published work within their local scientific communities.
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2015
\\n\\n
\\n\\t
Downloads milestone: More than 70 million downloads reached, more than doubling since the previous year.
\\n\\t
Publishing milestone: IntechOpen publishes its 2,500th book and 40,000th Open Access chapter, reaching 20,000 citations in Thomson Reuters ISI Web of Science.
\\n\\t
40 IntechOpen authors are included in the top one per cent of the world’s most-cited researchers.
\\n\\t
Thomson Reuters’ ISI Web of Science Book Citation Index begins indexing IntechOpen’s books in its database.
\\n
\\n\\n
2016
\\n\\n
\\n\\t
IntechOpen is identified as a world leader in Simba Information’s Open Access Book Publishing 2016-2020 report and forecast. IntechOpen came in as the world’s largest Open Access book publisher by title count.
\\n
\\n\\n
2017
\\n\\n
\\n\\t
Downloads milestone: IntechOpen reaches more than 100 million downloads
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Publishing milestone: IntechOpen publishes its 3,000th Open Access book, making it the largest Open Access book collection in the world
We started by publishing journals and books from the fields of science we were most familiar with - AI, robotics, manufacturing and operations research. Through our growing network of institutions and authors, we soon expanded into related fields like environmental engineering, nanotechnology, computer science, renewable energy and electrical engineering, Today, we are the world’s largest Open Access publisher of scientific research, with over 4,200 books and 54,000 scientific works including peer-reviewed content from more than 116,000 scientists spanning 161 countries. Our authors range from globally-renowned Nobel Prize winners to up-and-coming researchers at the cutting edge of scientific discovery.
\n\n
In the same year that IntechOpen was founded, we launched what was at the time the first ever Open Access, peer-reviewed journal in its field: the International Journal of Advanced Robotic Systems (IJARS).
\n\n
The IntechOpen timeline
\n\n
2004
\n\n
\n\t
Intech Open is founded in Vienna, Austria, by Alex Lazinica and Vedran Kordic, two PhD students, and their first Open Access journals and books are published.
\n\t
Alex and Vedran launch the first Open Access, peer-reviewed robotics journal and IntechOpen’s flagship publication, the International Journal of Advanced Robotic Systems (IJARS).
\n
\n\n
2005
\n\n
\n\t
IntechOpen publishes its first Open Access book: Cutting Edge Robotics.
\n
\n\n
2006
\n\n
\n\t
IntechOpen publishes a special issue of IJARS, featuring contributions from NASA scientists regarding the Mars Exploration Rover missions.
\n
\n\n
2008
\n\n
\n\t
Downloads milestone: 200,000 downloads reached
\n
\n\n
2009
\n\n
\n\t
Publishing milestone: the first 100 Open Access STM books are published
\n
\n\n
2010
\n\n
\n\t
Downloads milestone: one million downloads reached
\n\t
IntechOpen expands its book publishing into a new field: medicine.
\n
\n\n
2011
\n\n
\n\t
Publishing milestone: More than five million downloads reached
\n\t
IntechOpen publishes 1996 Nobel Prize in Chemistry winner Harold W. Kroto’s “Strategies to Successfully Cross-Link Carbon Nanotubes”. Find it here.
\n\t
IntechOpen and TBI collaborate on a project to explore the changing needs of researchers and the evolving ways that they discover, publish and exchange information. The result is the survey “Author Attitudes Towards Open Access Publishing: A Market Research Program”.
\n\t
IntechOpen hosts SHOW - Share Open Access Worldwide; a series of lectures, debates, round-tables and events to bring people together in discussion of open source principles, intellectual property, content licensing innovations, remixed and shared culture and free knowledge.
\n
\n\n
2012
\n\n
\n\t
Publishing milestone: 10 million downloads reached
\n\t
IntechOpen holds Interact2012, a free series of workshops held by figureheads of the scientific community including Professor Hiroshi Ishiguro, director of the Intelligent Robotics Laboratory, who took the audience through some of the most impressive human-robot interactions observed in his lab.
\n
\n\n
2013
\n\n
\n\t
IntechOpen joins the Committee on Publication Ethics (COPE) as part of a commitment to guaranteeing the highest standards of publishing.
\n
\n\n
2014
\n\n
\n\t
IntechOpen turns 10, with more than 30 million downloads to date.
\n\t
IntechOpen appoints its first Regional Representatives - members of the team situated around the world dedicated to increasing the visibility of our authors’ published work within their local scientific communities.
\n
\n\n
2015
\n\n
\n\t
Downloads milestone: More than 70 million downloads reached, more than doubling since the previous year.
\n\t
Publishing milestone: IntechOpen publishes its 2,500th book and 40,000th Open Access chapter, reaching 20,000 citations in Thomson Reuters ISI Web of Science.
\n\t
40 IntechOpen authors are included in the top one per cent of the world’s most-cited researchers.
\n\t
Thomson Reuters’ ISI Web of Science Book Citation Index begins indexing IntechOpen’s books in its database.
\n
\n\n
2016
\n\n
\n\t
IntechOpen is identified as a world leader in Simba Information’s Open Access Book Publishing 2016-2020 report and forecast. IntechOpen came in as the world’s largest Open Access book publisher by title count.
\n
\n\n
2017
\n\n
\n\t
Downloads milestone: IntechOpen reaches more than 100 million downloads
\n\t
Publishing milestone: IntechOpen publishes its 3,000th Open Access book, making it the largest Open Access book collection in the world
\n
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