Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

This chapter is devoted to an application of a finite element method formulation to forecast the static and mechanical behavior of catenary-like structures subject to general force distributions, whose development was motivated by the need of installing assemblies of containment structures, called log boom lines, upstream a hydroelectric power plant to protect its integrity from the threats that logs carried through the river pose on it. Each log boom is modeled by a tridimensional truss element and the entire lines by assemblages of trusses. While the external forces, modeled with the aid of both simulations from computational fluid dynamics and experiments from a towing tank, originate from both the river stream and the logs that accumulate through the extension of the lines, the internal forces are calculated from classic expressions of solid mechanics; hence, the numerical method imposes equilibrium between them, which ultimately defines the geometry assembly. Verification and validation were performed at both model and prototype scales, and the results corroborated the accuracy of the tool for a series of flow conditions.


Introduction
The operation of a hydroelectric power plant at the Madeira River is compromised due to the presence of many logs carried through the water stream that can damage the power plant machinery and reduce its production efficiency. To prevent such damage, nine assemblies of containment structures, called log booms (Figures 1 and 2), were installed near the power plant and across the river to retain and deflect these logs. Nevertheless, as the influence of logs and stream can exert large loads over the assemblies, they are under the risk of structural failure; so a numerical tool based on the finite element method (FEM) was developed to predict the distributions of loads and equilibrium configurations of the lines, therefore assisting their surveillance and maintenance. The numerical tool was part of a research and development (R&D) project developed by the Institute for Technological Research (IPT), and it can be extended to similar problems in which the structures behave likewise, as is the case of fishing nets and cleaning structures [1]. Moreover, while commercial software could be used for the same purpose, the intrinsic drawbacks of large computation time, automation, and customization difficulties were generally not justified by the potential increase in accuracy, as the tool would eventually be integrated in a larger numerical simulator.

Numerical model
The mathematical model used in the present formulation is based on the works of Felippa and Gavin [2][3][4], whereas the numerical implementation was developed within the Matrix Laboratory (MATLAB) environment [5]. The material deformations are assumed elastic only (i.e., no plasticity or yielding) as each log boom is  mainly composed of steel, making the unit respond linearly until approximately 80% of the yield stress [6], but the displacements are geometrically nonlinear, as the forces, the geometric stiffness, and the assembly equilibrium configuration depend on one another.

Mathematical model
Consider the three-dimensional truss element composed by six degrees of freedom (DoF) that represent all the possible translation directions its nodes have. Such element is used to model the structural behavior of each log boom ( Figure 1).
With this numbering, the element constitutive stiffness matrix K Ee is (1) while its geometric stiffness matrix K Ge is These matrices represent the stiffness of the truss due to its constitutiveness (as K Ee depends on the stiffness module E e , on the cross-sectional area A ϕ , and on the undeformed length L 0 ) and existence of external influence (counterbalanced by the tension T). Consequently, the tangential element matrix K Te , defined as the overall stiffness, is simply their sum: An assembly is then modeled by the joint of N trusses, representing its N log booms, as in Figure 2; thus, to account for the assembly overall stiffness K T , each K Te is linearly transformed from the truss local coordinate systems to the global assembly counterpart, and they are later superimposed. Now, considering a function g u ð Þ that represents the unbalance between internal and external forces (F and R, respectively), for a given geometric configuration u, there is a relation between g u ð Þ and K T : The equilibrium configuration u * is achieved when g u ð Þ¼0, and in order to find it, g u * ð Þ can be expanded into a Taylor series about an arbitrary initial configuration u j : Truncating Eq. (5) to the first order, u * can be approximated by the unbalance and its first derivative: which, however, is a linear approximation of this nonlinear formulation. Given such nonlinearity, the problem must be solved numerically, and all the equations presented insofar are part of an iterative scheme, in which j (j¼0, 1, …) is the j th iteration. Therefore, numerically speaking, successive expansions must be taken about the consecutive values of u until a satisfactory equilibrium u * Num is achieved within a specified tolerance. So, from Eq. (6), at each iteration an improved equilibrium configuration u jþ1 is obtained. Also, rearranging Eq. (4) in terms of the same improved equilibrium configuration, one obtains: As expected, by similarity, u¼u j , i.e., the tangential matrix is a function of u j . So, the numerical method consists of starting from an initial geometry, calculating the external and internal forces, calculating the assembly tangential matrix, and updating the geometric configuration. This process is repeated until the internal and external forces are sufficiently close.

Force calculations
The loads acting on the assembly originate from the hydrodynamic interaction and from the log accumulation and must be counterbalanced by the internal forces, which are a function of the mechanical properties of the material. Although the formulation is tridimensional, for the purposes by which the model was made, it is sufficient to obtain the equilibrium configuration on the XZ plane. Hence, the calculations presented next will only be on this plane.

External loads
To estimate the external forces, a database was created with the aid of computational fluid dynamics (CFD) (Figure 3). The software Siemens Star-CCM+, version 12.02.011, was used for these numerical simulations: both water and air were admitted incompressible, and the flow field was modeled using the unsteady Reynolds-averaged Navier-Stokes equations implicitly (at the first iterations, the movements were frozen to ensure fluid stabilization and then gradually released, still seeking stabilization); the interaction between the fluids was represented by the volume of fluid Eulerian multiphase model, and the multiphase conditions were defined by the flat wave model with a numerical damping, added in the momentum equation on the vertical direction, on both the inlet and outlet boundary conditions to minimize the effects of reflection and ensure better convergence; finally, the k-ω SST model was adopted.
A series of velocity magnitudes V and incidence angles β were simulated such that normal and tangential coefficients (C z and C x , respectively) were calculated [7][8][9]. From them, fitting curves in the form of Eq. (8) were adjusted, whose values of K depend on V ( Figure 4): Each part of the log boom that is free to rotate was assigned with a pair of coefficients (i.e., the grids and the chassis) such that the total hydrodynamic force acting on the trusses is retrieved based on them. So, for the i th log boom (i ¼ 1, …, NÞ,  A CN and A GN are the chassis and grid front areas, and ρ is the water density. The V and β combinations used in the simulations were arithmetically averaged from velocimetry data measured in situ along a line s that connects the assemblies' mooring points ( Figure 5).
Once the hydrodynamic coefficients were obtained, the contribution from the logs that accumulate along the lines was assumed to be increments on the normal coefficients [10] and estimated with the aid of data from experiments at the towing tank of the IPT ( Figure 6). Thus, considering already such increments, Eq. (9) accounts for the total external forces acting on the center of the element, which are equally distributed to its nodes and further transformed from local to global coordinates as well, according to the angle θ i between the two reference frames: So, R Xi =2 and R Zi =2 act on the X and Z directions of each node element.

Internal loads
The tension distribution throughout the trusses is calculated based on classic linear relations of solid mechanics, and it acts on the axial direction of the element. Example of the variation of the velocity magnitude and incidence angle along a projected line that connects the mooring points. The angles are measured taking the north coordinate of reference; thus, it is not β itself, but a function of it. Therefore, for the same i th log boom, stretched to a new length L i (thus having new coordinates u i ), the internal tension acting upon it is calculated as in Eq. (11): This tension must be transformed to internal forces F i that act on the nodes and are represented on the global coordinate system. Considering the same angle θ i , the internal forces are obtained through a similar transformation:

Initial condition
Originally, the solution strategy was to adopt the line completely stretched along the X direction and then gradually move the leftmost mooring point by small increments Δ ! j toward its final location, while the rightmost mooring point would start at its correct location. Then, at each Δ ! j the iterative scheme was run until convergence ( Figure 7). However, such quasi-static approach is computationally expensive because of the repetitiveness of the iterative procedure. Additionally, Δ ! j has to be carefully chosen to avoid numerical divergence. Thus, an improved initial condition was proposed in which the assembly was initially approximated by a catenary whose "weight vector" was obtained from the average of the hydrodynamic forces along s ( Figure 8). Then, with the initial geometry determined, the external and internal forces could be calculated based on node equilibrium ( Figure 9), as in Eq. (13): are not known a priori (note that there is no "element 0," but T the mooring points is performed, as (13) has only one unknown at any of these locations. The internal forces are then calculated using Eq. (12), and the iterative scheme starts by either stretching the geometry by a small amount δ ! from the catenary geometry or, equivalently, by creating a small unbalance g.

Static condensation
An intrinsic characteristic of structural problems is the presence of fixed degrees of freedom (F) in addition to the free ones (L). Thus, a common practice in FEM, known as static condensation, is to renumber the DoF seeking to reorder and split the linear system into two sub-systems. So, the displacement array can be written as: and the unbalance array can be similarly split:  For the log boom assembly, the fixed DoF are related to the mooring points, so Δu F are their known displacements, and g u F ð Þ are the unknown reaction forces, whereas the free DoF are related to the unknown displacement of the rest of the line; Δu L and g u L ð Þ are the known difference between the external hydrodynamic and internal loads. Thus, the linear system (Eq. (4)) can be rewritten as: Additionally, the matrix K T can be interpreted as follows: Thus, from Eq. (4), These sub-systems are solved recursively: the first of Eq. (18) is solved for Δu j L and, sequentially, the reaction forces as calculated from the second one. Moreover, static condensation is needed regardless of the initial condition used.

Sub-relaxation
Several times, the initial condition itself is not sufficient to ensure convergence because the average velocity used to calculate the catenary "weight" might not create a representative starting geometry. So, whenever the direct method fails, a sub-relaxation parameter is applied to the force unbalance as a way to prevent large deflections through the line, which is especially interesting during the first iterations; the restrained deflections are generally sufficient to adequately converge the simulation. The sub-relaxation Ω, 0 < Ω ≤ 1 is applied in the updating process of g u ð Þ such that Ω is usually a function of the iteration, but it must necessarily achieve unity prior to final convergence. An example of how this parameter varies with the iteration is shown in Figure 10.

Post-processing
After numerical convergence, a series of variables can be outputted, such as the final geometry, the velocity magnitude and incidence angle acting over each log boom, the reaction forces, and the tension distribution along the length of the line L A . These variables, especially the last one, are of great importance, once they help predict whether or not the simulated condition threatens the integrity of the line. Examples of converged line and tension distribution are presented on Figures 11  and 12.

Results
In order to verify and validate the tool, thus assessing the numerical method developed, many simulations were compared to theoretical, experimental, and numerical data. As the ASME Verification & Validation Committee defines [11], validation determines the degree of accuracy that a numerical model has in representing the real world from its perspective, whereas verification evaluates its intrinsic errors and uncertainties. So, the committee states that verification precedes validation, as they must be performed to ensure the numerical reliability of the solution. In this work code and solution verifications were done by systematically refining the discretization and comparing the outputs with theoretical data from the problem of a catenary [12]. Following, the validation was done for two major cases: by comparing the solution with experimental results from a 1:10 scale model tested in the towing tank at the Naval and Ocean Engineering Laboratory of the IPT [13] and by comparing the solution with numerical simulations from SIMPACK® Multi-Body Simulation software in prototype scale [12,14]; the later comparison had to be done against numerical data because it was unfeasible to instrument the in situ assemblies. Additionally, only hydrodynamic cases (i.e., without logs) were compared.

Verification from a theoretical catenary model
While evaluation of the error was performed through comparison with the catenary model, estimation of the error was done through classical Richardson extrapolation [12]. Code and solution verifications were performed by systematic grid refinement, with the coarsest mesh having 100 elements, the finest 800, and in between these two, successive meshes were created such that the number of trusses was consecutively increased by a value of approximately ffiffiffi 2 p , as recommended by the committee [11]. For the analysis, a refinement ratio r that is a dimensionless form of representing the number of elements was defined: h N is a representative grid size that, for the present formulation, assumes the form of Eq. (21): Thus, according to Eq. (20), r¼1 represents the finest grid and r¼0 an extrapolation in which the number of elements would hypothetically tend to infinity, as their length would decrease accordingly. In the convergence region, the behavior of the solution takes the form of Eq. (23), in which f ext is the extrapolated value, p the order of convergence, and C a constant to be determined. From this expression, a fitting curve is plotted against r for the assessment of the convergence behavior: The relative error is simply evaluated by calculating the difference between the theoretical and numerical values: and the uncertainty σ estimated through a grid convergence index (GCI): F S and ψ are assigned, 1.25 and 1.1, respectively, so a safe value for σ is estimated. The catenary used for verification has a total length of L C ¼ 200 m, and the mooring points are at 0; 200 ð Þand 10; 20 ð Þ. A vertical force per unit length F 0 ¼ 617:32 N=m acts throughout, and the minimum and maximum forces over the catenary are used as variables of verification, f h N ð Þ; they are both functions of the curve parameter a, which in turn is obtained by solving a transcendental equation that depends on the geometric parameters and on the force distribution: l is the length of a segment measured from one of the catenary vertices. Figure 13 shows the trends of the numerical forces as a function of the refinement ratio along with the theoretical values, which are F min ≈ 110:793 kN and F min ≈ 133:492 kN. On both cases p is close to 1, i.e., a convergence accuracy of first order. Furthermore, the extrapolated values differ from the theoretical ones by no more than 0:2%, and for the finest grid, the error is within the numerical uncertainty. Finally, all the values are close to the fitting curve, indicating that even for the poorest grids they seem to be within the convergence region: in fact, if the uncertainties remain of the same order regardless of the level of discretization, the disparity between the poorest grid and theoretical value is acceptable [12].
Finally, Figure 14 shows a comparison between numerical and theoretical geometries; as they practically overlap, the verification corroborates the precision of the proposed model. As a note, in order not to be much close to the correct Figure 13. Convergence of the numerical catenary maximum and minimum forces versus refinement ratio. Values are normalized by the theoretical ones. catenary geometry, the verification was conducted with the line completely stretched along the X direction.

Model scale validation
Reduced model experiments were conducted at the facilities of the IPT in a 1:10 scale with the purpose of studying the hydrodynamic behavior of such structures [13]. To do so, the model, developed mainly in polycarbonate and polymer, was constructed with the aid of a laser cutting machine and a 3D printer; the assembly was ballasted by adding lead stripes along the structure but without changing its front area as to minimize drag interference; to test both symmetric and asymmetric setups, log booms were added from a five-unit symmetric configuration, while the left mooring point was offset by distances ΔZ along the flow direction, since the model width was limited by the basin; flat plates were attached on both extremities to reduce the interference of the structures and to better control the flow incidence.
Four S-Type S9M uniaxial load cells, produced by HBM, were used at each extremity, each of which with a nominal measure limit of 500 N, so the tensile forces acting on the leftmost and rightmost log booms could be acquired. Additionally, to understand how the stain behaves on each log boom and also how it is transferred throughout the structure, water strain gages, manufactured by Kyowa Electronic Instruments, were placed on one of the modules, as shown schematically in Figure 15. These locations were appropriately chosen with the aid of numerical simulations from finite element method: the criterion was to locate the stress concentration area that increased the measurement sensitivity while avoided the high strain gradients, since the latter can contribute to unreliable measurements. Figures 16-18 compare experimental and numerical results for three different configurations as the function of the velocity. "LE" indicates "left extremity" and "RE" "right extremity," while "Exp" and "Num" refer to experimental and numerical values. No numerical uncertainty was provided since the number of trusses was maintained equal the number of log booms.
The results agree satisfactorily: the differences are more evident as the free stream velocity increases; the numerical method reproduces the experimental observation that the tensile force is larger at the leftmost log boom, an expected trend since tensile loads are higher for elements that are parallel to the stream; the number of log booms seems not to affect the accuracy of the results, as the error remains practically the same regardless of the configuration; the larger differences are within 6:0% and are possibly due to reasons such as the model material properties, the modeling of the external forces, and the few number of elements. For the   later reason, better agreement could result from finer grids, although the one-toone representation would be lost.

Prototype-scale validation
A prototype-scale validation is presented, and the results were compared to simulations from SIMPACK® Multi-Body Simulation software. The variables of comparison are again the reaction forces at the leftmost and rightmost anchor points, and the relative differences u LE and u RE use the values of SIMPACK® for normalization. In the following tables, SP refers to the results from SIMPACK®, while FEM to the results from the current method. Three validation cases were performed: line 02, line 12, and the set of line 13.

Line 02
The first prototype validation was performed for line 02 with a variable velocity profile, similar to the one presented in Figure 5. An example of converged solution for line 02 is presented in Figure 19, in which the equilibrium configuration, projected line, and velocity distribution are depicted.
The results from SIMPACK® and the current method are presented in [14]: the first simulation considers a 20% increase in the normal external force (C z ), while the second a 50% increase.

Line 12
The second validation was performed for line 12 with several velocity profiles, also similar to the one presented in Figure 5. Analogously, Figures 19 and 20 depict an example of a converged solution. Table 2 presents the comparison between SIMPACK® and the developed formulation. For this set of simulations, the maximum percentage difference happens at the LE, and its value is approximately 6.6%, while at the RE this difference is about 5.2%; the values are comparable to those for line 02, except that the extremities at which the maximum occurs are swapped.

Lines 13
The last validation was conducted for the group of lines 13, i.e., for lines 13A, 13B, 13C, and 13D. Figure 21 is an example of a converged simulation for line 13C/13D.  Particularly for line 13, the magnitude and incidence angle remained constant throughout them, varying from simulation to simulation. In Table 3 these parameters are summarized: the first columns indicate the velocity range (VR) used in each simulation for the choice of the coefficients of Eq. (8), even though the actual velocity magnitude was sometimes not within that range. Table 4 presents the comparison between SIMPACK® and the developed formulation.
The maximum percentage difference of the simulations happens again at the LE and is approximately 7.0% for line 13A/13B and 7.1% for line 13C/13D. At the RE, the difference is about 5.5% for line 13A/13B and 6.3% for line 13C/13D.

Conclusions
A simple truss-based finite element method was proposed to simulate the load distributions and geometric configurations of assemblies that behave in a catenarylike manner, subject to external, variable loads. The formulation was tailored to the particular problem of log booms (structures that retain logs from reaching the machinery of hydropower plants) under the influence of river streams and the logs they convey. The nonlinear formulation imposes equilibrium between internal and external forces so that an iterative scheme must be numerically solved. The method was verified by comparison against analytical results from a theoretical catenary model: the relative error and uncertainty for the maximum and minimum forces were within 0.2%, while the mesh refinement order of convergence was close to 1. The tool was later validated against experimental model-scale data from the towing tank at the Institute for Technological Research [12] and prototype-scale numerical data from commercial software SIMPACK®, and all the results agree adequately [14]: for the experimental data validation, the numerical method was capable of reproducing the observations of the experiments, and the maximum relative discrepancy observed was about 6%. The differences are invariant to the increase in the assembly length, but they seem sensible to variations of the free stream. Likewise, the prototype-scale validations all show adequate agreement, regardless of the line configurations, and a maximum relative error, considering SIMPACK® as the reference, of less than 8%. These percentages corroborate the adequacy of the method for the purpose by which it was developed: to a have a fast, yet reliable, tool, to forecast the tension distributions throughout the lines such that it can be used as a simulator for the assessment of the structural safety of log boom assemblies.