The degree of a complete polynomial and its corresponding degrees of freedom are listed. Correspondingly, the number of uniform segments on each edge and the number of subelements are also listed.
In this chapter, a new efficient high-order finite volume method for 3D elastic modelling on unstructured meshes is developed. The stencil for the high-order polynomial reconstruction is generated by subdividing the relative coarse tetrahedrons. The reconstruction on the stencil is performed by using cell-averaged quantities represented by the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, the new method has a very local property like the discontinuous Galerkin method. Furthermore, it can be written as an inner-split computational scheme which is beneficial to reducing computational amount. The reconstruction matrix is invertible and remains unchanged for all tetrahedrons, and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. The high-order accuracy in time is obtained by the Runge-Kutta method. Numerical computations including a 3D real model with complex topography demonstrate the effectiveness and good adaptability to complex topography.
- numerical solutions
- computational seismology
- 3D elastic wave
- wave propagation
- high-order finite volume method
- unstructured meshes
Wave propagation based on wave equations has important applications in geophysics. It is usually used as a powerful tool to detect the structures of reservoir. Thus solving wave equations efficiently and accurately is always an important research topic. There are several types of numerical methods to solve wave equations, for example, the finite difference (FD) method [1, 2], the pseudo-spectral (PS) method [3, 4], the finite element (FE) method [5, 6, 7, 8, 9], the spectral element (SE) method [10, 11, 12, 13, 14], the discontinuous Galerkin (DG) method [15, 16, 17, 18], and the finite volume (FV) method [19, 20, 21, 22]. Each numerical method has its own inherent advantages and disadvantages. For example, the FD method is efficient and relatively easy to implement, but the inherent restriction of using regular meshes limits its application to complex topography. The FE method has good adaptability to complex topography, but it has huge computational cost. In this chapter, the FV method is the key consideration.
In order to simulate wave propagation on unstructured meshes efficiently, the FV method is a good choice due to its high computational efficiency and good adaptability to complex geometry. In this chapter an efficient FV method for 3D elastic wave simulation on unstructured meshes is developed. It incorporates some nice features from the DG and FV methods [15, 16, 17, 19, 20, 23] and the spectral FV (SFV) method [24, 25, 26]. In our method, the computational domain is first meshed with relative coarse tetrahedral elements in 3D or triangle elements in 2D. Then, each element is further divided as a collection of finer subelements to form a stencil. The high-order polynomial reconstruction is performed on this stencil by using local cell-averaged values on the finer elements. The resulting reconstruction matrix on all coarse elements remains unchanged, and it can be pre-computed before time evolution. Moreover, the method can be written as an inner-split computational scheme. These two advantages of our method are very beneficial to enhancing the parallelization and reducing computational cost.
The rest of this chapter is organized as follows. In Section 2, the theory is described in detail. In Section 3, numerical results are given to illustrate the effectiveness of our method. Finally, the conclusion is given in Section 4.
2.1 The governing equation
where , , and are the wavefield of particle velocities in , , and directions, respectively; and are the Lamé coefficients and is the density; are the known sources; , , and are the normal stress components while , , and are the shear stresses. For the convenient of discussion, we rewrite Eq. (1) as the following compact form:
where ,, and the matrices , , and are all matrices and can be obtained obviously .
The propagation velocities of the elastic waves are determined by the eigenvalues of matrices , , and and are given by
are the velocities of the compression () wave and the shear () wave velocities, respectively.
2.2 The generation of a stencil
Suppose that the 3D computational domain is meshed by conforming tetrahedral elements :
In practical computations, the integrals in the FV scheme on physical tetrahedral element are usually changed to be computed on its reference element. Figure 1 shows a physical tetrahedron in the physical system, and is transformed into a reference element in the reference system . Let for be the coordinates of physical element . The transformations between system and system will be given in the final subsection of Section 2. For convenience, let and . And denote the transformation from system to system by
and its corresponding inverse transformation by
Inside each the solutions of Eq. (2) are approximated numerically by using a linear combination of polynomial basis functions and the time-dependent coefficients :
where is the degree of freedom of a complete polynomial.
In order to construct a high-order polynomial, we need to choose a stencil. Traditionally, the elements being adjacent to the element are selected to form a stencil. In  three types of stencils, i.e., the central stencil, the primary sector stencil, and the reverse stencil, are investigated. These stencils usually choose neighbors for the 3D reconstruction. Here is the degree of a complete polynomial. Due to geometrical issues, the reconstruction matrix resulting from these stencils may be not invertible. This may happen when all elements are aligned in a straight line . In the following, we propose to partition or in fact its corresponding reference element into finer subelements to form a stencil. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. Moreover, this algorithm is easy to implement especially in 3D and for all elements whether they are internal or boundary elements.
Let be the number of subelements in after subdividing. For a complete polynomial of degree in 3D, a reconstruction requires at least subelements, where
In our algorithm, we guarantee is always greater than . As shown in Figure 2, we divide each edge of the reference element into uniform segments. Thus we have tetrahedral subelements in . Note that a small subcubic in consists of six tetrahedrons. With the transformations of Eqs. (6) and (7), we denote all subelements in for a fixed by for . In Table 1, the degree of a complete polynomial and its corresponding degrees of freedom are listed. Correspondingly, the number of and are also listed in Table 1. This algorithm for generating the stencil is easily implemented for all coarse tetrahedrons. Moreover, the reconstruction matrix resulting from this stencil is always invertible and remains unchanged for all elements for . Note that the reconstruction matrix may be not invertible if all elements are aligned on a straight line . However, this will not happen here for our algorithm.
2.3 The high-order polynomial reconstruction
The high-order polynomial is reconstructed in each element or . For the stencil designed above, we have
where is the index for subelements in . The FV method will use the cell-averaged quantities, i.e.,
to reconstruct a high-order polynomial, where represents the volume of the subelement . The time variable in is omitted for discussion convenience. The reconstruction requires integral conservation for in each subelement , i.e.,
To solve the reconstruction problem, inspired by the DG method [15, 16, 17, 23, 28, 29], we use hierarchical orthogonal basis functions. The basis functions of a complete polynomial of degree () in the reference coordinate system can be found in . We remark that the basis functions are orthonormal and satisfy the following property:
where is in fact the reference element and is the transformed element corresponding to the subelement .
The integration in Eq. (14) over in system can be computed efficiently if it is performed over its reference element in a second reference system . Denote the transformation from to and its inverse by and , respectively. Transforming Eq. (14) into system and rewriting the result as a compact form, we have
where is the matrix with entries given by
We need at least subelements in the stencil since the reconstructed number of degrees of freedom is . As listed in Table 1, subelements are used to form the stencil. Note that is definitely larger than , which is helpful to improve the reconstruction robustness [20, 21]. Thus Eq. (15) is an overdetermined problem. We use the constrained least squared technique to solve it.
where is the Lagrangian multiplier and both and are matrices:
2.4 The spatial discrete formulation
Using Eq. (8) and integration by parts yield
where is the index number of coarse tetrahedral element neighboring subelement . The notation denotes applying the absolute value operator of the eigenvalues given in Eq. (3), i.e.,
where is the matrix and its columns are made up of the eigenvectors associated with eigenvalues in Eq. (3), i.e.,
And is the rotation matrix given by
where is the normal vector of the face and and are the two tangential vectors. denotes the inverse of .
where is the area of the -th face of subelement . and are the left flux matrix and the right state flux matrix, respectively, which are given by
where and are the face parameters. The transformation of the face parameters and to the face parameters and in the neighbor tetrahedron depends on the orientation of the neighbor face with respect to the local face of the considered tetrahedron. And the mapping is given in Table 2. For a given tetrahedral mesh with the known indices and , there are only 4 of 12 possible matrices per element [15, 20]. Comparing with the traditional FV method, the method with the splitting form described above has much less computations of face integrations. Note that only our proposed FV method can be written as a splitting form. Theoretical analysis shows our method can save about half computational time under the condition of the same number of elements .
2.5 The time discretization
Equation (27) is in fact a semi-discrete ordinary differential equation (ODE) system. In order to solve it formally, we denote the spatial semi-discrete part in Eq. (27) by a linear operator . Then Eq. (27) can be written as a concise ODE form:
Traditionally, the classic fourth-order explicit RK (ERK) method
can be applied to advance from to . Here is the time step. Now we use the low-storage version of ERK (LSERK) to solve Eq. (32):
where is the wave velocity and is the minimum diameter of the circumcircles of tetrahedral elements. This condition is a necessary condition for discrete stability, and a bit more restrictive form is actually used in numerical computations.
The absorbing boundary conditions (ABCs) in computations are required as the computational domain is finite. There are two typical ABCs to be adopted here. One is flux type ABCs [16, 19]. That is to say, the following numerical flux in Eq. (23) at all tetrahedral faces that coincide with domain boundary
which allows only for outgoing waves and is equivalent to the first order ABCs. Though the absorbing effects of this method vary the angles of incidence, it is still effective in many cases . The advantage of this type ABCs is that it merged into the FVM framework naturally and there is almost no additional computational cost. Another type is the perfectly matched layer (PML) technique originally developed by , which is very popular in recent more 10 years.
2.6 Coordinate transformation
The transformation between different coordinate systems is frequently used. For ease of reading, we present the formulations here. Let for be the coordinates of a physical element. The transformation from system to system is defined by
then the transformation from system to system can be solved for and from Eq. (37) by the Cramer ruler, i.e.,
Note that is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the tetrahedron element .
The coordinate transformation from the second reference coordinate to system is defined by
then the transform from system to system can be solved for from Eq. (41) by Cramer ruler similarly. Denote
which is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the subelement for . In Eqs. (41) and (42), for , denote the vertex coordinates of in system.
3. Numerical computations
In this section we give three numerical examples to illustrate the performance of the developed method above. The convergence test of the proposed method can be found in . Though the method is developed for the 3D case, it can be simplified to 2D without essential difficulty. The principle is the same. The first example is a test for a 2D model with uneven topography. The other two examples are for two 3D models.
Example 1. The first example is a two-layered model with the inclined interface shown in Figure 3a. The range of the model is and . The surface of the model is uneven to imitate the real topography. The and velocities are and in the upper layer and and in the lower layer, respectively. The densities are and in the upper and lower layer, respectively. Figure 3b is the coarser triangular meshes for this model. A coarser version of the mesh is shown here as the finest mesh in computations cannot be seen clearly. The triangular meshes can fit the curve topography very well. Note that none triangular element crosses the interface. In computations the polynomial reconstruction is applied. The computational domain is meshed by 113472 coarse elements. Each coarse element is subdivided into 25 subelements further. So there are 2,836,800 fine elements totally. The time step is . The source is located at with time history
where is the main frequency. In order to simulate point source excitation, a spatial local distribution function defined by
is applied, where are positions of the source center. The source is added to the component; that is to say, all source terms except in Eq. (1) are all zero. Figure 4 is the snapshots of and components at propagation time . Figure 5 is the snapshots of and components at propagation time . We can see the wave and wave propagate toward out of the model. The reflected and transmitted waves due to the tilted physical interface are also very clear. These are the expected physical phenomena of wave propagation in elastic media.
Example 2. The second example is a cuboid model. The physical size of the model is . The model and its unstructured tetrahedral meshes are shown in Figure 6. There are totally 836,612 coarse tetrahedrons to mesh the model. A coarser mesh is shown as the actual mesh in computations is too fine to see clearly. Each coarse tetrahedron is subdivided into subelements as we adopt polynomial reconstruction. The parameters for , , and are , , and . The time step in computations is . The source is located in the center of the model with time history given by
It is applied to the component. The 3D snapshots of , , and components at propagation time are shown in Figure 7. From these figures, we can clearly see two types of waves, i.e., the compressive wave and the shear wave. The splitting PML in nonconvolutional form is adopted here , and the boundary reflections are absorbed obviously and effectively. The message passing interface (MPI) parallelization based on spatial domain decomposition is applied. The CPU time for extrapolation 1000 time steps is about with 128 processors each with 2.6 GHz main frequency.
Example 3. The third example is a real geological model in China. As shown in Figure 8a, it has a very complex topography. The physical scope of the model is , , and . The corresponding 3D mesh is shown in Figure 8b. A coarser version of the mesh is given as the actual mesh in computations is too fine to see clearly in the figure. The model is meshed with 210,701 relative coarse tetrahedral elements. Each coarse tetrahedron is subdivided into subelements as we adopt polynomial reconstruction, and thus there are 13,484,864 fine elements totally. The time step is . The source is situated at with the same time history in Eq. (45). The media velocities of and are and . The MPI parallelization based on spatial domain decomposition is applied. The nonconvolutional splitting PML  is adopted. The 3D snapshots of , , and components at propagation time are shown in Figure 9. The CPU time for extrapolation 10,000 time steps is with 256 processors each with 2.6 GHz main frequency. From Figure 9, we can see clearly the propagation of wave and wave.
A new efficient high-order finite volume method for the 3D elastic wave simulation on unstructured meshes has been developed. It combines the advantages of the DG method and the traditional FV method. It adapts irregular topography very well. The reconstruction stencil is generated by refining each coarse tetrahedron which can be implemented effectively for all tetrahedrons whether they are internal or boundary elements. The hierarchical orthogonal basis functions are exploited to perform the high-order polynomial reconstruction on the stencil. The resulting reconstruction matrix remains unchanged for all tetrahedrons and can be pre-computed and stored before time evolution. The method preserves a very local property like the DG method, while it has high computational efficiency like the FV method. These advantages facilitate 3D large-scale parallel computations. Numerical computations including a 3D real physical model show its good performance. The method also can be expected to solve other linear hyperbolic equations without essential difficulty.
I appreciate Dr. Y. Zhuang, Prof. Chung, and Dr. L. Zhang very much for their important help and cooperation. This work is supported by the National Natural Science Foundation of China under the grant number 11471328 and 51739007. It is also partially supported by the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.