2:00 PM - 2:15 PM
[STT39-02] Application of Hierarchical matrices to 3D dynamic stress wave kernel: Origin of errors and computational efficiency
Keywords:High performance computing, Boundary element method, H-matrices, Elastodynamics
The spatiotemporal boundary integral equation method is a powerful simulation tool for its easiness to treat complex boundary surface geometries numerically and its accuracy originated in semi-analytical approach, therefore it is widely used for dynamic rupture simulations. However, there is a common issue in the computational sciences in the large calculation time and memory usage of the integral kernel matrix-vector products, which are proportional to the square of the number of elements N (O(N^2 )). While "Sato and Ando" ("2021" ) developed a quasi-O(N) method called Fast Domain Partitioning=Hierarchical matrices method (FDP=H-matrices method), the application is limited to 2"D" problems and the implementation without the MPI parallelization.
In this study, we first developed the code that extends FDP=H matrices method with Lattice H matrices method (LH-matrices method)(Ida,2018), which is highly efficient in large parallel computers. We also made improvements of the algorithms, including the formulations of the temporal convolution integral that replaces the previous approximated one to the rigorous one. In the first half, the algorithm of this new method called FDP=LH matrices method is described.
In the latter half, we present about numerical experiments in which FDP=LH matrices method was applied to 3"D" stress wave propagation problems and evaluated the dependence of parameters on calculation errors and efficiency performance. As a result of the numerical experiments, it is found that the numerical compressibility by the low rank approximation is less efficient for the integral kernel of the "" dynamic stress wave field due to its strong orientation dependence between source-receiver points (radiation pattern) than that of the static stress field with the weaker orientation dependence. However, we find that the error attributed to the low rank approximation is negligibly smaller than that of the representative point approximation method used to approximate the time dependent terms of the kernels and the travel times"" , therefore in practice, the low rank approximation works quite well even for the dynamic kernels in reducing the number of numerical operations and memory consumption . This error factor was not apparent in 2"D" problems of the previous research due to the small azimuthal dependence and is quantitatively revealed by the present 3"D" analysis in this study. It is also found that the error strongly depends on the parameter η, which control the admissibility condition of the hierarchical matrices approximation, , and higher accuracy is possible with smaller η, giving the smaller sizes of the hierarchical sub-matrices. The measurements of calculation time and memory consumption with 100 MPI processes confirm that the costs are of O(NlogN) as predicted by the theory. It is shown that this method is effective in large scale calculation.
In this study, we first developed the code that extends FDP=H matrices method with Lattice H matrices method (LH-matrices method)(Ida,2018), which is highly efficient in large parallel computers. We also made improvements of the algorithms, including the formulations of the temporal convolution integral that replaces the previous approximated one to the rigorous one. In the first half, the algorithm of this new method called FDP=LH matrices method is described.
In the latter half, we present about numerical experiments in which FDP=LH matrices method was applied to 3"D" stress wave propagation problems and evaluated the dependence of parameters on calculation errors and efficiency performance. As a result of the numerical experiments, it is found that the numerical compressibility by the low rank approximation is less efficient for the integral kernel of the "" dynamic stress wave field due to its strong orientation dependence between source-receiver points (radiation pattern) than that of the static stress field with the weaker orientation dependence. However, we find that the error attributed to the low rank approximation is negligibly smaller than that of the representative point approximation method used to approximate the time dependent terms of the kernels and the travel times"" , therefore in practice, the low rank approximation works quite well even for the dynamic kernels in reducing the number of numerical operations and memory consumption . This error factor was not apparent in 2"D" problems of the previous research due to the small azimuthal dependence and is quantitatively revealed by the present 3"D" analysis in this study. It is also found that the error strongly depends on the parameter η, which control the admissibility condition of the hierarchical matrices approximation, , and higher accuracy is possible with smaller η, giving the smaller sizes of the hierarchical sub-matrices. The measurements of calculation time and memory consumption with 100 MPI processes confirm that the costs are of O(NlogN) as predicted by the theory. It is shown that this method is effective in large scale calculation.