Study on the numerical solution of huge boundary value problems in earthquake engineering

在1998年，我们研究了弹性静力裂纹问题的公式和数值分析。我们已经表明，一个Galerkin制定允许分析的三维问题与几十万自由度与一台PC。然而，我们发现，从精度的观点来看，原来计划用于弹性动力学的对角形式的使用是不可取的。在1999年，我们开始研究亥姆霍兹方程，特别注意使用维格纳3-j符号。然后，我们继续在2和3维弹性动力学的FMM配方。新开发的配方使用2多极矩在2D问题，4在3D问题，与Galerkin的向量，要求4或6的时刻，在2D或3D问题，分别与方法。这种改进使我们能够开发更有效的实现弹性动力学FMM，无论是在2D和3D中，相比之前提出的。接下来，我们考虑了使用MPI和PC集群的静态代码的并行化。证明了代码的可扩展性，并尝试扩展到弹性动力学。最后，我们考虑了基于拉普拉斯方程指数展开的新的FMM。当问题的几何形状复杂时，新的公式比原来的FMM更有效。

In 1998 we have investigated formulations and numerical analyses for elastostatic crack problems. We have shown that a Galerkin formulation allows analyses of three dimensional problems with several hundreds of thousands of DOF with one PC. We have found, however, that the use of the diagonal form originally planned for elastodynamics is not desirable from the point of view of the accuracy. In 1999, we started the investigation with the Helmholtz equation, paying particular attention to the use of the Wigner 3-j symbols. We then proceeded to the FMM formulations in 2 and 3 dimensional elastodynamics. The newly developed formulation uses 2 multipole moments in 2D problems, and 4 in 3D problems, in contrast to the approach with Galerkin's vector which calls for 4 or 6 moments in 2D or 3D problems, respectively. This improvement enabled us to develop more efficient implementations for the elastodynamic FMM, both in 2D and 3D, compared to those proposed earlier. We next considered the parallelisation of the code in statics, using MPI and a PC cluster. The scalability of the code has been proved, and the extension to elastodynamics was attempted. We finally considered the new FMM based on the exponential expansion for Laplace's equation. The new formulation was found to be more efficient than the original FMM when the geometry of the problem is complicated.