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

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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.