Fast BEM in Time Domain

Wave propagation phenomena are mostly interesting in unbounded domains, e.g., in soil. Applications in visco- or poroelastic media can be found in geomechanics. As the Boundary Element Method (BEM) fulfills the radiation condition it is the preferred method. However, wave propagation problems should be treated in time domain to observe the waves as they evolve. The usual BEM time stepping procedure requires to calculate and to store for every time step a matrix comparable to one static calculation. Especially, in inelastic problems there is no cutoff at some time. Hence, techniques must be developed to establish a data sparse matrix approximation of the overall matrix not only with respect to the spatial variable but also with respect to the time. Unfortunately, both variables are connected in the retarded potentials of the governing integral equation. Further, aiming in the long-term perspective on poroelastic wave propagation the kernels are highly complicated and, therefore, a technique requiring an analytical kernel decomposition seems to be not promising. Here, the Adaptive Cross Approximation (ACA) and/or the so-called `black box` Panel Clustering based on an interpolation of the kernel will be applied. Beside this sparse technique, a fast solution of the final equation system has to be explored, where either iterative solvers or hierarchical LU-solvers will be studied. Also an efficient preconditioner is essential for a fast solution.

Wave propagation phenomena are mostly interesting in unbounded domains, e.g., in soil. Applications in visco- or poroelastic media can be found in geomechanics. As the Boundary Element Method (BEM) fulfills the radiation condition it is the preferred method. However, wave propagation problems should be treated in time domain to observe the waves as they evolve. In the current project, two versions of a convolution quadrature (CQM) based BE formulation have been studied and accelerated. Using the CQM as inverse transformation for a calculation in Laplace domain allows to apply known fast techniques from elliptic problems. Here, the fast multipole method (FMM) with a Chebyschev interpolation of the kernel has been used. With these ingredients a fast BE formulation with an almost linear complexity in storage and computing time has been established. However, the overhead of the FMM and high iteration numbers of the equation solver requires very large problems to be faster than a classical technique. The alternative way to use the CQM directly in time domain has also been studied and improved. Using the representation of the integration weights with a Taylor series expansion allows to identify a sparsity structure of the matrix corresponding to the wave fronts. Combining this storage reduction with a Spline interpolation of the series coefficients gives, finally, a fast method. Summarizing, two ways to accelerate a CQM based BEM have been developed. However, it must be stated that both approaches pay only for very large problem sizes.