Performance analysis of parallel Krylov methods for solving boundary integral equations in electromagnetism

Abstract

SUMMARYWith the advent of high-performance parallel computers, boundary integral methods have received an increasing interest for the solution of electromagnetic scattering problems of electrically large objects. The Galerkin discretization of integral equations leads to dense and complex linear systems whose size increases linearly with the dimension of the scatterer and quadratically with the frequency of the illuminating radiation. For solving realistic high-frequency problems, iterative Krylov methods can be a viable alternative to out-of-core direct methods provided they are used in combination with fast algorithms for the matrix-vector products and robust parallel preconditioners. In this paper, we present experiments with iterative Krylov solvers combined with an approximate inverse preconditioner and the fast multipole method for the matrix-vector products. Numerical results on a set of problems arising from realistics radar cross-section calculations in industry illustrate the potential of the proposed strategy for solving large-scale problems in electromagnetism. The research described in this paper has been developed in the framework of a joint collaboration between CERFACS (Toulouse), EADS-CCR (Toulouse), INRIA-CERMICS (Sophia Antipolis).

Thermal and mechanical numerical modelling of electric discharge machining process

“…Supposing that the conductivity is constant, then a finite difference first-order approximation for Equation (4) can be expressed as Writing Equation (5) for each interior node and taking into account the boundary conditions give a linear algebraic system of equations [A]{T } = {b}, where the matrix [A] is a diagonally dominant one. Gauss-Seidel method convergence is known to be very slow; generalized minimal residual (GMRES) method [13] is a method that has shown its robustness for many problems [14] and it can be supplied for a faster convergence of the resolution of the linear algebraic system; and then the general algorithm taking into account the temperature dependence of the thermal conductivity can be presented as follows: (5)- (9)) using N Gauss-Seidel method iterations and then using GMRES method. 3.…”