Berkeley Lab Scientific Computing Seminar

Historically, all our simulation codes have been based on standard numerical methods such as Boundary Integral Equations (BIE) and Partial Differential Equations (PDE). For example, our code ARLENE is a fully BIE code based on a classical Finite Element approximation of surface Integral Equations such as EFIE and CFIE formulations. These formulations lead to linear systems with full matrices that are complex, non-hermitian but symmetric. Although this code can deal with an anisotropic tensor impedance, a better approach to take into account anisotropic media is a hybrid of PDE and BIE methods, as is done in our code ARLAS. In ARLAS, the formulation leads to a linear system with a full submatrix part arising from the BIE method and a very sparse submatrix part arising from the PDE method. Both of these codes use direct methods to solve the linear systems. ARLENE uses our parallel Cholesky-Crout solver, developed in collaboration with IBM; ARLAS uses a Schur complement method--the full submatrix is solved by the same solver as ARLENE, and the solution of the sparse submatrix is solved with a high performance parallel software called EMILIO (using the high performance sparse direct solver PaStiX), which was developed in collaboration with the INRIA ScAlApplix team. The drawback of this approach is the large CPU time for the construction of the Schur complement. This time is due to the fact that we have to perform as many solutions as the number of Degrees of Freedom on the outer boundary.

For our recent applications, these numerical methods tend to be inappropriate for large numbers of Degrees of Freedom: diffculty in constructing the Schur complement, and the limiting size of the full matrix that can be solved with a direct method. Our aim is to have more than 10 millions of unknowns for the inner problem and 1 million for the free space domain on the coupling interface. In order to solve this problem size, we have combined our 3D code Odyssee, an original numerical method based on a Domain Decomposition Method with our code EMILIO with new features, e.g., a new Boundary Integral Equation Method called EID and a parallel Multi-Level Fast Multipole Algorithm.