Parallel Algorithms for Orthotropic Problems

Finite element meshes and nodenumberings suitable for parallel solution with equally loaded processors are presented for linear orthotropic elliptic partial differential equations. These problems are of great importance for instance in oil and airfoil industry. The linear systems of equations are solved by the conjugate gradient method preconditioned by modified incomplete factorization, MIC. The basic method presented, is based on fronts of uncoupled nodes and unlike earlier methods it has the advantage of no requirement of a specific orientation of the mesh. This method is however in general restricted to small degree of anisotropy in the differential equation. Another method, which does not suffer from this limitation, uses rotation of the differential equation and spectral equivalence. The rotation is made in such a way that in the new coordinate system, the basic method is applicable. The spectral equivalence property is used for estimation of the condition number of the preconditioned system. Both methods are suitable for implementation on parallel computers. The computer architecture could be single instruction multiple data (SIMD) as well as multiple instruction multiple data (MIMD) with shared or distributed memory. Implementation of the basic method on a shared memory parallel computer shows a significant improvement by use of the MIC method compared to the diagonal scaling preconditioning method.