Improved Parallel Preconditioners for Multiphysics Topology Optimizations

Topology optimization, also known as layout optimization, involves multiple physics encompassing structural, thermal, fluidic electrical and electromechanical systems including coupled phenomena such as solids and fluids as in convective cooling systems, aerodynamical systems, electronics, actuators and motors. In the root of topology optimization is the repeated solution of the finite element equations Au = f representing the physics of the problem at hand such as elasticity, heat transfer, fluid flow and electromagnetics, where A is the coefficient matrix, which is highly sparse, u is the vector of physical unknowns (displacements, temperature or velocity, etc.), f is the known source (load) vector. Numerical difficulties associated with these equations are two folds. Firstly, they are very large systems, due to large number of cells (finite elements) needed for defining a topology, requiring the use of the iterative solvers such as the conjugate gradient algorithm instead of the direct solvers. Even with that, the solution times tend to become very high, which is often tried to be resolved by parallel solution strategies. Secondly, they are extremely ill-conditioned due to highly heterogeneous material distributions that evolve during the course of topology formations, with material properties of elements varying from nearly zero values in empty regions to very large values in full regions, which is usually addressed by matrix preconditioning. When simple preconditioners, such as diagonal and incomplete LU factorization preconditioners, are used parallel efficiency deteriorate very fast due to increased number of iterations needed for convergence as more processors are used. In this paper, we modify our previous parallel topology solver implementation by introducing new matrix reordering and scaling schemes to improve the scalability of iterative solvers. We use here more sophisticated preconditioners in order to improve the parallel scalability of the iterative algorithm, even for ill-conditioned cases; not only due to heterogeneous material properties, but also due to condition number deterioration in the equations of coupled multiphysics media. The developed solver was tested for accuracy and parallel efficiency of extreme cases, demonstrating high parallel scalability