Paper

Multigrid methods are asymptotically optimal algorithms ideal for large-scale simulations. But, they require making numerous algorithmic choices that significantly influence their efficiency. Unlike recent approaches that learn optimal multigrid components using machine learning techniques, we adopt a complementary strategy here, employing evolutionary algorithms to construct efficient multigrid cycles from available individual components.

This technology is applied to finite element simulations of the laser beam welding process. The thermo-elastic behavior is described by a coupled system of time-dependent thermo-elasticity equations, leading to nonlinear and ill-conditioned systems. The nonlinearity is addressed using Newton’s method, and iterative solvers are accelerated with an algebraic multigrid (AMG) preconditioner using hypre BoomerAMG interfaced via PETSc. This is applied as a monolithic solver for the coupled equations.

To further enhance solver efficiency, flexible AMG cycles are introduced, extending traditional cycle types with level-specific smoothing sequences and non-recursive cycling patterns. These are automatically generated using genetic programming, guided by a context-free grammar containing AMG rules. Numerical experiments demonstrate the potential of these approaches to improve solver performance in large-scale laser beam welding simulations.

Scalar wave propagation analysis is one of the fundamental types of analysis used in many fields and has been the subject of much research. As measurement data accumulates, the need for faster and more accurate analysis using more detailed models has arisen. This paper proposes tetrahedral and voxel finite elements based on orthogonal discontinuous functions that enable fast and accurate analysis. Through accuracy and cost analysis on recent computers with actual implementations, we show that the cost of analysis can be significantly reduced and that faster and more accurate wave analysis can be expected as shown in the application example. In addition, many problems lead to operations with a large number of relatively small matrix-vector products like the problem in this paper. This paper showed that such computation can be handled efficiently by implementations taking advantage of recent computers, and is expected to provide insight for problems with similar operations.