Towards Automated Algebraic Multigrid Preconditioner Design Using Genetic Programming for Large-Scale Laser Beam Welding Simulations

TL;DR

The paper tackles the challenge of designing efficient algebraic multigrid preconditioners for large-scale coupled thermo-elastic simulations of laser beam welding. It introduces grammar-guided genetic programming (G3P) to automatically synthesize flexible AMG cycles, implemented via the EvoStencils framework and applied to BoomerAMG in PETSc GMRES as a monolithic preconditioner. The results show that GP-generated cycles yield Pareto-optimal solvers that outperform default and hand-tuned configurations in both solve time and convergence across time steps and problem sizes, with notable improvements up to ~60% faster solves on the largest tested problem. The work highlights the potential of automated solver design for HPC-based FEM simulations and discusses scaling challenges that motivate future enhancements, including block-triangular preconditioners and more problem-aware fitness measures.

Abstract

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.

Figures (10)

• Figure 1: Triangulated surface representing the geometry of the melting pool bakir2018numerical and an example of a discrete representation of it using hexahedral elements.
• Figure 2: A simplified illustration of the evolution of AMG cycles using G3P.
• Figure 3: Representation of the temperature field of a fully coupled thermo-elasticity laser beam welding simulation at time 1.0 s, computed on 128 MPI processes with 1.37 million DOFs.
• Figure 4: Processor topology for MPI parallelism with 89 100 DOFs on 8 processes, where each tile of the welding plate is assigned to a separate process.
• Figure 5: Evolution of AMG-preconditioned GMRES programs (blue dots) with respect to solve time per iteration and convergence, progressing from an initial random population to the final generation (clockwise from top left: initial population, generation 1, generation 10, generation 100).