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.

翻译：多重网格方法是一种渐近最优算法，特别适用于大规模数值仿真。然而，该方法需要做出大量算法选择，这些选择对其计算效率有显著影响。与近期利用机器学习技术学习最优多重网格组件的方法不同，本文采用一种互补策略，即利用进化算法从现有独立组件中构建高效的多重网格循环。该技术被应用于激光束焊接过程的有限元仿真中。热弹性行为由时变热弹性方程耦合系统描述，导致非线性且病态的系统。非线性问题通过牛顿法处理，迭代求解器则通过代数多重网格（AMG）预条件子加速，该预条件子采用通过PETSc接口调用的hypre BoomerAMG实现，并作为耦合方程组的整体求解器使用。为进一步提升求解器效率，本文引入了柔性AMG循环，通过层级特定的平滑序列和非递归循环模式扩展了传统循环类型。这些循环结构由包含AMG规则的上下文无关文法引导，通过遗传编程自动生成。数值实验证明了这些方法在提升大规模激光束焊接仿真求解器性能方面的潜力。