Monolithic overlapping Schwarz preconditioners for nonlinear finite element simulations of laser beam welding processes

TL;DR

This work addresses solving a nonlinear, nonsymmetric saddle-point thermoelastic system arising in laser beam welding, tackled by Newton-Krylov methods with monolithic two-level Schwarz preconditioners. The authors introduce a Generalized Dryja-Smith-Widlund (GDSW) coarse-space preconditioner, $\hat{B}^{-1}_{GDSW} = \Phi K_0^{-1} \Phi^T + \sum_{i=1}^N R_i^T K_i^{-1} R_i$, along with an economic variant that reduces coarse-space effort by using uncoupled subdomain solves. The coarse space is built from interface-driven extensions of basis functions, with $4M$ coarse modes on the interface and careful removal of off-diagonal blocks to handle Dirichlet boundaries in temperature; an economic variant further enforces a block-diagonal structure. Numerical experiments on a moving-melting-pool welded plate show that one-level preconditioning does not scale, while the two-level GDSW implementations maintain low GMRES iteration counts and demonstrate scalability under both weak and strong tests, albeit with challenges from the large temperature jump and geometry. Overall, the study provides scalable, HPC-friendly preconditioning strategies for high-fidelity laser welding simulations using thermoelasticity with Newton-GMRES solvers.

Abstract

Highly resolved finite element simulations of a laser beam welding process are considered. The thermomechanical behavior of this process is modeled with a set of thermoelasticity equations resulting in a nonlinear, nonsymmetric saddle point system. Newton's method is used to solve the nonlinearity and suitable domain decomposition preconditioners are applied to accelerate the convergence of the iterative method used to solve all linearized systems. Since a onelevel Schwarz preconditioner is in general not scalable, a second level has to be added. Therefore, the construction and numerical analysis of a monolithic, twolevel overlapping Schwarz approach with the GDSW (Generalized Dryja-Smith-Widlund) coarse space and an economic variant thereof are presented here.

Figures (3)

• Figure 1: Example of temperature field (left) and norm of the displacements (right) computed after 100 time step iterations.
• Figure 2: Parameters of the material austenitic chrome nickel steel(1.4301) at $20^{\circ}$C richter2010physikalischen.
• Figure 3: Triangulated surface representing the geometry of the melting pool bakir2018numerical.