Highly Scalable Two-level Monolithic Overlapping Schwarz Preconditioners for Thermo-elastoplastic Laser Beam Welding Problems

TL;DR

The paper addresses scalable solution of the nonlinear, nonsymmetric saddle-point systems arising from coupled thermo-elasto-plastic LBW simulations by introducing monolithic two-level Schwarz preconditioners with GDSW-type coarse spaces. It systematically analyzes GDSW, GDSW*, and RGDSW variants, including combinations across fields, implemented in PETSc within the FE2TI framework, with selective coarse-space recycling to reduce setup costs. Numerical experiments on 3D cube and 2D plate geometries demonstrate strong and weak scalability, highlight the impact of coarse-space choices and rotations on GMRES convergence, and validate the approach on a physically motivated LBW plate. The findings indicate that a tunable mix of coarse-space variants — notably GDSW*(T+R)-RGDSW or GDSW(T)-RGDSW — coupled with recycling, delivers robust performance and enables efficient large-scale fully coupled LBW simulations with practical runtimes.

Abstract

A thermo-elastoplastic finite element approach is used to perform the simulation of a laser beam welding (LBW) process. This results in a nonlinear, nonsymmetric saddle point multiphysics system, for which the nonlinearity is handled via the Newton method. The iterative solution of the arising linear system is accelerated by using robust and highly scalable, overlapping Schwarz domain decomposition preconditioners. It is well-known that a one-level method of this type is not scalable and therefore a second level has to be added. Therefore, the construction and numerical analysis of monolithic, two-level overlapping Schwarz preconditioners with different variants of the GDSW (Generalized Dryja-Smith-Widlund) coarse space are presented here. A new and parallel efficient implementation of several variants of GDSW, that is, GDSW, RGDSW, and GDSW*, in PETSc, is introduced, which is usable for multiphysics problems, as, for example, the thermo-mechanical LBW problem considered here. Different combinations of the GDSW variants for the different fields (temperature and displacements) are compared and parallel scalability for realistic problems is shown.

Figures (5)

• Figure 1: Section of temperature field (left) and norm of the displacements (right) for the cube test.
• Figure 1: Strong and weak scalability for GDSW*-RGDSW with and without recycling option over 5 time iterations.
• Figure 2: Sketch of the thin plate geometry and displacements boundary conditions. In red the melting pool, initially placed with the center on the edge of the plate.
• Figure 3: Strong scalability for GDSW(T)-RGDSW with recycling option over 130 time iterations.
• Figure 4: Distribution of the temperature field $\theta$, the norm of the displacements $\Vert \bm{u} \Vert_2$ and the strain component into the direction of the external strain applied ${\varepsilon }_{22}$ at the end of the simulation.