Patch‐smoother and multigrid for the dual formulation for linear elasticity

Abstract

The dual formulation for linear elasticity, in contrast to the primal formulation, is not affected by locking, as it is based on the stresses as main unknowns. Thus it is quite attractive for nearly incompressible and incompressible materials. Discretization with mixed finite elements will lead to—possibly large—linear saddle point systems. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity for nearly incompressible materials, we propose a multigrid method that is also stable in the incompressible limit. There are two main challenges in constructing a multigrid method for the dual formulation for linear elasticity. First, in the incompressible limit, the matrix block related to the stress is positive semidefinite. Second, the stress belongs to Hdiv and standard smoothers, working for H1 regular problems, cannot be applied. We present a novel patch‐based smoother for the dual formulation for linear elasticity. We discuss different types of local boundary conditions for the patch subproblems. Based on our patch‐smoother, we build a multigrid method for the solution of the resulting saddle point problem and investigate its efficiency and robustness. Numerical experiments show that Dirichlet and Robin conditions work best and eventually lead to textbook multigrid performance.

Multigrid for the dual formulation of the frictionless Signorini problem

“…In this section, we examine the strong formulation of the frictionless Signorini problem for a homogeneous and isotropic body that can come into contact with a rigid obstacle. With respect to the linear elastic case presented in Reference 12, we must take into account also the contact conditions, prescribed on the surface of the contact boundary. For this reason, we first recall the volumetric equations, that are the same as for the linear elastic case, and then discuss the ones on the boundary.…”

“…They could be directly applied on the coarser problems, but would be redundant and make the method suboptimal. Instead, we project them on the coarser levels, as done in Reference 12. However, since the fine global constraints cannot be satisfied anymore on the coarser levels, we cannot ensure sequential energy minimization.…”

“…To fulfill the box‐constraints on the fine level, we could use the standard projected Gauss‐Seidel method, that acts on mono‐dimensional subspaces. However, this smoother is not able to handle the issues explained in Reference 12. First, the stress variable $$$ \boldsymbol{\sigma} $$$ belongs to the non‐Sobolev space $$$ {\mathbf{H}}_{\mathrm{div}} $$$ and divergence‐free components of the error have to be properly smoothed.…”

“…Third, the global equality constraints have to be fulfilled. All these features are solved by the monolithic patch smoother with Robin boundary conditions of parameter $$$ \alpha $$$, introduced in Reference 12 for the dual formulation of linear elasticity. In addition, a fourth difficulty has to be taken into account: the box‐constraints.…”

“…Therefore, on the fine level, we consider a monolithic patch smoother that solves for local corrections that satisfy the box‐constraints and that fulfill local Robin boundary conditions. On coarser levels, we only project the global equality constraints but not the box‐constraints, so we can reuse the linear monolithic patch smoother of Reference 12. As we will see, even though the coarser problems are linear, the nonlinear information at the fine level is transferred to the coarser levels by means of the truncation strategy.…”