An Algebraic Multigrid Method for Linear Elasticity

Abstract

Abstract. We present an algebraic multigrid (AMG) method for the efficient solution of linear (block-)systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block-interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well-known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio ν. We obtain rates ρ < 0.4 for ν < 0.4. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

“…Most commonly used is the point-based or node-based approach, where all variables discretized at a common spatial node are treated together in the coarsening and interpolation processes. This approach was first proposed for linear elasticity in [39] and has been extended in [24,31]. An extension framework to improve AMG for elasticity problems was proposed in [8], which uses a hybrid approach with nodal coarsening, but interpolation based on the unknown-based approach.…”

Improving algebraic multigrid interpolation operators for linear elasticity problems

“…This approach is outlined in early AMG papers (see, e.g. [3,6]) and studied more recently in [7] and extensively in [8]. Specifically, if each of the p unknowns is discretized on the same n grid points, then we have p variables at each point and can write A with a point-wise ordering:…”

“…[7,8,14]), though other useful metrics are possible (see examples in [8]). The scalar entries {c i j } of the 'condensed' matrix C are then used to determine the strength of connection as in classical AMG.…”

“…It is not clear that this technique could be easily extended to 3D problems. In [7], the authors investigate the node-based AMG approach for linear elasticity problems. They show that for a specific simple mesh and set of boundary conditions, the RBMs are reproduced by a node-based interpolation scheme.…”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…Controllability methods have been proposed for both Helmholtz and Navier problems in [17,18]. Multigrid methods have been considered for acoustic and elastic problems in [19,20,21,22]. With multigrid methods, it is difficult to define a stable and sufficiently accurate coarse grid problem and smoother for it.…”