Shifted Laplacian multigrid for the elastic Helmholtz equation

The shifted Laplacian multigrid method is a well known approach for preconditioning the indefinite linear system arising from the discretization of the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth's subsurface is the elastic medium. The elastic Helmholtz equation is much harder to solve than its acoustic counterpart, partially because it is three times larger, and partially because it models more complicated physics. Despite this, there are very few solvers available for the elastic equation compared to the array of solvers that are available for the acoustic one. In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We demonstrate the efficiency and properties of our solver using numerical experiments for problems with heterogeneous media in two and three dimensions

On an Uzawa smoother in multigrid for poroelasticity equations

A poroelastic saturated medium can be modeled by means of Biot''s theory of consolidation. It describes the time-dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure. It shall be interpreted as a combination of the symmetric Gauss-Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme

Robust block preconditioners for biot’s model

In this paper, we design robust and efficient block preconditioners for the two-field formulation of Biot’s consolidation model, where stabilized finite-element discretizations are used. The proposed block preconditioners are based on the well-posedness of the discrete linear systems. Block diagonal (norm-equivalent) and block triangular preconditioners are developed, and we prove that these methods are robust with respect to both physical and discretization parameters. Numerical results are presented to support the theoretical results

Robust preconditioners for a new stabilized discretization of the poroelastic equations

In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [45]. The discretization is proved to be well-posed with respect to the physical and discretization parameters, and thus provides a framework to develop preconditioners that are robust with respect to such parameters as well. We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization that leads to a smaller overall problem after static condensation. Numerical tests for both two- and three-dimensional problems confirm the robustness of the block preconditioners with respect to the physical and discretization parameters

LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation

We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.Comment: 20 page

Multigrid Solution of the 3D Elastic Subsurface Stress Field for Heterogeneous Materials in Contact Mechanics

The need to increase efficiency, stimulates the development of new materials tailored to specific applications and thermal/mechanical loading conditions, e.g. by controlling the property variations on a local scale: layered, graded, granular, porous and fibre-reinforced. For design and optimization of such materials the response to specific load conditions must be predicted which requires computer simulations. For applications in contact mechanics and lubrication failure criteria need to be developed which require the stress fields inside the (strongly heterogeneous) material induced by surface loading. The geometrical complexity of the subsurface topography and the need of an accurate solution require the use of a very fine discretization with a large number of elements, especially for three-dimensional problems. This requires optimally efficient numerical algorithms. In this paper the authors demonstrate the capability of Multigrid techniques to compute displacement and stress fields with great detail in strongly heterogeneous materials subject to surface loading, and in a contact mechanics application. Results are presented for a ceramic application and a contact problem of material with multiple inclusions. The efficiency of the method will allow extensive parameter studies with limited computational means. Moreover, it can efficiently be used to derive macroscopic stress-strain relations by simulations of microscopic problems. Also the method can be used for computational diagnostics of materials with specific heterogeneitie