The Schwarz alternating method in solid mechanics

Journal of Computational Physics, 2010

In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in , an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid-solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of nonconformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle. (D. Nordsletten). The level of under/over refinement seen is relative to the governing physics and how one chooses to best approximate them.

Lecture Notes in Computer Science, 2001

Edge-based data structures are used to improve computational efficiency of Inexact Newton methods for solving finite element nonlinear solid mechanics problems on unstructured meshes. Edge-based data structures are employed to store the stiffness matrix coefficients and to compute sparse matrix-vector products needed in the inner iterative driver of the Inexact Newton method. Numerical experiments on threedimensional plasticity problems have shown that memory and computer time are reduced respectively by factors of 4 and 6, compared with solutions using element-by-element storage and matrix-vector products.

Low-order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill-conditioning, when the material approaches the incompressible limit, if only the displacement variables are used. Mixed finite elements using both displacement and pressure variables provide a well-known remedy, but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging. Two-level overlapping Schwarz preconditioners for the almost incompressible system of linear elasticity, discretized by mixed finite elements with discontinuous pressures, are constructed and analyzed. The preconditioned systems are accelerated either by a GMRES (generalized minimum residual) method applied to the resulting discrete saddle point problem or by a PCG (preconditioned conjugate gradient) method applied to a positive definite, although extremely ill-conditioned, reformulation of the problem obtained by eliminating all pressure variables on the element level. A novel theoretical analysis of the algorithm for the positive definite reformulation is given by extending some earlier results by Dohrmann and Widlund. The main result of the paper is a bound on the condition number of the algorithm which is cubic in the relative overlap and grows logarithmically with the number of elements across individual subdomains but is otherwise independent of the number of subdomains, their diameters and mesh sizes, the incompressibility of the material, and possible discontinuities of the material parameters across the subdomain interfaces. Numerical results in the plane confirm the theory and also indicate that an analogous result should hold for the saddle point formulation, as well as for spectral element discretizations.

Contemporary Mathematics, 1994

Domain decomposition methods based on the Schwarz framework were originally proposed for the h-version nite element method for elliptic problems. In this paper, we consider instead the p-version, in which increased accuracy is achieved by increasing the degree of the elements while the mesh is xed. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the nite element discretization. For a class of overlapping additive Schwarz methods, we prove a constant bound, independent of the degree p and the number of elements N, for the condition number of the iteration operator. This optimal result holds in two and three dimensions for additive and multiplicative schemes, as well as variants on the interface. We then study local re nement for the same class of overlapping methods in two dimensions. A constant bound holds under certain hypotheses on the re nement region, while in general an almost optimal bound with logarithmic growth in p is obtained.

2005

This paper extends previous results on nonlinear Schwarz preconditioning ([4]) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in [8]. Then, the algebraic construction from [9] of the corresponding nonlinear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. Numerical illustration is also provided.

Lecture Notes in Computational Science and Engineering, 2014

Numerische Mathematik

We propose two variants of the overlapping additive Schwarz method for the finite element discretization of scalar elliptic problems in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces. In the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges. The functions that constitute the coarse spaces are chosen adaptively, and they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.

Computer Methods in Applied Mechanics and Engineering, 2007

The uniform convergence of finite element approximations based on a modified Hu-Washizu formulation for the nearly incompressible linear elasticity is analyzed. We show the optimal and robust convergence of the displacement-based discrete formulation in the nearly incompressible case with the choice of approximations based on quadrilateral and hexahedral elements. These choices include bases that are well known, as well as newly constructed bases. Starting from a suitable threefield problem, we extend our α-dependent three-field formulation to geometrically nonlinear elasticity with Saint-Venant Kirchhoff law. Additionally, an α-dependent three-field formulation for a general hyperelastic material model is proposed. A range of numerical examples using different material laws for small and large strain elasticity is presented.

2016

We propose two variants of the overlapping additive Schwarz method for the finite element discretization of the elliptic problem in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problem on the faces, and in the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and on subdomain edges. The functions that are used to build the coarse spaces are chosen adaptively, they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases, is shown to be independent of the variations in the...

2020

A nonlinear block-coupled Finite Volume methodology is developed for large displacement and large strain regime. The new methodology uses the same normal and tangential face derivative discretisations found in the original fully coupled cell-centred Finite Volume solution methodology for linear elasticity, meaning that existing block-coupled implementations may easily be extended to include finite strains. Details are given of the novel approach, including use of the Newton-Raphson procedure on a residual functional defined using the linear momentum equation. A number of 2-D benchmark cases have shown that, compared with a segregated procedure, the new approach exhibits errors with many orders of magnitude smaller and a much higher convergence rate.