Dual-primal FETI Methods Research Articles

ADVANCED DOMAIN DECOMPOSITION METHOD BY LOCAL AND MIXED LAGRANGE MULTIPLIERS

This paper presents development of an improved domain decomposition method for large scale structural problem that aims to provide high computational efficiency. In the previous researches, we developed the domain decomposition algorithm based on augmented Lagrangian formulation and proved numerical efficiency under both serial and parallel computing environment. In this paper, new computational analysis by the proposed domain decomposition method is performed. For this purpose, reduction in computational time achieved by the proposed algorithm is compared with that obtained by the dual-primal FETI method under serial computing condition. It is found that the proposed methods significantly accelerate the computational speed for a linear structural problem.

국부 및 혼합 Lagrange 승수법을 이용한 영역분할 기반 유한요소 구조해석 기법 개발

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre–Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results.

Robust FETI-DP methods for heterogeneous three dimensional elasticity problems

Iterative substructuring methods with Lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses. In particular, results for algorithms belonging to the family of dual-primal FETI methods are presented. The core issue of these algorithms is the construction of an appropriate global problem, in order to obtain a robust method which converges independently of the material discontinuities. In this article, several necessary and sufficient conditions arising from the theory are numerically tested and confirmed. Furthermore, results of numerical experiments are presented for situations which are not covered by the theory, such as curved edges and material discontinuities not aligned with the interface, and an attempt is made to develop rules for these cases.

Preconditioners for the dual-primal FETI methods on nonmatching grids: Numerical study

The FETI-DP method is a substructuring method that uses Lagrange multipliers to match the continuity condition on the subdomain boundaries. For the FETI-DP method on non-matching grids, two different formulations are known with respect to how to employ the mortar matching condition. Keeping step with the developments of the FETI-DP methods, a variety of preconditioners for the FETI-DP operator have been developed. However, there has not been any numerical study for the FETI-DP method, which compares those preconditioners on nonmatching grids while there have been a few papers for numerical study on the comparison of FETI preconditioners. Therefore, we present the numerical study of four different preconditioners for two-dimensional elliptic problems. The numerical results confirm the superiority of the preconditioner by Kim and Lee [1] for noncomparably nonmatching grids, while the superiority of the preconditioner by Dryja and Widlund [2] is confirmed for matching grids and comparably nonmatching grids.

A FETI-DP Preconditioner with A Special Scaling for Mortar Discretization of Elliptic Problems with Discontinuous Coefficients

We consider two-dimensional elliptic problems with discontinuous coefficients discretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a dual-primal FETI method. In this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds. We show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters. Extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations (distribution of subregions, grid assignment, grid ratios, number of subregions) for both continuous and discontinuous problems.

Dual-primal FETI methods for linear elasticity

Dual-primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. These methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them. The algorithms are described in terms of a change of basis that has proven to be quite robust in practice. A theoretical analysis is provided for the case of linear elasticity, and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain. The strategies have already proven quite successful in large-scale implementations of these iterative methods. © 2006 Wiley Periodicals, Inc.

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method.

On the convergence of a dual-primal substructuring method

In the Dual-Primal FETI method, introduced by Farhat et al. (1999), the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by $C(1+\log^2(H/h))$ for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model.