Iterative Substructuring Methods Research Articles

Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems I: Compressible Linear Elasticity

An iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiffness matrix is symmetric and positive definite. The proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse, global solver related to the wire basket of the elements. As in the scalar case, the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree.

A unified framework for accelerating the convergence of iterative substructuring methods with Lagrange multipliers

The FETI algorithms are a family of numerically scalable substructuring methods with Lagrange multipliers that have been designed for solving iteratively large-scale systems of equations arising from the finite element discretization of structural engineering, solid mechanics, and structural dynamics problems. In this paper, we present a unified framework that simplifies the interpretation of several of the previously presented FETI concepts. This framework has enabled the improvement of the robustness and performance of the transient FETI method, and the design of a new family of coarse operators for iterative substructuring algorithms with Lagrange multipliers. We report on both of these new developments, discuss their impact on the iterative solution of large-scale finite element systems of equations by the FETI method, and illustrate them with a few static and dynamic structural analyses on an IBM SP2 parallel processor. © 1998 John Wiley & Sons, Ltd.

An iterative substructuring method for the $p$ -version of the boundary element method for hypersingular integral operators in three dimensions

We study preconditioners for the \(p\)-version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by \(C(1+\log p)\). Here, \(p\) is the polynomial degree of the ansatz functions and \(C\) is a constant which is independent of \(p\) and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.

Iterative substructuring methods for spectral elements: Problems in three dimensions based on numerical quadrature

Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for p-version finite element methods based on continuous, piecewise Q p functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using Gauss-Lobatto-Legendre quadrature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p) 2. These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory.

Parallel penalty combinations for singularity problems

An embedding technique is presented in this paper to implement penalty combinations into parallel by the iterative substructuring method. Penalty combinations using a penalty integral are much simpler than the direct nonconforming constraints to match different admissible functions. In the penalty combinations, the Ritz-Galerkin method is used in the singular subdomains Ω + where there exist solution singularities. The singular particular solutions are chosen to be admissible functions so that only a few of them are needed, to cope well with the difference grids in the finite difference methods used in the rest of the solution domain. Consequently, while applying the iterative substructuring methods, we may attain the singular domain Ω + to the interface of two subregions in the domain decomposition methods, and regard Ω + as a ‘fat’ interface or call an interface ‘zone’. The matrix contribution resulting from Ω + can be included in the preconditioner matrix due to a few of unknown coefficients to be sought. The new embedding technique may reform penalty combinations easily into parallel computing by the existing, domain decomposition methods. Such a technique has been proven to be effective, by a brief analysis and numerical experiments of Motz's problem given in this paper.

A Polylogarithmic Bound for an Iterative Substructuring Method for Spectral Elements in Three Dimensions

Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. A p-version finite element method based on continuous, piecewise $Q_p $ functions is considered for second-order elliptic problems in three dimensions; this special method can also be viewed as a conforming spectral element method. An iterative method is designed for which the condition number of the relevant operator grows only in proportion to $(1 + \log p)^2 $. This bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions. Numerical results are also reported which support the theory.

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted\(L^{2}\) -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.

Schwarz Analysis of Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions

Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange of information that can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A general theoretical framework has previously been developed. In this paper, these techniques are used in an analysis of iterative substructuring methods for elliptic problems in three dimensions. A special emphasis is placed on the difficult problem of designing good coarse models and obtaining robust methods for which the rate of convergence is insensitive to large variations in the coefficients of the differential equation. Domain decomposition algorithms can conveniently be built from modules that represent local and global components of the preconditioner. In this paper, a number of such possibilities are explored, and it is demonstrated how a great variety of fast algorithms can be designed and analyzed.

Towards the ultimate iterative substructuring method: Combined numerical and parallel scalability, and multiple load cases

First, we overview the FETI substructuring method and discuss its numerical scalability properties. Next, we present an extension to structural analysis with multiple load cases, and to the solution of systems with repeated right hand sides. Such systems arise, among others, in linear transient analysis and in the solution of eigenvalue problems. We show that this extension prompts the design of an efficient parallel coarse grid solver which transcends the numerical scalability of the FETI method into a massively parallel scalability. We illustrate these ultimately desirable properties and demonstrate their impact on computational performance with the static and transient analyses via the FETI method of two large-scale three-dimensional structural problems on a 128-processor iPSC-860 system.

An Optimal Domain Decomposition Preconditioner for the Finite Element Solution of Linear Elasticity Problems

For linear elasticity problems the finite element method is an extremely successful method to model complicated structures. The successful implementation requires the solution of very large, sparse, positive definite linear systems of algebraic equations. A new technique for solving these systems using the preconditioned conjugate gradient method is proposed. Using ideas from both additive Schwarz methods and iterative substructuring methods, it is proven that the condition number of the resulting system does not grow as the substructures are made smaller and the mesh is refined. This result holds for two and three dimensions. Numerical experiments have been performed to demonstrate the power of this method. For linear elasticity problems in two dimensions the condition numbers are observed numerically to be less than four when using a regular mesh.

A domain decomposition algorithm for elliptic problems in three dimensions

Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions. A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.

An iterative substructuring algorithm for equilibrium equations

The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.