A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations

The standard SSOR preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the Biot's consolidation equations. In this paper, a modified block SSOR preconditioner combined with the Eisenstat‐trick implementation is proposed. For actual implementation, a pointwise variant of this modified block SSOR preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness. Numerical experiments show that the proposed modified SSOR preconditioned symmetric QMR solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime. Moreover, the proposed modified SSOR preconditioners can be generalized to non‐symmetric Biot's systems. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations

Finite element simulations of very large‐scale soil–structure interaction problems (e.g. excavations, tunnelling, pile‐rafts, etc.) typically involve the solution of a very large, ill‐conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large‐scale Biot's finite element equations using the symmetric quasi‐minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element‐by‐element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three‐dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

Localized Jacobian ILU preconditioners for hydraulic fractures

We discuss the properties of a class of sparse localized approximations to the Jacobian operator that arises in modelling the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. We demonstrate that an incomplete LU factorization of these localized Jacobians yields an efficient preconditioner for the fully populated, stiff, non‐symmetric system of algebraic equations that need to be solved multiple times for every growth increment of the fracture. The performance characteristics of this class of preconditioners is explored via spectral analysis and numerical experiment. Copyright © 2005 John Wiley & Sons, Ltd.     

Multifrontal incomplete factorization for indefinite and complex symmetric systems

A new class of preconditioners based on the adaptive threshold incomplete multifrontal factorization for indefinite and complex symmetric systems is developed. Numerical experiments consisting of the 3D Helmholtz equations, fluid–structure interaction and localization problems demonstrate the excellent performance of the preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.     

Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure

The direct methods for the solution of systems of linear equations with a symmetric positive‐semidefinite (SPS) matrix A usually comprise the Cholesky decomposition of a nonsingular diagonal block A _???? of A and effective evaluation of the action of a generalized inverse of the corresponding Schur complement. In this note we deal with both problems, paying special attention to the stiffness matrices of floating structures without mechanisms. We present a procedure which first identifies a well‐conditioned positive‐definite diagonal block A _???? of A , then decomposes A _???? by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A _????. The Schur complement S is typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank of A or a lower bound on the nonzero eigenvalues of A are known, then the SVD can be implemented without any ‘epsilon’. Moreover, if the kernel of A is known, then the SVD can be replaced by effective regularization. The results of numerical experiments show that the proposed method is useful for effective implementation of the FETI‐based domain decomposition methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

A fast,memory efficient and robust sparse preconditioner based on a multifrontal approach with applications to finite‐element matrices

In this article, we introduce a fast, memory efficient and robust sparse preconditioner that is based on a direct factorization scheme for sparse matrices arising from the finite‐element discretization of elliptic partial differential equations. We use a fast (but approximate) multifrontal approach as a preconditioner and use an iterative scheme to achieve a desired accuracy. This approach combines the advantages of direct and iterative schemes to arrive at a fast, robust, and accurate preconditioner. We will show that this approach is faster (～2×) and more memory efficient (～2–3×) than a conventional direct multifrontal approach. Furthermore, we will demonstrate that this preconditioner is both faster and more effective than other preconditioners such as the incomplete LU preconditioner. Specific speedups depend on the matrix size and improve as the size of the matrix increases. The preconditioner can be applied to both structured and unstructured meshes in a similar manner. We build on our previous work and utilize the fact that dense frontal and update matrices, in the multifrontal algorithm, can be represented as hierarchically off‐diagonal low‐rank matrices. Using this idea, we replace all large dense matrix operations in the multifrontal elimination process with O(N) hierarchically off‐diagonal low‐rank operations to arrive at a faster and more memory efficient factorization scheme. We then use this direct factorization method at low accuracies as a preconditioner and apply it to various real‐life engineering test cases. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficients

Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse‐grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd.     

Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics

Multiconjugate adaptive optics (MCAO) systems with 10(4)-10(5) degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of adaptive optics degrees of freedom. We develop scalable open-loop iterative sparse matrix implementations of minimum variance wave-front reconstruction for telescope diameters up to 32 m with more than 10(4) actuators. The basic approach is the preconditioned conjugate gradient method with an efficient preconditioner, whose block structure is defined by the atmospheric turbulent layers very much like the layer-oriented MCAO algorithms of current interest. Two cost-effective preconditioners are investigated: a multigrid solver and a simpler block symmetric Gauss-Seidel (BSGS) sweep. Both options require off-line sparse Cholesky factorizations of the diagonal blocks of the matrix system. The cost to precompute these factors scales approximately as the three-halves power of the number of estimated phase grid points per atmospheric layer, and their average update rate is typically of the order of 10(-2) Hz, i.e., 4-5 orders of magnitude lower than the typical 10(3) Hz temporal sampling rate. All other computations scale almost linearly with the total number of estimated phase grid points. We present numerical simulation results to illustrate algorithm convergence. Convergence rates of both preconditioners are similar, regardless of measurement noise level, indicating that the layer-oriented BSGS sweep is as effective as the more elaborated multiresolution preconditioner.     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems

Several domain decomposition methods with Lagrange multipliers have been recently designed for solving iteratively large‐scale systems of finite element equations. While these methods differ typically by implementational details, they share in most cases the same substructure based preconditioners that were originally developed for the FETI method. The success of these preconditioners is due to the fact that, for homogeneous structural mechanics problems, they ensure a computational performance that scales with the problem size. In this paper, we address the suboptimal behaviour of these preconditioners in the presence of material and/or discretization heterogeneities. We propose a simple and virtually no‐cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations. We consider several intricate structural analysis problems, and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities. Copyright © 1999 John Wiley & Sons, Ltd.     

The importance of diagonal dominance in the iterative solution of equations generated from the boundary element method

The unsymmetric matrix equations generated from the boundary element method (BEM) can be solved iteratively, with convergence to the correct solution guaranteed, if the boundary element system of equations can be first transformed into an equivalent, diagonally dominant system. A transformation is presented which selectively annihilates terms in the coefficient matrix of the system Ax = b until an equivalent, row diagonally dominant system, if available, is obtained. The new, row diagonally dominant system is well suited for use with Jacobi and Gauss-Seidel point iterative equation solvers. The diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the BEM. A simple Laplacian problem is used to examine the structure of the BEM equations and to introduce the diagonal dominizing transformation. The importance of diagonal dominance is shown by comparing iterative convergence of positive-definite, symmetric positive-definite and diagonally dominant systems of BEM equations obtained from a plane strain elasticity problem.     

A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non‐symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N³). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub‐sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Incomplete factorization‐based preconditionings for solving the Helmholtz equation

Preconditioning techniques based on incomplete factorization of matrices are investigated, to solve highly indefinite complex‐symmetric linear systems. A novel preconditioning is introduced. The real part of the matrix is made positive definite, or less indefinite, by adding properly defined perturbations to the diagonal entries, while the imaginary part is unaltered. The resulting preconditioning matrix, which is obtained by applying standard methods to the perturbed complex matrix, turns out to perform significantly better than classical incomplete factorization schemes. For realistic values of the GMRES restart parameter, spectacular reduction of iteration counts is observed. A theoretical spectral analysis is provided, in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a Stieltjes matrix extracted from the indefinite matrix. Results of numerical experiments are reported, which display the efficiency of the new preconditioning. Copyright © 2001 John Wiley & Sons, Ltd.     

A FETI‐DP method for the parallel iterative solution of indefinite and complex‐valued solid and shell vibration problems

The dual‐primal finite element tearing and interconnecting (FETI‐DP) domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form ( K ?σ² M ) u = f , and a class of complex problems of the form ( K ?σ² M +iσ D ) u = f , where K , M , and D are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems, i is the imaginary complex number, and σ is a real positive number. A key component of this extension is a new coarse problem based on the free‐space solutions of Navier's equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI‐H method. For this reason, it is named here the FETI‐DPH method. For a practically large σ range, FETI‐DPH is shown numerically to be scalable with respect to all of the problem size, substructure size, and number of substructures. The CPU performance of this iterative solver is illustrated on a 40‐processor computing system with the parallel solution, for various σ ranges, of several large‐scale, indefinite, or complex‐valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Performance and robustness of block constraint preconditioners in finite element coupled consolidation problems

Block constraint preconditioners are a most recent development for the iterative solution to large‐scale, often ill‐conditioned, coupled consolidation problems. A major limitation to their practical use, however, is the somewhat difficult selection of a number of user‐defined parameters (at least 4) in a more or less optimal way. The present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters. A theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters. A number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner (MCP) combined with the symmetric quasi‐minimal residual (SQMR) solver and the MCP triangular variant (T‐MCP) combined with the bi‐conjugate gradient stabilized (Bi‐CGSTAB) are efficient and robust tools for the solution to difficult Finite Element‐discretized coupled consolidation problems. Moreover, the practical selection of the user‐defined parameters is relatively easy as a stable behavior is observed for a wide range of fill‐in degree values. The theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-iterative solution of finite-element equations in incompressible solids

Summary. Finite-element equations for incompressible and near-incompressible media give rise to a matrix with a diagonal block of zeroes or very small numbers. The matrices are not amenable to conventional techniques involving pivoting on diagonal entries. Uzawa methods have been applied to the associated linear systems. They are iterative and converge when the matrix is nonsingular. In the current study an alternate form of the matrix is used which is amenable to a solution without iteration. It likewise is applicable whenever the matrix is nonsingular. The solution process consists of a block LU factorization, followed by Cholesky decomposition of a positive definite diagonal block together with several forward and backward substitution operations. Two illustrative examples are developed.     

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight‐adjusted inner products are easily invertible approximations to weighted L² inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.     

Crack growth behavior and fatigue-life prediction based on worst-case near-threshold data of a large crack for 9310 steel

ABSTRACT Both experimental and analytical investigations were conducted to study crack initiation and growth of small cracks, near‐threshold growth behavior of large cracks at constant R‐ratio/decreasing ΔK and constant K_max/decreasing ΔK, respectively, for 9310 steel. The results showed that a pronounced small‐crack effect was not observed even at R = ?1, small cracks initiated by a slip mechanism at strong slip sites. Worst‐case near‐threshold testing results for large cracks under several K_max values showed that an effect of K_max on the near‐threshold behavior does not exist in the present investigation. A worst‐case near‐threshold test for a large crack, i.e. constant K_max/decreasing ΔK test, can give a conservative prediction of growth behavior of naturally initiated small cracks. Using the worst‐case near‐threshold data for a large crack and crack‐tip constraint factor equations defined in the paper, Newman's total fatigue‐life prediction method was improved. The fatigue lives predicted by the improved method were in reasonable agreement with the experiments. A three‐dimensional (3D) weight function method was used to calculate stress‐intensity factors for a surface crack at a notch of the present SENT specimen (with r/w = 1/8) by using a finite‐element reference solution. The results were verified by limited finite‐element solutions, and agreed well with those calculated by Newman's stress‐intensity factor equations when the stress concentration factor of the present specimen was used in the equations.     

Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport

This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton–Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behaviour of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non‐zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two‐level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large‐scale distributed‐memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two‐ and three‐level preconditioners are demonstrated to be scalable to 1024 processors. Copyright © 2006 John Wiley & Sons, Ltd.