Abstract
SummarySequentially linear analysis (SLA), an event‐by‐event procedure for finite element (FE) simulation of quasi‐brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λcrit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low‐rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two‐dimensional (2D) and three‐dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury‐based direct solver over its counterparts, especially for large 3D problems.
Highlights
Linear analysis (SLA), wherein the nonlinear response of a structure in a displacement-based finite element (FE) framework is approximated as a series of linear analyses*, has in the past few years been improved to enable2128 wileyonlinelibrary.com/journal/nmeInt J Numer Methods Eng. 2020;121:2128–2146.structural-level applications in the civil engineering field
Thereafter, the computational performance of these reference models are compared against those solved with the Woodbury identity-based method and the preconditioned CG (PCG)
The first benchmark considered is that of an unreinforced brick masonry wall, 1.35m x 1.1m in size and clamped along the top and bottom edges, firstly subject to an overburden/precompression of 0.6 MPa followed by a quasi-static lateral load
Summary
Linear analysis (SLA), wherein the nonlinear response of a structure in a displacement-based finite element (FE) framework is approximated as a series of linear analyses*, has in the past few years been improved to enable. Inspired by remarks made in, for example[17,18] this motivated the need for a tailor-made solver for SLA To address this issue, and efficiently make use of previous stiffness matrix factorisations and solutions, two solution strategies are proposed in this article. Alternative methods combining a traditional incremental-iterative technique and the total approach of SLA are available in the literature addressing the need for a practical alternative,[8] but the focus of this work is to solely improve the performance of SLA with regards to solving the system of linear equations. An adapted direct solution technique based on the Woodbury matrix identity is proposed This identity, the generalisation of the Sherman-Morrison formula (to find the inverse of a rank-1 corrected matrix) to a rank-r correction, allows for cheaper numerical computation of the inverse of a low-rank corrected matrix by avoiding the matrix factorisation every analysis step.
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