An element-based displacement preconditioner for linear elasticity problems

Augarde, Charles E.; Ramage, Alison; Staudacher, Jochen

doi:10.1016/j.compstruc.2006.08.057

Cited by 23 publications

(35 citation statements)

References 16 publications

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“…However, iterative solver (Hestenes and Stiefel 1952; Augarde et al 2006) can solve large-scale problems efficiently. To this end, line 72 is replaced by a built-in MATLAB function pcg, called preconditioned conjugate gradients method, as shown in the following Direct solver is a special case by setting the preconditioner (line 72c) to Table 3 gives the comparison of two different finite element analysis solvers.…”

Section: Iterative Solvermentioning

confidence: 99%

An efficient 3D topology optimization code written in Matlab

Tovar

2014

Struct Multidisc Optim

504

243

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This paper presents an efficient and compact MATLAB code to solve three-dimensional topology optimization problems. The 169 lines comprising this code include finite element analysis, sensitivity analysis, density filter, optimality criterion optimizer, and display of results. The basic code solves minimum compliance problems. A systematic approach is presented to easily modify the definition of supports and external loads. The paper also includes instructions to define multiple load cases, active and passive elements, continuation strategy, synthesis of compliant mechanisms, and heat conduction problems, as well as the theoretical and numerical elements to implement general non-linear programming strategies such as SQP and MMA. The code is intended for students and newcomers in the topology optimization. The complete code is provided in Appendix C and it can be downloaded from http://top3dapp. com.

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Section: Iterative Solvermentioning

confidence: 99%

An efficient 3D topology optimization code written in Matlab

Tovar

2014

Struct Multidisc Optim

504

243

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“…In this element-by-element (EBE) approach the element stiffness matrix K e and the topology matrix T e occur, see e.g. [5,16,24,38,46]. The T e consist of a few columns of the identity matrix of order n representing the mapping of the local to the global node numbers.…”

Section: Introductionmentioning

confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

132

106

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Abstract. The recent advances in microarchitectural bone imaging are disclosing the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element (µFE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction.Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting µFE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the µFE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-by-element techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, it has modest memory requirements, while being at the same time robust and scalable.Using the proposed methods, we have solved in less than 10 minutes a model of trabecular bone composed of 247'734'272 elements, leading to a matrix with 1'178'736'360 rows, using only 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, the spine and the wrist.

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“…In this paper we focus on element-based methods, that is, methods in which connectivity information and the element stiffness matrices are stored but never combined together to form K or P. We have previously presented a survey of the efficiency of element-based methods for linear elasticity problems in [13]: a short summary of some of the element-based techniques studied in that and other papers is given in Section 2.3. In Section 2.4, we combine these ideas with those in Section 2.2 to obtain a new element-based preconditioner for linear elastic problems and present some theoretical and numerical results which demonstrate its potential efficiency in terms of reducing iteration counts.…”

mentioning

confidence: 99%

Element‐based preconditioners for elasto‐plastic problems in geotechnical engineering

Augarde

Ramage²,

Staudacher

2006

Numerical Meth Engineering

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SUMMARYIterative solvers are widely regarded as the most efficient way to solve the very large linear systems arising from finite element models. Their memory requirements are small compared to those for direct solvers. Consequently, there is a major interest in iterative methods and particularly the preconditioning necessary to achieve rapid convergence. In this paper we present new element-based preconditioners specifically designed for linear elasticity and elasto-plastic problems. The study presented here is restricted to simple associated plasticity but should find wide application in other plasticity models used in geotechnics.

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An element-based displacement preconditioner for linear elasticity problems

Cited by 23 publications

References 16 publications

An efficient 3D topology optimization code written in Matlab

An efficient 3D topology optimization code written in Matlab

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Element‐based preconditioners for elasto‐plastic problems in geotechnical engineering

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