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

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

doi:10.1002/nme.1947

Cited by 13 publications

(11 citation statements)

References 38 publications

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“…Efficient implicit nonlinear methods are generally built around one of the numerous variants of Newton's method [3,6]. Newton's method is also wellknown to exhibit a convergence rate that is independent of spatial resolution in systems arising from elliptic-like PDEs.…”

Section: Conclusion and Future Developmentsmentioning

confidence: 99%

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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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Section: Conclusion and Future Developmentsmentioning

confidence: 99%

“…This increases the number of colors, and therefore the applications of K 0 and the memory requirements. For these reasons, in the following we will just focus on (6).…”

mentioning

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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“…In and , the soil is assumed to follow a von Mises model; and hence, the elastoplastic stiffness matrix K ep is symmetric. Element‐by‐element method is used in , and element‐based preconditioners are proposed while this paper considers the preconditioners for the global stiffness matrix. Augarde et al .…”

Section: Introductionmentioning

confidence: 99%

“…Augarde et al . show that when the preconditioner is constructed from K e , the iterative solver requires more iteration to converge but the iteration time is reduced. Borja proposes the use of the whole matrix K e as a preconditioner.…”

Section: Introductionmentioning

confidence: 99%

Preconditioned IDR(s) iterative solver for non‐symmetric linear system associated with FEM analysis of shallow foundation

Tran

Toh

Phoon

2013

Num Anal Meth Geomechanics

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SUMMARYNon-associated flow rule is essential when the popular Mohr-Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes nonsymmetric when this non-associated flow rule is applied. Efficient solution of this large-scale nonsymmetric linear system is of practical importance. The standard Krylov solver for a non-symmetric solver is Bi-CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s = 6) is more efficient than Bi-CGSTAB when the preconditioner is the incomplete factorization with zero fill-in of global stiffness matrix K ep (ILU(0)-K ep ). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)-K ep . To further reduce computational cost, the global stiffness matrix K ep is divided into two parts. The first part is the linear elastic stiffness matrix K e , which is formed only once at the beginning of solution step. The second part is a low-rank matrix Δ, which is re-formed at each Newton-Raphson iteration. Numerical studies show that IDR(s = 6) with this ILU(0)-K e preconditioner is more time effective than IDR(s = 6) with ILU(0)-K ep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used.

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“…Much work has been devoted to the development of efficient preconditioners for several geomechanical applications exploiting the intimate structure of the stiffness matrix. For example, multi‐grid methods 7 and element‐based techniques 8 have been borrowed from the structural mechanics, while several variants of the block constrained approach, initially developed for optimization problems 9, have been successfully applied to the prediction of soil consolidation 10, 11 and the mechanics of fractured media 12. Recently a novel and promising multilevel incomplete factorization (MIF) has been advanced for geomechanical simulations 13, taking into account the stiffness matrix subdivision naturally generated by a proper ordering of the model unknowns.…”

Section: Introductionmentioning

confidence: 99%

Parallel solution to ill‐conditioned FE geomechanical problems

Ferronato¹,

Janna²,

Pini³

2011

Num Anal Meth Geomechanics

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Parallel computers are potentially very attractive for the implementation of large size geomechanical models. One of the main difficulties of parallelization, however, relies on the efficient solution of the frequently ill-conditioned algebraic system arising from the linearization of the discretized equilibrium equations. While very efficient preconditioners have been developed for sequential computers, not much work has been devoted to parallel solution algorithms in geomechanics. The present study investigates the state-of-the-art performance of the factorized sparse approximate inverse (FSAI) as a preconditioner for the iterative solution of ill-conditioned geomechanical problems. Pre-and post-filtration strategies are experimented with to increase the FSAI efficiency. Numerical results show that FSAI exhibits a promising potential for parallel geomechanical models mainly because of its almost ideal scalability. With the present formulation, however, at least 4 or 8 processors are required in the selected test cases to outperform one of the most efficient sequential algorithms available for FE geomechanics, i.e. the multilevel incomplete factorization (MIF). Further research is needed to improve the FSAI efficiency with a more effective selection of the preconditioner non-zero patter

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

Cited by 13 publications

References 38 publications

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

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

Preconditioned IDR(s) iterative solver for non‐symmetric linear system associated with FEM analysis of shallow foundation

Parallel solution to ill‐conditioned FE geomechanical problems

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