Large-scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients

Hughes, Thomas J.R.; Ferencz, Robert M.; Hallquist, J.O.

doi:10.1016/0045-7825(87)90005-3

Cited by 166 publications

(59 citation statements)

References 9 publications

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“…The conjugate gradient iterative solving technique, with diagonal scaling as a preconditioner (Hughes et al, 1987;Strang, 1986), was used to calculate the unknown displacement vector u. Each iteration requires a multiplication of the global stiffness matrix K with a direction-vector p. The memory required to solve this problem was reduced by three separate strategies.…”

Section: 381602mentioning

confidence: 99%

“…First, the entire global stiffness matrix is never actually computed. Instead, an element by element (EBE) approach (Hughes et al, 1987) is applied which allows for the multiplication at element level Kp=~Kepe, where Ke is the element stiffness matrix (order 24) and p~ the decomposed vector p. Second, because all elements are identical in size and have the same orientation, the global structure can be described by the element connectivity alone. Hence, no nodal coordinate data have to be stored.…”

Section: 381602mentioning

confidence: 99%

“…To reduce the CPU time, vectorization schemes as described by Hughes et al (1987) and Hayes and Devloo (1986) were applied to the global stit~ess matrix-vector product, taking full advantage of the CRAY-YMP computer which was used for the calculations. To monitor the convergence of the process the norm of the residual force vector II f,®, II was calculated every increment.…”

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confidence: 99%

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A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models

Rietbergen

Weinans

Huiskes

et al. 1995

Journal of Biomechanics

765

264

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Abstract--The apparent mechanical behavior of trabecular bone depends on properties at the tissue or trabecular level. Many investigators have attempted to determine trabecular tissue properties and loading. However, accuracy and applicability of all methods reported are limited. The small size of the trabeculae and a possible size effect are complicating factors when using traditional testing methods on single trabeculae. Other methods reported, using models that describe the trabecular structure, are of limited value because they consider bone as a repetitive structure in order to describe a reasonably large region of bone.The present study introduces a new finite-element method strategy that enables analysis of reasonably large regions of trabecular bone in full detail. The method uses three-dimensional serial reconstruction techniques to construct a large-scale FE model, by directly converting voxels to elements. A 5 mm cube of trabecular bone was modeled in this way, resulting in a FE model that consists of 296,679 elements. Special strategies were developed to solve the set of equations that results from the FE approach.Using this model in combination with experimental apparent data taken from the literature, the upper and lower boundaries for the tissue modulus were calculated to be 10.1 and 2.23 GPa, respectively. From the local stress and strain distributions it was concluded that the deformation mode of the trabeculae in the present cube was predominantly in bending. It was concluded that the method developed offers new perspectives for the study of trabecular bone.

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Section: 381602mentioning

confidence: 99%

Section: 381602mentioning

confidence: 99%

mentioning

confidence: 99%

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A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models

Rietbergen

Weinans

Huiskes

et al. 1995

Journal of Biomechanics

765

264

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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%

“…Therefore, the emphasis has been on the use of preconditioners derived from simplications of the problem for which explicit matrices can be readily constructed. Widely used preconditioners for matrix-free environments are Jacobi preconditioning (diagonal scaling) or EBE preconditioning [24]. The analysis of Wathen [29,53] shows that these methods are spectrally equivalent to the unpreconditioned system and hence are not suited for large scale problems.…”

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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Analysis of piled rafts

Smith

Wang

1998

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYAnalysis of piled raft foundations, taking account of their full three-dimensional complexity, can be accomplished by modern finite element analysis techniques. The characteristics of the (preconditioned conjugate gradient) numerical method applied to this problem are analysed, and then the method is used in a field problem of a raft subjected to very rapidly varying loading patterns.1998 John Wiley & Sons, Ltd.

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Large-scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients

Cited by 166 publications

References 9 publications

A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models

A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models

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

Analysis of piled rafts

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