Ultrascalable Implicit Finite Element Analyses in Solid Mechanics with over a Half a Billion Degrees of Freedom

Adams, Mark F.; Bayraktar, Harun H.; Keaveny, Tony M.; Papadopoulos, Panayiotis

doi:10.1109/sc.2004.62

Cited by 106 publications

(127 citation statements)

References 32 publications

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“…This preconditioner has modest requirements in terms of CPU time, is only slightly slower than smoothed aggregation preconditioners for matrix-ready problems, is scalable up to thousands of processors and billions of unknowns, is orders of magnitude faster than the widely adopted Jacobi or element-by-element preconditioner, and reduces the memory consumption with respect to AMG preconditioners for matrix-ready problems by a factor of about 3 to 3.5, while converging in a comparable number of iterations. This memory savings is the main improvement over the approach presented by Adams [3] and later implemented by us in the Trilinos framework [4,33]. The proposed method can also be applied in other areas in which K exists only in the form of a matrix-vector multiplication routine provided that the graph of the K is available or can be cheaply generated.…”

Section: Introductionmentioning

confidence: 87%

“…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%

“…In [3], the algebraic multigrid method based on aggregation has been successfully used for the solution of linear systems arising in conventional analysis of bone strength. Aggregation-based methods are also supported by theoretical results [51], and have been proven scalable up to thousands of processors [3,4,31] for a variety of applications. However, to the best of our knowledge, no article has yet shown the applicability of AMG preconditioners to matrix-free approaches in bone analysis.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 87%

Section: Conclusion and Future Developmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…All analyses were run on an IBM Power4 supercomputer (IBM corporation, Armonk, NY) using a maximum of 440 processors in parallel and 900 GB memory, and a custom code with a parallel mesh partitioner and algebraic multigrid solver [26], requiring a total of approximately 4300 CPU hours. To simulate compressive loading of each vertebra, an apparent level compressive strain of 1.0% was applied to each model by using different displacement magnitudes based on the height of each model.…”

Section: Methodsmentioning

confidence: 99%

Locations of bone tissue at high risk of initial failure during compressive loading of the human vertebral body

Eswaran

Gupta

Keaveny

2007

Bone

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Knowledge of the location of initial regions of failure within the vertebra -cortical shell, cortical endplates vs. trabecular bone, as well as anatomic location -may lead to improved understanding of the mechanisms of aging, disease and treatment. The overall objective of this study was to identify the location of the bone tissue at highest risk of initial failure within the vertebral body when subjected to compressive loading. Toward this end, micro-CT based 60-micron voxel-sized, linearly elastic, finite element models of a cohort of thirteen elderly (age range: 54-87 years, 75 ± 9 years) female whole vertebrae without posterior elements were virtually loaded in compression through a simulated disc. All bone tissue within each vertebra having either the maximum or minimum principal strain beyond its 90 th percentile was defined as the tissue at highest risk of initial failure within that particular vertebral body. Our results showed that such high-risk tissue first occurred in the trabecular bone and that the largest proportion of the high-risk tissue also occurred in the trabecular bone. The amount of high-risk tissue was significantly greater in and adjacent to the cortical endplates than in the mid-transverse region. The amount of high-risk tissue in the cortical endplates was comparable to or greater than that in the cortical shell regardless of the assumed Poisson's ratio of the simulated disc. Our results provide new insight into the micromechanics of failure of trabecular and cortical bone within the human vertebra, and taken together, suggest that during strenuous compressive loading of the vertebra, the tissue near and including the endplates is at the highest risk of initial failure.

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“…The FETI and AMG methods are also robust but are often much less expensive than direct solution methods and have been discussed in [23] and [49]. As a comparison to DPCG, we focus on the best AMG adaptation, smoothed aggregation (SA), as it has been demonstrated to be a successful parallel preconditioner for a number of structural mechanics applications [2,4,14]. The two most relevant studies of SA to the simulations considered here are those of [4,7], both of which focus on micro-FE modeling of bone deformation, based on micro-CT scans of human bones.…”

Section: Introductionmentioning

confidence: 99%

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

et al. 2011

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Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. T. B. Jönsthövel (B)·

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Ultrascalable Implicit Finite Element Analyses in Solid Mechanics with over a Half a Billion Degrees of Freedom

Cited by 106 publications

References 32 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

Locations of bone tissue at high risk of initial failure during compressive loading of the human vertebral body

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

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