Deformation simulation using cubic finite elements and efficient p-multigrid methods

Weber, Daniel; Mueller-Roemer, Johannes Sebastian; Altenhofen, Christian; Sourander, André; Fellner, Dieter W.

doi:10.1016/j.cag.2015.06.010

Cited by 16 publications

(7 citation statements)

References 27 publications

(41 reference statements)

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“…For simulating general nonlinear materials, a lot of effort has been put into the eigen-analysis of the energy Hessian in order to fix indefiniteness and use the conjugate gradient method Smith et al 2018Smith et al , 2019Teran et al 2005]. Advanced techniques like higher-order elements [Bargteil and Cohen 2014;Weber et al 2011] or multi-resolution solvers [Weber et al 2015;Zhu et al 2010] were also employed in computer graphics. To address the issues with volume mesh generation, some authors embedded the original surface mesh into a lattice of hexahedra with special quadrature rules [McAdams et al 2011;Nesme et al 2009;Patterson et al 2012].…”

Section: Related Work 31 Finite Element Methodsmentioning

confidence: 99%

“…In this article, we choose to use only linear tetrahedral meshes as they are widely used in computer graphics. Extensions to higher order can be made using numerical quadrature or analytically using Bernstein-Bézier polynomials as in Roth et al [1998], Roth [2002], and Frâncu et al [2019], and similar to standard FEM [Bargteil and Cohen 2014;Weber et al 2011Weber et al , 2015].…”

Section: Linear Tetrahedral Elementsmentioning

confidence: 99%

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Locking-Proof Tetrahedra

et al. 2021

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The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

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Section: Related Work 31 Finite Element Methodsmentioning

confidence: 99%

Section: Linear Tetrahedral Elementsmentioning

confidence: 99%

Locking-Proof Tetrahedra

et al. 2021

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“…The CPU backends call conventionally compiled functions to apply residuals (10) and Jacobians (11) at quadrature points, thereby enabling a rich debugging experience. CPU backends implement the element action B using tensor contractions with architecture-specific vectorization (e.g., AVX intrinsics, LIBXSMM [33]).…”

Section: Portability and Productivitymentioning

confidence: 99%

“…Meanwhile, algebraic multigrid (AMG) setup costs increase due to sparse matrix-matrix products, and the resulting solvers are observed to converge more slowly for highorder discretizations [10]- [12], even when using specialized methods [13]. A practical alternative to algebraic multigrid is p-multigrid [14], which is observed to be robust for finite element discretizations on unstructured meshes [11], [12] and pairs naturally with efficient matrix-free data structures [15].…”

Section: Introductionmentioning

confidence: 99%

Performance Portable Solid Mechanics via Matrix-Free $p$-Multigrid

Brown¹,

Barra²,

Beams³

et al. 2022

Preprint

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Finite element analysis of solid mechanics is a foundational tool of modern engineering, with low-order finite element methods and assembled sparse matrices representing the industry standard for implicit analysis. We use performance models and numerical experiments to demonstrate that highorder methods greatly reduce the costs to reach engineering tolerances while enabling effective use of GPUs. We demonstrate the reliability, efficiency, and scalability of matrix-free p-multigrid methods with algebraic multigrid coarse solvers through large deformation hyperelastic simulations of multiscale structures. We investigate accuracy, cost, and execution time on multi-node CPU and GPU systems for moderate to large models using AMD MI250X (OLCF Crusher), NVIDIA A100 (NERSC Perlmutter), and V100 (LLNL Lassen and OLCF Summit), resulting in order of magnitude efficiency improvements over a broad range of model properties and scales. We discuss efficient matrix-free representation of Jacobians and demonstrate how automatic differentiation enables rapid development of nonlinear material models without impacting debuggability and workflows targeting GPUs.

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“…In a recent conference paper, Altenhofen et al presented an approach to generate tetrahedral simulation meshes directly from volumetric Catmull‐Clark models [ASSF17]. As Altenhofen et al already use a GPU‐based FEM solver (originally presented by Weber et al [WMA∗15]), their approach would benefit significantly from performing subdivision and mesh operations directly on the GPU.…”

Section: Background and Related Workmentioning

confidence: 99%

Ternary Sparse Matrix Representation for Volumetric Mesh Subdivision and Processing on GPUs

Mueller-Roemer

Altenhofen

Sourander

2017

Computer Graphics Forum

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In this paper, we present a novel volumetric mesh representation suited for parallel computing on modern GPU architectures. The data structure is based on a compact, ternary sparse matrix storage of boundary operators. Boundary operators correspond to the first‐order top‐down relations of k‐faces to their (k − 1)‐face facets. The compact, ternary matrix storage format is based on compressed sparse row matrices with signed indices and allows for efficient parallel computation of indirect and bottom‐up relations. This representation is then used in the implementation of several parallel volumetric mesh algorithms including Laplacian smoothing and volumetric Catmull‐Clark subdivision. We compare these algorithms with their counterparts based on OpenVolumeMesh and achieve speedups from 3× to 531×, for sufficiently large meshes, while reducing memory consumption by up to 36%.

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Deformation simulation using cubic finite elements and efficient p-multigrid methods

Cited by 16 publications

References 27 publications

Locking-Proof Tetrahedra

Locking-Proof Tetrahedra

Performance Portable Solid Mechanics via Matrix-Free $p$-Multigrid

Ternary Sparse Matrix Representation for Volumetric Mesh Subdivision and Processing on GPUs

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