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Optimizing Cubature for Efficient Integration of Subspace Deformations

Identifieur interne : 001346 ( Ncbi/Merge ); précédent : 001345; suivant : 001347

Optimizing Cubature for Efficient Integration of Subspace Deformations

Auteurs : Steven S. An ; Theodore Kim ; Doug L. James

Source :

ACM transactions on graphics [ 0730-0301 ] ; 2009.

RBID : PMC:2731311

Abstract

We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an r-dimensional deformation subspace with O(r) cubature points, our method is able to evaluate subspace forces at O(r²) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.

CR Categories:

I.6.8 [Simulation and Modeling]: Types of Simulation—Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling G.1.4 [Mathematics of Computing]: Numerical Analysis—Quadrature and Numerical Differentiation

Url:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2731311

DOI: 10.1145/1409060.1409118
PubMed: 19956777
PubMed Central: 2731311

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PMC:2731311

Le document en format XML

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<author><name sortKey="An, Steven S" sort="An, Steven S" uniqKey="An S" first="Steven S." last="An">Steven S. An</name>
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<author><name sortKey="Kim, Theodore" sort="Kim, Theodore" uniqKey="Kim T" first="Theodore" last="Kim">Theodore Kim</name>
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<author><name sortKey="James, Doug L" sort="James, Doug L" uniqKey="James D" first="Doug L." last="James">Doug L. James</name>
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<author><name sortKey="Kim, Theodore" sort="Kim, Theodore" uniqKey="Kim T" first="Theodore" last="Kim">Theodore Kim</name>
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<author><name sortKey="James, Doug L" sort="James, Doug L" uniqKey="James D" first="Doug L." last="James">Doug L. James</name>
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<series><title level="j">ACM transactions on graphics</title>
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<front><div type="abstract" xml:lang="en"><p id="P1">We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an <italic>r</italic>
-dimensional deformation subspace with <italic>O</italic>
(<italic>r</italic>
) cubature points, our method is able to evaluate subspace forces at <italic>O</italic>
(<italic>r</italic>
<sup>2</sup>
) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.</p>
<sec id="S1"><title>CR Categories:</title>
<p id="P2">I.6.8 [Simulation and Modeling]: Types of Simulation—Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling G.1.4 [Mathematics of Computing]: Numerical Analysis—Quadrature and Numerical Differentiation</p>
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  <front><journal-meta><journal-id journal-id-type="nlm-journal-id">101152234</journal-id>
<journal-id journal-id-type="pubmed-jr-id">31873</journal-id>
<journal-id journal-id-type="nlm-ta">ACM Trans Graph</journal-id>
<journal-title>ACM transactions on graphics</journal-title>
<issn pub-type="ppub">0730-0301</issn>
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<title-group><article-title>Optimizing Cubature for Efficient Integration of Subspace Deformations</article-title>
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<contrib-group><contrib contrib-type="author"><name><surname>An</surname>
<given-names>Steven S.</given-names>
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<contrib contrib-type="author"><name><surname>Kim</surname>
<given-names>Theodore</given-names>
</name>
</contrib>
<contrib contrib-type="author"><name><surname>James</surname>
<given-names>Doug L.</given-names>
</name>
</contrib>
<aff id="A1">Cornell University</aff>
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<pub-date pub-type="nihms-submitted"><day>16</day>
<month>5</month>
<year>2009</year>
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<pub-date pub-type="ppub"><day>1</day>
<month>12</month>
<year>2009</year>
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<pub-date pub-type="pmc-release"><day>1</day>
<month>12</month>
<year>2009</year>
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<volume>27</volume>
<issue>5</issue>
<fpage>165</fpage>
<lpage>165</lpage>
<abstract><p id="P1">We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an <italic>r</italic>
-dimensional deformation subspace with <italic>O</italic>
(<italic>r</italic>
) cubature points, our method is able to evaluate subspace forces at <italic>O</italic>
(<italic>r</italic>
<sup>2</sup>
) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.</p>
<sec id="S1"><title>CR Categories:</title>
<p id="P2">I.6.8 [Simulation and Modeling]: Types of Simulation—Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling G.1.4 [Mathematics of Computing]: Numerical Analysis—Quadrature and Numerical Differentiation</p>
</sec>
</abstract>
<kwd-group><kwd>Dimensional model reduction</kwd>
<kwd>reduced-order modeling</kwd>
<kwd>subspace integration</kwd>
<kwd>quadrature</kwd>
<kwd>subspace dynamics</kwd>
<kwd>dynamic deformations</kwd>
<kwd>nonlinear solid mechanics</kwd>
<kwd>real-time simulation</kwd>
</kwd-group>
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<contract-sponsor id="EB1">National Institute of Biomedical Imaging and Bioengineering : NIBIB</contract-sponsor>
</article-meta>
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<tree><noCountry><name sortKey="An, Steven S" sort="An, Steven S" uniqKey="An S" first="Steven S." last="An">Steven S. An</name>
<name sortKey="James, Doug L" sort="James, Doug L" uniqKey="James D" first="Doug L." last="James">Doug L. James</name>
<name sortKey="Kim, Theodore" sort="Kim, Theodore" uniqKey="Kim T" first="Theodore" last="Kim">Theodore Kim</name>
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Optimizing Cubature for Efficient Integration of Subspace Deformations

Optimizing Cubature for Efficient Integration of Subspace Deformations

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Abstract

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