A time-integration method for stable simulation of extremely deformable hyperelastic objects

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This paper presents a time integration method for realtime simulation of extremely deformable objects subject to geometrically nonlinear hyperelasticity. In the presented method, the equation of motion of the system is discretized by the backward Euler method, and linearly approximated through the first-order Taylor expansion. The approximate linear equation is solved with the quasi-minimal residual method (QMR), which is an iterative linear equation solver for non-symmetric or indefinite matrices. The solution is then corrected considering the nonlinear term that is omitted at the Taylor expansion. The method does not demand the constitutive law to guarantee the positive definiteness of the stiffness matrix. Experimental results show that the presented method realizes stable behavior of the simulated model under such deformation that the tetrahedral elements are almost flattened. It is also shown that QMR outperforms the biconjugate gradient stabilized method (BiCGStab) in this application.

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    This result is as expected because QMR performs two matrix-vector multiplications per iteration while CG performs only one.

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    The criteria \(e_{1,k}\) and \(e_{2,k}\) were not computed in the experiments of Fig. 3, but was performed in another session because the computation of \(e_{1,k}\) and \(e_{2,k}\) resulted in nonnegligible computational time. This computation resulted in roughly 10 % reduction in the number of QMR iterations.


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The surface meshes of Stanford Armadillo and the “bunny” were obtained from the Web site of Stanford Computer Graphics Laboratory ( The surface mesh of the “dinosaur” was obtained courtesy of an unknown creator from the AIM@SHAPE Shape Repository ( They were converted into tetrahedral meshes by using Adventure TetMesh provided by the ADVENTURE Project (

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Correspondence to Ryo Kikuuwe.

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This work was supported in part by Grant-in-Aid for Exploratory Research, No. 25540044, from Japan Society for the Promotion of Science.

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Kikuuwe, R. A time-integration method for stable simulation of extremely deformable hyperelastic objects. Vis Comput 33, 1335–1346 (2017) doi:10.1007/s00371-016-1225-0

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  • QMR
  • Finite elements
  • Interactive simulation
  • Hyperelasticity