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Multi-Resolution Isotropic Strain Limiting

Huamin Wang, James F. O'Brien, Ravi Ramamoorthi

SIGGRAPH AsiaAcademicCFX

Coordinate-invariant strain limiting via SVD of the deformation gradient at multiple resolutions, enforcing isotropic inextensibility for cloth and shells.

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Abstract

In this paper we describe a fast strain-limiting method that allows stiff, incompliant materials to be simulated efficiently. Unlike prior approaches, which act on springs or individual strain components, this method acts on the strain tensors in a coordinate-invariant fashion allowing isotropic behavior. Our method applies to both two-and three-dimensional strains, and only requires computing the singular value decomposition of the deformation gradient, either a small 2x2 or 3x3 matrix, for each element. We demonstrate its use with triangular and tetrahedral linear-basis elements. For triangulated surfaces in three-dimensional space, we also describe a complementary edge-angle-limiting method to limit out-of-plane bending. All of the limits are enforced through an iterative, non-linear, Gauss-Seidel-like constraint procedure. To accelerate convergence, we propose a novel multi-resolution algorithm that enforces fitted limits at each level of a non-conforming hierarchy. Compared with other constraint-based techniques, our isotropic multi-resolution strain-limiting method is straightforward to implement, efficient to use, and applicable to a wide range of shell and solid materials.

How to read this

Category: Method: a strain-limiting technique for cloth and shells
Contributions: A coordinate-invariant strain-limiting method acting on strain tensors (via SVD of the deformation gradient) to give isotropic behavior, unlike spring- or component-based prior approaches
Applicability to both 2D and 3D strains with triangular and tetrahedral linear elements, plus a complementary edge-angle method to limit out-of-plane bending
A novel multi-resolution algorithm that enforces fitted limits at each level of a non-conforming hierarchy to accelerate the iterative Gauss-Seidel-like constraint convergence
Context: Extends constraint-based strain limiting toward coordinate-invariant isotropy, related to continuum-based strain limiting (Thomaszewski et al.) and using per-element SVD of the deformation gradient.Builds on: Continuum-based Strain Limiting
Correctness: Assumes that limiting the singular values of the deformation gradient adequately enforces isotropic inextensibility and that the Gauss-Seidel-like iteration converges; readers should note enforcement is iterative and approximate, with the multi-resolution scheme mainly improving convergence rather than guaranteeing exact limits.
Clarity: Moderately technical but stated as straightforward to implement; a first pass conveys the SVD-based isotropic idea, do a second pass for the multi-resolution hierarchy and the constraint iteration.
How to read it: Focus on why coordinate-invariance matters and the per-element SVD limit; a second pass on the multi-resolution hierarchy pays off if convergence speed or implementation is your goal.

Builds on

Continuum-based Strain Limiting 2009

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Related work

Strain Based Dynamics 2014 / SCA
Projective Dynamics: Fusing Constraint Projections for Fast Simulation 2014 / SIGGRAPH
Continuum-based Strain Limiting 2009 / Eurographics
Dynamic Deformables: Implementation and Production Practicalities 2020 / SIGGRAPH Courses

Keywords

strain limiting multi-resolution singular value decomposition deformation gradient finite element method cloth simulation thin shells tetrahedral elements triangular elements principal strains bending angle limiting gauss-seidel iteration velocity filter isotropic strain mesh hierarchy green's strain tensor corotational strain

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