An efficient multigrid method for the simulation of high-resolution elastic solids

Zhu, Yongning; Sifakis, Eftychios; Teran, Joseph; Brandt, Achi

doi:10.1145/1731047.1731054

Cited by 98 publications

(92 citation statements)

References 36 publications

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“…Thus, a large part of work has focused on improving it, e.g. with fast iterative solvers [Zhu et al 2010;Ferstl et al 2014] and by introducing flexible ways to couple multiple grids [English et al 2013]. A variety of different discretizations have been proposed over the years, many of them using tetrahedral meshes [Klingner et al 2006;Batty et al 2010], Elcott and colleagues [2007] proposed a discretization based on generic simplices while focusing on preserving circulations in the flow.…”

Section: Related Workmentioning

confidence: 99%

A stream function solver for liquid simulations

2015

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This paper presents a liquid simulation technique that enforces the incompressibility condition using a stream function solve instead of a pressure projection. Previous methods have used stream function techniques for the simulation of detailed single-phase flows, but a formulation for liquid simulation has proved elusive in part due to the free surface boundary conditions. In this paper, we introduce a stream function approach to liquid simulations with novel boundary conditions for free surfaces, solid obstacles, and solidfluid coupling.Although our approach increases the dimension of the linear system necessary to enforce incompressibility, it provides interesting and surprising benefits. First, the resulting flow is guaranteed to be divergence-free regardless of the accuracy of the solve. Second, our free-surface boundary conditions guarantee divergence-free motion even in the un-simulated air phase, which enables two-phase flow simulation by only computing a single phase. We implemented this method using a variant of FLIP simulation which only samples particles within a narrow band of the liquid surface, and we illustrate the effectiveness of our method for detailed two-phase flow simulations with complex boundaries, detailed bubble interactions, and two-way solid-fluid coupling.

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Section: Related Workmentioning

confidence: 99%

A stream function solver for liquid simulations

2015

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“…As previously noted by Zhu et al [2010] reconstructing the embedded object surface using trilinear interpolation from lattice nodes yields visible grid artifacts due to discontinuous gradients at voxel boundaries. Consequently, our examples employed tricubic interpolation [Lekien and Marsden 2005] to generate the embedded object surface for rendering, with the exception of Figure 5 where standard trilinear interpolation was used, to demonstrate this effect and emphasize that the trilinear basis is still capable of rapid convergence under refinement when coupled with a sub-voxel accurate quadrature scheme.…”

Section: Examplesmentioning

confidence: 70%

“…The most relevant graphics work [Zhu et al 2010] also employs a pressure-based mixed formulation, yet is explicitly limited to linear and corotated materials. Furthermore, their formulation is based on finite-differences and voxel-accurate only.…”

Section: Mixed Formulationmentioning

confidence: 99%

Simulation of complex nonlinear elastic bodies using lattice deformers

2012

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Lattice deformers are a popular option for modeling the behavior of elastic bodies as they avoid the need for conforming mesh generation, and their regular structure offers significant opportunities for performance optimizations. Our work expands the scope of current lattice-based elastic deformers, adding support for a number of important simulation features. We accommodate complex nonlinear, optionally anisotropic materials while using an economical one-point quadrature scheme. Our formulation fully accommodates near-incompressibility by enforcing accurate nonlinear constraints, supports implicit integration for large time steps, and is not susceptible to locking or poor conditioning of the discrete equations. Additionally, we increase the accuracy of our solver by employing a novel high-order quadrature scheme on lattice cells overlapping with the model boundary, which are treated at sub-cell precision. Finally, we detail how this accurate boundary treatment can be implemented at a minimal computational premium over the cost of a voxel-accurate discretization. We demonstrate our method in the simulation of complex musculoskeletal human models.

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“…These methods use fast matrix inversion techniques and other insights about the algebra of the finite element method to increase its performance. Simulators based on the multigrid method [Peraire et al 1992;Zhu et al 2010;McAdams et al 2011] and Krylov subspace techniques [Patterson et al 2012] have yielded impressive performance increases. Other hierarchical numerical approaches, as well as highly parallel techniques, have also been applied to improve the time required to perform complex simulations [Farhat and Roux 1991;Mandel 1993].…”

Section: Related Workmentioning

confidence: 99%

Data-driven finite elements for geometry and material design

Chen¹,

Levin

Sueda

et al. 2015

ACM Trans. Graph.

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Figure 1: Examples produced by our data-driven finite element method. Left: A bar with heterogeneous material arrangement is simulated 15x faster than its high-resolution counterpart. Left-Center: Our fast coarsening algorithm dramatically accelerates designing this shoe sole (up to 43x). Right-Center: A comparison to 3D printed results. Right: We repair a flimsy bridge by adding a supporting arch (8.1x) speed-up. We show a High-Res Simulation for comparison. AbstractCrafting the behavior of a deformable object is difficult-whether it is a biomechanically accurate character model or a new multimaterial 3D printable design. Getting it right requires constant iteration, performed either manually or driven by an automated system. Unfortunately, previous algorithms for accelerating three-dimensional finite element analysis of elastic objects suffer from expensive precomputation stages that rely on a priori knowledge of the object's geometry and material composition. In this paper we introduce Data-Driven Finite Elements as a solution to this problem. Given a material palette, our method constructs a metamaterial library which is reusable for subsequent simulations, regardless of object geometry and/or material composition. At runtime, we perform fast coarsening of a simulation mesh using a simple table lookup to select the appropriate metamaterial model for the coarsened elements. When the object's material distribution or geometry changes, we do not need to update the metamaterial library-we simply need to update the metamaterial assignments to the coarsened elements. An important advantage of our approach is that it is applicable to non-linear material models. This is important for designing objects that undergo finite deformation (such as those produced by multimaterial 3D printing). Our method yields speed gains of up to two orders of magnitude while maintaining good accuracy. We demonstrate the effectiveness of the method on both virtual and 3D printed examples in order to show its utility as a tool for deformable object design.

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An efficient multigrid method for the simulation of high-resolution elastic solids

Cited by 98 publications

References 36 publications

A stream function solver for liquid simulations

A stream function solver for liquid simulations

Simulation of complex nonlinear elastic bodies using lattice deformers

Data-driven finite elements for geometry and material design

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