Abstract
Barrier functions are crucial for maintaining an intersection- and inversion-free simulation trajectory but existing methods, which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second-order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3× speedup over the standard Incremental Potential Contact (IPC) barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.
- 2010. Volume contact constraints at arbitrary resolution. In Proceedings of the ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH’10). 1–10.Google ScholarDigital Library .
- 2022. Contact and friction simulation for computer graphics. In Proceedings of the ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH’22). Article
2 , 124 pages.Google ScholarDigital Library . - 2014. Fast and simple agglomerative LBVH construction. In Proceedings of the Conference on Theory and Practice of Computer Graphics, and (Eds.). Eurographics Association, 41–44.Google Scholar .
- 1998. Large steps in cloth simulation. In Proceedings of the ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH’98), , , and (Eds.). ACM, 43–54.Google ScholarDigital Library .
- 2002. Robust treatment of collisions, contact and friction for cloth animation. In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques. 594–603.Google ScholarDigital Library .
- 2020. Binary ostensibly implicit trees for fast collision detection. Comput. Graph. Forum 39, 2 (2020), 509–521.Google ScholarCross Ref .
- 2005. Bounding volume hierarchies. In Real-Time Collision Detection, (Ed.). Morgan Kaufmann, San Francisco, CA, 235–284.Google ScholarCross Ref .
- 2018. Methodology for assessing mesh-based contact point methods. ACM Trans. Graph. 37, 3 (2018), 1–30.Google ScholarDigital Library .
- 2021. Guaranteed globally injective 3D deformation processing. ACM Trans. Graph. 40, 4, Article
75 (2021).Google ScholarDigital Library . - 2021. Intersection-free rigid body dynamics. ACM Trans. Graph. 40, 4, Article
183 (2021).Google ScholarDigital Library . - 2018. GPU optimization of material point methods. ACM Trans. Graph. 37, 6 (2018).Google Scholar .
- 2022. Eigen v3.4. Retrieved from http://eigen.tuxfamily.orgGoogle Scholar .
- 2008. Robust treatment of simultaneous collisions. ACM Trans. Graph. 27, 3 (2008), 1–4.Google ScholarDigital Library .
- 2022. Differentiable solver for time-dependent deformation problems with contact. Retrieved from https://arxiv:cs.GR/2205.13643Google Scholar .
- 2004. Invertible finite elements for robust simulation of large deformation. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA’04). Eurographics Association, Goslar, DEU, 131–140.Google ScholarDigital Library .
- 2017. Simplicial complex augmentation framework for bijective maps. ACM Trans. Graph. 36, 6 (2017).Google ScholarDigital Library .
- 2000. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49, 10 (2000), 1295–1325.Google ScholarCross Ref .
- 1999. Finite element analysis of nonsmooth contact. Comput. Methods Appl. Mech. Eng. 180, 1 (1999), 1–26.Google ScholarCross Ref .
- 2012. Maximizing parallelism in the construction of BVHs, octrees, and k-d trees. In Proceedings of the ACM SIGGRAPH Symposium on High Performance Graphics (HPG’12), , , and (Eds.). 33–37.Google Scholar .
- 2008. Staggered projections for frictional contact in multibody systems. In Proceedings of ACM SIGGRAPH Asia. 1–11.Google ScholarDigital Library .
- 2020. A Finite Element Formulation of Baraff-Witkin Cloth. Eurographics Association, Goslar, DEU.Google ScholarDigital Library .
- 2019. Anisotropic elasticity for inversion-safety and element rehabilitation. ACM Trans. Graph. 38, 4, Article
69 (July 2019), 15 pages.Google ScholarDigital Library . - 2020. Dynamic deformables: Implementation and production practicalities. In Proceedings of the ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH’20).Google ScholarDigital Library .
- 2009. Tensor decompositions and applications. SIAM Rev. 51, 3 (
Sept. 2009), 455–500.Google ScholarDigital Library . - 2022a. Affine body dynamics: Fast, stable & intersection-free simulation of stiff materials. Retrieved from https://2201.10022Google Scholar .
- 2022b. Penetration-free projective dynamics on the GPU. ACM Trans. Graph. 41, 4, Article
69 (July 2022), 16 pages.Google ScholarDigital Library . - 2021. Medial IPC: Accelerated incremental potential contact with medial elastics. ACM Trans. Graph. 40, 4, Article
158 (July 2021), 16 pages.Google ScholarDigital Library . - 2009. Fast BVH construction on GPUs. Comput. Graph. Forum 28, 2 (2009), 375–384.Google ScholarCross Ref .
- 2010. gProximity: Hierarchical GPU-based operations for collision and distance queries. In Comput. Graph. Forum, Vol. 29. Wiley Online Library, 419–428.Google Scholar .
- 2020. P-cloth: Interactive complex cloth simulation on multi-gpu systems using dynamic matrix assembly and pipelined implicit integrators. ACM Trans. Graph. 39, 6 (2020), 1–15.Google ScholarDigital Library .
- 2020a. Incremental potential contact: Intersection- and inversion-free, large-deformation dynamics. ACM Trans. Graph. 39, 4, Article
49 (2020).Google ScholarDigital Library . - 2020b. Technical supplement to incremental potential contact: Intersection- and inversion-free, large-deformation dynamics. ACM Trans. Graph. 39, 4 (2020).Google ScholarDigital Library .
- 2021. Codimensional incremental potential contact. ACM Trans. Graph. (SIGGRAPH) 40, 4, Article
170 (2021).Google ScholarDigital Library . - 2015. Deformable objects collision handling with fast convergence. In Computer Graphics Forum, Vol. 34. Wiley Online Library, 269–278.Google Scholar .
- 2022. Isotropic ARAP energy using cauchy-green invariants. ACM Trans. Graph. 41, 6, Article
275 (Nov. 2022), 14 pages.Google ScholarDigital Library . - 2019. Non-smooth newton methods for deformable multi-body dynamics. ACM Trans. Graph. 38, 5, Article
140 (Oct. 2019), 20 pages.Google ScholarDigital Library . - 2019. AMASS: Archive of motion capture as surface shapes. In Proceedings of the International Conference on Computer Vision. 5442–5451.Google ScholarCross Ref .
- 2021. A survey on bounding volume hierarchies for ray tracing. Comput. Graph. Forum 40, 2 (2021), 683–712.Google ScholarCross Ref .
- 2003. Particle-based fluid simulation for interactive applications. In Proceedings of the 4th International Conference on Smart City Applications (SCA’03). The Eurographics Association, 154–159.Google Scholar .
- 2015. Air meshes for robust collision handling. ACM Trans. Graph. 34, 4 (2015), 1–9.Google ScholarDigital Library .
- 2006. Numerical Optimization (2nd ed.). Springer, New York, NY.Google Scholar .
- 2009. Implicit contact handling for deformable objects. In Computer Graphics Forum, Vol. 28. Wiley Online Library, 559–568.Google Scholar .
- 2010. Fast and scalable cpu/gpu collision detection for rigid and deformable surfaces. In Computer Graphics Forum, Vol. 29. Wiley Online Library, 1605–1612.Google Scholar .
- 2020. Analytic Eigensystems for Isotropic Membrane Energies. Retrieved from https://arxiv.org/abs/2008.10698.
DOI: Google ScholarCross Ref . - 1997a. Collision and self-collision handling in cloth model dedicated to design garments. In Proceedings of the Conference on Computer Animation and Simulation, and (Eds.). Springer Vienna, Vienna, 177–189.Google ScholarCross Ref .
- 2023. A unified analysis of penalty-based collision energies. Proc. ACM Comput. Graph. Interact. Tech. 6, 3, Article
41 (Aug. 2023), 19 pages.Google ScholarDigital Library . - 2012. FEM simulation of 3D deformable solids: A practitioner’s guide to theory, discretization and model reduction. In Proceedings of the ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH’12). Article
20 , 50 pages.Google ScholarDigital Library . - 2008. Globally coupled collision handling using volume preserving impulses. In Proceedings of Symposium on Computer Animation.Google Scholar .
- 2005. Automatic determination of facial muscle activations from sparse motion capture marker data. ACM Trans. Graph. 24, 3 (
July 2005), 417–425.Google ScholarDigital Library . - 2018. Stable neo-Hookean flesh simulation. ACM Trans. Graph. 37, 2, Article
12 (Mar. 2018), 15 pages.Google ScholarDigital Library . - 2019. Analytic eigensystems for isotropic distortion energies. ACM Trans. Graph. 38, 1, Article
3 (Feb. 2019), 15 pages.Google ScholarDigital Library . - 2018a. PSCC: Parallel self-collision culling with spatial hashing on GPUs. Proc. ACM Comput. Graph. Interact. Techn. 1, 1 (2018), 1–18.Google ScholarDigital Library .
- 2016. CAMA: Contact-aware matrix assembly with unified collision handling for GPU-based cloth simulation. In Proceedings of the Computer Graphics Forum, Vol. 35. Wiley Online Library, 511–521.Google ScholarCross Ref .
- 2018b. I-Cloth: Incremental collision handling for GPU-based interactive cloth simulation. ACM Trans. Graph. 37, 6 (2018), 1–10.Google ScholarDigital Library .
- 2005. Robust quasistatic finite elements and flesh simulation. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA’05). Association for Computing Machinery, New York, NY, 181–190.Google ScholarDigital Library .
- 2019. Iterative Methods for Solving Linear Systems on Massively Parallel Architectures. Thesis. Sorbonne Université. Retrieved from https://theses.hal.science/tel-02428348Google Scholar .
- 2019. Efficient and accurate collision response for elastically deformable models. ACM Trans. Graph. 38, 2 (2019), 1–20.Google ScholarDigital Library .
- 2021. A large-scale benchmark and an inclusion-based algorithm for continuous collision detection. ACM Trans. Graph. 40, 5 (2021), 188:1–188:16.Google ScholarDigital Library .
- 2021. GPU-based simulation of cloth wrinkles at submillimeter levels. ACM Trans. Graph. 40, 4 (2021), 1–14.Google ScholarDigital Library .
- 2018. Efficient BVH-based collision detection scheme with ordering and restructuring. Comput. Graph. Forum 37, 2 (2018), 227–237.Google ScholarCross Ref .
- 2022. A GPU-based multilevel additive schwarz preconditioner for cloth and deformable body simulation. ACM Trans. Graph. 41, 4 (2022), 1–14.Google ScholarDigital Library .
Index Terms
- GIPC: Fast and Stable Gauss-Newton Optimization of IPC Barrier Energy
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