Zizhou Huang
Zachary Ferguson
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We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code.
We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.
Supplemental Material
- . 2020. A recursive algebraic coloring technique for hardware-efficient symmetric sparse matrix-vector multiplication. ACM Trans. Parallel Comput. 7, 3, Article
19 (June 2020), 37 pages. Google Scholar
Digital Library
- . 2021. Chapter 1 - Shape and topology optimization. In Geometric Partial Differential Equations - Part II, and (Eds.).
Handbook of Numerical Analysis , Vol. 22. Elsevier, 1–132.DOI: Google ScholarCross Ref
- . 2015. The FEniCS project version 1.5. Archive of Numerical Software 3 (2015).
DOI: Google ScholarCross Ref - . 2021. Design and control of soft robots using differentiable simulation. Current Robotics Reports (2021), 1–11.Google Scholar
- . 2018. Geodesic convolutional shape optimization. In International Conference on Machine Learning. PMLR, 472–481.Google Scholar
- . 2000. Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Ltd.Google Scholar
- . 2014. Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient. SIAM Journal on Control and Optimization 52, 5 (
Jan. 2014), 3371–3400.DOI: Google ScholarDigital Library - . 2019. Trajectory optimization for cable-driven soft robot locomotion. In Robotics: Science and Systems XV, Vol. 1. Robotics: Science and Systems Foundation.
DOI: Google ScholarCross Ref - . 2020. Soft robot control with a learned differentiable model. In 2020 3rd IEEE International Conference on Soft Robotics (RoboSoft ’20). IEEE, 417–423.
DOI: Google ScholarCross Ref - . 2000. Computing derivatives of computer programs. In Modern Methods and Algorithms of Quantum Chemistry: Proceedings, Second Edition, (Ed.).
NIC Series , Vol. 3. NIC-Directors, Jülich, 315–327. http://hdl.handle.net/2128/6053Google Scholar - . 2019. Large-scale sparse inverse covariance matrix estimation. SIAM Journal on Scientific Computing 41, 1 (2019), A380–A401.
DOI: arXiv:https://doi.org/10.1137/17M1147615 Google ScholarDigital Library - . 2020. State-of-the-art sparse direct solvers. (2020), 3–33. Google ScholarCross Ref
- . 2002. Robust treatment of collisions, contact and friction for cloth animation. ACM Trans. on Graph. 21 (
05 2002).Google ScholarDigital Library - . 1999. Nonsmooth Mechanics. Springer-Verlag.Google ScholarCross Ref
- . 2018. Accurate dissipative forces in optimization integrators. ACM Trans. Graph. 37, 6, Article
282 (Dec. 2018), 14 pages.DOI: Google ScholarDigital Library - . 2016. A compositional object-based approach to learning physical dynamics. arXiv preprint arXiv:1612.00341 (2016).Google Scholar
- . 2020. Design of dielectric elastomer actuators using topology optimization on electrodes. Smart Mater. Struct. 29, 7 (
June 2020), 075029.DOI: Google ScholarCross Ref - . 2011. A hybrid iterative solver for robustly capturing Coulomb friction in hair dynamics. ACM Trans. on Graph. 30 (
12 2011).Google ScholarDigital Library - . 2019. Material point method after 25 years: Theory, implementation and applications. Submitted to Advances in Applied Mechanics (2019), 1.Google Scholar
- . 2007. Structural rigidity optimization with frictionless unilateral contact. International Journal of Solids and Structures 44, 3 (
Feb. 2007), 1132–1144.DOI: Google ScholarCross Ref - . 2020. Automatic Shape Derivatives for Transient PDEs in FEniCS and Firedrake. (2020).
arxiv:math.OC/2001.10058 Google Scholar - . 2021. DiffPD: Differentiable projective dynamics. ACM Trans. Graph. 41, 2, Article
13 (Nov. 2021), 21 pages.DOI: Google ScholarDigital Library - . 2005. Unilateral Contact Problems: Variational Methods and Existence Theorems. CRC Press.
DOI: Google ScholarCross Ref - Zachary Ferguson and others. 2020. IPC Toolkit. Retrieved from https://github.com/ipc-sim/ipc-toolkitGoogle Scholar
- . 2021. Intersection-free rigid body dynamics. ACM Transactions on Graphics (SIGGRAPH) 40, 4, Article
183 (2021).Google ScholarDigital Library - . 2020. Computational design of cold bent glass FaçAdes. ACM Trans. Graph. 39, 6, Article
208 (Nov. 2020), 16 pages.DOI: Google ScholarDigital Library - . 2020. ADD: Analytically differentiable dynamics for multi-body systems with frictional contact. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1–15.Google ScholarDigital Library
- . 2009. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Internat. J. Numer. Methods Engrg. 79, 11 (2009), 1309–1331.
DOI: arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.2579 Google ScholarCross Ref - . 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Vol. 105. SIAM.
DOI: Google ScholarCross Ref - . 2019. X-CAD: Optimizing CAD models with extended finite elements. ACM Trans. Graph. 38, 6, Article
157 (Nov. 2019), 15 pages.DOI: Google ScholarDigital Library - . 2019. Real2Sim: Visco-elastic parameter estimation from dynamic motion. ACM Trans. Graph. 38, 6, Article
236 (Nov. 2019), 13 pages.DOI: Google ScholarDigital Library - . 2009. Asynchronous contact mechanics. In ACM Trans. on Graph. (TOG ’09), Vol. 28. ACM.Google ScholarDigital Library
- . 2008. Robust treatment of simultaneous collisions. SIGGRAPH (ACM Trans. on Graph.) 27, 3 (2008).Google Scholar
- . 1986. Shape optimization in contact problems based on penalization of the state inequality. Aplikace Matematiky 31, 1 (1986), 54–77. https://eudml.org/doc/15435Google Scholar
- . 2021. DiSECt: A differentiable simulation engine for autonomous robotic cutting. In Proceedings of Robotics: Science and Systems. Virtual.
DOI: Google ScholarCross Ref - . 2020. NeuralSim: Augmenting differentiable simulators with neural networks. arXiv preprint arXiv:2011.04217 (2020).Google Scholar
- . 2000. Contact shape optimization: A bilevel programming approach. Structural and Multidisciplinary Optimization 20, 3 (
Nov. 2000), 214–221.DOI: Google ScholarDigital Library - . 2019. Vibration-minimizing motion retargeting for robotic characters. ACM Trans. Graph. 38, 4 (
July 2019), 1–14.DOI: Google ScholarDigital Library - . 2022. A general two-stage initialization for sag-free deformable simulations. ACM Trans. Graph. 41, 4, Article
64 (Jul. 2022), 13 pages.DOI: Google ScholarDigital Library - . 2019a. DiffTaichi: Differentiable programming for physical simulation. In International Conference on Learning Representations.Google Scholar
- . 2019b. ChainQueen: A real-time differentiable physical simulator for soft robotics. In 2019 International Conference on Robotics and Automation (ICRA ’19). IEEE, 6265–6271.Google ScholarDigital Library
- . 2020. Fast tetrahedral meshing in the wild. ACM Trans. Graph. 39, 4, Article
117 (July 2020), 18 pages.DOI: Google ScholarDigital Library - . 2010. Mitsuba Renderer. (2010). http://www.mitsuba-renderer.orgGoogle Scholar
- . 2021. gradSim: Differentiable simulation for system identification and visuomotor control. International Conference on Learning Representations (ICLR) (2021). https://openreview.net/forum?id=c_E8kFWfhp0Google Scholar
- . 2020. Bijective projection in a shell. ACM Trans. Graph. 39, 6, Article
247 (Nov. 2020), 18 pages.DOI: Google ScholarDigital Library - . 1988. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods.
SIAM Studies in App. and Numer. Math. , Vol. 8. Society for Industrial and Applied Mathematics.Google ScholarCross Ref - . 2001. Algebraic mesh quality metrics. SIAM Journal on Scientific Computing 23, 1 (2001), 193–218.
DOI: arXiv:https://doi.org/10.1137/S1064827500371499 Google ScholarDigital Library - . 2020. Incremental potential contact: Intersection- and inversion-free large deformation dynamics. ACM Transactions on Graphics 39, 4 (2020).Google ScholarDigital Library
- . 2023a. Convergent Incremental Potential Contact. (2023).
arxiv:math.NA/2307.15908 Google Scholar - . 2022. DiffCloth: Differentiable cloth simulation with dry frictional contact. ACM Trans. Graph. (
Mar. 2022).DOI: Just Accepted. Google ScholarDigital Library - . 2023b. DiffFR: Differentiable SPH-based fluid-rigid coupling for rigid body control. ACM Trans. Graph. 42, 6, Article
179 (Dec. 2023), 17 pages.DOI: Google ScholarDigital Library - . 2019. Differentiable cloth simulation for inverse problems. Neural Information Processing Systems (2019).Google Scholar
- . 2018. Inverse elastic shell design with contact and friction. ACM Trans. Graph. 37, 6, Article
201 (Dec. 2018), 16 pages.DOI: Google ScholarDigital Library - . 2021. Automated routing of muscle fibers for soft robots. IEEE Trans. Robot. 37, 3 (
June 2021), 996–1008.DOI: Google ScholarCross Ref - . 2019. A review of automatic differentiation and its efficient implementation. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 9, 4 (2019), e1305.
DOI: Google ScholarCross Ref - . 2017. Shape Optimisation with the Level Set Method for Contact Problems in Linearised Elasticity. (
Jan. 2017). https://hal.archives-ouvertes.fr/hal-01435325Google Scholar - . 2004. Fluid control using the adjoint method. ACM Transactions on Graphics / SIGGRAPH 2004 23, 3 (
Aug. 2004).Google Scholar - . 2019. dolfin-adjoint 2018.1: Automated adjoints for FEniCS and Firedrake. Journal of Open Source Software 4, 38 (2019), 1292.
DOI: Google ScholarCross Ref - . 2022. Scalable automatic differentiation of multiple parallel paradigms through compiler augmentation. In Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC ’22). IEEE Press, Article
60 , 18 pages.Google ScholarCross Ref - . 2012. The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation. Vol. 24. SIAM.
DOI: Google ScholarCross Ref - . 2009. Implicit contact handling for deformable objects. Comp. Graph. Forum 28 (
04 2009).Google ScholarCross Ref - . 2017. Worst-case stress relief for microstructures. ACM Transactions on Graphics 36, 4 (2017).
DOI: Google ScholarDigital Library - . 2015. Elastic textures for additive fabrication. ACM Trans. Graph. 34, 4, Article
135 (July 2015), 12 pages.Google ScholarDigital Library - . 2020. Scalable differentiable physics for learning and control. In International Conference on Machine Learning. PMLR, 7847–7856.Google Scholar
- . 2017. Scalable locally injective mappings. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1.Google ScholarDigital Library
- . 2021. Differentiable implicit soft-body physics. arXiv preprint arXiv:2102.05791 (2021).Google Scholar
- . 2018. SPNets: Differentiable fluid dynamics for deep neural networks. In Proceedings of The 2nd Conference on Robot Learning (Proceedings of Machine Learning Research), , , , and (Eds.), Vol. 87. PMLR, 317–335. https://proceedings.mlr.press/v87/schenck18a.htmlGoogle Scholar
- . 2019. PolyFEM. https://polyfem.github.io/ (2019).Google Scholar
- . 2020. Simulation-ready characterization of soft robotic materials. IEEE Robot. Autom. Lett. 5, 3 (
July 2020), 3775–3782.DOI: Google ScholarCross Ref - . 2018. Set-in-stone: Worst-case optimization of structures weak in tension. ACM Trans. Graph. 37, 6, Article
252 (Dec. 2018), 13 pages.DOI: Google ScholarDigital Library - . 2015. Multistable architected materials for trapping elastic strain energy. Advanced Materials (Deerfield Beach, Fla.) 27 (
06 2015).DOI: Google ScholarCross Ref - . 2018. Stress-based topology optimization using spatial gradient stabilized XFEM. Structural and Multidisciplinary Optimization 57, 1 (2018), 17–38.Google ScholarDigital Library
- . 2013. Computational design of actuated deformable characters. ACM Trans. Graph. 32, 4, Article
82 (Jul. 2013), 10 pages.DOI: Google ScholarDigital Library - . 2001. Finite-dimensional contact mechanics. Phil. Trans. R. Soc. Lond. A 359 (2001).Google ScholarCross Ref
- . 2010. Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation. Computer Methods in Applied Mechanics and Engineering 199, 33 (
July 2010), 2165–2176.DOI: Google ScholarCross Ref - . 2020. MakeSense: Automated sensor design for proprioceptive soft robots. Soft Rob. 7, 3 (
June 2020), 332–345.DOI: Google ScholarCross Ref - . 2020. A low-parametric rhombic microstructure family for irregular lattices. ACM Trans. Graph. 39, 4, Article
101 (Jul. 2020), 20 pages.DOI: Google ScholarDigital Library - . 2021. Optimizing contact-based assemblies. ACM Trans. Graph. 40, 6, Article
269 (Dec. 2021), 19 pages.DOI: Google ScholarDigital Library - . 2005. Review of options for structural design sensitivity analysis. Part 1: Linear systems. Computer Methods in Applied Mechanics and Engineering 194, 30 (2005), 3213–3243.
DOI: Structural and Design Optimization. Google ScholarCross Ref - . 2019. Efficient and accurate collision response for elastically deformable models. ACM Trans. on Graph. (TOG) 38, 2 (2019).Google ScholarDigital Library
- . 2016. CppOptimizationLibrary. https://github.com/PatWie/CppNumericalSolversGoogle Scholar
- . 1995. Finite element algorithms for contact problems. Archives of Comp. Meth. in Eng. 2 (
12 1995).Google ScholarCross Ref - . 2022. Accelerated Policy Learning with Parallel Differentiable Simulation. (2022).
DOI: Google ScholarCross Ref - . 2016. Data-driven bending elasticity design by shell thickness. Computer Graphics Forum (Proceedings of Symposium on Geometry Processing) 35, 5 (2016), 157–166.Google ScholarCross Ref
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- Differentiable solver for time-dependent deformation problems with contact
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