Using Nesterov's Method to Accelerate Multibody Dynamics with Friction and Contact

which was introduced to solve the rigid multibody dynamics problem in the presence of friction and contact. The discretized equations of motion obtained here are identical to the ones in Tasora and Anitescu 2011 ["A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics," Comput. Methods Appl. Mech. Eng., 200(5-8), pp. [439][440][441][442][443][444][445][446][447][448][449][450][451][452][453]; what is different is the process of obtaining these equations. Instead of using maximum dissipation conditions as the basis for the Coulomb friction model, the approach detailed uses complementarity conditions that combine with contact unilateral constraints to augment the classical index-3 differential algebraic equations of multibody dynamics. The resulting set of differential, algebraic, and complementarity equations is relaxed after time discretization to a cone complementarity problem (CCP) whose solution represents the first-order optimality condition of a quadratic program with conic constraints. The method discussed herein has proven reliable in handling large frictional contact problems. Recently, it has been used with promising results in fluid-solid interaction applications. Alas, this solution is not perfect, and it is hoped that the detailed account provided herein will serve as a starting point for future improvements.

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“…(5c) as B j ʯk ðlþ1Þ j ? À b j 2 B j (8)which is the "bilateral constraint" CCP analog of the condition in Eq (7)…”

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confidence: 99%

Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

Negrut

Serban

Tasora

2017

Journal of Computational and Nonlinear Dynamics

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“…Prior work on simulating smooth friction models has used proximal-map projection operators [Erleben 2017;Jean 1999;Jourdan et al 1998] which work by projecting contact forces to the friction cone one contact at a time until convergence. There has been considerable work to address the slow convergence of relaxation methods, Mazhar et al [2015] use a convexification of the frictional contact problem [Anitescu and Hart 2004] to obtain a cone complementarity problem (CCP) and solve it using an accelerated version of projected gradient descent. Silcowitz et al [2009; developed a method for solving LCPs based on non-smooth nonlinear conjugate gradient (NNCG) applied to a PGS iteration.…”

Section: Related Work 21 Contactmentioning

confidence: 99%

Non-smooth Newton Methods for Deformable Multi-body Dynamics

et al. 2019

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Fig. 1. The Fetch robot picking up and transferring a tomato to a mechanical scale. The tomato is modeled using tetrahedral FEM, while the robot and working mechanical scale are modeled as rigid bodies connected by revolute and prismatic joints. Our method provides full two-way coupling that allows for stable grasping and force sensing on the gripper. The robot is controlled by a human operator in real-time. Model provided courtesy of Fetch Robotics, Inc.We present a framework for the simulation of rigid and deformable bodies in the presence of contact and friction. Our method is based on a non-smooth Newton iteration that solves the underlying nonlinear complementarity problems (NCPs) directly. This approach allows us to support nonlinear dynamics models, including hyperelastic deformable bodies and articulated rigid mechanisms, coupled through a smooth isotropic friction model. The fixed-point nature of our method means it requires only the solution of a symmetric linear system as a building block. We propose a new complementarity preconditioner for NCP functions that improves convergence, and we develop an efficient GPU-based solver based on the conjugate residual (CR) method that is suitable for interactive simulations. We show how to improve robustness using a new geometric stiffness approximation and evaluate our method's performance on a number of robotics simulation scenarios, including dexterous manipulation and training using reinforcement learning.

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“…• Projected gradient descent methods like Accelerated Projected Gradient Descent [MHNT15], Barzilai -Borwein [BB88] and the Kucera and Preconditioned spectral projected gradient with fallback (P-SPG-FB) methods in [HATN13]. • Krylov subspace methods: Gradient projected minimum residual (GPMINRES) in [HATN13].…”

Section: Solving the Optimization Problemmentioning

confidence: 99%

Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

Corona

Gorsich

Jayakumar

et al. 2019

Applied Mechanics Reviews

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In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred a renewed interest in the discrete element method (DEM) modeling of frictional contact. Force penalty methods, while economic and widely accessible, introduce artificial stiffness, requiring small time steps to retain numerical stability. Optimization-based methods, which enforce contacts geometrically through complementarity constraints leading to a differential variational inequality problem (DVI), allow for the use of larger time steps at the expense of solving a nonlinear complementarity problem (NCP) each time step. We review the latest efforts to produce solvers for this NCP, focusing on its relaxation to a cone complementarity problem (CCP) and solution via an equivalent quadratic optimization problem with conic constraints. We distinguish between first order methods, which use only gradient information and are thus linearly convergent and second order methods, which rely on a Newton type step to gain quadratic convergence and are typically more robust and problem-independent. However, they require the approximate solution of large sparse linear systems, thus losing their competitive advantages in large scale problems due to computational cost.In this work, we propose a novel acceleration for the solution of Newton step linear systems in second order methods using low-rank compression based fast direct solvers, leveraging on recent direct solver techniques for structured linear systems arising from differential and integral equations. We employ the Quantized Tensor Train (QTT) decomposition to produce efficient approximate representations of the system matrix and its inverse. This provides a versatile and robust framework to accelerate its solution using this inverse in a direct or a preconditioned iterative method. We demonstrate compressibility of the Newton step matrices in Primal Dual Interior Point (PDIP) methods as applied to the multibody dynamics problem. Using a number of numerical tests, we demonstrate that this approach displays sublinear scaling of precomputation costs, may be efficiently updated across Newton iterations as well as across simulation time steps, and leads to a fast, optimal complexity solution of the Newton step. This allows our method to gain an order of magnitude speedups over state-of-the-art preconditioning techniques for moderate to large-scale systems, hence mitigating the computational bottleneck of second order methods.

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Using Nesterov's Method to Accelerate Multibody Dynamics with Friction and Contact

Cited by 80 publications

References 55 publications

Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

Non-smooth Newton Methods for Deformable Multi-body Dynamics

Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

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