Using krylov subspace and spectral methods for solving complementarity problems in many-body contact dynamics simulation

International Journal for Numerical Methods in Engineering

Volumn 95, Issue 7, 2013, Pages 541-561

  Heyn, Toby ^a   Anitescu, Mihai ^b   Tasora, Alessandro ^c   Negrut, Dan ^a  

^a University of Wisconsin Madison   (United States)

^b ARGONNE NATIONAL LABORATORY   (United States)

^c UNIVERSITY OF PARMA   (Italy)

Author keywords

Contact; Differential equations; Multibody dynamics; Solids

Indexed keywords

COMPLEMENTARITY PROBLEMS; CONTACT DYNAMICS SIMULATIONS; DIFFERENTIAL VARIATIONAL INEQUALITY; FRICTIONAL INTERACTIONS; GRADIENT PROJECTION METHODS; MULTI-BODY DYNAMIC; NUMERICAL SOLUTION; THREE-DIMENSIONAL FLOW;

CONTACTS (FLUID MECHANICS); DIFFERENTIAL EQUATIONS; EQUATIONS OF MOTION; SOLIDS; VARIATIONAL TECHNIQUES;

PROBLEM SOLVING;

EID: 84880633202     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.4513     Document Type: Article

References (58)

1. Pfeiffer F, Glocker C. Multibody Dynamics with Unilateral Contacts. John Wiley: New York, 1996.
2. SBEL. Multibody dynamics simulation movies, University of Wisconsin-Madison, 2012. (Available from: http://sbel.wisc.edu/Animations) [Accessed on 1 July 2012].
3. Mazhar H. Parallel multi-body dynamics on graphics processing unit (GPU) Cards. MS thesis, University of Wisconsin-Madison, 2012. (Available from: http://sbel.wisc.edu/documents/HammadMazharMSthesisFinal.pdf) [Accessed on 1 July 2012].
4. Negrut D, Tasora A, Anitescu M, Mazhar H, Heyn T, Pazouki A. Solving large multi-body dynamics problems on the GPU. GPU Gems 2011; 4:269-280.
5. Madsen J, Pechdimaljian N, Negrut D. Penalty versus complementarity-based frictional contact of rigid bodies: a CPU time comparison. Technical Report TR-2007-06, Simulation-Based Engineering Lab, University of Wisconsin, Madison, 2007.
6. Cundall P. Formulation of a three-dimensional distinct element model-Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 1988; 25(3):107-116.
7. Song P, Pang JS, Kumar V. A semi-implicit time-stepping model for frictional compliant contact problems. International Journal of Numerical Methods in Engineering 2004; 60(13):267-279.
8. Moreau JJ. Standard inelastic shocks and the dynamics of unilateral constraints. In Unilateral Problems in Structural Analysis, Piero GD, Macieri F (eds), CISM Courses and Lectures no. 288. Springer-Verlag: New York, 1983; 173-221.
9. Lotstedt P. Mechanical systems of rigid bodies subject to unilateral constraints. SIAM Journal of Applied Mathematics 1982; 42(2):281-296.
10. Marques MDPM. Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Progress in Nonlinear Differential Equations and Their Applications, vol.9. Birkhäuser Verlag: Basel, 1993.
11. Baraff D. Issues in computing contact forces for non-penetrating rigid bodies. Algorithmica 1993; 10:292-352.
12. Pang JS, Trinkle JC. Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction. Mathematical Programming 1996; 73(2):199-226.
13. Stewart DE, Trinkle JC. An implicit time-stepping scheme for rigid-body dynamics with inelastic collisions and Coulomb friction. International Journal for Numerical Methods in Engineering 1996; 39:2673-2691.
14. Anitescu M, Potra FA. Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dynamics 1997; 14:231-247.
15. Stewart DE. Rigid-body dynamics with friction and impact. SIAM Review 2000; 42(1):3-39.
16. Cottle RW, Dantzig GB. Complementary pivot theory of mathematical programming. Linear Algebra and Its Applications 1968; 1:103-125.
17. Baraff D. Fast contact force computation for nonpenetrating rigid bodies. Computer Graphics (Proceedings of SIGGRAPH), Orlando, 1994; 23-34.
18. Alart P, Iceta D, Dureisseix D. A nonlinear domain decomposition formulation with application to granular dynamics. Computer Methods in Applied Mechanics and Engineering 2012; 205:59-67.
19. Visseq V, Martin A, Iceta D, Azema E, Dureisseix D, Alart P. Dense granular dynamics analysis by a domain decomposition approach. Computational Mechanics 2012; 49(6):709-723.
20. Anitescu M. Optimization-based simulation of nonsmooth rigid multibody dynamics. Mathematical Programming 2006; 105(1):113-143. DOI: http://dx.doi.org/10.1007/s10107-005-0590-7.
21. Anitescu M, Tasora A. An iterative approach for cone complementarity problems for nonsmooth dynamics. Computational Optimization and Applications 2010; 47(2):207-235. DOI: 10.1007/s10589-008-9223-4.
22. Heyn T, Anitescu M, Tasora A, Negrut D. A comparison of several solvers for bound constraint quadratic programming within the context of frictionless multibody dynamics. Technical Report TR-2012-02, Simulation-Based Engineering Laboratory, University of Wisconsin-Madison, 2012.
23. Haug EJ. Computer-Aided Kinematics and Dynamics of Mechanical Systems Volume-I. Prentice-Hall: Englewood Cliffs, New Jersey, 1989.
24. Pang JS, Stewart DE. Differential variational inequalities. Mathematical Programming 2008; 113:1-80.
25. Stewart DE. Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painleve's problems. Archive Rational Mechanics and Analysis 1998; 145(3):215-260.
26. Tasora A. High performance complementarity solver for non-smooth dynamics. In Proceedings of the ECCOMAS Multibody Dynamics Conference, Milan, Italy, 2007; 1-10.
27. Anitescu M, Hart GD. A constraint-stabilized time-stepping approach for rigid multibodydynamics with joints, contact and friction. International Journal for Numerical Methods in Engineering 2004; 60(14):2335-2371.
28. Anitescu M, Hart GD. A fixed-point iteration approach for multibody dynamics with contact and friction. Mathematical Programming, Series B 2004; 101(1):3-32.
29. Tasora A, Anitescu M. A convex complementarity approach for simulating large granular flows. Journal of Computational and Nonlinear Dynamics 2010; 5(3):1-10. DOI: 10.1115/1.4001371.
30. Tasora A, Anitescu M. A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Computer Methods in Applied Mechanics and Engineering 2011; 200(5-8):439-453. DOI: 10.1016/j.cma.2010.06.030.
31. Moreau JJ, Jean M. Numerical treatment of contact and friction: the contact dynamics method. Proceedings of the Third Biennial Joint Conference on Engineering Systems and Analysis, Montpellier, France, 1996; 201-208.
32. Anitescu M, Potra FA. Time-stepping schemes for stiff multi-rigid-body dynamics with contact and friction. International Journal for Numerical Methods in Engineering 2002; 55(7):753-784.
33. Glocker C, Pfeiffer F. An LCP-approach for multibody systems with planar friction. Proceedings of the CMIS 92 Contact Mechanics International Symposium, Lausanne, Switzerland, 2006; 13-20.
34. Preclik TM, Iglberger K, Rüde U. Iterative rigid multibody dynamics. Proceeding of Multibody Dynamics ECCOMAS Thematic Conference, Warsaw, 2009; 1-17.
35. Shojaaee Z, Shaebani MR, Brendel L, Török J, Wolf DE. An adaptive hierarchical domain decomposition method for parallel contact dynamics simulations of granular materials. Journal of Computational Physics 2012; 231(2):612-628. DOI: 10.1016/j.jcp.2011.09.024. (Available from: http://www.sciencedirect.com/science/article/pii/S0021999111005675) [Accessed on 1 July 2012].
36. Delannay R, Louge M, Richard P, Taberlet N, Valance A. Towards a theoretical picture of dense granular flows down inclines. Nature Materials 2007; 6(2):99-108.
37. Preclik TM, Rüde U. Solution existence and non-uniqueness of coulomb friction. Technical Report 4, Friedrich-Alexander-University Erlangen-Nurnberg, Institut Fur Informatik, Nurnberg, Germany, 2011.
38. Dostál Z, Schöberl J. Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination. Computational Optimization and Applications 2005; 30(1):23-43.
39. Tasora A. Chrono::Engine, an open source physics-based dynamics simulation engine, 2006. (Available from www.chronoengine.info) [Accessed on 5 July 2012].
40. Moré JJ, Toraldo G. On the solution of large quadratic programming problems with bound constraints. SIAM Journal on Optimization 1991; 1(1):93-113. DOI: 10.1137/0801008. (Available from: http://link.aip.org/link/?SJE/1/93/1) [Accessed on 25 June 2012].
41. Benson SJ, McInnes LC, Moré JJ. A case study in the performance and scalability of optimization algorithms. ACM Transactions on Mathematical Software 2001; 27(3):361-376. DOI: 10.1145/502800.502805.
42. Burke J, Moré JJ. Exposing constraints. SIAM Journal on Optimization 1994; 4:573-595.
43. Saad Y. Iterative Methods for Sparse Linear Systems. Society for Industrial Mathematics: Philadelphia, 2003.
44. Barzilai J, Borwein JM. Two-point step size gradient methods. IMA Journal of Numerical Analysis 1988; 8(1):141-148. DOI: 10.1093/imanum/8.1.141. (Available from: http://imajna.oxfordjournals.org/content/8/1/141.abstract).
45. Raydan M. On the Barzilai and Borwein choice of steplength for the gradient method. IMA Journal of Numerical Analysis 1993; 13(3):321-326. DOI: 10.1093/imanum/13.3.321.
46. Raydan M. The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM Journal on Optimization 1997; 7(1):26-33.
47. Birgin EG, Martínez JM, Raydan M. Nonmonotone spectral projected gradient methods on convex sets. SIAM Journal on Optimization 1999; 10:1196-1211. DOI: http://dx.doi.org/10.1137/S1052623497330963.
48. Grippo L, Lampariello F, Lucidi S. A nonmonotone line search technique for Newton's method. SIAM Journal on Numerical Analysis 1986; 23:707-716. DOI: 10.1137/0723046.
49. Bello L, Raydan M. Preconditioned spectral projected gradient method on convex sets. Journal of Computational Mathematics 2005; 23:225-232.
50. Luengo F, Raydan M, Glunt W, Hayden T. Preconditioned spectral gradient method. Numerical Algorithms 2002; 30:241-258. DOI: 10.1023/A:1020181927999.
51. Fletcher R. On the Barzilai-Borwein method. In Optimization and Control with Applications, Vol.96, Qi L, Teo K, Yang X (eds), Applied Optimization. Springer: US, 2005; 235-256, DOI: 10.1007/0-387-24255-4_10.
52. Nocedal J, Wright SJ. Numerical Optimization, Vol.39. Springer: New York, 1999.
53. Dostál Z. Optimal Quadratic Programming Algorithms: with Applications to Variational Inequalities, Vol.23. Springer: New York, 2009.
54. Dostál Z, Domorádová M, Sadowská M. Superrelaxation and the rate of convergence in minimizing quadratic functions subject to bound constraints. Computational Optimization and Applications 2011; 48(1):23-44.
55. Tasora A, Negrut D, Anitescu M. Large-scale parallel multi-body dynamics with frictional contact on the Graphical Processing Unit. Journal of Multi-body Dynamics 2008; 222(4):315-326.
56. Tasora A, Negrut D, Anitescu M. GPU-based parallel computing for the simulation of complex multibody systems with unilateral and bilateral constraints: an overview. In Multibody Dynamics, Vol.23, Arczewski K, Blajer W, Fraczek J, Wojtyra M (eds), Computational Methods in Applied Sciences. Springer: Netherlands, 2011; 283-307, DOI: 10.1007/978-90-481-9971-6_14.
57. Negrut D, Tasora A, Mazhar H, Heyn T, Hahn P. Leveraging parallel computing in multibody dynamics. Multibody System Dynamics 2012; 27:95-117. DOI: 10.1007/s11044-011-9262-y.
58. Dostál Z, Kozubek T. An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications. Mathematical Programming 2012; 42:1-26.