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A constraint-stabilized time-stepping approach for piecewise smooth multibody dynamics

Hart, Gary D (2007) A constraint-stabilized time-stepping approach for piecewise smooth multibody dynamics. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Rigid multibody dynamics is an important area of mathematical modeling which attempts to predict the position and velocity of a system of rigid bodies. Many methods will use smooth bodies without friction. The task is made especially more difficult in the face of noninterpenetration constraints, joint constraints, and friction forces. The difficulty that arises when noninterpenetration constraints are enforced is directly related to the fact that the usual methods of computing the distance between bodies do not give any indication of the amount of penetration when two bodies interpenetrate. Because we wish to calculate vectors that are normal to contact, and because it is necessary to determine the amount of penetration, when it exists, the classical computation of the depth of penetration when applied to convex polyhedral bodies is inefficient.We hereby describe a new method of determining when two convex polyhedra intersect and of evaluating a measure of the amount of penetration, when it exists. Our method is much more efficient than the classic computation of the penetration depth since it can be shown that its complexity grows only linearly with the size of the problem. We use our method to construct a signed distance function and implement it for use with a method for achieving geometrical constraint stabilization for a linear-complementarity-based time-stepping scheme for rigid multibody dynamics with joints, contact, and friction which, before now, was not equipped to handle polyhedral bodies. During our analysis, we describe how to compute normal vectors at contact, despite the cases when the classic derivative fails to exist.We put this analysis into a time-stepping procedure that uses a convex relaxation of a mixed linear complementarity problem with a resulting fixed point iteration that is guaranteed to converge if the friction is not too large, the time step is not too large, and the initial solution is feasible. Finally, we construct an algorithm that achieves constraint stabilization with quadratic convergence.The numerical results proved to be quite satisfactory, implying that the constraint stabilization holds, and that quadratic convergence exists.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Hart, Gary Dgdhart@pitt.eduGDHART
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairAnitescu,
Committee CoChairLayton, William Jwjl@pitt.eduWJL
Committee MemberSchaefer, Andrew Jschaefer@engr.pitt.eduSCHAEFER
Committee MemberRiviere, Beatrice Mriviere@pitt.eduRIVIERE
Committee MemberYotov, Ivan Pyotov@math.pitt.eduYOTOV
Date: 20 June 2007
Date Type: Completion
Defense Date: 4 April 2007
Approval Date: 20 June 2007
Submission Date: 13 April 2007
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Constraint Stabilization; Depth of Penetration; Piecewise Smooth Multibody Dynamics
Other ID:, etd-04132007-012906
Date Deposited: 10 Nov 2011 19:36
Last Modified: 15 Nov 2016 13:40


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