A Generalized Maximum Dissipation Principle in an Impulse-velocity Time-stepping Scheme

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1 A Generalized Maximum Dissipation Principle in an Impulse-velocity Time-stepping Scheme T. Preclik, U. Rüde September 2, 213 Chair of Computer Science 1 (System Simulation) University of Erlangen-Nürnberg, Cauerstr. 11, 9158 Erlangen, Germany s: tobias.preclik@fau.de, web page:

friction (e.g. granular flow) particulate flows (e.g. fluidization and sedimentation) T.

2 Introduction The Chair of Computer Science 1 (System Simulation) amongst other topics deals with: rigid multibody dynamics with contact and friction (e.g. granular flow) particulate flows (e.g. fluidization and sedimentation) T. Preclik Chair for System Simulation, University of Erlangen-Nu rnberg 2

3 The Impulse-velocity Time-stepping Scheme Equations of motion. Semi-implicit Euler time discretization. Contact constraints act on relative contact velocities. Energy term. ( x v ϕ ) = δt ( Q( ϕ) w ) + ( x ϕ ) ( v w ) = δtm 1 f ( τ w diag k B (θ k (t)) w ) + ( v w ) E ( p) = U+ k B δv ( p) = A p b 1 2 m k v k ( p)t v k ( p)+1 2 w k ( p)t θ k w k ( p) E ( p) = δv ( p) T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 3

4 The One-Contact Problem Generalized maximum dissipation principle. Friction cone free from approximations. Inelastic contacts. Exclude contact impulses effecting an approaching velocity. Mathematical Program with Complementarity Constraints (MPCC) having a unique solution min E ( x) x F (A x b) n x n (A x) n T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 4

5 Numerical Method A contact separates iff b n < p = solves the MPCC Otherwise the contact persists (A x b) n = and x F define a conic section S: Either the unconstrained minimum of the objective function is inside the conic section S then the contact is static. Or the unconstrained minimum is outside S then the contact is dynamic and the energy minimum must be found on the boundary of S. T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 5

6 Numerical Method A contact separates iff b n < p = solves the MPCC Otherwise the contact persists (A x b) n = and x F define a conic section S: Either the unconstrained minimum of the objective function is inside the conic section S then the contact is static. Or the unconstrained minimum is outside S then the contact is dynamic and the energy minimum must be found on the boundary of S. x o x t 1/4 pi 1/2 pi 3/4 pi pi 5/4 pi 3/2 pi 7/4 pi 2 pi α T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 5

7 Numerical Method Univariate, scalar, non-linear, periodic function. Limit search interval to those points that have a direct line of sight to the unconstrained minimum. Use Golden Section Search to find minimum within unimodal range. Convergence is guaranteed (convergence rate is 1 ϕ.62 constantly). No derivative necessary. x o x t 1/4 pi 1/2 pi 3/4 pi pi 5/4 pi 3/2 pi 7/4 pi 2 pi α T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 6

8 The Multi-Contact Problem min E ( x) x i C F i (A x b) n x n (D x) n i C F i is the cartesian product of friction cones. D are the 3 3 diagonal blocks of A. Non-unique solution if problem is statically indeterminate. T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 7

9 Numerical Method Block non-linear Gauss-Seidel method (or Jacobi) relaxing the contacts successively. Solve each one-contact problem i C once per iteration: x (k+1) i = arg min E ( x), y i F i (A x b) i,n y i,n (D ii y i ) n where x is for Gauss-Seidel x = ( x (k+1) 1,..., x (k+1) i 1, y i, x (k) i+1,..., x(k) n ) T T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 8

10 Summary and Outlook A semi-implicit impulse-velocity time-stepping scheme, where a single MPCC is solved in each time step. Single contacts have unique solutions that can be computed reliably by an iterative method with constant convergence rate. The friction is modelled using a generalized maximum dissipation principle. Multi-contact problems can be solved with an iterative block non-linear Gauss-Seidel/Jacobi/SOR methods. Distributed-memory parallelization is work in progress. T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 9

11 Thank you for your attention! Questions? Comments? T. Preclik Chair for System Simulation, University of Erlangen-Nürnberg 1

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