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Moving Dirichlet boundary conditions∗

Published online by Cambridge University Press:  10 October 2014

Robert Altmann
Robert Altmann*
Affiliation:
Institut für Mathematik MA4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.. [email protected]
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Abstract

This paper develops a framework to include Dirichlet boundary conditions on a subset ofthe boundary which depends on time. In this model, the boundary conditions are weaklyenforced with the help of a Lagrange multiplier method. In order to avoid that the ansatzspace of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, whichmaps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition aswell as existence results are presented for a class of second order initial-boundary valueproblems. For the semi-discretization in space, a finite element scheme is presented whichsatisfies a discrete stability condition. Because of the saddle point structure of theunderlying PDE, the resulting system is a DAE of index 3.

Keywords

Dirichlet boundary conditionsoperator DAEinf-sup conditionwave equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1859 - 1876
Copyright
© EDP Sciences, SMAI 2014

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References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003).
Altmann, R.. Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. (ZAMM) 93 (2013) 648–664. Google Scholar
R. Altmann. Modeling flexible multibody systems by moving Dirichlet boundary conditions. In Proc. of Multibody Dynamics 2013 - ECCOMAS Thematic Conference, Zagreb, Croatia, July 1–4 (2013).
M. Arnold and B. Simeon, The simulation of pantograph and catenary: a PDAE approach. Preprint (1990), Technische Universität Darmstadt, Germany (1998).
Arnold, M. and Simeon, B., Pantograph and catenary dynamics: A benchmark problem and its numerical solution. Appl. Numer. Math. 34 (2000) 345–362. Google Scholar
Babuška, I., The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179–192. Google Scholar
Babuška, I. and Gatica, G.N., On the mixed finite element method with Lagrange multipliers. Numer. Meth. Part. D. E. 19 (2003) 192–210. Google Scholar
Ben Belgacem, F., The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–197. Doi:10.1007/s002110050468. Google Scholar
D. Braess, Finite Elements – Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, New York (2007).
Bramble, J.H., The Lagrange multiplier method for Dirichlet’s problem. Math. Comput. 37 (1981) 1–11. Google Scholar
S. C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
F.J. Cavalieri, A. Cardona, V.D. Fachinotti and J. Risso, A finite element formulation for nonlinear 3D contact problems. Mecánica Comput. XXVI(16) (2007) 1357–1372.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
E. Emmrich and D. Šiška, Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization. Technical report, University of Liverpool (2012).
L.C. Evans, Partial Differential Equations, 2nd edn. American Mathematical Society (AMS). Providence (1998).
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
M. Géradin and A. Cardona, Flexible Multibody Dynamics: A Finite Element Approach. John Wiley, Chichester (2001).
Griepentrog, J.A., Gröger, K., Kaiser, H.-C. and Rehberg, J., Interpolation for function spaces related to mixed boundary value problems. Math. Nachr. 241 (2002) 110–120. Google Scholar
B. Gustafsson, High Order Difference Methods for Time Dependent PDE. Springer-Verlag, Berlin (2008).
P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society (EMS), Zürich (2006).
J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968).
Lions, J.-L. and Strauss, W.A., Some non-linear evolution equations. Bull. Soc. Math. France 93 (1965) 43–96. Google Scholar
M.K. Lipinski, A posteriori Fehlerschätzer für Sattelpunktsformulierungen nicht-homogener Randwertprobleme. Ph.D thesis, Ruhr Universität Bochum (2004).
J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques. Masson et Cie, Éditeurs, Paris (1967).
Payne, L.E. and Weinberger, H.F., An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286–292. Google Scholar
Poetsch, G., Evans, J., Meisinger, R., Kortüm, W., Baldauf, W., Veitl, A. and Wallaschek, J., Pantograph/catenary dynamics and control. Vehicle System Dynamics 28 (1997) 159–195. Google Scholar
A.A. Shabana, Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005).
Simeon, B., On Lagrange multipliers in flexible multibody dynamics. Comput. Method. Appl. M 195 (2006) 6993–7005. Google Scholar
B. Simeon, Computational flexible multibody dynamics. A differential-algebraic approach. Differential-Algebraic Equations Forum. Springer-Verlag, Berlin (2013).
O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer-Verlag, New York (2008).
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart (1996).
J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987).
E. Zeidler, Nonlinear Functional Analysis and its Applications IIa: Linear Monotone Operators. Springer-Verlag, New York (1990).