This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.
Keywords: Dirichlet boundary conditions, operator DAE, inf-sup condition, wave equation
@article{M2AN_2014__48_6_1859_0,
author = {Altmann, Robert},
title = {Moving {Dirichlet} boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1859--1876},
year = {2014},
publisher = {EDP Sciences},
volume = {48},
number = {6},
doi = {10.1051/m2an/2014022},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2014022/}
}
TY - JOUR AU - Altmann, Robert TI - Moving Dirichlet boundary conditions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1859 EP - 1876 VL - 48 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2014022/ DO - 10.1051/m2an/2014022 LA - en ID - M2AN_2014__48_6_1859_0 ER -
Altmann, Robert. Moving Dirichlet boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1859-1876. doi: 10.1051/m2an/2014022
[1] and , Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003). | Zbl | MR
[2] . Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. (ZAMM) 93 (2013) 648-664. | MR
[3] . Modeling flexible multibody systems by moving Dirichlet boundary conditions. In Proc. of Multibody Dynamics 2013 - ECCOMAS Thematic Conference, Zagreb, Croatia, July 1-4 (2013).
[4] and , The simulation of pantograph and catenary: a PDAE approach. Preprint (1990), Technische Universität Darmstadt, Germany (1998).
[5] and , Pantograph and catenary dynamics: A benchmark problem and its numerical solution. Appl. Numer. Math. 34 (2000) 345-362. | Zbl | MR
[6] , The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | Zbl | MR
[7] and , On the mixed finite element method with Lagrange multipliers. Numer. Meth. Part. D. E. 19 (2003) 192-210. | Zbl | MR
[8] , The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. Doi:10.1007/s002110050468. | Zbl | MR
[9] , Finite Elements - Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, New York (2007). | Zbl
[10] , The Lagrange multiplier method for Dirichlet's problem. Math. Comput. 37 (1981) 1-11. | Zbl | MR
[11] and , The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008). | Zbl | MR
[12] and , Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | Zbl | MR
[13] , , and , A finite element formulation for nonlinear 3D contact problems. Mecánica Comput. XXVI(16) (2007) 1357-1372.
[14] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | Zbl | MR
[15] and , Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization. Technical report, University of Liverpool (2012).
[16] , Partial Differential Equations, 2nd edn. American Mathematical Society (AMS). Providence (1998). | Zbl
[17] and , Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | Zbl | MR
[18] and , Flexible Multibody Dynamics: A Finite Element Approach. John Wiley, Chichester (2001).
[19] , , and , Interpolation for function spaces related to mixed boundary value problems. Math. Nachr. 241 (2002) 110-120. | Zbl | MR
[20] , Difference Methods for Time Dependent PDE. Springer-Verlag, Berlin (2008). | Zbl | MR
[21] and , Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society (EMS), Zürich (2006). | Zbl | MR
[22] and , Problèmes aux Limites Non Homogènes et Applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968). | Zbl
[23] and , Some non-linear evolution equations. Bull. Soc. Math. France 93 (1965) 43-96. | Zbl | MR | Numdam
[24] , A posteriori Fehlerschätzer für Sattelpunktsformulierungen nicht-homogener Randwertprobleme. Ph.D thesis, Ruhr Universität Bochum (2004). | Zbl
[25] , Les Méthodes Directes en Théorie des Equations Elliptiques. Masson et Cie, Éditeurs, Paris (1967). | Zbl | MR
[26] and , An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286-292. | Zbl | MR
[27] , , , , , and , Pantograph/catenary dynamics and control. Vehicle System Dynamics 28 (1997) 159-195.
[28] , Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005). | Zbl
[29] , On Lagrange multipliers in flexible multibody dynamics. Comput. Method. Appl. M 195 (2006) 6993-7005. | Zbl | MR
[30] , Computational flexible multibody dynamics. A differential-algebraic approach. Differential-Algebraic Equations Forum. Springer-Verlag, Berlin (2013). | Zbl | MR
[31] , Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer-Verlag, New York (2008). | Zbl | MR
[32] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart (1996). | Zbl
[33] , Partial Differential Equations. Cambridge University Press, Cambridge (1987). | Zbl | MR
[34] , Nonlinear Functional Analysis and its Applications IIa: Linear Monotone Operators. Springer-Verlag, New York (1990). | Zbl | MR
Cité par Sources :
