Interface problems in elastoviscoplasticity
Authors:
Carsten Carstensen and Ernst P. Stephan
Journal:
Quart. Appl. Math. 53 (1995), 633-655
MSC:
Primary 73E60; Secondary 34G20, 35Q72, 73E50, 73Vxx
DOI:
https://doi.org/10.1090/qam/1359500
MathSciNet review:
MR1359500
Full-text PDF Free Access
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Abstract: This paper is concerned with three-dimensional interface (or transmission) problems in solid mechanics that consist of time-dependent nonlinear problems in a bounded Lipschitz domain and the homogeneous linear elasticity problem in an unbounded exterior domain. The exterior part of the interface problem is rewritten with integral operators on the interface boundary using the Poincaré-Steklov operator. This coupling approach uses the Calderón projector.
J. Bielak and R. C. MacCamy, An exterior interface problem in two-dimensional elastodynamics, Quart. Appl. Math. 41, 143–159 (1983)
F. Brezzi and C. Johnson, On the coupling of boundary integral and finite element methods, Calcolo 16, 189–201 (1979)
C. Carstensen, Interface problems in solid mechanics, Habilitation thesis, University of Hanover, 1993
C. Carstensen, Interface problem in holonomic elasto-plasticity, Math. Meth. Appl. Sci. 16, 819–835 (1993)
C. Carstensen, Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening, SIAM J. Numer. Anal. (1994), accepted for publication
M. Costabel, Symmetry’s methods for the coupling of finite elements and boundary elements, In: C. A. Brebia et al. (eds.), Boundary Elements IX, Vol. 1, Springer-Verlag, Berlin, 1987, pp. 411–420
M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19, 613–626 (1988)
M. Costabel, private communication, 1992
M. Costabel and E. P. Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Banach Center Publ. 15, 175–251 (1985)
M. Costabel and E. P. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106, 367–413 (1985)
M. Costabel and E. P. Stephan, Integral equations for transmission problems in linear elasticity, J. Integral Equations Appl. 2, 211–223 (1990)
M. Costabel and E. P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal. 27, 1212–1226 (1990)
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976
G. N. Gatica and G. C. Hsiao, The coupling of boundary element and finite element methods for a nonlinear exterior boundary value problem, Z. Anal. Anwendungen 8, 377–387 (1989)
G. N. Gatica and G. C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in $R^{2}$ , Numer. Math. 61, 171–214 (1992)
K. Gröger, Initial-value problems for elastoplastic and elasto-viscoplastic systems (S. Fučik and A. Kufner, eds.), Teubner, Leipzig, 1979
H. Han, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comput. Math. 8, 223–232 (1990)
G. C. Hsiao, E. P. Stephan, and W. L. Wendland, On the Dirichlet problem in elasticity for a domain exterior to an arc, Comput. Appl. Math. 34, 1–19 (1991)
C. Johnson and J. C. Nedelec, On the Coupling of Boundary Integral and Finite Element Methods, Math. Comp. 35 (152), 1063–1079 (1980)
V. D. Kupradze, editor, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Appl. Math. Mech. 25, North-Holland Publ. Co., Amsterdam, 1979
W. Rudin, Functional Analysis, McGraw-Hill, 1973
P. Le Tallec, Numerical analysis of viscoelastic problems, Recherches en Mathématiques Appliquées 15, Springer-Verlag, Berlin, 1990
W. L. Wendland, On asymptotic error estimates for combined BEM and FEM, in Finite and Boundary Element Techniques from a Mathematical and Engineering Point of View (E. Stein and W. L. Wendland, eds.), CISM Courses 301, Springer-Verlag, New York, 1988, pp. 273–331
E. Zeidler, Nonlinear Functional Analysis and its Applications II, Vols. A and B, Springer-Verlag, New York, 1990
E. Zeidler, Nonlinear Functional Analysis and its Applications III, Springer-Verlag, New York, 1985
E. Zeidler, Nonlinear Functional Analysis and its Applications IV, Springer-Verlag, New York, 1988
J. Bielak and R. C. MacCamy, An exterior interface problem in two-dimensional elastodynamics, Quart. Appl. Math. 41, 143–159 (1983)
F. Brezzi and C. Johnson, On the coupling of boundary integral and finite element methods, Calcolo 16, 189–201 (1979)
C. Carstensen, Interface problems in solid mechanics, Habilitation thesis, University of Hanover, 1993
C. Carstensen, Interface problem in holonomic elasto-plasticity, Math. Meth. Appl. Sci. 16, 819–835 (1993)
C. Carstensen, Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening, SIAM J. Numer. Anal. (1994), accepted for publication
M. Costabel, Symmetry’s methods for the coupling of finite elements and boundary elements, In: C. A. Brebia et al. (eds.), Boundary Elements IX, Vol. 1, Springer-Verlag, Berlin, 1987, pp. 411–420
M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19, 613–626 (1988)
M. Costabel, private communication, 1992
M. Costabel and E. P. Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Banach Center Publ. 15, 175–251 (1985)
M. Costabel and E. P. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106, 367–413 (1985)
M. Costabel and E. P. Stephan, Integral equations for transmission problems in linear elasticity, J. Integral Equations Appl. 2, 211–223 (1990)
M. Costabel and E. P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal. 27, 1212–1226 (1990)
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976
G. N. Gatica and G. C. Hsiao, The coupling of boundary element and finite element methods for a nonlinear exterior boundary value problem, Z. Anal. Anwendungen 8, 377–387 (1989)
G. N. Gatica and G. C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in $R^{2}$ , Numer. Math. 61, 171–214 (1992)
K. Gröger, Initial-value problems for elastoplastic and elasto-viscoplastic systems (S. Fučik and A. Kufner, eds.), Teubner, Leipzig, 1979
H. Han, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comput. Math. 8, 223–232 (1990)
G. C. Hsiao, E. P. Stephan, and W. L. Wendland, On the Dirichlet problem in elasticity for a domain exterior to an arc, Comput. Appl. Math. 34, 1–19 (1991)
C. Johnson and J. C. Nedelec, On the Coupling of Boundary Integral and Finite Element Methods, Math. Comp. 35 (152), 1063–1079 (1980)
V. D. Kupradze, editor, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Appl. Math. Mech. 25, North-Holland Publ. Co., Amsterdam, 1979
W. Rudin, Functional Analysis, McGraw-Hill, 1973
P. Le Tallec, Numerical analysis of viscoelastic problems, Recherches en Mathématiques Appliquées 15, Springer-Verlag, Berlin, 1990
W. L. Wendland, On asymptotic error estimates for combined BEM and FEM, in Finite and Boundary Element Techniques from a Mathematical and Engineering Point of View (E. Stein and W. L. Wendland, eds.), CISM Courses 301, Springer-Verlag, New York, 1988, pp. 273–331
E. Zeidler, Nonlinear Functional Analysis and its Applications II, Vols. A and B, Springer-Verlag, New York, 1990
E. Zeidler, Nonlinear Functional Analysis and its Applications III, Springer-Verlag, New York, 1985
E. Zeidler, Nonlinear Functional Analysis and its Applications IV, Springer-Verlag, New York, 1988
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Article copyright:
© Copyright 1995
American Mathematical Society
