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Stability of symmetric and nonsymmetric FEM–BEM couplings for nonlinear elasticity problems

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Abstract

We consider symmetric as well as non-symmetric coupling formulations of FEM and BEM in the frame of nonlinear elasticity problems. In particular, the Johnson-Nédélec coupling is analyzed. We prove that these coupling formulations are well-posed and allow for unique Galerkin solutions if standard discretizations by piecewise polynomials are employed. Unlike prior works, our analysis does neither rely on an interior Dirichlet boundary to tackle the rigid body motions nor on any assumption on the mesh-size of the discretization used.

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References

  1. Aurada, M., Feischl, M., Führer, T., Karkulik, M., Melenk, J.M., Praetorius, D.: Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity. Comput. Mech. 51(4), 399–419 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aurada, M., Feischl, M., Praetorius, D.: Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems. ESAIM Math. Model. Numer. Anal. 46(5), 1147–1173 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi, F., Johnson, C.: On the coupling of boundary integral and finite element methods. Calcolo 16(2), 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bielak, J., MacCamy, R.C.: An exterior interface problem in two-dimensional elastodynamics. Quart. Appl. Math. 41(1):143–159 (1983/84)

  5. Brink, U., Stephan, E.P.: Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity. Numer. Methods Partial Differ. Equ. 17(1), 79–92 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carstensen, C.: Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16(11), 819–835 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Carstensen, C.: Interface problems in viscoplasticity and plasticity. SIAM J. Math. Anal. 25(6), 1468–1487 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carstensen, C.: Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening. SIAM J. Numer. Anal. 33(1), 171–207 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Costabel, M., Ervin, V.J., Stephan, E.P.: Experimental convergence rates for various couplings of boundary and finite elements. Math. Comput. Model. 15(3–5), 93–102 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carstensen, C., Funken, S.A., Stephan, E.P.: On the adaptive coupling of FEM and BEM in \(2\)-d-elasticity. Numer. Math. 77(2), 187–221 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Costabel, M.: A symmetric method for the coupling of finite elements and boundary elements. The Mathematics of Finite Elements and Applications. VI (Uxbridge, 1987), pp. 281–288. Academic Press, London (1988).

  12. Costabel, M., Stephan, E.P.: Coupling of finite elements and boundary elements for inhomogeneous transmission problems in \({ R}^3\). In: The Mathematics of Finite Elements and Applications. VI (Uxbridge, 1987), pp. 289–296. Academic Press, London (1988)

  13. Costabel, M., Stephan, E.P.: Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27(5), 1212–1226 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carstensen, C., Stephan, E.P.: Adaptive coupling of boundary elements and finite elements. RAIRO Modél. Math. Anal. Numér. 29(7), 779–817 (1995)

    MATH  MathSciNet  Google Scholar 

  15. Carstensen, C., Zarrabi, D., Stephan, E.P.: On the \(h\)-adaptive coupling of FE and BE for viscoplastic and elastoplastic interface problems. J. Comput. Appl. Math. 75(2), 345–363 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gatica, G.N., Hsiao, G.C.: The coupling of boundary element and finite element methods for a nonlinear exterior boundary value problem. Z. Anal. Anwendungen 8(4), 377–387 (1989)

    MATH  MathSciNet  Google Scholar 

  17. Gatica, G.N., Hsiao, G.C.: On a class of variational formulations for some nonlinear interface problems. Rend. Mat. Appl. (7) 10(4), 681–715 (1990)

  18. Gatica, G.N., Hsiao, G.C.: On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in \({ R}^2\). Numer. Math. 61(2), 171–214 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gatica, G.N., Hsiao, G.C.: Boundary-field equation methods for a class of nonlinear problems. In: Pitman Research Notes in Mathematics Series, vol. 331 Longman, Harlow (1995)

  20. Gatica, G.N., Hsiao, G.C., Sayas, F.-J.: Relaxing the hypotheses of Bielak-MacCamy’s BEM-FEM coupling. Numer. Math. 120(3), 465–487 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. De Han, H.: A new class of variational formulations for the coupling of finite and boundary element methods. J. Comput. Math. 8(3), 223–232 (1990)

    MATH  MathSciNet  Google Scholar 

  22. Hsiao, G.C., Wendland, W.L.: Boundary integral equations. In: Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)

  23. Johnson, C., Nédélec, J.-C.: On the coupling of boundary integral and finite element methods. Math. Comp. 35(152), 1063–1079 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  24. Alois, K., Oldřich, J., Svatopluk, F.: Function spaces. In: Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden; Academia, Prague (1977)

  25. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  26. Of, G., Steinbach, O.: Is the one-equation coupling of finite and boundary element methods always stable? Z. Angew. Math. Mech. 93, 476–484 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. Polizzotto, C.: A symmetric-definite BEM formulation for the elastoplastic rate problem (invited contribution). In: Boundary Elements IX, vol. 2 (Stuttgart, 1987), pp. 315–334. Computational Mechanics, Southampton (1987)

  28. Sayas, F.-J.: The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces. SIAM J. Numer. Anal. 47(5), 3451–3463 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stephan, E.P.: Coupling of finite elements and boundary elements for some nonlinear interface problems. Comput. Methods Appl. Mech. Eng. 101(1–3), 61–72 (1992)

    Article  MATH  Google Scholar 

  30. Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, New York (2008)

    Book  MATH  Google Scholar 

  31. Steinbach, Olaf: A note on the stable one-equation coupling of finite and boundary elements. SIAM J. Numer. Anal. 49(4), 1521–1531 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Steinbach, O.: On the stability of the non-symmetric BEM/FEM coupling in linear elasticity. Comput. Mech. 51(4), 421–430 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Steinbach, O., Wendland, W.L.: On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262(2), 733–748 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wolfgang, L.: Wendland. On asymptotic error estimates for combined BEM and FEM. In: Finite element and boundary element techniques from mathematical and engineering point of view. CISM Courses and Lectures, vol. 301, pp. 273–333. Springer, Vienna (1988)

  35. Zeidler, E.: Nonlinear Functional Analysis and its Applications. IV. Springer, New York (1988)

    Book  MATH  Google Scholar 

  36. Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/B. Springer, New York (1990)

    Book  Google Scholar 

  37. Zienkiewicz, O.C., Kelly, D.W., Bettess, P.: Marriage à la mode-the best of both worlds (finite elements and boundary integrals). In: Energy Methods in Finite Element Analysis, pp. 81–107. Wiley, Chichester (1979)

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Acknowledgments

The research of the authors is supported through the FWF research project Adaptive Boundary Element Method, see http://www.asc.tuwien.ac.at/abem/, funded by the Austrian Science Fund (FWF) under grant P21732, as well as through the Innovative Projects Initiative of Vienna University of Technology. This support is thankfully acknowledged. The authors thank Ernst P. Stephan (University of Hannover) and Heiko Gimperlein (Heriot-Watt University Edinburgh) for fruitful discussions and careful revisions of earlier versions of this manuscript. Moreover, we thank Francisco-Javier Sayas (University of Delaware) for some hint on the simplification of the proof of Theorem 2.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040, Wien, Austria

    M. Feischl, T. Führer & D. Praetorius

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    M. Karkulik

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  1. M. Feischl
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  2. T. Führer
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  3. M. Karkulik
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  4. D. Praetorius
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Correspondence to T. Führer.

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Feischl, M., Führer, T., Karkulik, M. et al. Stability of symmetric and nonsymmetric FEM–BEM couplings for nonlinear elasticity problems. Numer. Math. 130, 199–223 (2015). https://doi.org/10.1007/s00211-014-0662-9

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  • DOI: https://doi.org/10.1007/s00211-014-0662-9

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