Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T06:50:02.080Z Has data issue: false hasContentIssue false
  • English
  • Français

A Numerical Method for Solving Elasticity Equations with Interfaces

Published online by Cambridge University Press:  20 August 2015

Songming Hou ,
Zhilin Li ,
Liqun Wang  and
Wei Wang
Songming Hou*
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA
Zhilin Li*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; and Nanjing Normal University, China
Liqun Wang*
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA
Wei Wang*
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

Solving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner.

Keywords

65N30Elasticity equationsnon-body fitted meshfinite element methodjump condition
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 595 - 612
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Babuška., I.The finite element method for elliptic equations with discontinuous coefficients. Computing, 5:207–213, 1970.Google Scholar
[2]B”ansch, E., Hausser, F., Lakkis, O., Li, B., and Voigt, A.. Finite element method for epitaxial growth with attachment-detachment kinetics. J. Comput. Phys., 194:409–434, 2004.Google Scholar
[3]Bramble, J. and King, J.. A finite element method for interface problems in domains with smooth boundaries and interfaces. Advances in Comput. Math., 6:109–138, 1996.CrossRefGoogle Scholar
[4]Burton, W. K., Cabrera, N., and Frank, F. C.. The growth of crystals and the equilibrium of their surfaces. Phil. Trans. Roy. Soc. London Ser. A, 243(866):299–358, 1951.Google Scholar
[5]Chan, K., Zhang, K., Liao, X., J.Zou, and Schubert, G.. A three-dimensional spherical nonlinear interface dynamo. Astrophy. J., 596:663–679, 2003.Google Scholar
[6]Chen, Z. and Zou, J.. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math., 79:175–202, 1998.Google Scholar
[7]Deng, S., Ito, K., and Li, Z.. Three dimensional elliptic solvers for interface problems and applications. J. Comput. Phys., 184:215–243, 2003.Google Scholar
[8]Gong, Y., Li, B., and Li, Z.. Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions. SIAM J. Numer. Anal., 46:472–495,2008.Google Scholar
[9]Gong, Y. and Li, Z.. Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-homogeneous Jump Conditions Numerical Mathematics: Theory, Methods and Applications, in press.Google Scholar
[10]Grisvard, P.. Elliptic Problems in Nonsmooth Domains - Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, ISSN 0743-0329, 1985.Google Scholar
[11]Hou, S. and Liu, X.-D.. A numerical method for solving variable coefficient elliptic equations with interfaces. J. Comput. Phys., 202:411–445, 2005.Google Scholar
[12]Hou, S., Wang, W., Wang, L.Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces. J. Comput. Phys., 229:7162–7179, 2010.Google Scholar
[13]LeVeque, R. J. and Li, Z.. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal, 31:1019–1044, 1994.Google Scholar
[14]Li, Z.. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal., 35:230–254, 1998.Google Scholar
[15]Li, Z.. The immersed interface method using a finite element formulation. Applied Numer. Math., 27:253–267, 1998.Google Scholar
[16]Li, Z., Lin, T., and Wu, X.. New Cartesian grid methods for interface problem using finite element formulation. Numer. Math., 96:61–98, 2003.Google Scholar
[17]Liu, X.-D., Fedkiw, R. P., and Kang, M.. A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys, 160 (1):151–178, 2000.CrossRefGoogle Scholar
[18]Liu, X.-D. and Sideris, T.. Convergence of the ghost fluid method for elliptic equations with interfaces. Math. Comp, 72:1731–1746, 2003.Google Scholar
[19]Necas, J.. Introduction to the theory of nonlinear elliptic equations. Teubner-Texte zur Mathematik, Band 52, ISSN 0138-502X, 1983.Google Scholar
[20]Yu, S., Zhou, Y., and Wei, G. W.. Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys., 224:729–756, 2007.CrossRefGoogle Scholar
[21]Oevermann, M., Scharfenberg, C., and Klein, R.A sharp interface finite volume method for elliptic equations on Cartesian grids. J. Comput. Phys., 228:5184-5206, 2009.CrossRefGoogle Scholar
[22]Yang, X.. Immersed interface method for elasticity problems with interfaces. PhD thesis, North Carolina State University, 2004.Google Scholar
[23]Yang, X., Li, B., and Li, Z.. The immersed interface method for elasticity problems with interface. Dynamics of Continuous, Discrete and Impulsive Systems, 10:783–808, 2003.Google Scholar
[24]Bedrossian, J., Brecht, J.Von, Zhu, S., Sifakis, E., and Teran, J.A Second Order Virtual Node Method for Poisson Interface Problems on Irregular Domains J. Comput. Phys., 229:6405–6426, 2010Google Scholar