An Interior-Constraint BEM for Regularization of Problems with Improper Boundary Conditions

Document Type : Regular Article

Authors

1 Assistant Professor, Department of Civil Engineering, Penn State Harrisburg, Middletown, PA 17057, USA

2 Associate Professor, Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA

3 Professor, Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA

Abstract

A well-posed problem in the analysis of elastic bodies requires the definition of appropriate constraints of the boundary to prevent rigid body motion. However, one is sometimes presented with the problem of non-self-equilibrated tractions on an elastic body that will cause rigid body motion, while the boundary should remain unconstrained. One such case is the analysis of multi-particle dynamics where the solution is obtained in a quasi-static approach. In such cases, the motion of the particles is governed by the dynamic equilibrium while the contact forces between particles may be computed from elastostatic solutions. This paper presents two regularization methods of Interior-Constraint Boundary Element techniques for elastostatic analysis with improper boundary supports. In the proposed method rigid body modes are eliminated by imposing constraints on the interior of an elastic body. This is accomplished through simultaneously solving the governing Boundary Integral Equation and Somigliana’s Identity. The proposed method is examined through assessment and verification studies where it is demonstrated, that for all considered problems rigid body motion is successfully constrained with minimal effects on body deformations.

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References

[1]     Cundall PA, Strack ODL. A discrete numerical model for granular assemblies. Geotechnique 1979;29:47–65.
[2]     Poschel T, Schwager T. Computational Granular Dynamics: Models and Algorithms. 2005.
[3]     Mathews IV GF. Developments for the advancement of the discrete element method. University of South Carolina, 2015.
[4]     Vable M. Importance and use of rigid body mode in boundary element method. Int J Numer Methods Eng 1990;29:453–72. doi:10.1002/nme.1620290302.
[5]     BLÁZQUEZ A, MANTIČ V, PARÍS F, CAÑAS J. ON THE REMOVAL OF RIGID BODY MOTIONS IN THE SOLUTION OF ELASTOSTATIC PROBLEMS BY DIRECT BEM. Int J Numer Methods Eng 1996;39:4021–38. doi:10.1002/(SICI)1097-0207(19961215)39:233.0.CO;2-Q.
[6]     Vodička R, Mantič V, París F. Note on the removal of rigid body motions in the solution of elastostatic traction boundary value problems by SGBEM. Eng Anal Bound Elem 2006;30:790–8. doi:10.1016/j.enganabound.2006.04.002.
[7]     Vodička R, Mantič V, París F. On the removal of the non-uniqueness in the solution of elastostatic problems by symmetric Galerkin BEM. Int J Numer Methods Eng 2006;66:1884–912. doi:10.1002/nme.1605.
[8]     Fredholm I. Sur une classe d’équations fonctionnelles. Acta Math 1903;27:365–90. doi:10.1007/BF02421317.
[9]     Asadollahi P, Tonon F. Coupling of BEM with a large displacement and rotation algorithm. Int J Numer Anal Methods Geomech 2011;35:749–60. doi:10.1002/nag.916.
[10]    Rump SM. Inversion of extremely Ill-conditioned matrices in floating-point. Jpn J Ind Appl Math 2009;26:249–77. doi:10.1007/BF03186534.
[11]    Lutz E, Ye W, Mukherjee S. Elimination of rigid body modes from discretized boundary integral equations. Int J Solids Struct 1998;35:4427–36. doi:10.1016/S0020-7683(97)00261-8.
[12]    Sapountzakis EJ, Dikaros IC. Advanced 3D beam element of arbitrary composite cross section including generalized warping effects. Int J Numer Methods Eng 2015;102:44–78. doi:10.1002/nme.4849.
[13]    Chen JT, Chen WC, Lin SR, Chen IL. Rigid body mode and spurious mode in the dual boundary element formulation for the Laplace problems. Comput Struct 2003;81:1395–404. doi:10.1016/S0045-7949(03)00013-0.
[14]    Xiao Y-X, Zhang P, Shu S. An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity. J Comput Appl Math 2007;200:637–52. doi:10.1016/j.cam.2006.01.021.
[15]    Ko YY, Chen CH. Application of symmetric Galerkin boundary element method on elastostatic neumann problems. Int Assoc Comput Methods Adv Geomech 2008:146–53.
[16]    Rizzo FJ. An integral equation approach to boundary value problems of classical elastostatics. Q Appl Math 1967;25:83–95. doi:10.1090/qam/99907.
[17]    Wagdy M, Rashed YF. Boundary element analysis of multi-thickness shear-deformable slabs without sub-regions. Eng Anal Bound Elem 2014;43:86–94. doi:10.1016/j.enganabound.2014.03.011.
[18]    Lamé G. Leçons sur la théorie mathématique de l’elasticité des corps solides par G. Lamé. Gauthier-Villars; 1852.
[19]    Brebbia CA, Dominguez J. Boundary elements: an introductory course. WIT press; 1994.
[20]    Cauchy AL. Sur un nouveau genre de calcul analogue au calcul infinitésimal. Oeuvres Complet d’Augustin Cauchy, Gauthier-Villars, Paris 1826.
[21]    Kelvin, Lord. Note on the integration of the equations of equilibrium of an elastic solid. Cambridge Dublin Math J 1848;3:87–9.
[22]    Kronecker L. Vorlesungen über die Theorie der Determinanten: Erste bis Einundzwanzigste Vorlesungen. vol. 2. BG Teubner; 1903.
[23]    Mineur H, Berthod-Zaborowski M, Henri B, Jean M. Techniques de calcul numérique: à l’usage des mathématiciens, astronomes, physiciens et ingénieurs. Dunod; 1952.
[24]    Hadamard J. Le problème de Cauchy et les équations aux derivées partielles linéaires hyperboliques 1932.
[25]    Dassault. Abaqus 6.8 Program 2008.
[26]    Slaughter WS. Variational Methods. Linearized Theory Elast., Boston, MA: Birkhäuser Boston; 2002, p. 387–429. doi:10.1007/978-1-4612-0093-2_10.

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History

  • Receive Date: 28 November 2017
  • Revise Date: 11 January 2018
  • Accept Date: 11 January 2018

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