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

Mathews IV, Grady; Rizos, Dimitris; Mullen, Robert

doi:10.22115/scce.2018.108597.1036

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	An Interior-Constraint BEM for Regularization of Problems with Improper Boundary Conditions
Article 1, Volume 2, Issue 2 - Serial Number 4, Spring 2018, Page 1-18 PDF (2056 K)
Document Type: Regular Article
DOI: 10.22115/scce.2018.108597.1036
Authors
Grady Mathews IV¹; Dimitris Rizos ²; Robert Mullen³
¹Assistant Professor, Department of Civil Engineering, Penn State Harrisburg, Middletown, PA 17057, USA
²Associate Professor, Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA
³Professor, Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA
Receive Date: 28 November 2017, Revise Date: 11 January 2018, Accept Date: 11 January 2018
Abstract
A well-posed problem in analysis of elastic bodies requires the definition of appropriate constrains 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 constrains 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.
Highlights
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Keywords
Boundary element method; Rigid body motion constrains; Regularization; Approximate solutions
Main Subjects
Structural Optimization
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References
[1] Cundall PA and Strack ODL. A discrete numerical model for granular assemplies. Geotechnique 29(1), 1979, 47-65. [2] Pöschel T and Thomas S. Computational Granular Dynamics: Models and Algorithums.: Springer; 2005. [3] Mathews GF. Developments for the Advancement of the Discrete Element Method. PhD Dissertation. Department of Civil and Environmental Engineering, 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 the 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 (On a class of functional equations). 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., Goa, India: 2008, p. 146–53. [16] Rizzo FJ. An integral equation approach to boundary value problems of classical elastostatics. Q Appl Math 1967;25:83–95. [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’élasticité des corps solides (Lessons on the mathematical theory of elastic solids). Paris: Gauthier-Villars; 1852. [19] Brebbia CA, Dominguez J. Boundary Elements: An Introductory Course. WIT Press; 1996. [20] Cauchy A. Sur un nouveau genre de calcul analogue au calcul infinitesimal (On a new type of calculus analogous to the infinitesimal calculus). Oeuvres Complet d’Augustin Cauchy, Gauthier-Villars, Paris 1826. [21] Thompson (Lord Kelvin) W. Note on the integration of the equations of equilibrium of an elastic solid. Cambridge Dublin Math J 1848:87–9. [22] Kronecker L. Vorlesungen über die Theorie der Determinanten: Erste bis Einundzwanzigste Vorlesungen (Lectures on the theory of determinants : First to Twenty-first lectures). B. G. Teubner; 1903. [23] Banerjee P. The Boundary Element Methods in Engineering. London: McGraw-Hill; 1994. [24] Mineur H, Berthod-Zaborowski H, Bouzitat J, Mayot M. Techniques de calcul numérique à l’usage des mathématiciens, astronomes, physiciens et ingénieurs (Numerical Computation techniques to use of Mathematicians, Astronomers, Physicists and Engineers). Liège Libr Polytech Béranger, Paris 1952. [25] Hadamard J. Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques (The Cauchy problem for linear equations and hyperbolic partial differential). Paris: Hermann & Cie.; 1932. [26] Dassault. Abaqus 6.8 Program 2008. [27] Slaughter W. The linearized theory of elasticity. Birkhäuser; 2002.
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