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Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

Published online by Cambridge University Press:  07 May 2020

Y. -Y. Ko Open the ORCID record for Y. -Y. Ko [Opens in a new window]
Y. -Y. Ko*
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan
*
*Corresponding author ([email protected])
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Abstract

When the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

Keywords

SGBEMNeumann problemnon-uniquenesshalf-space problem
Type
Research Article
Information
Journal of Mechanics , Volume 36 , Issue 6 , December 2020 , pp. 749 - 761
Copyright
Copyright © 2020 The Society of Theoretical and Applied Mechanics

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References

REFERENCES

Bonnet, M., Maier, G. and Polizzotto, C., “Symmetric Galerkin Boundary Element Methods,” ASME Applied Mechanics Reviews, 51(11), pp. 669–704 CrossRefGoogle Scholar
Hong, H.-K. and Chen, J.-T., “Derivations of Integral Equations of Elasticity,” ASCE Journal of Engineering Mechanics, 114(6), pp. 1028–1044 (1988).CrossRefGoogle Scholar
Frangi, A. and Novati, G. “BEM-FEM Coupling for 3D Fracture Mechanics Applications,” Computational Mechanics, 32(4-6), pp. 415–422 (2003).Google Scholar
Lamp, U., Schleicher, T., Stephan, E. and Wendland, W., “Theoretical and Experimental Asymptotic Convergence of The Boundary Integral Method for A Plane Mixed Boundary Value Problem,” in Boundary Element Methods in Engineering, ed. Brebbia, C. A., Springer-Verlag, Berlin, Germany, pp. 3–17 (1982).CrossRefGoogle Scholar
Chen, J.-T. and Hong, H.-K., “Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series,” Applied Mechanics Reviews, ASME, 52(1), pp. 17–33 (1999)CrossRefGoogle Scholar
Haas, M. and Kuhn, G., “A Symmetric Galerkin BEM Implementation for 3D Elastostatic Problems with An Extension to Curved Elements,” Computational Mechanics, 28(3-4), pp. 250–259 (2002).CrossRefGoogle Scholar
Frangi, A. and Novati, G., Springhetti, R. and Rovizzi, M., “3D Fracture Analysis by The Symmetric Galerkin BEM,” Computational Mechanics, 28(3-4), pp. 220–232 (2002).CrossRefGoogle Scholar
Ko, Y.-Y., “Symmetric Galerkin BEM and FE~BE Coupling for 3D Elastostatic Problems,” PhD Thesis, National Taiwan University, Taipei, Taiwan (2006) (in Chinese).Google Scholar
Frangi, A. and Novati, G. “Symmetric BE Method in Two-Dimensional Elasticity Evaluation of Double Integrals for Curved Elements,” Computational Mechanics, 19(2), pp. 58–68 (1996).CrossRefGoogle Scholar
Polizzotto, C. “A Symmetric Galerkin Boundary/Domain Element Method for Finite Elastic Deformations,” Computer Methods in Applied Mechanics and Engineering, 189(2), pp. 481–514 CrossRefGoogle Scholar
Verchery, G., “Régularisation du Système de L’équilibre des Structures Élastiques Discrètes,” Comptes Rendus de l’Académie des Sciences - Series II, 311, pp. 585–589 (1990).Google Scholar
Lutz, E., Ye, W.-J. and Mukherjee, S. “Elimination of Rigid Body Modes from Discretized Boundary Integral Equations,” International Journal of Solids and Structures, 35(33), 4427–4436 (1998).CrossRefGoogle Scholar
Blázquez, A., Mantie, V., París, F. and Cañas, J. “On the Removal of Rigid Body Motions in the Solution of Elastostatic Problems by Direct BEM,” International Journal for Numerical Methods in Engineering, 39(23), pp. 4021–4038 (1996).Google Scholar
Vodieka, R., Mantie, V. and París, F. “Note on the Removal of Rigid Body Motions in the Solution of Elastostatic Traction Boundary Value Problems by SGBEM,” Engineering Analysis with Boundary Elements, 30(9), pp. 790–798 (2006).Google Scholar
Pan, L., Rizzo, F. and Martin, P.A. “Some Efficient Boundary Integral Strategies for Time-Harmonic Wave Problems in An Elastic Halfspace,” Computer Methods in Applied Mechanics and Engineering, 164(1-2), pp. 207–221 (1998).CrossRefGoogle Scholar
Duffy., M. G., “Quadrature over a Pyramid or Cube of Integrals with a Singularity at a Vertex,” SIAM Journal on Numerical Analysis, 19(6), pp. 1260–1262 (1982).CrossRefGoogle Scholar
Freddi, F. and Royer-Carfagni, G., “Symmetric Galerkin BEM for Bodies with Unconstrained Contours,” Computer Methods in Applied Mechanics and Engineering, 195(9-12), pp. 961–981 (2006).CrossRefGoogle Scholar
Chen, J.-T., Chen, W.-C., Lin, S.-R. and Chen, I.-L. “Rigid Body Mode and Spurious Mode in The Dual Boundary Element Formulation for The Laplace Problems,” Computers and Structures, 81(13), pp. 1395–1404 (2003).CrossRefGoogle Scholar
Chen, J.-T., Huang, W.-S., Tu, Y.-C. and Lee, J.-W., 2014, A Self-Regularized Approach for Deriving the Free-Free Stiffness and Flexibility Matrices, Computers and Structures, 145, pp. 12–22 (2014).CrossRefGoogle Scholar
Chen, J.-T., Lee, Y.-T., Chang, Y.-L. and Jian, J., “A Self-Regularized Approach for Rank-Deficient Systems in The BEM of 2D Laplace Problems,” Inverse Problems in Science and Engineering, 25(1), pp. 89–13 (2017).CrossRefGoogle Scholar
Chen, J.-T., Chang, Y.-L, Leu, S.-Y. and Lee, J.-W., “Static Analysis of the Free-Free Trusses by Using a Self Regularization Approach, Journal of Mechanics, 34(4), pp. 505–518 (2018).CrossRefGoogle Scholar
Chen, G. and Zhou, J., Boundary Element Methods, Press, Academic, London, UK (1992).Google Scholar
Corona, E., Greengard, L., Rachh, M. and Veerapaneni, S., “An Integral Equation Formulation for Rigid Bodies in Stokes Flow in Three Dimensions,” Journal of Computational Physics, 332, pp. 504–519 (2017).CrossRefGoogle Scholar
Timoshenko, S. P. and Goodier, J. N. Theory of Elasticity 3rd ed., McGraw-Hill, New York, USA (1970).CrossRefGoogle Scholar
Gao, X. W. and Davies, T. G., Boundary Element Programming in Mechanics, Cambridge University Press, Cambridge, UK (2002).Google Scholar
Ahlvin, R. G. and Ulery, H. H., “Tabulated Values for Determining the Complete Pattern Of Stresses, Strains, and Deflections Beneath a Uniform Load on a Homogeneous Half Space,” Highway Research Board Bulletin, 342, 1–13 (1962).Google Scholar
Muki, R., “Asymmetric Problems of The Theory of Elasticity for a Semi-Infinite Solid and a Thick Plate” in Progress in Solid Mechanics, Vol. 1, ed. N. Sneddon and R. Hill, North-Holland, Amsterdam, Netherland, pp. 401–439. (1960).Google Scholar
Mattes, N. S. and Poulos, H. G. “Settlement of Single Compressive Pile,” Journal of the Soil Mechanics and Foundations Division, 95(1), pp. 189–207 (1969).Google Scholar
Zhou, K., Chen, W. W., Keer, L. M., and Wang, Q. J., 2009, “A Fast Method for Solving ThreeDimensional Arbitrarily Shaped Inclusions in a Half Space,” Computer Methods in Applied Mechanics and Engineering, 198(9-12), pp. 885–892 (2009).CrossRefGoogle Scholar
Wang, Z., Jin, X., Zhou, Q., Ai, X., Keer, L.M. and Wang, Q. “An Efficient Numerical Method with a Parallel Computational Strategy for Solving Arbitrarily Shaped Inclusions in Elastoplastic Contact Problems,” Journal of Tribology, 135(3): 031401 (2013).CrossRefGoogle Scholar