Lower bound limit analysis by BEM: convex optimization problem and incremental approach. (English) Zbl 1297.74152

Summary: The lower bound limit approach of the classical plasticity theory is rephrased using the Multidomain Symmetric Galerkin Boundary Element Method, under conditions of plane and initial strains, ideal plasticity and associated flow rule. The new formulation couples a multidomain procedure with nonlinear programming techniques and defines the self-equilibrium stress field by an equation involving all the substructures (bem-elements) of the discretized system. The analysis is performed in a canonical form as a convex optimization problem with quadratic constraints, in terms of discrete variables, and implemented using the Karnak.sGbem code coupled with the optimization toolbox by MatLab. The numerical tests, compared with the iterative elastoplastic analysis via the Multidomain Symmetric Galerkin Boundary Element Method, developed by some of the present authors, and with the available literature, prove the computational advantages of the proposed algorithm.


74S15 Boundary element methods applied to problems in solid mechanics
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
65N38 Boundary element methods for boundary value problems involving PDEs
74P10 Optimization of other properties in solid mechanics
90C25 Convex programming
90C90 Applications of mathematical programming
Full Text: DOI


[1] Panzeca, T; Cucco, F; Terravecchia, S., Symmetric boundary element method versus finite element method, Comput Meth Appl Mech Eng, 191, 3347-3367, (2002) · Zbl 1101.74370
[2] Panzeca, T; Salerno, M; Terravecchia, S., Domain decomposition in the symmetric boundary element method analysis, Comput Mech, 28, 191-201, (2002) · Zbl 1076.74568
[3] Boyd, S; Vandeberghe, L., Convex optimization, (2004), Cambridge University press New York, US, 152-160
[4] Maier G, Polizzotto C. A boundary element approach to limit analysis, In: Fifth international conference on boundary elements, Hiroshima, Japan, Springer, Berlin, 1983: pp. 265-277.
[5] Panzeca, T., Shakedown and limit analysis by the boundary integral equation method, Eur J Mech A Solids, 11, 685-699, (1992) · Zbl 0765.73073
[6] Zhang, X; Liu, Y; Zhao, Y; Cen, Z., Lower bound limit analysis by the symmetric Galerkin boundary element method and complex method, Comput Meth Appl Mech Eng, 191, 1967-1982, (2002) · Zbl 1098.74730
[7] Chen, S; Liu, Y; Li, J; Cen, Z., Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming, Comput Meth Appl Mech Eng, 197, 3911-3921, (2008) · Zbl 1194.74514
[8] Chen, S; Liu, Y; Li, J; Cen, Z., Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures, Eur J Mech A Solids, 30, 183-194, (2011) · Zbl 1261.74034
[9] Maier, G; Diligenti, M; Carini, A., A variational approach to boundary element elastodynamic analysis and extension to multidomain problems, Comput Meth Appl Mech Eng, 92, 192-213, (1991) · Zbl 0744.73043
[10] Gray, LJ; Paulino, GH., Symmetric Galerkin boundary integral formulation for interface and multi-zone problems, Int J Numer Methods Eng, 40, 3085-3101, (1997) · Zbl 0905.73077
[11] Layton, JB; Ganguly, S; Balakrishna, C; Kane, JH., A symmetric Galerkin multi-zone boundary element formulation, Int J Numer Methods Eng, 40, 2913-2931, (1997) · Zbl 0909.73084
[12] Ganguly, S; Layton, JB; Balakrishna, C; Kane, JH., A fully symmetric multi-zone Galerkin boundary element method, Int J Numer Methods Eng, 44, 991-1009, (1999) · Zbl 0966.74075
[13] Vodicka, R; Mantic, V; Paris, F., Symmetrical variational formulation of BIE for domain decomposition problems in elasticity—an SGBEM approach for nonconforming discretizations of curved interfaces, Comput Model Eng Sci, 17, 173-2003, (2007) · Zbl 1184.65111
[14] Perez-Gavilan, JJ; Aliabadi, MH., A symmetric Galerkin BEM for multi-connected bodies: a new approach, Eng Anal Boundary Elem, 25, 633-638, (2001) · Zbl 1065.74631
[15] Freddi, F; Royer-Carfagni, G., Symmetric Galerkin BEM for bodies with unconstrained contours, Comput Meth Appl Mech Eng, 195, 961-981, (2006) · Zbl 1121.74060
[16] Vodicka, R; Mantic, V; Paris, F., On the removal of the non-uniqueness in the solution of the elastostatic problems by symmetric Galerkin BEM, Int J Solids Struct, 66, 1884-1912, (2006) · Zbl 1110.74867
[17] Panzeca, T; Salerno, M., Macro-elements in the mixed boundary value problems, Comput Mech, 26, 437-446, (2000) · Zbl 0993.74078
[18] Sloan, SW., Lower bound limit analysis using finite elements and linear programming, Int J Numer Methods Eng, 12, 61-77, (1988) · Zbl 0626.73117
[19] Makrodimopoulos, A., Computational formulation of shakedown analysis as a conic quadratic optimization problem, Mech Res Commun, 33, 72-83, (2006) · Zbl 1192.74302
[20] Cucco, F; Panzeca, T; Terravecchia, S., The program karnak.sgbem release 2.1, (2002), Palermo University
[21] MathWorks Inc, 2008; MatLab version 7.6.0. Natick, MA, USA.
[22] Zito, L; Cucco, F; Parlavecchio, E; Panzeca, T., Incremental elastoplastic analysis for active macro-zones, Int J Numer Methods Eng, 91, 1365-1385, (2012)
[23] Wolfram, S., Mathematica version 5.0. the Mathematica book, (2003), Cambridge University Press
[24] Maier, G; Polizzotto, CA., Galerkin approach to boundary element elastoplastic analysis, Comput Meth Appl Mech Eng, 60, 175-194, (1987) · Zbl 0602.73081
[25] Panzeca, T; Terravecchia, S; Zito, L., Computational aspects in 2D SBEM analysis with domain inelastic actions, Int J Numer Methods Eng, 82, 184-204, (2010) · Zbl 1188.74077
[26] Zito, L; Parlavecchio, E; Panzeca, T., On the computational aspects of a symmetric multidomain BEM approach for elastoplastic analysis, J Strain Anal Eng Des, 46, 103-120, (2011)
[27] Gao, XW., The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng Anal Boundary Elem, 26, 905-916, (2002) · Zbl 1130.74461
[28] Gao, XW., A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems, ASME J Appl Mech, 69, 154-160, (2002) · Zbl 1110.74446
[29] Holzer, S., How to deal with hypersingular integrals in the symmetric BEM, Commun Numer Meth Eng, 9, 219-232, (1993) · Zbl 0781.65091
[30] Bonnet, M., Regularized direct and indirect symmetric varational BIE formulation for three-dimensional elasticity, Eng Anal Boundary Elem, 15, 93-102, (1995)
[31] Guiggiani, M., Hypersingular boundary integral equation have an additional free term, Comput Mech, 16, 245-248, (1995) · Zbl 0840.65117
[32] Frangi, A; Novati, G., Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curved elements, Comput Mech, 19, 58-68, (1996) · Zbl 0888.73069
[33] Gao, XW., An effective method for numerical evaluation of general 2D and 3D high-order singular boundary integrals, Comput Meth Appl Mech Eng, 199, 2856-2864, (2010) · Zbl 1231.65236
[34] Terravecchia, S., Closed form coefficients in the symmetric boundary element approach, Eng Anal Boundary Elem, 30, 479-488, (2006) · Zbl 1195.74263
[35] Panzeca T, Fujita Yashima H, Salerno M Mathematical aspects and applications of the symmetric Galerkin boundary element method, in CD-ROM IV WCCM; 1998: n.437, Vol. I., Idelsohn SR, Onate E, Dvorkin EN (Eds.), Buenos Aires.
[36] Panzeca T, Salerno M, Terravecchia S Symmetric Galerkin boundary element method for 3-D analysis. In: International conference on boundary element techniques; 1999: Aliabadi MH (Ed.), London, pp. 477-486.
[37] Garcea, G; Armentano, G; Petrolo, S; Casciaro, R., Finite element shakedown analysis of two-dimensional structures, Int J Numer Methods Eng, 63, 1174-1202, (2005) · Zbl 1084.74052
[38] Zhang, X; Liu, Y; Cen, Z., Boundary element methods for lower bound limit and shakedown analysis, Eng Anal Boundary Elem, 28, 905-917, (2004) · Zbl 1130.74477
[39] Gao, XW; Davies, TG., An effective boundary element algorithm for 2D and 3D elastoplastic problem, Int J Solids Struct, 37, 4987-5008, (2000) · Zbl 0970.74077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.