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Maximum-entropy meshfree method for compressible and near-incompressible elasticity. (English) Zbl 1231.74491
Summary: Numerical integration errors and volumetric locking in the near-incompressible limit are two outstanding issues in Galerkin-based meshfree computations. In this paper, we present a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction to machine precision. Secondly, a locking-free small-strain elasticity formulation for meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. In this study, maximum-entropy basis functions are used; however, the generality of our approach permits the use of any meshfree approximation. Various benchmark problems in two-dimensional compressible and near-incompressible small strain elasticity are presented to demonstrate the accuracy and optimal convergence in the energy norm of the maximum-entropy meshfree formulation.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74B05 Classical linear elasticity
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[1] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications, Inc Mineola, NY
[2] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and non-linear media, International journal for numerical methods in engineering, 15, 9, 1413-1418, (1980) · Zbl 0437.73053
[3] Malkus, D.S.; Hughes, T.J.R., Mixed finite element methods — reduced and selective integration techniques: a unification of concepts, Computer methods in applied mechanics and engineering, 15, 1, 63-81, (1978) · Zbl 0381.73075
[4] Simo, J.C.; Rifai, S., A class of mixed assumed strain methods and the method of incompatible modes, International journal for numerical methods in engineering, 29, 8, 1595-1638, (1990) · Zbl 0724.73222
[5] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Computational mechanics, 10, 307-318, (1992) · Zbl 0764.65068
[6] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, International journal for numerical methods in engineering, 37, 2, 229-256, (1994) · Zbl 0796.73077
[7] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, International journal for numerical methods in engineering, 20, 8-9, 1081-1106, (1995) · Zbl 0881.76072
[8] Liszka, T.J.; Duarte, C.A.; Tworzydlo, W.W., hp-meshless cloud method, Computer methods in applied mechanics and engineering, 139, 1-4, 263-288, (1996) · Zbl 0893.73077
[9] Atluri, S.N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational mechanics, 22, 2, 117-127, (1998) · Zbl 0932.76067
[10] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, International journal for numerical methods in engineering, 43, 5, 839-887, (1998) · Zbl 0940.74078
[11] Sukumar, N.; Moran, B.; Semenov, A.Y.; Belikov, V.V., Natural neighbour Galerkin methods, International journal for numerical methods in engineering, 50, 1, 1-27, (2001) · Zbl 1082.74554
[12] De, S.; Bathe, K.J., The method of finite spheres, Computational mechanics, 25, 4, 329-345, (2000) · Zbl 0952.65091
[13] Arroyo, M.; Ortiz, M., Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, International journal for numerical methods in engineering, 65, 13, 2167-2202, (2006) · Zbl 1146.74048
[14] Huerta, A.; Fernández-Méndez, S., Locking in the incompressible limit for the element-free Galerkin method, International journal for numerical methods in engineering, 51, 11, 1361-1383, (2001) · Zbl 1065.74635
[15] Dolbow, J.; Belytschko, T., Volumetric locking in the element free Galerkin method, International journal for numerical methods in engineering, 46, 6, 925-942, (1999) · Zbl 0967.74079
[16] González, D.; Cueto, E.; Doblaré, M., Volumetric locking in natural neighbour Galerkin methods, International journal for numerical methods in engineering, 61, 4, 611-632, (2004) · Zbl 1124.74328
[17] Vidal, Y.; Villon, P.; Huerta, A., Locking in the incompressible limit: pseudo-divergence-free element free Galerkin, Communications in numerical methods in engineering, 19, 9, 725-735, (2003) · Zbl 1112.74545
[18] Recio, D.P.; Jorge, R.M.N.; Dinis, L.M.S., Locking and hourglass phenomena in an element-free Galerkin context: the B-bar method with stabilization and an enhanced strain method, International journal for numerical methods in engineering, 68, 13, 1329-1357, (2006) · Zbl 1129.74050
[19] Beissel, S.; Belytschko, T., Nodal integration of the element-free Galerkin method, Computer methods in applied mechanics and engineering, 139, 1, 49-74, (1996) · Zbl 0918.73329
[20] Chen, J.S.; Wu, C.T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, International journal for numerical methods in engineering, 50, 2, 435-466, (2001) · Zbl 1011.74081
[21] Chen, J.S.; Hu, W.; Puso, M., Orbital HP-clouds for solving Schrödinger equation in quantum mechanics, Computer methods in applied mechanics and engineering, 196, 37-40, 3693-3705, (2007) · Zbl 1173.81301
[22] Puso, M.A.; Chen, J.S.; Zywicz, E.; Elmer, W., Meshfree and finite element nodal integration methods, International journal for numerical methods in engineering, 74, 3, 416-446, (2008) · Zbl 1159.74456
[23] Kucherov, L.; Tadmor, E.B.; Miller, R.E., Umbrella spherical integration: a stable meshless method for non-linear solids, International journal for numerical methods in engineering, 69, 13, 2807-2847, (2007) · Zbl 1194.74528
[24] Belytschko, T.; Guo, Y.; Liu, W.K.; Xiao, S.P., A unified stability analysis of meshless particle methods, International journal for numerical methods in engineering, 48, 9, 1359-1400, (2000) · Zbl 0972.74078
[25] Belytschko, T.; Xiao, S., Stability analysis of particle methods with corrected derivatives, Computers and mathematics with applications, 43, 3-5, 329-350, (2002) · Zbl 1073.76619
[26] Fries, T.P.; Belytschko, T., Convergence and stabilization of stress-point integration in mesh-free and particle methods, International journal for numerical methods in engineering, 74, 7, 1067-1087, (2008) · Zbl 1158.74525
[27] Dolbow, J.; Belytschko, T., Numerical integration of Galerkin weak form in meshfree methods, Computational mechanics, 23, 3, 219-230, (1999) · Zbl 0963.74076
[28] Griebel, M.; Schweitzer, M.A., A particle-partition of unity method. part II: efficient cover construction and reliable integration, SIAM journal on scientific computing, 23, 5, 1655-1682, (2002) · Zbl 1011.65069
[29] Gerstner, T.; Griebel, M., Numerical integration using sparse grids, Numerical algorithms, 18, 3-4, 209-232, (1998) · Zbl 0921.65022
[30] Riker, C.; Holzer, S.M., The mixed-cell-complex partition-of-unity method, Computer methods in applied mechanics and engineering, 198, 13-14, 1235-1248, (2009) · Zbl 1157.65492
[31] Atluri, S.N.; Kim, H.G.; Cho, J.Y., A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods, Computational mechanics, 24, 5, 348-372, (1999) · Zbl 0977.74593
[32] De, S.; Bathe, K.J., The method of finite spheres with improved numerical integration, Computers and structures, 79, 22-25, 2183-2196, (2001)
[33] Duflot, M.; Nguyen-Dang, H., A truly meshless Galerkin method based on a moving least squares quadrature, Communications in numerical methods in engineering, 18, 6, 441-449, (2002) · Zbl 1008.74082
[34] Zou, W.; Zhou, J.X.; Zhang, Z.Q.; Li, Q., A truly meshless method based on partition of unity quadrature for shape optimization of continua, Computational mechanics, 39, 4, 357-365, (2007) · Zbl 1178.74189
[35] Zeng, Q.-H.; Lu, D.-T., Galerkin meshless methods based on partition of unity quadrature, Applied mathematics and mechanics, 26, 7, 893-899, (2005) · Zbl 1144.65320
[36] Carpinteri, A.; Ferro, G.; Ventura, G., The partition of unity quadrature in meshless methods, International journal for numerical methods in engineering, 54, 7, 987-1006, (2002) · Zbl 1028.74047
[37] Balachandran, G.R.; Rajagopal, A.; Sivakumar, S.M., Mesh free Galerkin method based on natural neighbors and conformal mapping, Computational mechanics, 42, 6, 885-905, (2008) · Zbl 1163.74562
[38] Liu, G.R.; Tu, Z.H., An adaptive procedure based on background cells for meshless methods, Computer methods in applied mechanics and engineering, 191, 1, 1923-1943, (2002) · Zbl 1098.74738
[39] Babuška, I.; Banerjee, U.; Osborn, J.E.; Li, Q.L., Quadrature for meshless methods, International journal for numerical methods in engineering, 76, 9, 1434-1470, (2008) · Zbl 1195.65165
[40] Schembri, P.; Crane, D.L.; Reddy, J.N., A three-dimensional computational procedure for reproducing meshless methods and the finite element method, International journal for numerical methods in engineering, 61, 6, 896-927, (2004) · Zbl 1075.74676
[41] Bonet, J.; Burton, A.J., A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications, Communications in numerical methods in engineering, 14, 5, 437-449, (1998) · Zbl 0906.73060
[42] Dohrmann, C.R.; Heinstein, M.W.; Jung, J.; Key, S.W.; Witkowski, W.R., Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, International journal for numerical methods in engineering, 47, 9, 1549-1568, (2000) · Zbl 0989.74067
[43] Taylor, R.L., A mixed-enhanced formulation for tetrahedral finite elements, International journal for numerical methods in engineering, 47, 1-3, 205-227, (2000) · Zbl 0985.74074
[44] Bonet, J.; Marriot, M.; Hassan, O., An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications, Communications in numerical methods in engineering, 17, 8, 551-561, (2001) · Zbl 1154.74307
[45] Andrade Pires, F.M.; de Souza Neto, E.A.; de la Cuesta Padilla, J.L., An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains, Communications in numerical methods in engineering, 20, 7, 569-583, (2004) · Zbl 1302.74173
[46] Puso, M.A.; Solberg, J., A stabilized nodally integrated tetrahedral, International journal for numerical methods in engineering, 67, 6, 841-867, (2006) · Zbl 1113.74075
[47] Irving, G.; Schroeder, C.; Fedkiw, R., Volume conserving finite element simulations of deformable models, ACM transactions on graphics, 26, 3, 13.1-13.6, (2007)
[48] Krysl, P.; Zhu, B., Locking-free continuum displacement finite elements with nodal integration, International journal for numerical methods in engineering, 76, 7, 1020-1043, (2008) · Zbl 1195.74182
[49] Simo, J.C.; Hughes, T.J.R., On the variational foundations of assumed strain methods, Journal of applied mechanics, 53, 1, 51-54, (1986) · Zbl 0592.73019
[50] Sukumar, N.; Wright, R.W., Overview and construction of meshfree basis functions: from moving least squares to entropy approximants, International journal for numerical methods in engineering, 70, 2, 181-205, (2007) · Zbl 1194.65149
[51] Shannon, C.E., A mathematical theory of communication, The Bell systems technical journal, 27, 379-423, (1948) · Zbl 1154.94303
[52] Jaynes, E.T., Information theory and statistical mechanics, Physical review, 106, 4, 620-630, (1957) · Zbl 0084.43701
[53] Sukumar, N., Construction of polygonal interpolants: a maximum entropy approach, International journal for numerical methods in engineering, 61, 12, 2159-2181, (2004) · Zbl 1073.65505
[54] Shore, J.E.; Johnson, R.W., Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE transactions on information theory, 26, 1, 26-36, (1980) · Zbl 0429.94011
[55] Li, S.; Liu, W.K., Meshfree and particle methods and their applications, Applied mechanics reviews, 55, 1, 1-34, (2002)
[56] Fries, T.P.; Matthies, H.G., Classification and overview of meshfree methods, tech. rep. informatikbericht-nr. 2003-03, ()
[57] Fernández-Méndez, S.; Huerta, A., Imposing essential boundary conditions in mesh-free methods, Computer methods in applied mechanics and engineering, 193, 12-14, 1257-1275, (2004) · Zbl 1060.74665
[58] Yaw, L.L.; Sukumar, N.; Kunnath, S.K., Meshfree co-rotational formulation for two-dimensional continua, International journal for numerical methods in engineering, 79, 8, 979-1003, (2009) · Zbl 1171.74469
[59] Cyron, C.J.; Arroyo, M.; Ortiz, M., Smooth, second order, non-negative meshfree approximants selected by maximum entropy, International journal for numerical methods in engineering, 79, 13, 1577-1702, (2009) · Zbl 1176.74208
[60] Nakshatrala, K.B.; Masud, A.; Hjelmstad, K.D., On finite element formulations for nearly incompressible linear elasticity, Computational mechanics, 41, 4, 547-561, (2008) · Zbl 1162.74472
[61] Yvonnet, J.; Villon, P.; Chinesta, F., Natural element approximations involving bubbles for treating mechanical models in incompressible media, International journal for numerical methods in engineering, 66, 7, 1125-1152, (2006) · Zbl 1110.74873
[62] Sukumar, N.; Malsch, E.A., Recent advances in the construction of polygonal finite element interpolants, Archives of computational methods in engineering, 13, 1, 129-163, (2006) · Zbl 1101.65108
[63] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1970), McGraw-Hill NY · Zbl 0266.73008
[64] Arnold, D.N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984) · Zbl 0593.76039
[65] de Souza Neto, E.A.; Andrade Pires, F.M.; Owen, D.R.J., F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. part I: formulation and benchmarking, International journal for numerical methods in engineering, 62, 3, 353-383, (2005) · Zbl 1179.74159
[66] Hauret, P.; Kuhl, E.; Ortiz, M., Diamond elements: a finite element/discrete-mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity, International journal for numerical methods in engineering, 72, 3, 253-294, (2007) · Zbl 1194.74406
[67] Kikuchi, N., Remarks on 4CST-elements for incompressible materials, Computer methods in applied mechanics and engineering, 37, 1, 109-123, (1983) · Zbl 0487.73084
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