## Galerkin-based finite strain analysis with enriched radial basis interpolation.(English)Zbl 07526205

Summary: We present an enriched radial basis function (RBF) interpolation, which is combined with a symmetric Galerkin method to solve finite strain problems. Specifically, Wendland compact support radial basis functions (RBF) are enriched with a quadratic polynomial, whose parameters are condensed-out during the shape-function construction phase. A total Lagrangian technique is adopted and for finite strain plasticity the formalism with the Mandel stress is adopted. These recent developments are especially convenient from the implementation perspective, as RBF formulations for finite strain plasticity have been limited by interpolation updating. We also note that, although tetrahedra are adopted for quadrature in the undeformed configuration, mesh deformation is of no consequence for the results. Four benchmark tests are successfully solved, with extensive convergence studies and showing clear advantages with respect to finite-element in terms of robustness and with respect to classical Element-Free Galerkin (EFG) formulations in terms of efficiency.

### MSC:

 74-XX Mechanics of deformable solids 65-XX Numerical analysis
Full Text:

### References:

 [1] Lee, N. S.; Bathe, K. J., Effects of element distortions on the performance of isoparametric elements, Internat. J. Numer. Methods Engrg., 36, 3553-3576 (1993) · Zbl 0800.73465 [2] Peng, X.; Atroshchenko, E.; Kerfriden, P.; Bordas, S. P.A., Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Comput. Methods Appl. Math., 316, 151-185 (2017) · Zbl 1439.74370 [3] Bathe, K.-J., Finite Element Procedures (1996), Prentice-Hall [4] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077 [5] Bourantas, G.; Zwick, B. F.; Joldes, G. R.; Wittek, A.; Miller, K., Simple and robust element-free Galerkin method with almost interpolating shape functions for finite deformation elasticity, Appl. Math. Model., 96, 284-303 (2021) · Zbl 1481.74073 [6] V. Bayona, Comparison of moving least squares and RBF＋poly for interpolation and derivative approximation, 81 (2019) 486-512. · Zbl 1462.65020 [7] Chen, J.-S.; Hillman, M.; Chi, S.-W., Meshfree methods: Progress made after 20 years, J. Eng. Mech., 143, 4, Article 04017001 pp. (2017) [8] Kansa, E. J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I surface approximations and partial derivative estimates, Comput. Math. Appl., 19, 8-9, 127-145 (1990) · Zbl 0692.76003 [9] Buhmann, M. D., (Radial Basis Functions: Theory and Implementation. Radial Basis Functions: Theory and Implementation, Cambridge Monographs on Applied and Computational Mathematics (2004), Cambridge University Press: Cambridge University Press Cambridge, United Kingdom) [10] Wu, Z., Compactly supported positive definite radial functions, Adv. Comput. Math., 4, 283-292 (1995) · Zbl 0837.41016 [11] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4, 389-396 (1995) · Zbl 0838.41014 [12] Liu, G. R.; Zhang, G. Y.; Gu, Y. T.; Wang, Y. Y., A meshfree radial point interpolation method (RPIM) for three-dimensional solids, Comput. Mech., 36, 421-430 (2005) · Zbl 1138.74420 [13] Liew, K. M.; Chen, X. L.; Reddy, J. N., Mesh-free radial basis function method for buckling analysis of non-uniform loaded arbitrary shaped shear deformable plates, Comput. Methods Appl. Math., 193, 205-224 (2004) · Zbl 1075.74700 [14] Wang, J. G.; Liu, G. R., A point interpolation meshless method based on radial basis functions, Internat. J. Numer. Methods Engrg., 54, 1623-1648 (2002) · Zbl 1098.74741 [15] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117-127 (1998) · Zbl 0932.76067 [16] Atluri, S. N.; Kim, H.-G.; Cho, J. Y., A critical assessment of truly meshless local Petrov-Galerkin (MLPG) and local boundary integral equation (LBIE) methods, Comput. Mech., 24, 348-372 (1999) · Zbl 0977.74593 [17] Hu, D. A.; Long, S. Y.; Li, G. Y., A meshless local Petrov-Galerkin method for large deformation analysis of elastomers, Eng. Anal. Bound. Elem., 31, 657-666 (2007) · Zbl 1195.74284 [18] Safarpoor, M.; Takhtabnoos, F.; Shirzadi, A., A localized RBF-MLPG method and its application to elliptic PDEs, Eng. Comput., 36, 171-183 (2020) [19] Liu, Z.; Wei, G.; Qin, S.; Wang, Z., The elastoplastic analysis of functionally graded materials using a meshfree RRKPM, Appl. Math. Comput., 413, Article 126651 pp. (2022) · Zbl 07427474 [20] Rossi, R.; Alves, M. K., On the analysis of an EFG method under large deformations and volumetric locking, Comput. Mech., 39, 381-399 (2007) · Zbl 1178.74185 [21] Areias, P.; Rabczuk, T.; Ambrósio, J., Extrapolation and $$c_e$$-based implicit integration of anisotropic constitutive behavior, Internat. J. Numer. Methods Engrg., 122, 1218-1240 (2021) [22] Hosseini, S.; Rahimi, G.; Shahgholian-Ghahfarokhi, D., A meshless collocation method on nonlinear analysis of functionally graded hyperelastic plates using radial basis function, ZAMM Z. Angew. Math. Mech., 102, 2 (2022) [23] Ho, P. L.H.; Le, C. V., A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures, Comput. Struct., 228, Article 106157 pp. (2020) [24] Wriggers, P., Nonlinear Finite Element Methods (2008), Springer · Zbl 1153.74001 [25] Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The Finite Element Method. Its Basics & Fundamentals, volume 1 (2013), Butterworth-Heinemann, Elsevier: Butterworth-Heinemann, Elsevier Oxford, OX5 1GB UK · Zbl 1307.74005 [26] Belytschko, T.; Liu, W. K.; Moran, B., Nonlinear Finite Elements for Continua and Structures (2000), John Wiley & Sons · Zbl 0959.74001 [27] Areias, P.; César de Sá, J. M.A.; Conceição António, C. A.; Fernandes, A. A., Analysis of 3D problems using a new enhanced strain hexahedral element, Internat. J. Numer. Methods Engrg., 58, 1637-1682 (2003) · Zbl 1032.74661 [28] Areias, P.; Tiago, C.; Carrilho Lopes, J.; Carapau, F.; Correia, P., A finite strain Raviart-Thomas tetrahedron, Eur. J. Mech. A Solids, 80, Article 103911 pp. (2020) · Zbl 1472.74195 [29] Mandel, J., Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Struct., 9, 725-740 (1973) · Zbl 0255.73004 [30] Eidel, B.; Gruttmann, F., Elastoplastic orthotropy at finite strains: Multiplicative formulation and numerical implementation, Comput. Mater. Sci., 28, 732-742 (2003) [31] Kröner, E., Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Arch. Ration. Mech. Anal., 4, 273-334 (1960) · Zbl 0090.17601 [32] Lee, E. H., Elasto-plastic deformation at finite strains, Trans. ASME, J. Appl. Mech., 36, 1-6 (1969) · Zbl 0179.55603 [33] Lubliner, J., Plasticity Theory (1990), Macmillan · Zbl 0745.73006 [34] Gurtin, M. E., (An Introduction to Continuum Mechanics. An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, vol. 158 (1981), Academic Press: Academic Press 111 Fifth Avenue, New York, New York 10003) · Zbl 0559.73001 [35] Mandel, J., Thermodynamics and plasticity, (Foundations of Continuum Thermodynamics (1974), MacMillan: MacMillan London), 283-304 [36] Hill, R., A theory of yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond., 193, 281-297 (1948) · Zbl 0032.08805 [37] J. Zhu, Y. Xia, G. Gu, Q. Zhou, Influence of flow rule and calibration approach on plasticity characterization of DP780 steel sheets using Hill48 model, 89 (2014) 148-157. [38] P. Areias, Simplas. http://www.simplassoftware.com. Portuguese Software Association (ASSOFT) registry number 2281/D/17. [39] Wolfram Research Inc, Mathematica (2007) [40] Korelc, J., Multi-language and multi-environment generation of nonlinear finite element codes, Eng. Comput., 18, 4, 312-327 (2002) [41] Areias, P., Galerkin RBF (2021), https://github.com/PedroAreiasIST/RBF [42] Timoshenko, S.; Goodier, J. N., Theory of Elasticity (1951), McGraw-Hill Book Company, Inc: McGraw-Hill Book Company, Inc New-York · Zbl 0045.26402 [43] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, XXI, IV, 337-344 (1984) · Zbl 0593.76039 [44] Cook, R. D., Improved two-dimensional finite element, ASCE J. Struct. Div., 100, 9, 1851-1863 (1974) [45] Schröder, J.; Wick, T.; et al Reese, S., A selection of benchmark problems in solid mechanics and applied mathematics, Arch. Comput. Methods Eng., 28, 713-751 (2021) [46] Simo, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 33, 1413-1449 (1992) · Zbl 0768.73082 [47] Reese, S.; Wriggers, P.; Reddy, B. D., A new locking-free brick element technique for large deformation problems in elasticity, Comput. Struct., 75, 291-304 (2000) [48] Caylak, I.; Mahnken, R., Stabilization of mixed tetrahedral elements at large deformations, Internat. J. Numer. Methods Engrg., 90, 218-242 (2012) · Zbl 1242.74100 [49] Puso, M. A.; Solberg, J., A stabilized nodally integrated tetrahedral, Internat. J. Numer. Methods Engrg., 67, 841-867 (2006) · Zbl 1113.74075
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.