An improved meshless method for finite deformation problem in compressible hyperelastic media

Nha Thanh Nguyen, Minh Ngoc Nguyen, Thien Tich Truong, Tinh Quoc Bui


Hyperelastic materials are considered as special category of elastic solid materials because of their nonlinear complicated constitutive laws. Due to large strain state, the behaviour of such materials is often considered in finite deformation analysis. The nonlinear large deformation behavior of such materials is important. In this study, a novel meshless radial point interpolation method (RPIM) enhanced by Cartesian transformation method (CTM), an effective numerical integration, is presented for nonlinear behavior of hyperelastic media under finite deformation state with total Lagrange formulation. Unlike the mesh-based approaches, the meshless methods have shown their advantages in analysis of large deformation problems. The developed CTM-based RPIM is thus free from the need for background cells, which are often used for numerical integration in many conventional meshfree approaches. The developed meshfree method owns some desirable features of an effective technique in solving large deformation, which will be illustrated through the numerical experiments in which our computed results are validated against reference solutions derived from other approaches. 


CTM-based meshless RPIM; hyperelastic materials; large deformation

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A. M. Maniatty, Y. Liu, O. Klaas, and M. S. Shephard. Higher order stabilized finite element method for hyperelastic finite deformation. Computer Methods in Applied Mechanics and Engineering, 191, (13-14), (2002), pp. 1491–1503.

A. A. Ramabathiran and S. Gopalakrishnan. Automatic finite element formulation and assembly of hyperelastic higher order structural models. Applied Mathematical Modelling, 38, (11-12), (2014), pp. 2867–2883.

A. Nomoto, H. Yasutaka, S. Oketani, and A. Matsuda. 2-dimensional homogenization FEM analysis of hyperelastic foamed rubber. Procedia Engineering, 147, (2016), pp. 431–436.

A. Angoshtari, M. F. Shojaei, and A. Yavari. Compatible-strain mixed finite element methods for 2D compressible nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering, 313, (2017), pp. 596–631.

T. Belytschko, Y. Y. Lu, and L. Gu. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 37, (2), (1994), pp. 229–256.

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 139, (1), (1996), pp. 3–47.

N. S. Atluri and T. Zhu. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 22, (2), (1998), pp. 117–127.

J. G.Wang and G. R. Liu. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 54, (11), (2002), pp. 1623–1648.

L. Gu. Moving kriging interpolation and element-free Galerkin method. International Journal for Numerical Methods in Engineering, 56, (1), (2002), pp. 1–11.

Y. T. Gu, Q. X.Wang, and K. Y. Lam. A meshless local Kriging method for large deformation analyses. Computer Methods in Applied Mechanics and Engineering, 196, (9-12), (2007), pp. 1673–1684.

D. A. Hu, S. Y. Long, X. Han, and G. Y. Li. A meshless local Petrov–Galerkin method for large deformation contact analysis of elastomers. Engineering Analysis with Boundary Elements, 31, (7), (2007), pp. 657–666.

D. Hu, Z. Sun, C. Liang, and X. Han. A mesh-free algorithm for dynamic impact analysis of hyperelasticity. Acta Mechanica Solida Sinica, 26, (4), (2013), pp. 362–372.

Y. Zhang, W. Ge, Y. Zhang, Z. Zhao, and J. Zhang. Topology optimization of hyperelastic structure based on a directly coupled finite element and element-free Galerkin method. Advances in Engineering Software, 123, (2018), pp. 25–37.

E. Khosrowpour, M. R. Hematiyan, and M. Hajhashemkhani. A strong-form meshfree method for stress analysis of hyperelastic materials. Engineering Analysis with Boundary Elements, 109, (2019), pp. 32–42.

A. Khosravifard and M. R. Hematiyan. A new method for meshless integration in 2D and 3D Galerkin meshfree methods. Engineering Analysis with Boundary Elements, 34, (1), (2010), pp. 30–40.

A. Khosravifard, M. R. Hematiyan, and L. Marin. Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method. Applied Mathematical Modelling, 35, (9), (2011), pp. 4157–4174.

T. Q. Bui, A. Khosravifard, C. Zhang, M. R. Hematiyan, and M. V. Golub. Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method. Engineering Structures, 47, (2013), pp. 90–104.

N. T. Nguyen, T. Q. Bui, M. N. Nguyen, and T. T. Truong. Meshfree thermomechanical crack growth simulations with new numerical integration scheme. Engineering Fracture Mechanics, (2020), p. 107121.

R. Hassani, R. Ansari, and H. Rouhi. Large deformation analysis of 2D hyperelastic bodies based on the compressible nonlinear elasticity: A numerical variational method. International Journal of Non-Linear Mechanics, 116, (2019), pp. 39–54.

J. P. Pascon. Large deformation analysis of plane-stress hyperelastic problems via triangular membrane finite elements. International Journal of Advanced Structural Engineering, 11, (3), (2019), pp. 331–350.

G. R. Liu, G. Y. Zhang, Y. T. Gu, and Y. Y.Wang. A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Computational Mechanics, 36, (6), (2005), pp. 421–430.

G. R. Liu. Mesh free methods - Moving beyon the finite element method. CRC Press, (2009).

P. Wriggers. Nonlinear Finite Element Methods. Springer Berlin Heidelberg, (2008).

H. D. Huynh, P. Tran, X. Zhuang, and H. Nguyen-Xuan. An extended polygonal finite element method for large deformation fracture analysis. Engineering Fracture Mechanics, 209, (2019), pp. 344–368.

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