Abstract
This chapter provides a brief account of numerical integration schemes used to approximately evaluate definite integrals of arbitrary functions. Numerical integration schemes are required to evaluate the integrals that appear in the Galerkin weak forms presented in the preceding chapter for both mesh-based and meshfree methods. First, the classical Gauss quadrature scheme is explained before the more modern stabilized conforming nodal integration (SCNI) scheme is derived. Stabilized conforming nodal integration is a more advanced domain integration scheme that relies on a modification of the Galerkin weak form. A firm theoretical foundation for the modification of the Galerkin weak form used in stabilized conforming nodal integration is provided by the enhanced assumed strain (EAS) method. This method was originally introduced to alleviate volumetric locking in the finite element method. In this chapter, it is demonstrated how the enhanced assumed strain method can be used as a basis for a nodal integration scheme that can be applied to both Galerkin mesh-based and meshfree methods.
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References
Braess, D.: Enhanced assumed strain elements and locking in membrane problems. Comput. Methods Appl. Mech. Engrg. 165, 155–174 (1998)
Chen, J.S., Wang, D.: A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates. Int. J. Numer. Meth. Engng. 68, 151–172 (2006)
Chen, J.S., Wu, C.T., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 50, 435–466 (2001)
Chen, J.S., Yoon, S., Wu, C.T.: Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 53, 2587–2615 (2002)
Chen, J.S., Hu, W., Puso, M.A., Wu, Y., Zhang, X.: Strain smoothing for stabilization and regularization of Galerkin meshfree methods. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations III, pp. 57–75. Springer, Berlin (2007)
Chen, J.S., Hillman, M., Rüter, M.: An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int. J. Numer. Meth. Engng. 95, 387–418 (2013a)
Chen, J.S., Hillman, M., Rüter, M., Hu, H.Y., Chi, S.W.: The role of quadrature in meshfree methods: Variational consistency in Galerkin weak form and collocation in strong form. IACM Expr. 34, 11–16 (2013b)
Duan, Q., Li, X., Zhang, H., Belytschko, T.: Second-order accurate derivatives and integration schemes for meshfree methods. Int. J. Numer. Meth. Engng. 92, 399–424 (2012)
Guan, P.C., Chi, S.W., Chen, J.S., Slawson, T.R., Roth, M.J.: Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int. J. Impact Eng. 38, 1033–1047 (2011)
Hu, H.-C.: On some variational principles on the theory of elasticity and the theory of plasticity. Sci. Sinica 4, 33–54 (1955)
Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)
Liu, G.R., Nguyen-Thoi, T., Nguyen-Xuan, H., Lam, K.Y.: A node-based smoothed finite element method for solid problems (NS-FEM). Comput. & Struct. 87, 14–26 (2009)
Puso, M.A., Zywicz, E., Chen, J.S.: A new stabilized nodal integration approach. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations III, pp. 207–217. Springer, Berlin (2007)
Reese, S., Wriggers, P.: A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Meth. Engng. 48, 79–109 (2000)
Rüter, M.O., Chen, J.S.: An enhanced-strain error estimator for Galerkin meshfree methods based on stabilized conforming nodal integration. Comput. Math. Appl. 74, 2144–2171 (2017)
Simo, J.C., Rifai, M.S.: A class of assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 29, 1595–1638 (1990)
Wang, D., Chen, J.S.: Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput. Methods Appl. Mech. Engrg. 193, 1065–1083 (2004)
Washizu, K.: On the variational principles of elasticity and plasticity. Technical Report 25-18, Aeroelastic and Structures Research Laboratory, MIT, Cambridge, Massachusetts (1955)
Wriggers, P., Reese, S.: A note on enhanced strain methods for large deformations. Comput. Methods Appl. Mech. Engrg. 235, 201–209 (1996)
Xuan, Z.C., Lassila, T., Rozza, G., Quarteroni, A.: On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method. Int. J. Numer. Meth. Engng. 83, 174–195 (2010)
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Rüter, M.O. (2019). Numerical Integration. In: Error Estimates for Advanced Galerkin Methods. Lecture Notes in Applied and Computational Mechanics, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-06173-9_5
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DOI: https://doi.org/10.1007/978-3-030-06173-9_5
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