A new coupling technique for the combination of wavelet-Galerkin method with finite element method in solids and structures

Abstract

In this paper, a coupling technique is developed for the combination of the wavelet-Galerkin method (WGM) with the finite element method (FEM). In this coupled method, the WGM and FEM are respectively used in different sub-domains. The WGM sub-domain and the FEM sub-domain are connected by a transition region that is described by B-spline basis functions. The basis functions of WGM and FEM are modified in the transition region to ensure the basic polynomial reconstruction condition and the compatibility of displacements and their gradients. In addition, the elements of FEM and WGM are not necessary to conform to the transition region. This newly developed coupled method is applied to the stress analysis of 2D and 3D elasticity problems in order to investigate its performance and study parameters. Numerical results show that the present coupled method is accurate and stable. The new method has a promising potential for practical applications and can be easily extended for coupling of other numerical methods. COUPLING TECHNIQUE WAVELET-GALERKIN AND FINITE ELEMENTFor approximation in irregular domain Ω, only those basis functions that influence the given region should be used [14], that is, supp ∩ Ω ≠ 0The size of each element used for computation in the 2D approximation is (2 −i , 2 −k ).

Wavelets for Differential Equations and Numerical Operator Calculus

“…Moreover, nonlinear equations can turn out to be difficult to handle. Nevertheless, there have been many successful examples in the application to elliptic, hyperbolic, and parabolic PDE [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Wavelet-based collocation methods, where wavelet functions are used as shape functions, also registered some success.…”

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

“…for the standard finite element method as well as for the material point method (MPM) to model boundaries of any inclination in which the problem domain boundary does not coincide with background grid element edges. Liu et al [5] developed a coupling technique for combining the wavelet-Galerkin method with the finite element method and established its accuracy and stability for 2D and 3D elasticity problems. Dekker et al [6] proposed an extended finite element model (XFEM) in an adaptive environment to capture the fatigue crack propagation and crack growth retardation under mixed-mode loading and overloading.…”

A wavelet domain numerical method for the prediction of transient dynamic response in composite structures with a focus on wave propagation

“…B-spline functions are used for solving two dimensional elastic problems [78]. Liu, et al [79] have proposed the combination of the wavelet-Galerkin method and the FEM, which was achieved by exploiting a bridging transition domain. An algorithm based on B-spline technique is developed to describe the transition domain and construct a weight function.…”