Triangular grid method for stress-wave propagation in 2-D orthotropic materials

Abstract

A triangular grid method is presented to calculate propagation problems of elastic stress waves in 2-D orthotropic materials. This method is based on the dynamic equilibrium equations of the computational cells formed among the auxiliary triangular grids. The solution is obtained by calculating alternately the nodal displacements and the central point stresses of the spatial grids. The numerical results are compared with the corresponding solutions of the finite element method. Comparisons show that the triangular grid method yields a higher calculational speed than the finite element method. The stress concentrations are investigated from wave-field analyses when the stress wave propagates within an orthotropic plate with a hole. Finally, the presented numerical method is used to study the features of wave propagation and diffraction in a square orthotropic plate with a hole when an impact load is applied to the top of the plate. IntroductionIt is well known that the methods for numerically simulating propagation problems of elastic stress waves are generally classified into finite element [1, 2], finite difference [3, 4] and boundary element methods [5]. The grid method [6, 7] provides a new approach for stress-wave propagation in elastic material with simple programming. This method incorporates the advantages of the finite element and finite difference methods and is similar to the finite element method in the discretization of a numerical mesh, and to the staggered grid difference scheme [8] in computation. Thus, there are no requirements for calculating and storing the stiffness matrix, so computational cost and memory requirements are small. The numerical results indicate that the grid method is an accurate and efficient numerical method for analyzing wave propagation problems.In this paper, we aim to develop an accurate, efficient and versatile numerical scheme for analyzing stress-wave propagation in orthotropic materials due to an impact load. We extend the grid method for modeling elastic stress-wave propagation in 2-D orthotropic materials. The angle , defined between the x axis of the global coordinates and the principal direction of an orthotropic material, can have any value. The algorithm formulation based on the triangular grids is discussed in detail. Numerical simulations are performed for the stress-wave

Quadrangular grid method for stress wave propagation in 2D orthotropic materials

“…(7) and (8) can be obtained at time t according to equation (7) through equation (12). The velocity components and the displacement components can be obtained by using time integration.…”

“…Following the literature [12] , the equations for the two dimensional problem of wave propagation are …”

“…Following Liu's work [12], the stability criterion for quadrangular grid method is also adopted as follows:…”

“…Finite element method [7][8][9] and Boundary Element Method [10][11] are effective methods for studying phenomena of wave propagation in anisotropic media. A kind of triangular grid method [12] has been proposed for stress wave propagation in 2D orthotropic media.…”

Thirty years of the finite volume method for solid mechanics

“…An additional form of vertex-centred approach is the so-called grid method, where an explicit solution algorithm is employed. The approach was initially proposed by Zhang and Liu [208], where the main intended application was geomechanical wave propagation problems; subsequently, the method has been extended to three dimensions, anisotropy, fluid-solid interaction and multi-CPU parallelisation [226,234,237,239,244]. Although the originators of the method, Zhang and Liu [208], argue for the distinction between the grid method and the vertex-based finite volume method, the authors of the current article believe this distinction is unwarranted: like other finite volume methods, the grid method starts from the governing momentum equation in strong integral form and approximates the forces over the boundary of control volumes; although there are minor differences in the techniques used to approximate the surface forces, for example, comparing Dormy and Tarantola [185] and Zhang and Liu [226], the grid method still remains a form of vertex-centred finite volume method.…”