Local Discontinuous Galerkin Method for the Impact-Induced Wave in a Slender Cylinder Composed of a Non-Convex Elastic Material

Communications in Mathematics and Statistics, Jan 2014

In this paper, we present a numerical scheme based on the local discontinuous Galerkin (LDG) method for the wave propagation of phase transition in a slender cylinder by introducing new temporal auxiliary variables. The stability for the LDG scheme is presented. In order to verify the validity of the LDG scheme, we give the errors and accuracy order of a numerical example. Due to the interaction between the dispersion and the material nonlinearity, some interesting wave patterns occur for different pre-strains and impacts, such as the pattern with transformation front and solitary wave and the pattern with rarefaction wave and solitary wave. We also investigate the interaction of the transformation fronts and rarefaction waves, and demonstrate this interesting wave phenomena.

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Local Discontinuous Galerkin Method for the Impact-Induced Wave in a Slender Cylinder Composed of a Non-Convex Elastic Material

Jinfeng Jiang Yan Xu In this paper, we present a numerical scheme based on the local discontinuous Galerkin (LDG) method for the wave propagation of phase transition in a slender cylinder by introducing new temporal auxiliary variables. The stability for the LDG scheme is presented. In order to verify the validity of the LDG scheme, we give the errors and accuracy order of a numerical example. Due to the interaction between the dispersion and the material nonlinearity, some interesting wave patterns occur for different pre-strains and impacts, such as the pattern with transformation front and solitary wave and the pattern with rarefaction wave and solitary wave. We also investigate the interaction of the transformation fronts and rarefaction waves, and demonstrate this interesting wave phenomena. - 65M60 74M20 35L05 1 Introduction It is known in the nonlinear elasticity theory that the one-dimensional dynamics of phase transitions in solids can be modeled by the basic field equations [hyperbolic elliptic partial differential equations (PDEs)]. For example, in the stress-induced dynamic of phase transitions in one-dimensional bar, the model is governed by the following system of conservation laws Fig. 1 A diagram of the strainstress function ( )/ where , u, , and represent the strain, velocity, density, and stress, respectively. If the strainstress relation ( ) is non-monotonic (see Fig. 1), the system (1.1) is hyperbolic when ( ) > 0 and elliptic when ( ) < 0. Then we get a mixed hyperbolicelliptic system. However, when the solution occurs in the elliptic region, the system is not well-posed. More specifically, the uniqueness of solution is difficult to determine. Fortunately, many authors [1,2,6,15,17] (and the references therein) have carried out studies and analysis to ensure the uniqueness of solution by imposing additional conditions. For example, in [2], Abeyaratne neglected the effects of physical information (e.g., the viscosity, capillarity, and heat conditions) to get the model (1.1). However, Slemrod [15] reconsidered these neglected effects and first proposed the viscositycapillarity (VC) approach for Riemann solutions to ensure the uniqueness of the solution. Abeyaratne and Knowles [1] introduced a kinetic relation which is an additional condition to conquer the non-uniqueness. Vainchtein [17] considered two dissipation mechanisms (heat conduction and the internal viscous dissipation of kinetic origin) in a finite bar and added a viscoelastic term (ux xt , where is the viscosity coefficient) for dissipation to the Eq. (1.1) to ensure the uniqueness of the solution. Dai [6] considered the effects of the radial deformation and traction-free condition on the lateral surface when studying a slender circular cylinder, and introduced a dispersive term to ensure the uniqueness of the solution. The experiments in [10 13] also suggested that the lateral movement and the radial deformations for phase transitions did play an important part. For the mixed-type system of conservation laws (1.1), certain numerical schemes have been used to compute the solutions based on the VC approach (see [3,16] and the references therein). Cockburn and Gau [3] implemented a convergence study for different viscosity and capillarity coefficients. Recently, Tian et al. [16] used the local discontinuous Galerkin (LDG) method to solve the propagation of phase transition with different viscosity and capillarity coefficients in solids and fluids and presented the error estimate for the LDG scheme. Moreover, Dai and Xu [7], based on the model in Dai [6], made some approximation to the model by omitting the fourth-order terms and obtained a high-order nonlinear equations. Then they applied the fifth-order finite difference weighted essentially non-oscillatory (WENO) scheme to approximate the convective term, and the higher-order central scheme to approximate the high-order and nonlinear terms. In this paper, we will reinvestigate the model in Dai [6] and obtain a system (see the Eq. (3.1)) by introducing new temporal auxiliary variables rather than by asymptotic approximations in [6]. For this system, we apply the LDG method to carry out numerical simulation and to demonstrate some interesting wave phenomena. The discontinuous Galerkin (DG) method discussed here is a class of finite element methods. It uses completely discontinuous piecewise polynomial space as the numerical solution space and the test function space in the spatial variables. Using DG method is beneficial for parallel computing and maintaining the high-order accuracy due to this completely local, element-based DG discretization. Additionally, the DG method has the advantage of well hp-adaptation, which consists of local mesh refinement and/or the adjustment of the polynomial order in individual elements. Furthermore, it shares the excellent provable nonlinear stability. The DG method was generalized to the LDG method by Cockburn and Shu [4] to s (...truncated)


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Jinfeng Jiang, Yan Xu. Local Discontinuous Galerkin Method for the Impact-Induced Wave in a Slender Cylinder Composed of a Non-Convex Elastic Material, Communications in Mathematics and Statistics, 2014, pp. 393-415, Volume 1, Issue 4, DOI: 10.1007/s40304-013-0022-6