Discontinuous Galerkin Methods For Wave Propagation In Elastic Materials And Hyper-elastic Materials

In this thesis, we mainly focus on three types of materials:nonlinearly elastic composite bar, non-convex elastic slender cylinder and compressible hyper-elastic rod, and study the wave catching-up and collision phenomena and the phase transition in wave propagation in these materials. We use (local) discontinuous Galerkin method and adaptive Runge-Kutta discontinuous Galerkin method for numerical simulation.There are four parts in this thesis. In the first part, we consider nonlinearly e-lastic composite bar, in which the model equations is conservation laws. The model equations contain a discontinuous flux and a source term with a δ-function due to the materials’composition. These set difficulties for the design of numerical schemes. Some attention should be paid to deal with the numerical fluxes for the discontinuity of the flux. In addition, the waves can reflect and transmit at the physical interface, and the solutions contain many shock waves, which also increase the difficulty for the problem. From the point of physics, we first introduce the concept of dissipation rate and physically admissible solution. Then, by virtue of the dissipation rate, we develop a dissipation-rate reserving discontinuous Galerkin method and give numerical simula-tions to demonstrate the advantage of the scheme. In the end, we use the dissipation-rate reserving discontinuous Galerkin scheme to simulate the details of the interactions of waves at the propagation process, which include the interactions between tensile shock wave, compressive shock and rarefaction wave. We give the detailed times and po-sitions for the interactions. We also examine how material parameters influence the reduction ratio by four different group of parameters. These results confirm that when the stress-strain relation of the second nonlinearly elastic material is convex, the tensile wave can indeed catch the previously transmitted compressive wave. The simulation results by the discontinuous Galerkin method show that the catching-up phenomena can reduce the magnitude of the tensile wave by more than400%. They also show that the non-linear elastic materials have excellent impact resistance.In the second part, we focus on the impact-induced wave in a slender cylinder composed of a non-convex elastic material. This system is not well-posed due to the non-convexity of the stress-strain relation. A new model equation can be obtained by considering the effects of the radial deformation and traction-free condition on the lateral surface. There are higher order time derivative terms that increase the difficulty for solving. We introduce new time auxiliary variables into the equation so that the model equations can be written as a system with first-order time derivative. Then we present the stability of the local discontinuous Galerkin scheme. The method with pk elements (denoting all polynomials of degree at most k) gives a uniform (k+1)-th order of accuracy forγ in both Ll and L∞norms through one example. In the end, we use local discontinuous Galerkin scheme for a series of simulation, there are some interesting wave patterns, due to the effect of the dispersion and the non-convexity of material, such as the wave pattern with transformation front and solitary wave, the wave pattern with rarefaction wave and solitary wave. We also investigate the interaction of transformation fronts and rarefaction waves, and demonstrate the interesting wave phenomena.In the third part, we consider the travelling waves in a compressible hyper-elastic rod. The model equation (compressible hyper-elastic rod wave equation) for the ma-terial is similar to the form of Camassa-Holm equation, and there is another constant coefficientλfor the nonlinear term. Therefore, we can refer to the numerical scheme of Camassa-Holm equation for that of this model equation. The Camassa-Holm equation is a completely integrable equation. Unfortunately, the property of completely inte-grable isn’t enjoyed by the current model equation when λ≠1. But interestingly, the model equation when λ<0shares traveling waves such as compacton waves, soliton waves. We will focus on these traveling wave, and give numerical simulation for inter-action between these wave.Some unexpected phenomena are found. For example, the compacton and soliton wave are stable with a Gaussian perturbation. After the collision of the two compacton waves, two soliton waves occur. After the collision of the three compactons, three solitons occur. After the collision of compacton wave and soliton wave, the profiles of the finally separated wave from them are likely with that of one soliton wave.In the fourth part, we continue to consider the wave catching-up phenomena in a nonlinearly elastic composite bar. The solutions contain many shock waves. We use a large-number mesh to capture the information of shock wave, but which unfortunate-ly increases the cost of computation. The discontinuous Galerkin method shares the advantage of easy adaptive implementation since there is no need of continuity on cel-1interface. Therefore, we use adaptive Runge-Kutta discontinuous Galerkin method with KXRCF indicator and TVB indicator to simulate. We consider the effects of the TVB reconstruction, the parameter M, the discontinuous Galerkin scheme with dissi-pation and Pk element, etc. Overall, The results by TVB indicator are better than that by KXRCF indicator, because there is less oscillation. However, there occurs some s-mearing phenomena. Therefore, we need to select an appropriate parameter M to make the results more accurate.