**7. Concluding remarks**

14 Will-be-set-by-IN-TECH

(a) (b)

(a) (b)

numerical accuracy, (2) waves propagation in a plane of an arbitrary orientation to assess the non-reflective boundary condition, and (3) waves propagation in a plane composed of three

In Case 1, two-dimensional numerical simulations are performed to calculate the density of the total energy, which is the summation of the kinetic energy and the strain energy normalized by area. For elastic solids, the group velocity is identical to the energy velocity [2]. Here, we consider wave propagation in 100 (*x*2-*x*3) or 010 (*x*1-*x*3) plane of the hexagonal

**Figure 4.** Comparison between the analytical solutions of the group velocities (in SH, qS, and qL polarization) and the calculated energy profiles for beryl at *t* = 91 *μ*s, including: (a) The analytical

**Figure 3.** Two-dimensional mesh (2.2 million triangular elements). (a) close look at the mesh around

origin. (b) decomposed 16 sub-domains.

solution, and (b) The calculated energy density profiles.

blocks of beryl in different lattice orientations.

In this chapter, we have presented the first-order velocity-stress equations as an alternative to the conventional second-order wave equations for modeling wave propagation in anisotropic

**Author details**

CFL number = 0.95.

Sheng-Tao John Yu, Yung-Yu Chen and Lixiang Yang

*The Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA*

Velocity-Stress Equations for Wave Propagation in Anisotropic Elastic Media 217

(a) (b)

(c) (d)

**Figure 7.** Waves propagation in a heterogeneous domain. The total energy density (*e*/ max(*e*)) are plotted at different times: (a) *t* = 72 *μ*s, (b) *t* = 96 *μ*s, (c) *t* = 108 *μ*s, (d) *t* = 180 *μ*s. Δ*t* = 60 ns, and

**Figure 6.** The computational domain is divided into three regions. The orientations of the solids in the central, left and right regions are *θ* = 0◦ and *φ* = 0◦, *θ* = 0◦ and *φ* = 60◦, and *θ* = 0◦ and *φ* = 30◦, respectively.

elastic solids. The eigen structure of the equations has been thoroughly studied by analyzing the composed Jacobian matrix of the first-order equations, i.e., *B* = ∑<sup>3</sup> *<sup>μ</sup>*=<sup>1</sup> *<sup>h</sup>μA*(*μ*) . The hyperbolicity of the equations has been rigorously proved by showing the real spectrum of matrix *B* and diagonalizability of *B*. *B* has a degenerate null eigenvalue. For non-zero eigenvalues, *B* has at least one and at most three positive-negative pairs of real eigenvalues. Moreover, *B* has nine linearly independent eigenvectors, which can be used to diagonalize *B*. The eigenvalues represent the wave speeds, and the eigenvectors are related to the polarization of the waves. Since the first-order velocity-stress equations are hyperbolic, the Cauchy problem is well-posed [9]. For numerical simulation, the proven hyperbolicity justifies the use of modern finite-volume methods, originally developed for solving conservation laws, to solve the velocity-stress equations. For upwind methods [12] and discontinuous Galerkin methods [18], information about eigenvalues and eigenvectors of matrix *B* are critically important. The derived characteristic relations of the governing equations can be directly used to derive the Riemann solver and the flux function as the building blocks of the methods. Moreover, by clearly defining the local and global coordinate systems, we also show that matrix *B* is directly connected to the classic Christoffel matrix. Therefore, nine coupled first-order velocity-stress equations are shown to be directly related to the classic second-order elastodynamic equations. To demonstrate the application of the velocity-stress equations, we apply the space-time CESE method to solve the equations for modeling wave propagation in a block of beryl. The calculated energy profiles compare well with the analytical solution. We also simulate wave propagation in a heterogeneous solid composed of three blocks of beryl with different lattice orientations. Numerical results show complex wave/interface interactions.
