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This thesis reports the results of my M.S. research work about applying the space-time Conservation Element and Solution Element (CESE) method to calculate the wave propagation in the human brain. I am concerned with theoretical and numerical simulations of wave propagation based on two models: Fung’s model and Iatridis’ model. The thesis is divided into two parts.First, the governing equations which include the equation of motion, the viscoelastic constitutive relations, and the equations of internal variables are cast into vector-matrix form, and eigenvalues of the Jacobian matrices of the governing equations are thoroughly derived. The system of equations is shown to be hyperbolic with real eigenvalues and diagonalizable eigenvector matrices in the equations. The treatment for source terms in the CESE method is also reported. The material response is modeled by Fung’s model and modified Fung’s model which is developed by Iatridis. I used parallel connected standard linear solid (SLS) models to discretize Fung’s model and the modified Fung’s model. The resultant relaxation functions formulated in an integral form are then transformed to be differential equations by using the method of internal variables. I then performed simulation of wave absorption in the human brain tissues. While I took conventional approach to determine the parameters in the relaxation functions by using experimental quasi-static longitudinal tests, I also employed the measured wave absorption coefficients to determine the relaxation functions. As such, the constructed relaxation functions are inherently suitable for wave dynamics.In the second part of the thesis, I applied the CESE method to solve newly developed model equations for the waves in the human brain. CESE method is a generic numerical method for high-fidelity simulation of first-order, coupled, linear or nonlinear hyperbolic Partial Differential Equations (PDEs). Originally developed for solving compressible flows with complex shock waves, theCESE method had been widely used to simulate various complex compressible flows, including detonations, hypersonic flows, and shock and acoustic wave interactions. The approach is validated by the simulation of an one-dimensional impact wave. The results in this thesis present a general theoretical framework of using the first-order, hyperbolic PDEs to model wave motion in brain. The model equations are presented in three-dimensional space. To demonstrate the capabilities of the model equations, I reported numerical solutions of one-dimensional and validated the result, and showed the two-dimensional, and three-dimensional results to create a general pictures of wave propagation in the human brain. |
