Finite difference method

Abstract: SUMMARY To simulate seismic wave propagation in the spherical Earth, the Earth’s curvature has to be taken into account. This can be done by solving the seismic wave equation in spherical coordinates by numerical methods. In this paper, we use an optimized, collocated-grid finite-difference scheme to solve the anisotropic velocity–stress equation in spherical coordinates. To increase the efficiency of the finite-difference algorithm, we use a non-uniform grid to discretize the computational domain. The grid varies continuously with smaller spacing in low velocity layers and thin layer regions and with larger spacing otherwise. We use stress-image setting to implement the free surface boundary condition on the stress components. To implement the free surface boundary condition on the velocity components, we use a compact scheme near the surface. If strong velocity gradient exists near the surface, a lower-order scheme is used to calculate velocity difference to stabilize the calculation. The computational domain is surrounded by complex-frequency shifted perfectly matched layers implemented through auxiliary differential equations (ADE CFS-PML) in a local Cartesian coordinate. We compare the simulation results with the results from the normal mode method in the isotropic and anisotropic models and verify the accuracy of the finite-difference method.

Abstract: Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) beams are analyzed in this research. All thermomechanical properties of the beam are assumed to be temperature and position dependent. Beam is subjected to thermal shock on the ceramic-rich surface whereas the metal-rich one is kept at reference temperature or thermally insulated. The one-dimensional transient heat conduction equation is established and solved via a hybrid iterative central finite difference method and Crank–Nicolson method. Total functional of the beam is obtained under the assumptions of uncoupled thermoelasticity laws, first order beam theory, and the von Karman type geometrical non-linearity. The conventional multi-term p-Ritz method appropriate for arbitrary in-plane and out-of-plane boundary conditions is applied to the total functional of the system which results in the matrix representation of the equations of motion. Non-linear coupled equations of motion are solved via the iterative Newton–Raphson method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of isotropic homogeneous beams. Some parametric studies are conducted to examine the influences of beam geometry, material composition, temperature dependency, in-plane and out-of-plane mechanical and thermal boundary conditions. It is shown that, thermally induced vibrations indeed exist especially for the case of sufficiently thin beams.

Abstract: Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank-Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999