Tham khảo tài liệu 'waves in fluids and solids part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 214 Waves in Fluids and Solids 2. Fourth-order RK method for solving ODEs Basic RK algorithm Consider the following ordinary differential equation du T L u . dt 1 Where u is an unknown function of time t and L is a known operator with respect to u at each spatial point i j k for the 3D case. Equation 1 can be solved as an ordinary equation using the following fourth-order Runge-Kutta method u 1 un 1 AtL un u 2 un -1 AtL u 1 u 3 un AtL u 2 un 1 3 -un u 1 2u 2 u 3 j 1 AtL u 3 . 2 Where At is the temporal increment un u nAt and u 1 u 2 and u 3 are the intermediate variables. Equation 2 shows that the RK algorithm needs to store these three intermediate variables at each time advancing step so the storage required for computer code is very large for 3D problems. To save storage we can equivalently change it into the following two-stage scheme u un 1 AtL un 1 At2L2 un un 1 3 un 2u j 1 AtL un 1 AtL u 1 At2L2 u . 3 Where L2 L L . Algorithm 3 uses only one intermediate variable u resulting in that the modified two-stage RK used in this chapter can effectively save the computer memory in the 3D wave propagation modeling. Transformations of 3D wave equations In a 3D anisotropic medium the wave equations describing the elastic wave propagation are written as d ij r d li sXj f - 4a 1 duk du . a a _cikl - v L- ij 2 ijkl dx dxk 4b where subscripts i j k and l take the values of 1 2 3 p p x y z is the density ui and fi denote the displacement component and the force-source component in the i-th direction 215 A Fourth-Order Runge-Kutta Method with Low Numerical Dispersion for Simulating 3D Wave Propagation and x1 x2 and X3 are x y and z directions respectively. ơịj are the second-order stress tensors Ciju are the fourth-order tensors of elastic constants which satisfy the symmetrical conditions Cjj Cjm Cijik Cktij and may be up to 21 independent elastic constants for a 3D anisotropic case. Specially for the isotropic and transversely isotropic case the 21 independent .
