The Numerical Simulation Of Discontinuous Phenomena Using SPH

With the rapid development of computer hardware and software, numerical simulation has become an important instrument in analyzing and solving the complex physical phenomena in recent years. Compared with traditional experiments, the numerical simulation, which does not involve in expensive, time-consuming and dangerous experiments in the laboratory, also gives more specific and complete information for physical problems that are difficult in calculation and observation, and provides data as the theoretical basis for explaining or even discovering new phenomena. The common numerical simulations are now primarily based on mesh and meshfree approaches. For many discontinuous phenomena such as dynamic crack propagation, material damage and failure, corrosion and infiltration, and large deformation etc, mesh methods are difficult to apply because they suffer from inherent deficiencies. However, the meshfree methods have incomparable advantages in dealing with discontinuous problems.As a meshfree method, smoothed particle hydrodynamics (SPH), which uses a series of randomly distributed Lagrangian particles, integral equations and partial differential equations with various boundary conditions, has become an effective numerical method applied widely in various areas of computational fluid. The kernel approximation in integral form and the particle approximation in discrete form constitute the foundation of SPH. The consistency conditions in integral and discrete form are the key to the reproduction of SPH. The kernel consistency conditions not only provide a generalized approach for constructing the smoothing kernel function, but also play an important role in SPH formula. In the same time, the particle consistency conditions not only provide a solution to the particle inconsistency in traditional SPH method, but also promote the development of CSPM (Corrective smoothed particle method) and RSPH (Restoring particle consistency in smoothed particle hydrodynamics).For the particle inconsistency of the traditional SPH method in discontinuous problems, this paper uses DCSPM as reference and puts forward a new formula (DRSPH) for simulating discontinuous physical phenomenon. DRSPH based on Taylor series expansion on either side of the discontinuities rather than in the entire domain is an extension of RSPH. The major part of the final kernel and particle approximations for a function and its derivatives are similar to those in RSPH. The remained part ensures the consistency of the kernel approximation for treating discontinuities.DRSPH remedies the boundary deficiency problem and restores the particle consistency in discontinuous regions without modifying kernel functions. Compared with DCSPM proposed by Liu in 2003, DRSPH does not need the normalization. The function and its first derivatives in DRSPH are calculated by solving the matrix at the same time with 2-order approximation accuracy. Through defining the particle consistency in discontinuous regions, it is demonstrated that DRSPH restores the particle consistency in discontinuous regions more effectively than DCSPM, and has first order particle consistency in approximating the function. DRSPH requires an efficient algorithm of discontinuity detection based on discrete functions in current time step. If an efficient algorithm for discontinuity detection is provided, DRSPH will be very useful in discontinuous regions, especially for the problems in a multi-dimensional space.