Study On The Coupling Algorithm Of Finite Element Method And Smoothed Particle Hydrodynamics

Large deformation and fracture of materials are involved in many engineering problems, such as penetration, explosion, metal forming, fluid flow, and so on. It is very useful to develop numerical methods to simulate this kind of problems. Finite Element Method (FEM) has good calculation accuracy and efficiency for small deformation problems, but its accuracy and efficiency decrease significantly due to the element distortion when it is used for problems involving large deformations and material fracture. Smoothed Particle Hydrodynamics (SPH) has strong ability in modeling large deformations and is easy to simulate material fragmentation and splash, but its calculation efficiency and ability in treating boundary conditions is not as good as FEM. Coupling FEM and SPH can make use of the advantages of the two methods and provides an effective approach for the simulation of large deformation problems. The large deformation regions are solved with SPH, and the other regions are solved with FEM. By doing so, not only can element distortion be avoided, but also good efficiency can be obtained. However, the study on coupling algorithm has just started, and SPH still has some deficiencies. Thus, there are many problems required to be further studied and solved for the coupling algorithm of FEM and SPH in aspects of accuracy, efficiency and stability.In this paper, some improvements for the efficiency and accuracy of SPH are presented. Based on this, fixed and adaptive coupling algorithms of FEM and SPH are systematically studied and applied to numerical simulations of high-velocity impact (HVI) problems. Moreover, the coupling algorithm is extended to the application of the simulation of fluid-structure interaction (FSI) problems. The research work done in this paper is given as follows:(1) The basic principle of SPH is described systematically, and some improvements are presented for it to remove tensile instability, treat contact and search neighboring particles. The artificial stress method for removing tensile instability is discussed. A calculation method of artificial stress for axisymmetric problems is suggested. It is verified by calculating a rubber ball impacting on a rigid wall. A particle contact algorithm based on penalty method is extended for two-dimensional axisymmetric and three-dimensional problems and is validated by some test examples. The particle contact algorithm is used for the simulation of normal and oblique penetration of aluminum plates. Simulation results by SPH with particle contact algorithm are in good agreement with experimental results and show much better accuracy than that of SPH without contact algorithm. Some usually used neighboring particle search algorithms are discussed. The tree and Point-In-Box (PIB) search algorithms are briefed. The problem and its reason in PIB search algorithm are analyzed, and an improved strip-like PIB search algorithm is proposed. Some test examples are calculated to compare the performance of the three algorithms. Calculated results show that strip-like PIB search algorithm is better than PIB and tree search algorithm and can improve the calculation efficiency of SPH markedly.(2) Two key problems in coupling algorithm, namely the treatment of contact and coupling between elements and particles, are studied. Firstly, an element-particle contact algorithm is obtained by introducing particle radius in contact algorithm of FEM and validated by calculating some test examples. A filtration algorithm based on neighboring particle search is presented to reduce the number of slave nodes and master segments required for contact search, and the efficiency of element-particle contact search is improved. Then, two different element-particle coupling algorithms are constructed. One is an algorithm based on attaching particles to element sides, in which a particle is fixed at a point of an element side so that they have the same velocities and displacements. Test examples are calculated to validate the algorithm and analyze the effect of particle density on accuracy. Calculated results indicate good coupling accuracy can be obtained when reasonable particle density is taken, but local stress oscillation at coupling interfaces is observed. The other one is an algorithm based on imaginary particles and equivalent tractions. In this algorithm, to calculate forces on particles from elements, elements near coupling interfaces are treated as imaginary particles and included in SPH calculations. To calculate forces on element nodes from particles, equivalent tractions are determined with particle stresses at coupling interfaces and transmitted to element nodes. The feasibility and validity of the algorithm are demonstrated through several test examples. The effect of smoothing length on calculation accuracy is investigated, and its reasonable values are obtained for the coupling cases with different particle distributions. The algorithm is compared with some existing algorithms and found to produce better accuracy than them.(3) A fixed coupling algorithm of FEM and SPH is constructed with the fundamental algorithms previously described. The corresponding program is developed, and a two-dimensional impact simulation software is developed. The fixed coupling algorithm is used to simulate Taylor bar experiment and normal penetration of aluminum plates and compared with SPH. The simulation results demonstrate the feasibility and validity of the fixed coupling algorithms in HVI simulation and indicate it gives similar accuracy but higher efficiency compared with SPH. Embedding a coupled constitutive model of viscoplasticity and ductile damage in the fixed coupling program, plugging failure during blunt projectiles penetrating into metal plates is simulated. Simulation results are compared with experiment results. It is found the fixed coupling algorithm can effectively predict residual velocities of projectiles, deformations of projectiles and targets, failure pattern of targets, and so on.(4) Based on the fixed coupling algorithm, an adaptive coupling algorithm of FEM and SPH is constructed by establishing a minimum interior angle conversion criterion and a group-based conversion manner to achieve the automatic conversion of distorted elements to particles. The algorithm is applied to simulate the Taylor bar experiment. Simulated results are compared with experiment results and other algorithm’s results, which demonstrate the feasibility and validity of the algorithm. The algorithm is further applied to simulate penetration problems, which include normal penetration of aluminum and concrete plates and oblique penetration of an aluminum plate by a cylinder. Simulated results indicate the algorithm can predict residual velocities of projectiles accurately, model material fracture effectively, and reproduce the fragmentation and splash phenomenon of brittle materials. It is also observed the algorithm has advantages over fixed coupling algorithm in calculation efficiency, especially for problems where material fracture regions are difficult to be estimated in advance.(5) Simulations of FSI problems using fixed coupling algorithm are presented. In the simulations, fluids and structures are solved with SPH and FEM, respectively. The coupling interfaces between fluids and structures are treated with element-particle contact algorithm. Elastic plate subjected to time-dependent water pressure and collapsed water column on an elastic structure are calculated. Calculated results are in good agreements with experimental and existing numerical results. This indicates the coupling algorithm is feasible and promising in FSI simulation.