Development of flat-top partition of unity based high-order discontinuous deformation analysis - Liu, Xiaoying (2018)
In this thesis, the flat-top partition of unity (PU) based high-order Discontinuous Deformation Analysis (DDA) is developed and applied in simulating different problems. In the proposed method, the continuous blocks in the problem domain are discretized by the flat-top PU mesh, and the interactions between blocks is simulated with the improved contact algorithm. For each flat-top PU unit in the mesh, high-order polynomial is assigned as local approximation, where the coefficients of monomials are denoted as unknowns. The global approximation can be obtained by multiplying the local approximation with the corresponding shape functions. With the principle of virtual work and simplex integration, the global equilibrium equations are established and solved. Thus, the displacement field and stress/strain field can be obtained. In flat-top PU based high-order DDA method, since the flat-top PU mesh and high-order approximation are utilized instead of discretizing the problem domain into numbers of small blocks, only the main discontinuities need to be considered. Therefore, the time-consuming iteration of contact detection, adding and removing stiff springs can be reduced. The construction of regularly patterned flat-top PU mesh is introduced based on the traditional PU mesh, which is easy to understand and straightforward to implement. In order to simulate the stress concentration/singularity and crack propagation problems with high cost-efficiency, the local refinement of flat-top PU mesh is introduced and its property of linear independence is proven from element level based on the equivalent p-version element and rank deficiency counting approach. During the generation of numerical model for discontinuous problem domain, in order to avoid ill-conditioned stiffness matrix in global equilibrium equations, modification for the numerical model is implemented based on the concept of percentage of effective area. During the numerical simulation, the contact algorithm in original DDA method is improved by determining the exact contact position. Therefore, the energy consumption is more accurate, especially for the collision problem. The improved formula for non-reflective boundary is coupled with flat-top PU based high-order DDA method to simulate stress wave propagation within infinite problem domain, where the numerical accuracy can be improved by deriving the analytical velocity of dashpots, instead of calculating the velocity with finite difference method.Based on the crack propagation modes, the tension failure criterion and the Mohr-Coulomb failure criterion are adopted to determine the crack initiation in flat-top PU based high-order DDA method. Within each time step, the criteria are applied at every node in the flat-top PU mesh. In order to avoid generating new unknowns and determining their initial values artificially, it is assumed that the crack propagates along the element sides. Correspondingly, the treatment of PU elements after crack propagation is introduced.Numerical examples are presented and analyzed regarding to each development of the flat-top PU based high-order DDA method.