The Numerical Manifold Method And Simulation Of Dynamic Fracture Propagation

This paper focuses in the Numerical Manifold Method (NMM), aims at the coupling with the Finite Element Method, and studies the implementation of the NMM on common-used FEM meshes. With techniques presented in this paper, it analyses 2-D dynamic fracture propagations and achieves satisfactory results. Firstly, this paper studies the relationship among several numerical methods. Not only the construction of the approximation space of the traditional FEM, also of the meshless methods and NMM, can fit into a general mold specified by the partition of unity method, in which global approximation is derived from local approximation using partition of unity functions. Secondly, this paper gives a suggestion on the choice of cover functions. Analysis show that the currently-used polynomial cover functions, which rely on global co-ordinates, will introduce entries with very large absolute-value into stiffness matrix if the order of cover functions greater than zero. As a result, stiffness matrix is badly conditioned and can damage the accuracy of solution. Therefore, localized cover function basis are preferable. This paper derives stiffness matrix formula of simplex integration with the first order cover functions on triangle meshes. Thirdly, this paper extends the NMM to practical and more efficient FEM meshes, imposing the essential boundary conditions with Lagrange multiplier. In order to couple with the FEM in which 2D eight-node element and 3D twenty-node element are widely used for efficiency, this paper presents the methods applying the NMM on these two kinds of meshes and derives the formulae dealing with essential boundary conditions with Lagrange multiplier. Linear dependency of stiffness matrix is discussed and a solution is presented. The accuracy and efficiency of the NMM on these two kinds of meshes are studied in detail; compared with other commonly-used FEM methods, they show better functionalities and turn out to be promising analysis tools. Fourthly, this paper studies dynamic analysis method that can dovetail with FEM. With Newmark presumption, this method has high accuracy of the fourth order truncation error. Under some easily-satisfied conditions, it's unconditionally stable and straightforward to implement. It enriches the NMM dynamic analysis methods. At last, this paper analyses 2D dynamic fracture propagation problems. Based on the partition of unity theory, using shape functions of FEM element as weighting functions, and borrowing the visibility criteria from meshless methods, this paper setup discontinuous partition-of-unity functions with the Shepard formula, thus the discontinuity of the global approximation is brought in, which can depict fracture problems. Using the dynamic analysis method in the previous chapter, stationary and propagating fracture problems are studied. Based on interaction energy integral, dynamic SIFs are calculated; in order to improve the accuracy and stability of numerical results, a new auxiliary function is presented. The numerical results coincide with the theoretical solution satisfactorily, which proves the reliability of this method.