The Development Of Elements And Application Of Mesh Adaptive In Large Increment Method

The finite element method is developed on the basis of displacement method, but the development of the traditional force method has been stagnant. The reason is that the force method is needed to select the redundant unknown quantity, but for the large-scale operation, it is not appropriate. For the general statically indeterminate truss problem, the equilibrium equation can be obtained according to the equilibrium relation of the nodes. However, the number of unknown quantities of this set of equilibrium equations is more than the number of equations, so there are many groups of solutions. According to the principle of virtual work, the coordinate equations can be obtained, and the correct solution is to satisfy the equilibrium equations and the coordinate equations. Then, there exists a kind of idea, which can find a set of solutions of the equilibrium equations to satisfy the coordinate equations. According to the introduction of the principle of generalized inverse matrix, the general solution formula of the equilibrium equation can be obtained. After getting a set of internal forces, the deformation is obtained by the constitutive relation. At this time we must find a way to determine whether the coordination equation is satisfied. According to the principle of generalized inverse matrix, the norm of the coordination error can be expressed. When the equation is not satisfied, the conjugate gradient method is used to search for better solutions.Adaptive mesh is a kind of method to determine the error and get the optimization mesh. When the adaptive mesh applicated in the large increment method, the ZZ method is used to compute the norm of the error, and the first class h adaptive method is used to refine the mesh, and three strategies is taked to optimize the quality of mesh. In the previous step calculation, the generalized internal force of new mesh can be obtained by the general force of old mesh, therefore, the application of adaptive mesh in the large increment method, can effectively use the results of the last step to improve the computational efficiency.The convergence and correctness of the 16 kinds of two dimensional solid elements have been verified by the patch test. The results show that in the presence of a linear displacement field, the calculation accuracy of large increment method elements can maintain equal with the finite element method, in the high order displacement field, the accuracy of the 8 node quadrilateral stress element is higher than the finite element method elements with exact integral.With the Mexwell stress force interpolation functions, three-dimensional solid stress format element can be established, and according satisfing the equilibrium differential equation and reduction of the number of generalized forces, reduced stress interpolation functions are used, and finally 12 kinds of three dimensional stress format element are established.By the definition of generalized internal forces in the node, based on the balance of the relationship, as well as the transformation between the local coordinate system and the global coordinate system, the balance equation and the constitutive relation can be obtained, and 2 kinds of three-dimensional force format element are established.The two-dimensional numerical examples are given to verify 16 2D solid elements in large increment method, and the 3D examples show that 16 3D solid elements in large increment method. In precision, the precision of the force format elements are the same as that of the finite elements, and the accuracy of stress elements are higher.