Local Differential Quadrature Method For The Elastic-plastic Torsion Problems

The elasto-plastic torsion problem appears widely in the fields of physics, mechanics, engineering, etc. The mathematical form can be formulated as the elliptic variational inequality with constraint condition, while the equivalent equation form is a free boundary problem. The differential quadrature method is adopted to solve the elasto-plasto torsion problem. First, the original problem is transformed into the saddle-point problem by the dual method. Two new Uzawa-LDQ algorithms are constructed to solve the elasto-plasto torsion problem based on the traditional Uzawa method. Comparing with the finite element method (FEM), it shows that our methods are effective. The effect of parameters in our method is discussed.The main work is as follows:1. We introduce the basic knowledge of differential quadrature method (DQM) and local differential quadrature method (LDQ). The derivative of higher order at a mesh point is given based on the Lagrange interpolation and the error analysis is discussed. Then by numerical examples, we compare DQM and LDQ. At last we discuss the effect of total number of meshless nodes, node distribution and the number of local meshless nodes to the solution of these methods.2. We construct the Uzawa-LDQ coupling method to solve a class of elliptic variational inequalities. By numerical experiments, and comparison of the results with the FEM, it shows the effectiveness of our algorithm. The new method should be superior to the traditional FEM methods, and has the advantage of programming simple, easy to implement, and no need of any mesh. It is a pure meshless method.3. We discussed the local differential quadrature method based on the radial basis function as the test function (LRBFDQ). Then we construct the Uzawa-LRBFDQ coupling method. Compared to the LDQ method, the weight coefficients can be determined by only once computation when the meshless nodes are fixed. The regular distribution of the meshless nodes is no necessary in this method. Numerical results verify the validity of the method. Finally, it discussed the effect of the parameters, such as MQ shape parameter, the number of total nodes and the number of local meshless nodes.