Low Order Finite Element Method Based On Hybrid Mesh Research

Flow phenomena is always accompany with boundary layer effects in nature. For boundary layer fluid problems, Construct the grid directly to design the finite element space of the large gradient basis function is not very economical and cause bad computational efficiency, meantime using conforming mesh is hard to meet the accuracy and unable to control the number of the mesh. Using hybrid mesh can overcome these problems.The conforming hybrid finite element method and the method of nonconforming finite element method are established in this paper, and the finite element method is verified by the Poisson equation which has real solution, The method is consisted of five parts:triangular mesh and rectangular mesh region, forming different types of grid interpolation basis functions, different types of grid element transformation, forming hybrid stiffness matrix and the right-hand-side vector, solve and verify the answer. After verifying the conforming hybrid finite element solver and nonconforming hybrid finite element solver, obtained the solution and found the solution satisfy the convergence orders in this paper, these two solver succeed in simulating most of the existing hybrid mesh algorithm.According to the theory of nonconforming linear rectangular finite element, the nonconforming rectangular method is modified to establish the nonconforming linear rectangular finite element and prove the correctness in this paper, Then a new linear hybrid finite element method is proposed, this new method can solve the locking problems. Using the nonconforming linear rectangular finite element and nonconforming triangular finite element, the new finite element method is verified to be convergent.In order to verify the advantages of the nonconforming linear hybrid finite element method for solving boundary layer problems, a new case with large boundary gradient layer is used to compare the accuracy and compute time in new algorithm with the existing hybrid element method. The results show that the new algorithm does not loss the accuracy compare with the old algorithm, and the speed of the computation is at least 60% higher than the old algorithm. This new algorithm greatly improves the efficiency of the large-scale solution of boundary layer problems, and provides a new solution of finite element method in engineering.