Berm?dez-Moreno Duality RBF-PS Method With Optimal Parameters

Based on generalized Berm?dez-Moreno duality methods, we propose the Berm?dez-Moreno duality RBF-PS method with optimal parameters to solve variational inequalities, called BM-RBF method for short, i.e. converting the original problems into iterative scheme, we can solve the corresponding partial differential equations or variational equalities by RBF-PS method or RBF-Galerkin method respectively. This method is applied to the static frictional contact problem and the dynamic viscoelastic frictional contact problem with damage.The main work is as follows: 1. It introduces the Berm?dez-Moreno duality method and generalized B-M method. Three methods with a choice of the parameters, namely constant parameters, vector-valued parameters and matrix-valued parameters are given. Next, we introduce the radial basis function pseudo-spectral method(RBF-PS), and present the Berm?dez-Moreno duality RBF-PS method with optimal parameters(BM-RBF). Compared with the finite element method, the effectiveness of the proposed method get confirmed in numerical examples. Meanwhile, we compare the convergence rate of the algorithm with three different parameters.2. It gives the basic mathematical model and the relevant variational inequality for the static frictional contact problems. Then we construct the coupling algorithm of B-M method with optimal constant parameters and the RBF-Galerkin method. This method is applied to solve the static frictional contact problems with two kinds of contact conditions,namely Coulomb and Tresca friction conditions. The numerical results indicate the effectiveness of our method. 3. It discusses the dynamic viscoelastic frictional contact problem with damage. The BM-RBF method of this problem is constructed. First, using the time semi-discrete scheme and decoupling the two coupled evolutional variational inequalities, we apply the BM-RBF method to first damage-induced the variational inequality. And for the second displacement-induced variational inequality, we propose two ways, one is the the coupling of iterative BM algorithm with optimal parameters to RBF-Galerkin method, another is the RBF-Galerkin method for the regularization of variational equations. Compared with the finite element method, the numerical examples illustrate the effectiveness and accuracy of the proposed method.