Finite difference method

Abstract: The Finite Difference Method (FDM), within the framework of the Meshless Local PetrovGalerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A “mixed” interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction boundary conditions are imposed in the stress equations directly, using a local coordinate system. Numerical examples show that the proposed MLPG mixed finite difference method is both accurate and efficient, and stable. keyword: Meshless method, Finite difference method, MLPG

Abstract: We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero. We prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. We discuss this problem in 1D and 2D in the interval and the square respectively. Our method of proof relies on discrete multiplier techniques.

Abstract: A two-dimensional numerical device-simulation system is presented. A novel discretization scheme, called "finite boxes," allows an optimal grid-point allocation and can be applied to nonrectangular devices. The grid is generated automatically according to the specified device geometry. It is adapted automatically during the solution process by equidistributing a weight function which describes the local discretization error. A modified Newton method is used for solving the discretized nonlinear system. To achieve high flexibility the physical parameters can be defined by user-supplied models. This approach requires numerical calculation of parts of the coefficients of the Jacobian. Supplementary algorithms speed up convergence and inhibit the commonly known Newton overshoot. The advantages and computer resource savings of the new method are described by the simulation of a 100-V diode. We also present results for thyristor and GaAs MESFET simulations.