Alternating Direction Implicit Method for Steady State Stress Analysis of Elastic Membrane

Abstract

Steady state stress analysis problem which is satisfies Laplace equation that is a stretched elastic membrane on a rectangular frame that has prescribed out of the plane displacement along the boundaries. Laplace equation is second-order partial differential equation (PDE) widely useful in physics because its solutions occur in problems of electrical, magnetic, and gravitational potentials, of steady-state temperatures, of hydrodynamics and of stress distribution. Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. It is virtually impossible to obtain analytical solution of the most of the partial differential equations that arise in mathematical models of physical phenomena. So, numerical methods are used to approximate the solution of such type of partial differential equation. The Alternating Direction Implicit (ADI) method has been used to solve the two-dimensional Laplace equations on regular (square and rectangular) region with Dirchlet boundary conditions. ADI method is an iterative and unconditionally stable method which is a popular method for solving the large matrix equations that arise in system theory and control theory and can be formulated to construct solutions in a memory-efficient, factored form.  The chosen body is square which is discretized into square grids. The obtained numerical results are compared with analytical solution. The study objective is to check the accuracy of ADI method for the numerical solutions of two-dimensional Laplace equations.

Keywords- Alternative Direction Alternative method, Dirichlet Boundary Condition, Laplace Equation, Regular domain.