Application of Newton and Quasi-Newton Techniques to Non-Linear Problems in Computational Mechanics

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Addis Ababa University


In the field of engineering, researches often come across strong nonlinear boundary value problems (BVPs) that cannot solve easily. Numerical convergence for many problems, typically solved by the Newton-Raphson algorithm, is sensitive to the initial guess and need computations of Jacobi and its inverse at each iteration. Emphasis in the present work is placed on the alternative approach, such as quasi-Newton, HM and optimization method. Many problems in applied mechanics are reduced to the solutions of systems of nonlinear algebraic, transcendental equations containing an explicit parameter. These are problems in the areas of thermo-fluids, gas dynamics, deformable solids, heat transfer, biomechanics, optimal control and others. A parameter found in these models is not unique, and may be easily identified artificially. An important aspect of these problems is a question of the variation of the solution when parameter is incrementally changed. The numerical solution of BVPs of ordinary differential equations (ODE‟s) relies heavily on methods for solving systems of algebraic equations. The choice of the optimal numerical method, which ensures the best convergence rate with minimum error for the corresponding system of nonlinear equations, is discussed. Some modifications of quasi-Newton‟s method for systems of ordinary nonlinear differential equations are apply and suggested. Effectiveness of the method is demonstrated by comparing the results with the analytic solution for model boundary value problem implemented using a MATLAB Program. The objective of the research is to investigate applicability of the method to the wide range of nonlinear boundary value problems in different areas of mechanics. Different problems of applied mechanics and physics with dominant nonlinearities due to constituent models, and others are analyzed and solve in the present work. In this paper, the Newton‟s Homotopy analysis method (NHAM) is also applied on nonlinear boundary value problems(BVPs) of mechanics problems. The result from the method prove NHAM with Runge-Kutta steps, were significantly reliable and more accurate however, computation cost is high.



Application of Newton, Quasi-Newton Techniques, Non-Linear Problems, Computational, Mechanics