Open Access

ARTICLE

A Time-Marching Algorithm for Solving Non-Linear Obstacle Problems with the Aid of an NCP-Function

Chein-Shan Liu¹

1 Department of Mechanical and Mechatronic Engineering, Computation and Simulation Center, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computers, Materials & Continua 2008, 8(2), 53-66. https://doi.org/10.3970/cmc.2008.008.053

Abstract

Proposed is a time-marching algorithm to solve a nonlinear system of complementarity equations: P_i(x_j) ≥ 0, Q_i(x_j) ≥ 0 , P_i(x_j)Q_i(x_j) = 0, i, j = 1,...,n, resulting from a discretization of nonlinear obstacle problem. We transform the above nonlinear complementarity problem (NCP) into a nonlinear algebraic equations (NAEs) system: F_i(x_j) = 0 with the aid of the Fischer-Burmeister NCP-function. Such NAEs are semi-smooth, highly nonlinear and usually implicit, being hard to handle by the Newton-like method. Instead of, a first-order system of ODEs is derived through a fictitious time equation. The time-stepping equations are obtained by applying a numerical integration on the resultant ODEs, which are derivative-free and do not need the inverse of any matrix. The computational cost is thus greatly reduced. The numerical examples of Bratu, von Karman and other elliptic equations are used to demonstrate that the new fictitious time integration method (FTIM) is highly efficient to calculate the obstacle problems.

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Keywords

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