Constrained Minimization Of Incomplete Quadratic Functions And Its Applications
Free (open access)
Q. Zheng & R. J. Gu
In an ordinary minimization problem, the quadratic function contains both quadratic and linear terms. Through minimization, a system of linear equations is yielded. The right-hand side vector of such a system is normally a known input. In a special type of elasticity problem in which coerced deformation of an elastic body is sought, the applied load becomes unknown. In other words, the quadratic function does not have a linear term. The objective function has two quadratic terms, and the minimization is conducted to satisfy some constraints. In another application where the amount of springback of a stamped metal is to be determined, the applied force giving the metal the least strain energy is sought. In this paper, the mathematical formulation as well as the approach to solve the problems is presented. Several numerical examples are presented. Keywords: constrained minimization, quadratic, coerced deformation, springback, minimum strain energy, Lagrange multipliers. 1 Introduction In some disciplines such as finding the displacements of a deformable elastic object, a solution xi is sought by finding the minimum of the following quadratic function. i x i b j x i x ij a i x U += 2 1 ) ( . (1)
constrained minimization, quadratic, coerced deformation, springback, minimum strain energy, Lagrange multipliers.