A Scalar Homotopy Method for Solving an Over ... - TechScience

52 Copyright © 2009 Tech Science Press CMES, vol.53, no.1, pp.47-71, 2009ODE for x as written in Eqs. (3) and (4). However, since the equation written inEq. (19) is a scalar equation and it cannot determine the ODE for x unless oneprescribes a certain form for dxdt. In plasticity theory, the normality condition, orequivalently the “stability” of the material “in the small” was originally derived byDrucker and Ilyushin, based on the inequality of the plastic work (that it should be≥0). A similar motivation for the use of the normality condition for dxdtis based onthe stability of solutions. We note that the form of dxdtneeds to be parallel to thegradient of the above scalar homotopy function, such that the trajectory of x can beequivalent to seeking of h(x, t)=0. We propose here to usedxdt = −q∂h ∂x , (20)in which the gradient of h is assigned to be the driving force to adjust x. In ananalogy to the theory of plasticity, Eq. (20) may be considered to be the “flowrule”,and we will introduce the similarity between them, later. It can be seen thatq =∥∂h∂t∥ ∂h∂x∥ 2 (21)by substituting Eq. (20) into Eq. (19), where∂h∂t = 1 [‖F(x)‖ 2 + ‖x − a‖ 2] , (22)2∂h∂x = tBT F − (1 −t)(x − a). (23)B := ∂F∂xis usually called the Jacobian matrix of the NAEs. Hence, we can writethe nonlinear ODEs for x as:ẋ = −∥∂h∂t∥ ∂h∂x∂h∥ 2 ∂x . (24)Now we will make an analogy to the plasticity theory. In the plasticity theory, wehave the associated flow rule given by [Liu and Chang (2004)]ė P =˙λ∂h∂x , (25)where ė P denotes the plastic strain rate, h is the yield function and x relates to thestress state. Then by assuming an unit elastic modulus, the evolution of stress x is