Elasticity with Arbitrarily Shaped Inhomogeneity

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a new, semi-analytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.Comment: (10 pages, 10 figures

Solving Variational Inequalities Defined on A Domain with Infinitely Many Linear Constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples for the proposed methods

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transformed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literatur

On the Efficient Solution of Variational Inequalities; Complexity and Computational Efficiency

In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach