Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.Comment: 40 pages using amsart style file, and 15 figure

Optimization techniques applied to passive measures for in-orbit spacecraft survivability

Spacecraft designers have always been concerned about the effects of meteoroid impacts on mission safety. The engineering solution to this problem has generally been to erect a bumper or shield placed outboard from the spacecraft wall to disrupt/deflect the incoming projectiles. Spacecraft designers have a number of tools at their disposal to aid in the design process. These include hypervelocity impact testing, analytic impact predictors, and hydrodynamic codes. Analytic impact predictors generally provide the best quick-look estimate of design tradeoffs. The most complete way to determine the characteristics of an analytic impact predictor is through optimization of the protective structures design problem formulated with the predictor of interest. Space Station Freedom protective structures design insight is provided through the coupling of design/material requirements, hypervelocity impact phenomenology, meteoroid and space debris environment sensitivities, optimization techniques and operations research strategies, and mission scenarios. Major results are presented

Dual methods and approximation concepts in structural synthesis

Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins

A Distributed Approach for the Optimal Power Flow Problem Based on ADMM and Sequential Convex Approximations

The optimal power flow (OPF) problem, which plays a central role in operating electrical networks is considered. The problem is nonconvex and is in fact NP hard. Therefore, designing efficient algorithms of practical relevance is crucial, though their global optimality is not guaranteed. Existing semi-definite programming relaxation based approaches are restricted to OPF problems where zero duality holds. In this paper, an efficient novel method to address the general nonconvex OPF problem is investigated. The proposed method is based on alternating direction method of multipliers combined with sequential convex approximations. The global OPF problem is decomposed into smaller problems associated to each bus of the network, the solutions of which are coordinated via a light communication protocol. Therefore, the proposed method is highly scalable. The convergence properties of the proposed algorithm are mathematically substantiated. Finally, the proposed algorithm is evaluated on a number of test examples, where the convergence properties of the proposed algorithm are numerically substantiated and the performance is compared with a global optimal method.Comment: 14 page