Control the Fibre Orientation Distribution at the Outlet of Contraction

A two dimensional model of the orientation distribution of fibres in a paper machine headbox is studied. The goal is to control the fibre orientation distribution at the outlet of contraction by changing its shape. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Then, the problem is expressed as an optimal control problem governed by variational forms. By using an embedding method, the class of admissible shapes is replaced by a class of positive Radon measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this linear programming problem. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to shape variation design to a one dimensional headbox. The usefulness of this idea is that the method is not iterative and it does not need any initial guess of the solution.      

Numerical instabilities in structural optimization – analogy between topology & shape design problems

The formulation of structural optimization problems on the basis of the finite–element–method often leads to numerical instabilities resulting in non–optimal designs, which turn out to be difficult to realize from the engineering point of view. In the case of topology optimization problems the formation of designs characterized by oscillating density distributions such as the well–known “checkerboard–patterns” can be observed, whereas the solution of shape optimization problems often results in unfavourable designs with non–smooth boundary shapes caused by high–frequency oscillations of the boundary shape functions. Furthermore a strong dependence of the obtained designs on the finite–element–mesh can be observed in both cases. In this context we have already shown, that the topology design problem can be regularized by penalizing spatial oscillations of the density function by means of a penalty–approach based on the density gradient. In the present paper we apply the idea of problem regularization by penalizing oscillations of the design variable to overcome the numerical difficulties related to the shape design problem, where an analogous approach restricting the boundary surface can be introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical aspects of X‐SIMP‐based topology optimization

The discretization of topology design problems on the basis of the finite‐element‐method results in general in large‐scale combinatorial optimization problems, which are usually relaxed by the introduction of a continuous material density function as design variable. To avoid optimal designs containing unfavourable microstructures such as the well‐known “checkerboard” patterns, the relaxed problem can be regularized by the X‐SIMP‐approach, which penalizes intermediate density values as well as high density gradients within the design domain. In this context we discuss numerical aspects of the X‐SIMP‐based regularization such as the discretization of the regularized problem, the formulation of the corresponding stiffness matrix and the numerical solution of the discretized problem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On optimal operating conditions for a data processing system: A stochastic approach

In this paper, we consider the problem of determining optimal operating conditions for a data processing system. The system is burned‐in for a fixed burn‐in time before it is put into field operation and, in field operation, it has a work size and follows an age‐replacement policy. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, the properties of optimal burn‐in time, optimal work size and optimal age‐replacement policy will be derived. It can be seen that this model is a generalization of those considered in the previous works, and it yields a better optimal operating conditions. This paper presents an analytical method for three‐dimensional optimization problem. An algorithm for determining optimal operating conditions is also given. Copyright © 2007 John Wiley & Sons, Ltd.     

Relaxation-based algorithms for minimax optimization problems with resource allocation applications

We consider minimax optimization problems where each term in the objective function is a continuous, strictly decreasing function of a single variable and the constraints are linear. We develop relaxation-based algorithms to solve such problems. At each iteration, a relaxed minimax problem is solved, providing either an optimal solution or a better lower bound. We develop a general methodology for such relaxation schemes for the minimax optimization problem. The feasibility tests and formulation of subsequent relaxed problems can be done by using Phase I of the Simplex method and the Farkas multipliers provided by the final Simplex tableau when the corresponding problem is infeasible. Such relaxation-based algorithms are particularly attractive when the minimax optimization problem exhibits additional structure. We explore special structures for which the relaxed problem is formulated as a minimax problem with knapsack type constraints; efficient algorithms exist to solve such problems. The relaxation schemes are also adapted to solve certain resource allocation problems with substitutable resources. There, instead of Phase I of the Simplex method, a max-flow algorithm is used to test feasibility and formulate new relaxed problems.Corresponding author.Work was partially done while visiting AT&T Bell Laboratories.     

On a variational,free-discontinuity formulation of brittle fracture: mean-field relaxation and numerical solution

In this paper we address the resolution of two important issues arising in the context of the relaxed variational formulation of the incremental free–boundary value problem of brittle fracture. First issue, is how by recasting the formulation into a discrete, minimum–maximum problem one can avoid the undesirable scale effects expressed in terms of the characteristic size and domain–shape dependence of the calculated minimum; second, how by a remeshing procedure in combination with a domain–shape update for tracking the propagating 0–th level set one can reconstruct the crack surface. We finally illustrate our approach by a geometrically linear 2–dimensional example for crack propagation in an initially isotropic brittle solid. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Level Set Representation Method for N-Dimensional Convex Shape and Applications

In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the first one which can work for high dimensional objects.Convexity prior is very useful for object completion in computer vision.It is a very challenging task to represent high dimensional convex objects.In this paper,we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function.Then,the second order condition of convex functions is used to characterize the shape convexity equivalently.We apply this new method to two applications:object segmentation with convexity prior and convex hull problem(especially with outliers).For both applications,the involved problems can be written as a general optimization problem with three constraints.An algorithm based on the alternating direction method of multipliers is presented for the optimization problem.Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.     

Relaxation Through Homogenization for Optimal Design Problems with Gradient Constraints

The problem of the relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of either a finite or infinite number of constraints. The relaxed formulation is used to identify minimizing sequences of configurations of phases.     

On the optimal control and relaxation of nonlinear infinite dimensional systems

Summary In this paper we study optimal control problems for infinite dimensional systems governed by a semilinear evolution equation. First under appropriate convexity and growth conditions, we establish the existence of optimal pairs. Then we drop the convexity hypothesis and we pass to a larger system known as the « relaxed system ». We show that this system has a solution and the value of the relaxed optimization problem is equal to the value of the original one. Next we restrict our attention to linear systems and establish two « bang-bang » type theorems. Finally we present some examples from systems governed by partial differential equations.Research supported by N.S.F. Grant-8602313.Work done while on leave at the « University of Thessaloniki, School of Technology, Mathematics Division, Thessaloniki 54006, Greece ».     

A new fictitious domain method in shape optimization

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from Mommer (IMA J. Numer. Anal. 26:503–524, 2006) to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method can be applied to three dimensional shape optimization problems.     

Global Optimization Problems in Optimal Design of Experiments in Regression Models

In this paper we show that optimal design of experiments, a specific topic in statistics, constitutes a challenging application field for global optimization. This paper shows how various structures in optimal design of experiments problems determine the structure of corresponding challenging global optimization problems. Three different kinds of experimental designs are discussed: discrete designs, exact designs and replicationfree designs. Finding optimal designs for these three concepts involves different optimization problems.     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On two different inverse form finding methods for hyperelastic and elastoplastic materials

This contribution focuses on two different developments of mechanical-computational methods for the optimal determination of the initial shape of formed functional components knowing the deformed configuration, the applied loads and the boundary conditions. The first method uses an inverse mechanical formulation and can be applied to materials with hyperelastic behaviors. For materials with elastoplastic properties this method is not advocated, without knowing the final plastic strains, due to the non uniqueness of the solution. The second method uses a shape optimization formulation in the sense of an inverse problem via successive iterations of the direct problem. For hyperelastic materials the inverse mechanical formulation is preferred for its velocity and the non exhibition of possible mesh distortions. In the shape optimization formulation mesh distortions can be avoided by an update of the reference configuration of the functional part. Both methods are using a formulation in the logarithmic strain space. A numerical example for materials with isotropic elastoplastic behaviors illustrates the shape optimization formulation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.     

A theoretical measure technique for determining 3D symmetric nearly optimal shapes with a given center of mass

In this paper, a new approach is proposed for designing the nearly-optimal three dimensional symmetric shapes with desired physical center of mass. Herein, the main goal is to find such a shape whose image in (r, θ)-plane is a divided region into a fixed and variable part. The nearly optimal shape is characterized in two stages. Firstly, for each given domain, the nearly optimal surface is determined by changing the problem into a measure-theoretical one, replacing this with an equivalent infinite dimensional linear programming problem and approximating schemes; then, a suitable function that offers the optimal value of the objective function for any admissible given domain is defined. In the second stage, by applying a standard optimization method, the global minimizer surface and its related domain will be obtained whose smoothness is considered by applying outlier detection and smooth fitting methods. Finally, numerical examples are presented and the results are compared to show the advantages of the proposed approach.     

A genetic algorithm for designing microarray experiments

Heuristic techniques of optimization can be useful in designing complex experiments, such as microarray experiments. They have advantages over the traditional methods of optimization, particularly in situations where the search space is discrete. In this paper, a search procedure based on a genetic algorithm is proposed to find optimal (efficient) designs for both one- and multi-factor experiments. A genetic algorithm is a heuristic optimization method that exploits the biological evolution to obtain a solution of the problem. As an example, optimal designs for \(3\times 2\) factorial microarray experiments are presented for different numbers of arrays and for various sets of research questions. Comparisons between different operators of the genetic algorithm are performed by simulation studies.     

An ant colony system for enhanced loop-based aisle-network design

The purpose of this paper is to develop a global optimization model, simplification schemes, and a heuristic procedure for the design of a shortcut-enhanced unidirectional loop aisle-network with pick-up and drop-off stations. The objective is to minimize the total loaded and empty trip distances. This objective is the main determinant for the fleet size of the vehicles, which in turn is the driver of the total life-cycle cost of vehicle-based unit-load transport systems. The shortcut considerably reduces the length of the trips while maintaining the simplicity of the system. The global model solves simultaneously for the loop design, stations’ locations and shortcut design. We then develop two simplifications each containing two serial phases. Phase-1 of the first simplification step focuses on both loaded and empty trips, while that of the second simplification focuses only on loaded trips. In phase-2, both designs are enhanced with a shortcut to minimize both loaded and empty trip distances. The quality and efficiency of the three alternative designs are tested for a set of problems with different layout size and product mix. While the solution time of the second simplification procedure is a small percentage of the global formulation, it generates satisfactory solutions. On this foundation, we then develop a heuristic procedure to replace phase-1 of the second simplification. The heuristic procedure is using ant colony system to generate feasible solutions and then we implement a local search algorithm to improve the results. The heuristic algorithm quickly generates close to optimal solutions for phase-1 of the second simplification. By applying phase-2 of the this second simplification on a set of loops generated by the heuristic, close to optimal solutions are also quickly obtained for the global model.     

An approximation technique for robust nonlinear optimization

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.     

Optimal design of the damping set for the stabilization of the wave equation

We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in RN, N=1,2. In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.     

Shape Optimization for Navier–Stokes Equations with Algebraic Turbulence Model: Existence Analysis

We study a shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to an optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by a generalized stationary Navier–Stokes system with nontrivial mixed boundary conditions. In this paper we prove the existence of solutions both to the generalized Navier–Stokes system and to the shape optimization problem.