Semi‐deterministic and genetic algorithms for global optimization of microfluidic protein‐folding devices

In this paper, we reformulate global optimization problems in terms of boundary‐value problems (BVP). This allows us to introduce a new class of optimization algorithms. Indeed, current optimization methods, including non‐deterministic ones, can be seen as discretizations of initial value problems for differential equations, or systems of differential equations. Furthermore, in order to reduce computational time approximate state and sensitivity evaluations are introduced during optimization. Lastly, we demonstrated the efficacy of two algorithms, included in the former class, on two academic test cases and on the design of a fast microfluidic protein‐folding device. The aim of the latter design is to reduce mixing times of proteins to microsecond time scales. Results are compared with those obtained with a classical genetic algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

An approach to multi‐start clustering for global optimization with non‐linear constraints

This paper describes a multi‐start with clustering strategy for use on constrained optimization problems. It is based on the characteristics of non‐linear constrained global optimization problems and extends a strategy previously tested on unconstrained problems. Earlier studies of multi‐start with clustering found in the literature have focused on unconstrained problems with little attention to non‐linear constrained problems. In this study, variations of multi‐start with clustering are considered including a simulated annealing or random search procedure for sampling the design domain and a quadratic programming (QP) sub‐problem used in cluster formation. The strategies are evaluated by solving 18 non‐linear mathematical problems and six engineering design problems. Numerical results show that the solution of a one‐step QP sub‐problem helps predict possible regions of attraction of local minima and can enhance robustness and effectiveness in identifying local minima without sacrificing efficiency. In comparison to other multi‐start techniques found in the literature, the strategies of this study can be attractive in terms of the number of local searches performed, the number of minima found, whether the global minimum is located, and the number of the function evaluations required. Copyright © 2002 John Wiley & Sons, Ltd.     

Studies of multi‐start clustering for global optimization

Global/multi‐modal optimization problems arise in many engineering applications. Owing to the existence of multiple minima, it is a challenge to solve the multi‐modal optimization problem and to identify the global minimum especially if efficiency is a concern. In this paper, variants of the multi‐start with clustering strategy are developed and studied for identifying multiple local minima in nonlinear global optimization problems. The study considers the sampling procedure, the use of Hessian information in forming clusters, the technique for cluster analysis and the local search procedure. Variations of multi‐start with clustering are applied to 15 multi‐modal problems. A comparative study focuses on the overall search effectiveness in terms of the number of local searches performed, local minima found and required function evaluations. The performance of these multi‐start clustering algorithms ranges from very efficient to very robust. Copyright © 2002 John Wiley & Sons, Ltd.     

Stress‐enhanced clonal selection algorithm for structural topology optimization

Recently, numerous modified versions of immune algorithms (IAs) have been adopted in both theoretical and practical applications. However, few have been proposed for solving structural topology optimization problems. In addition, the design connectivity handling and one‐node connected hinge prevention, which are vital in the application of population‐based methods with binary representation for structural topology optimization, have not been applied to IAs in the literature. A stress‐enhanced clonal selection algorithm (SECSA) incorporating an IA with a dominance‐based constraint‐handling technique and a new stress‐enhanced hypermutation operator is proposed to rectify those deficiencies. To demonstrate the high viability of the presented method, comparisons between the presented SECSA and genetic algorithm‐based methods were made on minimum compliance and minimum weight benchmark structural topology design problems in two‐dimensional, three‐dimensional, and multiloading cases. In each case, SECSA was shown to be competitive in terms of convergence speed and solution quality. The main goal of this study is not only to further explore the capabilities of IAs, but also to show that an IA with appropriate enhancements can lead to the development of attractive computational tools for global search in structural topology optimization. Copyright © 2011 John Wiley & Sons, Ltd.     

A modified version of a T‐Cell Algorithm for constrained optimization problems

In this paper, we present a heuristic inspired on the T‐Cell model of the immune system (i.e. an artificial immune system). The proposed approach (called T‐Cell) is used for solving constrained (numerical) optimization problems, and is validated using several test functions taken from the specialized literature on evolutionary optimization. Additionally, several engineering optimization problems are also used for assessing the performance of the proposed approach. The results are compared with respect to approaches representative of the state‐of‐the‐art in constrained evolutionary optimization. Copyright © 2010 John Wiley & Sons, Ltd.     

Hessian‐based model reduction for large‐scale systems with initial‐condition inputs

Reduced‐order models that are able to approximate output quantities of interest of high‐fidelity computational models over a wide range of input parameters play an important role in making tractable large‐scale optimal design, optimal control, and inverse problem applications. We consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions, and pose the task of determining appropriate training sets for reduced‐basis construction as a sequence of optimization problems. We show that, under certain assumptions, these optimization problems have an explicit solution in the form of an eigenvalue problem, yielding an efficient model reduction algorithm that scales well to systems with states of high dimension. Furthermore, tight upper bounds are given for the error in the outputs of the reduced models. The reduction methodology is demonstrated for a large‐scale contaminant transport problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Modelling topology optimization problems as linear mixed 0–1 programs

This paper deals with topology optimization of discretized continuum structures. It is shown that a large class of non‐linear 0–1 topology optimization problems, including stress‐ and displacement‐constrained minimum weight problems, can equivalently be modelled as linear mixed 0–1 programs. The modelling approach is applied to some test problems which are solved to global optimality. Copyright © 2003 John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Large strain phase‐field‐based multi‐material topology optimization

A multi‐material topology optimization scheme is presented. The formulation includes an arbitrary number of phases with different mechanical properties. To ensure that the sum of the volume fractions is unity and in order to avoid negative phase fractions, an obstacle potential function, which introduces infinity penalty for negative densities, is utilized. The problem is formulated for nonlinear deformations, and the objective of the optimization is the end displacement. The boundary value problems associated with the optimization problem and the equilibrium equation are solved using the finite element method. To illustrate the possibilities of the method, it is applied to a simple boundary value problem where optimal designs using multiple phases are considered. Copyright © 2015 John Wiley & Sons, Ltd.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A fast method for binary programming using first‐order derivatives,with application to topology optimization with buckling constraints

We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐level programming‐language implementation of topology optimization applied to steady‐state Navier–Stokes flow

We present a versatile high‐level programming‐language implementation of non‐linear topology optimization. Our implementation is based on the commercial software package FEMLAB, and it allows a wide range of optimization objectives to be dealt with easily. We exemplify our method by studies of steady‐state Navier–Stokes flow problems, thus extending the work by Borrvall and Petersson on topology optimization of fluids in Stokes flow (Int. J. Num. Meth. Fluids 2003; 41 :77–107). We analyse the physical aspects of the solutions and how they are affected by different parameters of the optimization algorithm. A complete example of our implementation is included as FEMLAB code in an appendix. Copyright © 2005 John Wiley & Sons, Ltd.     

Reliability optimization of complex systems using modified random‐to‐pattern search (MRPS) algorithm

The modified random‐to‐pattern search (MRPS) algorithm, developed by the authors for global optimization, is applied to find the global optimum of system cost of a complex system subject to constraints on system reliability. The global optimum solutions obtained by MRPS are compared with those obtained by employing gradient techniques as well as random search‐based methods from the literature to solve the same problems. Results clearly indicate that the proposed MRPS algorithm is more robust and efficient in overcoming difficulties associated with local optima and the need for a starting solution vector. Copyright © 1999 John Wiley & Sons, Ltd.     

Isogeometric shape optimization for quasi‐static processes

A framework to solve shape optimization problems for quasi‐static processes is developed and implemented numerically within the context of isogeometric analysis (IGA). Recent contributions in shape optimization within IGA have been limited to static or steady‐state loading conditions. In the present contribution, the formulation of shape optimization is extended to include time‐dependent loads and responses. A general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems. An adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of IGA. The methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi‐axial tension and pure shear. Under quasi‐static loading conditions, the method is validated using a passive control approach with an a priori known solution. Several applications of time‐dependent mechanical problems are shown to illustrate the capabilities of this approach. In particular, a problem is considered where an external load is allowed to move along the surface of a structure. The shape of the structure is modified in order to control the time‐dependent displacement of the point where the load is applied according to a pre‐specified target. Copyright © 2015 John Wiley & Sons, Ltd.     

Topology synthesis of large‐displacement compliant mechanisms

This paper describes the use of topology optimization as a synthesis tool for the design of large‐displacement compliant mechanisms. An objective function for the synthesis of large‐displacement mechanisms is proposed together with a formulation for synthesis of path‐generating compliant mechanisms. The responses of the compliant mechanisms are modelled using a total Lagrangian finite element formulation, the sensitivity analysis is performed using the adjoint method and the optimization problem is solved using the method of moving asymptotes. Procedures to circumvent some numerical problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

A constrained‐optimization methodology for the detection phase in contact mechanics simulations

The detection phase in computational contact mechanics can be subdivided into a global search and a local detection. When potential contact is detected by the former, a rigorous local detection determines which surface elements come or may come in contact in the current increment. We first introduce a rigorous definition of the closest point for non‐differentiable lower‐dimensional manifolds. We then simplify the detection by formulating an optimization problem subject to inequality constraints. The formulation is then solved using different techniques from the field of mathematical optimization, for both linear and quadratic finite element meshes. The resulting general and robust detection scheme is tested on a set of problems and compared with other techniques commonly used in computational geometry. Copyright © 2013 John Wiley & Sons, Ltd.