Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

Coupling of a mapping method and a genetic algorithm to optimize mixing efficiency in periodic chaotic flows

We present a new technique to globally optimize stirring and mixing in three-dimensional spatially periodic laminar flows. This technique includes a recently developed mapping method and the use of an optimization method. The method gives the opportunity of the statistics of scalar dissipation for multiple stirring protocols. Exhaustive tests, first conducted for a large number of stirring protocols show that finding a stirring protocol which produces a uniform mixture is rare for the specific geometries studied. The results indicate that a global approach is necessary to handle the problem of concentration homogenization. It is shown that Genetic Algorithms enable us to rapidly find the best stirring protocols. Thus, the method described in this paper could be an efficient computational tool for chaotic mixing optimization in other types of periodic flows (2D or 3D).     

Direct method for the design of optimal three-dimensional aerodynamic shapes

A direct optimization method for a broad class of three-dimensional aerodynamic shapes based on the approximation of the desired geometry by Bernstein-Bézier surfaces is developed. The high efficiency of the method is demonstrated by applying it to the design of an optimal supersonic section of an axisymmetric maximum-thrust de Laval nozzle. The method is also tested as applied to the design of a three-dimensional supersonic nozzle section in a dense multi-nozzle setup. In addition to three-dimensional supersonic nozzle sections with a circular throat, nozzles with a varying throat shape are considered. The results suggest that the method can be applied to various problems of 3D shape optimization.     

Space-Mapping and Optimal Shape Design

Optimal shape design is a common problem in practice. In most cases this means that a complex geometry has to be optimized such that it satisfies a desired quantity. The direct optimization is very difficult and time consuming, if it is possible at all. Therefore, often only simplified models are derived and optimized. In general this does not yield an optimal solution. For this reason the space–mapping method is introduced, which combines the results of the complex model, e.g. a 3D model, and a simplified model, e.g. a 1D or 2D model. The derivation of a simplified model of a filter device and the following optimization is presented. Finally, the advantages of the space–mapping method applied to this example is shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topology optimization based on the finite cell method

We present an extension of the Finite Cell Method (FCM) to topology optimization utilizing the Fully Stressed Design (FSD) approach. The proposed method is briefly explained in 2D and its potential is demonstrated by means of a linear elastic 3D problem. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization methodology assessment for the inlet velocity profile of a hydraulic turbine draft tube: part II—performance evaluation of draft tube model

Computational Fluids Dynamics (CFD) tools guide engineers and designers to estimate the performance of new designs. However, a CFD analysis can be very time-consuming depending mainly on the grid size and domain complexity. Thus, this paper aims to describe the tools used to evaluate and compare the performance of different 3D draft tube models for reducing the time-effort needed in an optimization procedure. The results presented here, are the second part of an overall research to establish a global optimization methodology to improve the performance of an hydraulic draft tube through the inlet velocity profile. Previously, three steps of optimization methodology to minimize the energy losses were studied: the inlet velocity profile parameterization, the numerical optimization set-up and the objective function validation. In the latter step, a global optimization method called Multi Island Genetic Algorithm (MIGA) was considered, which requires a large number of iterations before producing a reliable result. This step is able to identify an efficient inlet velocity profile to minimize the energy losses through the draft tube model. However, each iteration is expensive in terms of computational time due to the need for 3D Navier–Stokes (NS) computations to evaluate each profile’s fitness. Thus, in this work the methodology attempts to accelerate the optimization process with accurate results. In order to achieve the goal, the grid size of the 3D draft tube model was minimized, resulting in a much lower computational cost. Specifically, the draft tube calculations were performed on a sequence of five different grids each having approximately twice the number of elements compared to the previous. The measurements of the sensitivity of the draft tube performance quantities to the change of the inlet velocity parameters during the process showed that, in spite of the numerical difference between its performance, the results have the same tendency. Consequently, the 3D draft tube numerical model with a minimal grid size, is reliable and left record of its capabilities for being integrated in the optimization process.     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

A constrained optimization based method for acoustic finite element model updating of cavities using pressure response

A method based on constrained optimization for updating of an acoustic finite element model using pressure response is proposed in this paper. The constrained optimization problem is solved using sequential quadratic programming algorithm. Updating parameters related to the properties of the sound absorbers and the measurement errors are considered. Effectiveness of the method is demonstrated by numerical studies on a 2D rectangular cavity and a car cavity. It is shown that the constrained formulation, that includes lower and upper bounds on the updating parameters in the form of inequality constraints, is important for obtaining a correct updated model. It is seen that the proposed updating method is not only able to effectively update the model to obtain a close match between the finite element model pressure response and the reference pressure response, but is also able to identify the correction factors to the parameters in error with reasonable accuracy.     

Optimization methods for Dirichlet control problems

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.     

A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates

This paper presents a novel method of multi-objective optimization by learning automata (MOLA) to solve complex multi-objective optimization problems. MOLA consists of multiple automata which perform sequential search in the solution domain. Each automaton undertakes dimensional search in the selected dimension of the solution domain, and each dimension is divided into a certain number of cells. Each automaton performs a continuous search action, instead of discrete actions, within cells. The merits of MOLA have been demonstrated, in comparison with a multi-objective evolutionary algorithm based on decomposition (MOEA/D) and non-dominated sorting genetic algorithm II (NSGA-II), on eleven multi-objective benchmark functions and an optimal problem in the midwestern American electric power system which is integrated with wind power, respectively. The simulation results have shown that MOLA can obtain more accurate and evenly distributed Pareto fronts, in comparison with MOEA/D and NSGA-II.     

Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments.     

Control of a mechanical aeration process via topological sensitivity analysis

The topological sensitivity analysis method gives the variation of a criterion with respect to the creation of a small hole in the domain. In this paper, we use this method to control the mechanical aeration process in eutrophic lakes. A simplified model based on incompressible Navier–Stokes equations is used, only considering the liquid phase, which is the dominant one. The injected air is taken into account through local boundary conditions for the velocity, on the injector holes. A 3D numerical simulation of the aeration effects is proposed using a mixed finite element method. In order to generate the best motion in the fluid for aeration purposes, the optimization of the injector location is considered. The main idea is to carry out topological sensitivity analysis with respect to the insertion of an injector. Finally, a topological optimization algorithm is proposed and some numerical results, showing the efficiency of our approach, are presented.     

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

$${{\mathcal {D}(\mathcal {C})}}$$-optimization and robust global optimization

For solving global optimization problems with nonconvex feasible sets existing methods compute an approximate optimal solution which is not guaranteed to be close, within a given tolerance, to the actual optimal solution, nor even to be feasible. To overcome these limitations, a robust solution approach is proposed that can be applied to a wide class of problems called D(C){{\mathcal {D}(\mathcal {C})}}-optimization problems. DC optimization and monotonic optimization are particular cases of D(C){{\mathcal {D}(\mathcal {C})}}-optimization, so this class includes virtually every nonconvex global optimization problem of interest. The approach is a refinement and extension of an earlier version proposed for dc and monotonic optimization.     

A NURBS-based Multi-Material Interpolation (N-MMI) for isogeometric topology optimization of structures

In this paper, the main intention is to propose a new Multi-material Isogeometric Topology Optimization (M-ITO) method for the optimization of multiple materials distribution, where an improved Multi-Material Interpolation model is developed using Non-Uniform Rational B-splines (NURBS), namely the “NURBS-based Multi-Material Interpolation (N-MMI)”. In the N-MMI model, three key components are involved: (1) multiple Fields of Design Variables (DVFs): NURBS basis functions with control design variables are applied to construct DVFs with the sufficient smoothness and continuity; (2) multiple Fields of Topology Variables (TVFs): each TVF is expressed by a combination of all DVFs to present the layout of a distinct material in the design domain; (3) Multi-material interpolation: the material property at each point is equal to the summation of all TVFs interpolated with constitutive elastic properties. DVFs and TVFs are in the decoupled expression and optimized in a serial evolving mechanism. This feature can ensure the constraint functions are separate and linear with respect to TVFs, which can be beneficial to lower the complexity of numerical computations and eliminate numerical troubles in the multi-material optimization. Two kinds of multi-material topology optimization problems are discussed, i.e., one with multiple volume constraints and the other with the total mass constraint. Finally, several numerical examples in 2D and 3D are provided to demonstrate the effectiveness of the M-ITO method.     

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

A linear programming approach for linear multi-level programming problems

In an earlier paper, we proposed a modified fuzzy programming method to handle higher level multi-level decentralized programming problems (ML(D)PPs). Here we present a simple and practical method to solve the same. This method overcomes the subjectivity inherent in choosing the tolerance values and the membership functions. We consider a linear ML(D)PP and apply linear programming (LP) for the optimization of the system in a supervised search procedure, supervised by the higher level decision maker (DM). The higher level DM provides the preferred values of the decision variables under his control to enable the lower level DM to search for his optimum in a narrower feasible space. The basic idea is to reduce the feasible space of a decision variable at each level until a satisfactory point is sought at the last level.     

Decomposition in large system optimization using the method of multipliers

Although the method of multipliers can resolve the dual gaps which will often appear between the true optimum point and the saddle point of the Lagrangian in large system optimization using the Lagrangian approach, it is impossible to decompose the generalized Lagrangian in a straightforward manner because of its quadratic character. A technique using the linear approximation of the perturbed generalized Lagrangian has recently been proposed by Stephanopoulos and Westerberg for the decomposition. In this paper, another attractive decomposition technique which transforms the nonseparable crossproduct terms into the minimum of sum of separable terms is proposed. The computational efforts required for large system optimization can be much reduced by adopting the latter technique in place of the former, as illustrated by application of these two techniques to an optimization problem of a chemical reactor system.The authors would like to acknowledge the valuable comments given by Professor D. G. Luenberger of Stanford University.     

Vortex design problem

In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity. The problem is formulated as a PDE-constrained optimization in which the solutions are found using a gradient-based descent method. Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients. In explicit tracking of interfaces (vortex boundaries) this method offers an alternative to grid-based techniques, such as the level-set methods, and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique. We develop and validate this approach using the design of 2D equilibrium Euler flows with finite-area vortices as a model problem. It is also discussed how the proposed methodology can be applied to Euler flows featuring other vorticity distributions, such as vortex sheets, and to time-dependent phenomena.     

Bivariate Free Knot Splines

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.