Applied optimal shape design

This paper is a short survey of optimal shape design (OSD) for fluids. OSD is an interesting field both mathematically and for industrial applications. Existence, sensitivity, correct discretization are important theoretical issues. Practical implementation issues for airplane designs are critical too.The paper is also a summary of the material covered in our recent book, Applied Optimal Shape Design, Oxford University Press, 2001.     

An optimal shape design formulation for inhomogeneous dam problems

In this paper, the flow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with permeability allowing parametrization of the free boundary by a graph of continuous unidimensional function, is considered. We propose a new formulation on an optimal shape design problem. We show the existence of a solution of the optimal shape design problem. The finite element method is used to obtain numerical results which show the efficiency of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Measure theoretical approach for optimal shape design of a nozzle

In this paper we present a new method for designing a nozzle. In fact the problem is to find the optimal domain for the solution of a linear or nonlinear boundary value PDE, where the boundary condition is defined over an unspecified domain. By an embedding process, the problem is first transformed to a new shape-measure problem, and then this new problem is replaced by another in which we seek to minimize a linear form over a subset of linear equalities. This minimization is global, and the theory allows us to develop a computational method to find the solution by a finite-dimensional linear programming problem.     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993     

An iterative optimum-optimorum theory for the design of a globally optimal shape

In some previous papers, the author has developed the optimum-optimorum theory, which solves an enlarged variational problem with free boundaries, inside of a class of flying configurations (FCs), defined by some chosen common properties. This optimization strategy was used by the author for the inviscid, aerodynamical, global optimal design of three models, namely, Adela, a delta wing alone and of two fully-integrated wing-fuselage FCs, namely, Fadet I and Fadet II, which have all high values of L/D (lift to drag). A further enlargement of the optimization strategy is developed here, in form of an iterative optimum-optimorum theory, which uses the inviscid global optimized FC's shape as first step of iteration and the own developed reinforced Navier-Stokes solutions up the first computational checking and up the second step of iteration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A refined Newton’s mesh independence principle for a class of optimal shape design problems

Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

易拉罐形状和尺寸的最优设计

从用料最省的角度研究了易拉罐的形状和尺寸的优化设计问题,首先通过多次测量取平均值的方法得到了题目所需的数据．然后就问题二和问题三分别建立了优化模型,并借助数学软件进行了求解,得到了最优设计的尺寸．最后设计出了椭球形状的易拉罐作为自己的最优设计．     

Real-time computation of optimal control

A new approach to the real-time implementation of time-optimal control for linear systems with a bounded control is proposed. The computational costs are separated between preliminary computations and computations in the course of the control process. The preliminary computations are independent of the particular initial condition and are based on the approximation of sets reachable in different times by a collection of hyperplanes. Methods for constructing hyperplanes and selecting a supporting hyperplane are described. Methods are proposed for approximately finding the normalized vector of initial conditions of the adjoint system, the driving time, and the switching times of the time-optimal control, and an iterative method for their refinement is developed. The computational complexity of the method is estimated. The computational algorithm is described, and simulation and numerical results are presented.     

A general criterion for optimal structural design

A general criterion of structural optimality is presented and discussed. It applies to multipurpose structures subjected to multiple or movable loadings, the design of which is defined by several design functions. The cost is assumed to be a convex function of the various specific energies associated with the respective behavioral constraints. This criterion is shown to include most (if not all) criteria used up to now.     

Influence of various design parameters on the quality of optimal shape design in topology optimization analysis

The paper presents the discrete structural topology optimization problem. The analysis was made for the formulation of optimal design with a compliance minimization (as the objective functional) of statically loaded continuum structures with constraints forced on mass of the structure. Two different algorithms were compared (one taken from the literature and the second originally prepared) in order to study influence of design parameters formulation on the calculations results. Optimal topologies were presented and discussed for several 2D examples, with various optimization process parameters in order to get the best results in the shortest possible time. Other goal of presented research is to find out the possibilities of applying proposed algorithm in the designing process of bridge girders. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

LOOK: A Lazy Object-Oriented Kernel design for geometric computation

In this paper we describe and discuss a new kernel design for geometric computation in the plane. It combines different kinds of floating-point filter techniques and a lazy evaluation scheme with the exact number types provided by LEDA allowing for efficient and exact computation with rational and algebraic geometric objects.

It is the first kernel design which uses floating-point filter techniques on the level of geometric constructions.

The experiments we present—partly using the CGAL framework—show a great improvement in speed and—maybe even more important for practical applications—memory consumption when dealing with more complex geometric computations.     

On optimal shape design of systems governed by mixed Dirichlet-Signorini boundary value problems

A problem for finding optimal shape for systems governed by the mixed unilateral boundary value problem of Dirichlet-Signorini-type is considered. Conditions for the solvability of the problem are stated when a variational inequality formulation and when a penalty method is used for solving the state problem in question. The asymptotic relation of design problems based on these two formulations is presented. The optimal shape design problem is discretized by means of finite element method. The convergence results for the approximation are proved. The discretized versions are then formulated as a non-linear programming problem. Results of practical computations of the problem in question are reported.     

Existence analysis of an optimal shape design problem with non coercive state equation

In this paper, a shape optimization problem, modelling a welding process, governed by a second order non coercive PDE is considered. The well posedness of the shape optimal design problem is showed using the degree of Leray–Schauder. Then the existence of an optimal solution is proved.     

Evolutionary computation for optimal knots allocation in smoothing splines

In this paper, a novel methodology is presented for optimal placement and selections of knots, for approximating or fitting curves to data, using smoothing splines. It is well-known that the placement of the knots in smoothing spline approximation has an important and considerable effect on the behavior of the final approximation [1]. However, as pointed out in [2], although spline for approximation is well understood, the knot placement problem has not been dealt with adequately. In the specialized bibliography, several methodologies have been presented for selection and optimization of parameters within B-spline, using techniques based on selecting knots called dominant points, adaptive knots placement, by data selection process, optimal control over the knots, and recently, by using paradigms from computational intelligent, and Bayesian model for automatically determining knot placement in spline modeling. However, a common two-step knot selection strategy, frequently used in the bibliography, is an homogeneous distribution of the knots or equally spaced approach [3].     

A hybrid approach for optimal design of signalized road network

For a signalized road network with expansions of link capacity, the maximum possible increase in travel demands is considered while total delays for travelers are minimized. Using the concept of reserve capacity of signal-controlled junctions, the problem of finding the maximum possible increase in travel demand and determining optimal link capacity expansions can be formulated as optimization programs. In this paper, we present a new solution approach for simultaneously solving the maximum increase in travel demands and minimizing total delays of travelers. A projected Quasi-Newton method is proposed to effectively solve this problem to the KKT points. Numerical computations and comparisons are made on real data signal-controlled networks where obtained results outperform traditional methods.     

Far field boundary conditions for incompressible flows computation

Many far field boundary conditions are proposed in the literature to solve Navier-Stokes equations. It is necessary to distinguish the streamwise or outlet boundary conditions and the spanwise boundary conditions. In the first case the flow crosses the artificial frontier and it is required to avoid reflections that can change significantly the flow. In the second case the Navier-slip boundary condition is often used but if the frontier is not far enough the boundary is both inlet and outlet. Thus the Navier-slip boundary condition is not well suited as it imposes no flux through the frontier. The aim of this work is to compare some well-known boundary conditions, to quantify to which extend the artificial frontier can be close to the bodies in two- and three-dimensions and to take into account the flow rate through the spanwise directions.     

The optimal shape of a pipe

In shape optimization, recently the question arose, whether or not the cylindrical pipe has the optimal shape for the transport of an incompressible fluid. In this short note, a proof will be presented that a cylindrical pipe with Poiseuille’s flow inside indeed is optimal for the transportation of an incompressible fluid under the criterion “energy dissipated by the fluid.” The proof reduces the problem to the minimization of a two-dimensional Dirichlet’s integral. This simpler problem can be solved with a symmetrization argument.     

Linear-time computation of optimal subgraphs of decomposable graphs

A general problem in computational graph theory is that of finding an optimal subgraph H of a given weighted graph G. The matching problem (which is easy) and the traveling salesman problem (which is not) are well-known examples of this general problem. In the literature one can also find a variety of ad hoc algorithms for solving certain special cases in linear time. We suggest a general approach for constructing linear-time algorithms in the case where the graph G is defined by certain rules of composition (as are trees, series-parallel graphs, and outerplanar graphs) and the desired subgraph H satisfies a property that is “regular” with respect to these rules of composition (as do matchings, dominating sets, and independent sets for all the classes just mentioned). This approach is applied to obtain a linear-time algorithm for computing the irredundance number of a tree, a problem for which no polynomial-time algorithm was previously known.