Topological design of modular structures under arbitrary loading

A numerical method for the topological design of periodic continuous domains under general loading is presented. Both the analysis and the design are defined over a single cell. Confining the analysis to the repetitive unit is obtained by the representative cell method which by means of the discrete Fourier transform reduces the original problem to a boundary value problem defined over one module, the representative cell. The repeating module is then meshed into a dense grid of finite elements and solved by finite element analysis. The technique is combined with topology optimization of infinite spatially periodic structures under arbitrary static loading. Minimum compliance structures under a constant volume of material are obtained by using the densities of material as design variables and by satisfying a classical optimality criterion which is generalized to encompass periodic structures. The method is illustrated with the design of an infinite strip possessing 1D translational symmetry and a cyclic structure under a tangential point force. A parametric study presents the evolution of the solution as a function of the aspect ratio of the representative cell.     

Explicit exact analysis of infinite periodic structures under general loading

This paper deals with the explicit analysis of infinite periodic structures under arbitrary loadings. In the context or structural stiffness optimization, with its inherent problem of multiple reanalysis, the purpose is to obtain expressions for the stress resultants anywhere in the infinite structure as an explicit function of the stiffnesses of the elements. Following the method of the representative cell the analysis of an infinite structures is reduced to the analysis of single module under transformed loading and boundary conditions by using the discrete Fourier transform. This produces the equilibrium, strain-displacements and constitutive equations in terms of complex-valued displacements, generalized strains and generalized stresses transforms. Next an existing formula is used to write the stress resultants transforms explicitly in terms of the stiffnesses. Finally one computes the stress resultants wherever needed in the real structure by means of the inverse Fourier transform. The exact formula for the stress resultants is usually impractical due to the large number of terms involved in the analytical expressions. What makes the approach practical herein is the very reduced size of the repeating module that is to be analysed, which renders the analytical formula more tractable in many cases. The technique is illustrated with the explicit analysis of an infinite truss with 1D translational symmetry and of an infinite grid of orthogonal beams on elastic supports with 2D translational symmetry. Received December 4, 2000     

Optimal cross-section and configuration design of elastic-plastic structures subject to dynamic cyclic loading

This paper shows an optimal design problem with continuum variational formulation, applied to nonlinear elasticplastic structures subject to dynamic loading. The total Lagrangian procedure is used to describe the response of the structure. The direct differentiation method is used to obtain the sensitivities of the structural response that are needed to solve the optimization problem. Since unloading and reloading of the structure are allowed, the structural response is path-dependent and an additional step is needed to integrate the constitutive equations. It can be shown, consequently, that design sensitivity analysis is also path-dependent. A finite element method with implicit time integration is used to discretize the state and sensitivity equations.A mathematical programming approach is used for the optimization process. Numerical applications are performed on a 3-D truss structure, where cross-sectional areas and nodal point coordinates are treated as design variables. Optimal designs have been obtained and compared by using two different strategies: a twolevel strategy where the levels are defined according to the type of design variables, cross sectional areas or node coordinates, and optimizing simultaneously with respect to both types of design variables. Comparisons have also been made between optimal designs obtained by considering or not considering the inertial term of the structural equilibrium.     

Optimal limit design of elasto-plastic structures for time-dependent loading

A new approach is presented for the design of structures where the solution of the boundary value problem is approximated by an infinite function series of the state variables expressed in terms of time. Space discretization is used, but the state variables, ordered to the nodes, are continuous functions of time. In spite of a direct mathematical programming technique, the structural response to the time-dependent loading can be followed and the energy dissipation is considered. The design variables are the cross-sectional dimensions and the dissipative forces, and design constraints enforce upper bounds on the flexibilities. This paper was presented at WCSMO-6, Rio de Janeiro, Brazil.     

Design of absorbing material distribution for sound barrier using topology optimization

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0–1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton–Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.     

Convergence of topological patterns of optimal periodic structures under multiple scales

The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.     

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.     

Three-dimensional beam propagation method based on the variable transformed Galerkin''''s method

A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The infinite Cartesian x-y plane is mapped into a unit square by a tangent-type function transformation. Consequently, the infinite region problem is converted into the finite region problem. Thus, the boundary truncation is eliminated and the calculation accuracy is promoted. The three-dimensional BPM basic equation is reduced to a set of first-order ordinary differential equations through sinusoidal basis function, which fits arbitrary cladding optical waveguide, then direct solution of the resulting equations by means of the Runge-Kutta method. In addition, the calculation is efficient due to the small matrix derived from the present technique. Both z-invariant and z-variant examples are considered to test both the accuracy and utility of this approach.     

Shape design of elastostatic structures based on local perturbation analysis

This paper describes a non-gradient formulation for solving shape optimal design problems involving structures in plane stress or having an axially symmetric geometry. The minimization of the maximum von Mises stress value at a traction free boundary poses a non-linear optimization problem in which the design variables do not appear explicitly in the formulation. The most commonly used approach is to apply a standard non-linear programming technique. There exists in this field no universally accepted solution method. The major difficulty of shape optimization in connection with FEM is to perform an accurate and efficient sensitivity analysis. The perturbation analysis introduced here takes advantage of the character of the problem. It is based on methods from the theory of notches. The results are applied to an FE-model of the structural component. The iterative method with such a direction of search works efficiently even for a large number of design variables as shown by Schnack (1977b, 1978, 1979, 1980, 1983 and 1985). Using a dynamic programming formulation (see also Schnack and Spörl 1986), the existence of a solution for the shape optimal problem will be discussed. Examples of applications to structural components from mechanical engineering are presented to demonstrate the power of this approach.     

Optimal design of multi-stage structures: a nested decomposition approach

A new approach to the optimal design of multi-stage structures is presented. The problem considered is that of optimal sizing and configurations for minimal weight, using limit analysis under a single loading condition. It is formulated as a large-scale linear program with the staircase structure. Using the technique of nested decomposition, design problems too large for conventional methods can be solved. Computational results are reported on examples in planar truss design.     

Topology/shape optimisation of axisymmetric continuum structures – a metamorphic development approach

A novel topology/shape optimisation method for axisymmetric elastic solids, based on solid modeling and FE analysis, is presented. Optimal profiles of minimum-mass axisymmetric structures are sought by growing and degenerating simple initial structures subject to response constraints. The rates of the growth and degeneration are controlled based on the current objective and constraint functions of the optimisation problem under consideration. The optimal structures are developed metamorphically in specified infinite design domains using both quadrilateral and triangular axisymmetric finite elements that are ideally suited for modeling continua involving curved boundaries.The robustness of this fully automatic method is studied and validated with the first example of seeking the optimal shape of a centrally suspended axisymmetric object with minimum strain energy caused by self-weight. Then the method is applied to a practical industrial design problem: the design of a turbine disk. The variations of load and boundary conditions caused by shape change in these problems, including the gravitational and centrifugal loads, and temperature distribution are accommodated in the optimisation procedures. Thus, the design model closely resembles the real design problem. The results demonstrate the success of the method in generating optimal but realistic solutions to practical design problems.     

Topology optimization of pressure structures based on regional contour tracking technology

This article presents a new computational approach to solve the design-dependent loading problem in topology optimization of pressure structures. A simple algorithm based on digital image processing and regional contour tracking technology is proposed that generates the appropriate loading surface during the topology evolution. First, the topological layout produced during the optimization process is transformed into a compact image. Then, the regional contour tracking technology is used to represent the boundary of objects and extract pressure loading elements. At last, the pressures are transferred directly to corresponding element nodes. Due to the semi-automatically determined endpoints of the loading boundaries, the current scheme can deal with structures loaded by pressure from outside the domain, as well as pressure completely contained within the domain. Also, the calculation of the load sensitivities can be avoided in the current scheme. As a simple alternative computational strategy for compliance topology optimization of pressure structures, the current scheme is stable, flexible and efficient. Representative numerical examples are presented to show the validity and advantages of the proposed scheme. Especially, the design of closed containers and storage tanks indicates that it works well for the topology optimization of pressure structures.     

A level set based topology optimization method using the discretized signed distance function as the design variables

This paper deals with a new topology optimization method based on the level set method. In the proposed method, the discretized signed distance function, a kind of level set function, is used as the design variables, and these are then updated using their sensitivities. The signed distance characteristic of the design variables are maintained by performing a re-initialization at every update during the iterated optimization procedure. In this paper, a minimum mean compliance problem and a compliant mechanism design problem are formulated based on the level set method. In the formulations of these design problems, a perimeter constraint is imposed to overcome the ill-posedness of the structural optimization problem. The sensitivity analysis for the above structural optimization problems is conducted based on the adjoint variable method. The augmented Lagrangian method is incorporated to deal with multiple constraints. Finally, several numerical examples that include multiple constraints are provided to confirm the validity of the method, and it is shown that appropriate optimal structures are obtained.     

Optimal design of truss structures using parallel computing

Parallel design optimization of large structural systems calls for a multilevel approach to the optimization problem. The general optimization problem is decomposed into a number of non-interacting suboptimization problems on the first level. They are controlled from the second level through coordination variables. Thus, the solutions of the independent first-level subsystems are directed towards the overall system optimum. In the present paper, optimal design of truss structures using parallel computing technique is described. In this method, optimization of a large truss structure has been carried out by decomposing the structure into sub-domains and suboptimization tasks. Each sub-domain has independent design variables and a small number of behaviour constraints. The two-level sub-domain optimum design approach is summarized by several numerical examples with speedups and efficiencies of algorithms on message passing systems. It has been noticed that the efficiency of the algorithm for design optimization increases with the size of the structure.     

离散时滞系统的近似最优扰动抑制

研究了状态变量合有时滞的离散系统在外部扰动下的最优控制问题．通过引入一个灵敏度参数,将原系统的最优扰动抑制问题转化为一族不含超前项和时滞项的两点边值问题,并由此导出了最优扰动抑制控制器的这代近似设计方法．得到的最优扰动抑制控制律由解析的前馈一反馈项和伴随向量级数和形式的补偿项组成,截取伴随向量级数的有限和得到原系统的次优扰动抑制控制律．数值仿真表明该近似最优控制器对外部持续扰动具有良好的鲁棒性。     

Shape sensitivity analysis using the indirect boundary element method

The paper describes a novel formulation for the computation of the design sensitivities required for shape optimization problems using the indirect boundary element method. As a first stage, the system of equations that evaluate the fictitious traction sensitivities is differentiated with respect to shape design variables. The stress or displacement sensitivities are then evaluated by direct substitution of the fictitious traction sensitivities into the differentiated stress or displacement kernels. Two other finite difference-based techniques for the evaluation of the stress sensitivities, using the indirect boundary element method are also presented. The advantages and the drawbacks of each approach are discussed. These methods have been shown to be effective, accurate and can be incorporated in an existing BE code with much less programming effort than other BE-based techniques. The efficiency of the three methods is illustrated by optimizing the shape of a 90° V-notch. In all cases, convergence is achieved within three to four iterations.Various approximate techniques are suggested to minimize the computation cost of the optimization problem. These techniques are based on the fundamental features of the stress field, the differentiated kernels and the system of matrices of the optimization problem. Investigations have shown that employing these techniques yields more than a 50% reduction in computer time with insignificant loss of accuracy.     

飞行器上升段轨迹的优化设计

针对飞行器上升段轨迹优化求解困难的问题,提出一种基于正交配点的优化求解方法。该方法以第二类切比雪夫正交多项式的零点作为系统控制变量和状态变量的离散点,利用拉格朗日插值多项式对状态和控制变量进行拟合。通过对多项式的求导将动力学微分方程约束转化为代数约束,从而把无限维的最优控制问题转化为一个有限维的非线性规划(Nonlinear Programming,NLP)问题。随后,利用序列二次规划(Sequential Quadratic Program-ming,SQP)方法求解转化后的NLP问题,获得最优的飞行轨迹。最后,飞行器上的仿真结果验证了所提方法的有效性。研究成果可为飞行器的制导控制提供可行的飞行轨迹,有一定的工程应用价值。     

Infinite horizon optimal control for nonlinear interconnected large‐scale dynamical systems with an application to optimal attitude control

This paper presents a novel extended modal series method for solving the infinite horizon optimal control problem of nonlinear interconnected large‐scale dynamic systems. In this method, the infinite horizon nonlinear large‐scale two‐point boundary value problem (TPBVP), derived from Pontryagin's maximum principle, is transformed into a sequence of linear time‐invariant TPBVPs. Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of a uniformly convergent series. Moreover, in special cases, the proposed procedure facilitates the application of parallel processing, which improves its computational efficiency. In this study, an iterative algorithm is also presented, which has a low computational complexity and a fast convergence rate. Just a few iterations are required to obtain a suboptimal trajectory‐control pair. Finally, effectiveness of the proposed approach is verified by solving the optimal attitude control problem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society