Genetic algorithm for discrete‐sizing optimal design of trusses using the force method

In the process of discrete‐sizing optimal design of truss structures by Genetic Algorithm (GA), analysis should be performed several times. In this article, the force method is employed for the analysis. The advantage of using this method lies in the fact that the matrices corresponding to particular and complementary solutions are formed independently of the mechanical properties of members. These matrices are used several times in the process of the sequential analyses, increasing the speed of optimization. The second feature of the present method is the automatic nature of the prediction of the useful range of sections for a member from a list of profiles with a large number of cross‐sections. The third feature consists of a contraction process developed to increase the efficiency of the GA by which an optimal design for the first sub‐string associated with member cross‐sections is obtained. Improved designs are achieved in subsequent cycles by reducing the length of sub‐strings. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient algorithm for the optimum design of trusses with discrete variables

A method to efficiently solve the problem of minimum weight design of plane and space trusses with discrete or mixed variables is developed. The method can also be applied to continuous variables. The original formulation leads to a non-linear constrained minimization problem with inequality constraints, which is solved by means of a sequence of approximate problems using dual techniques. In the dual space, the objective function is to be maximized, depends on continuous variables, is concave and has first and second order discontinuities. In addition, the constraints deal simply with restricting the dual variables to be non-negative. To solve the problem an ad hoc algorithm from mathematical programming has been adapted. Some examples have been developed to show the effectiveness of the method.     

A basis change strategy for the reduced gradient method and the optimum design of large structures

This paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints. It ensures a non-singular basis matrix at every iteration. The same strategy can reliably be used within the generalized reduced gradient method for optimization under non-linear constraints. This method is applied to the minimum weight design of large structures under displacement and stress constraints, exploiting the sparsity of the constraint Jacobian matrix.     

Minimum-weight design of trusses by an optimality criteria method

This work presents an optimality criteria method which accurately and efficiently solves the structural optimization problem of trusses subject to deflection and stress constraints. The method is based upon the Newton–Raphson method and a first-order approximation both for deflection and stress constraints using the method of virtual work. It solves the problem rigorously and no serious difficulties have yet been encountered in spite of the drawbacks inherent in the Newton–Raphson method. Worked examples show its superiority to other methods and its ability to cope with trusses with many members and subject to multiple constraints.     

A steepest gradient method for optimum structural design

This paper describes a search procedure for finding the unconstrained maximum or minimum of a function of many independent variables. If this function represents the volume, cost, stiffness, etc. of a structure and each of the independent variables is associated with a parameter governing the geometry of the structure then designs of optimum geometry may be made in certain cases with the assistance of this search procedure. The procedure consists of steepest gradient calculations used in conjunction with a reverse Fibonacci location process. A ridge-following technique is included to speed convergence in addition to localized exploration in the region of an optimum. The search procedure has been programmed for computer use and an outline of its structural design applications is presented together with an example of its efficiency in a specific case.     

Topology optimization of trusses using genetic algorithm,force method and graph theory

In this article size/topology optimization of trusses is performed using a genetic algorithm (GA), the force method and some concepts of graph theory. One of the main difficulties with optimization with a GA is that the parameters involved are not completely known and the number of operations needed is often quite high. Application of some concepts of the force method, together with theory of graphs, make the generation of a suitable initial population well‐matched with critical paths for the transformation of internal forces feasible. In the process of optimization generated topologically unstable trusses are identified without any matrix manipulation and highly penalized. Identifying a suitable range for the cross‐section of each member for the ground structure in the list of profiles, the length of the substrings representing the cross‐sectional design variables are reduced. Using a contraction algorithm, the length of the strings is further reduced and a GA is performed in a smaller domain of design space. The above process is accompanied by efficient methods for selection, and by using a suitable penalty function in order to reduce the number of numerical operations and to increase the speed of the optimization toward a global optimum. The efficiency of the present method is illustrated using some examples, and compared to those of previous studies. Copyright © 2003 John Wiley & Sons, Ltd.     

Computational methods for optimum design of large complex systems

Computational procedures for optimal design of large complex systems are described. Requirements of a good algorithm are discussed. A general design optimization model applicable to several classes of problems is defined. Several optimization algorithms are outlined and differences between them are highlighted. Modern algorithms generate and use approximate Hessian of the Lagrange function to calculate the search direction. They are quite reliable and become extremely efficient when a potential constraint strategy is incorporated into them. Based on recent experience with them, they are recommended for general engineering design applications. Several other computational aspects are also discussed, such as robust implementation of algorithms, use of knowledge base in providing consulting and diagnostic support to the designer, interactive use of optimization, and role of a database and database management system in design optimization.     

A new implementation of the Lanczos method in linear problems

The Lanczos algorithm has proved to be a powerful solution method not only for finding the eigenvalues but for solving linear systems of equations. In this work a new implementation of the algorithm is presented for solving linear systems of equations with a sequence of right-hand sides. The versions of the method proposed in the past treat the right-hand side vectors successively by keeping the tridiagonal matrix and the orthonormal basis in fast or secondary storage. The new technique handles all approximations to the solution vectors simultaneously without the necessity for keeping the tridiagonal matrix or the orthonormal basis in fast or secondary storage. Thus, when the first solution vector has converged to a required accuracy good approximations to the remaining solution vectors have simultaneously been obtained. It then takes fewer iterations to reach the final accuracy by working separately on each of the remaining vectors.     

A fully automatic force method for the optimal shakedown design of frames

The paper presents a fully automatic way to handle the problem of the optimal shakedown design of planar frames. The evaluation of the elastic moments is essential for this design problem and due to the fact that they are design dependent, a classical iterative procedure is followed which updates these moments at the beginning of each iteration. A linear programming problem is then solved inside each iteration. The formulation adopted here is based on the force method which has computational advantages against the displacement method for this type of problems. Within the framework of the force method, the statical basis is provided by an easy to implement algorithm which selects a near minimal mesh basis for any planar graph. This basis is efficiently used, in a novel way, to find the flexibility matrix of the frame in a skyline form amenable to standard algorithms for its decomposition. The quickest way to the ground of each load is used to form the right hand side of the elastic compatibility equations for each load pattern. These equations are efficiently solved by standard back-substitution algorithms and the elastic bending moments to be introduced at the beginning of each iteration is established. Examples of application are also included.     

Optimality conditions for shakedown design of trusses

This paper deals with optimal shakedown design of truss structures constituted by elastic perfectly plastic material. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problem are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the maximum shakedown limit load design whose volume is assigned. The Kuhn-Tucker equations of the four problems here above mentioned are found by utilizing a variational approach; these equations prove the equivalence of the two types of design problem and provide useful information on the structure behaviour in optimality conditions. A suitable computational procedure of iterative type devoted to the reaching of the minimum volume design is presented. It is shown that the design obtained by this technique is the optimal one, since it satisfies the optimality conditions of the relevant search problem. In the typical step of this technique the dependency of the elastic response on the design variables is approximately taken into account. In the application stage a numerical example, aimed at utilizing this special technique, is presented.     

A theory of continua with projective microstructure as a model for large trusses

Summary A large truss is approximated by a continuum admitting projective transformations as microscopic deformations. A suitable set of parameters describing the microscopic deformation is extracted, and its spatial compatibility is investigated. Equations of equilibrium are derived by means of a variational principle. They define a special class of materials of grade 2. A complete solution is expressed in terms of potential functions in the three-dimensional isotropic field, which corresponds to a truss of random connection. Stresses around cylindrical and spherical cavities are analyzed to see the singular effects near the boundary surface.     

Reanalysis‐based optimal design of trusses

This paper presents a structural reanalysis method and its applications in optimal design of trusses. This reanalysis technique is derived primarily on the basis of a reduced basis formulation, and it has several advantages over previous reduced basis methods. In particular, the reduced system is uncoupled by using a Gram–Schmidt orthonormalization procedure and an error measure is introduced to adaptively monitor whether a good approximate solution is achieved. The latter aspect makes this reanalysis method suitable for use in optimal design problems because the changes in design variables usually vary during a design process. Discussions are presented on the implementation of this reanalysis method using both mathematical programming and optimality criteria‐based optimization schemes. Finally, several example problems of optimal truss design are used to validate the proposed reanalysis‐based design procedure. The presented numerical results indicate that the new reanalysis technique affects very slightly the accuracy of the optimal solutions and it does speed up the design process when the system analysed is large. Copyright © 2000 John Wiley & Sons, Ltd.     

A general methodological analysis for optimum design

Generalized applications of modern numerical analysis methods—while digital computers experienced a fast development—produced a first revolution in design techniques, allowing one to perform computations considered unapproachable until that time. Introduction of Computer Aided Design (CAD) techniques—while high-performance graphic peripherals experience a fast development—is actually producing a second revolution, by making easy and fast most routine design tasks. However, the introduction of Computer Aided OPTIMUM Design techniques has not yet produced the expected third revolution, in spite of the big amount of research and the interest of its potential applications. The authors think that this fact is due mainly to the dispersion of the optimum design research, and to the lack of a well established doctrine. In this paper we approach the design process from a general methodological perspective, suitable to be applied to a wide range of problems. The design process is organized in several related levels. This approach leads naturally to the concept of optimum design and to the statement of a general mathematical programming problem. The practical application of this methodology to any particular problem takes an efficient and modular form. First and second order sensitivity analysis techniques are introduced from the general formulation, and alternative techniques (adjoint state) of the direct differentiation method are discussed. DAO², a powerful and versatile computer aided optimum design system by the Finite Element Method, has been developed by the authors¹ according to this general methodology. The system can solve efficiently 2D and 3D structural fixed-geometry and shape optimization problems. The power and viability of this methodology is illustrated by the solution to a structural optimization problem. The shape of the central section of an arch dam is optimized. A linear elastic structural FEM analysis is simultaneously performed for plane stress and for radial symmetry—while constraints are imposed for several load cases—taking into account the construction and loading stages. It should be emphasized that the same optimum design is reached in a small number of iterations starting from two significantly different initial designs.     

桁架结构布局优化的并行子空间方法

由于结构布局优化存在设计变量类型众多和变量耦合等问题,采取合适的优化方法获得满足结构设计要求的最小质量的结构具有重要的工程意义.基于多学科设计优化方法中的并行子空间优化法,提出一种桁架结构布局优化的并行子空间优化方法.将结构布局设计问题按设计变量类型分为布局、形状和尺寸三个并行的子空间,设计变量在各自的子空间内单独优化,各子空间优化结束后,在系统级中协调3类设计变量,保持最小质量的子空间的优化设计变量不变,采用近似一维搜索的方法协调其他子空间的设计变量,然后进行下一次迭代直至收敛.2个算例表明该方法能够取得较好的优化结果,具有实际工程应用价值.     

Optimal design of plane trusses using material forces

The shape optimization of a plane truss is attained through a simultaneous equilibration of physical and material forces. For this reason, the potential energy of the structure is regarded as an explicit function of the joint displacements of the truss in current configuration (physical space) as well as the joint positions in reference configuration (design space). The equilibrium of physical forces leads to finding the optimal joint displacements in the “physical space” and the equilibrium of the material forces leads to finding the optimal initial joint positions of the truss in the “design space”. The two sets of equilibrium equations are derived and solved in a completely coupled manner.     

A new interval optimization method considering tolerance design

This study considers the design variable uncertainty in the actual manufacturing process for a product or structure and proposes a new interval optimization method based on tolerance design, which can provide not only an optimal design but also the allowable maximal manufacturing errors that the design can bear. The design variables' manufacturing errors are depicted using the interval method, and an interval optimization model for the structure is constructed. A dimensionless design tolerance index is defined to describe the overall uncertainty of all design variables, and by combining the nominal objective function, a deterministic two-objective optimization model is built. The possibility degree of interval is used to represent the reliability of the constraints under uncertainty, through which the model is transformed to a deterministic optimization problem. Three numerical examples are investigated to verify the effectiveness of the present method.     

A numerical study on the elements of shape optimum design

This paper discusses the main elements of shape optimization. The material derivative of a stress function using the continuum approach is derived by introducing an adjoint problem, which is then transformed into shape design sensitivity by replacing the velocity field with the change of the design variables. The difficulty related with the appearance of the concentrated adjoint loads is discussed, with two proposals for the modelling of the adjoint problem. A numerical example is used to demonstrate the accuracy of the proposed formulation for different adjoint loads.

Two shape optimization examples are used to investigate the numerical characteristics of the optimization process. Two kinds of design boundary modelling are employed, namely the linear and cubic spline boundary representation. The difference of the final design shapes under different design variables and mesh distributions are also studied.     

A pseudo-fully stressed design approach for optimum design of steel frames

An iterative approach based on the fully-stressed design concept employing scaling to find an efficient search path to the optimum design is developed for statically indeterminate elastic frames subjected to behavioural constraints on member stresses and nodal displacements, and side constraints on member sizes. In this approach the fully stressed solution, which recognizes only the stress constraints, is scaled to the boundary of the feasible design region by employing the other constraints on the structure. The inactive constraints in an iteration are identified by the Kuhn-Tucker conditions. The computed stresses of the inactive constraints, for the time being, are considered to be the allowable values in order to develop a pseudo-fully stressed solution with a new set of allowable stresses. Convergence takes places when the normalized change in the value of the objective function between the scaled and unsealed pseudo-fully stressed designs is less than a specified tolerance. The method employs the established relationships between sectional area, section modulus and moment of inertia of W-shapes to express the objective and constraint functions in terms of one design variable for each member. The efficiency and accuracy of the method in optimization of structural steel frames is demonstrated by sample problems designed for stress, displacement and minimum size constraints. The algorithm is verified against published results.     

New method of selecting the optimum geometrical parameters in the design of superconducting solenoids

A new method of selecting the optimum geometrical parameters in the design of superconducting solenoids has been proposed. The volume of superconducting winding can be much reduced and the homogeneity of the magnetic field is improved. A design example has shown that the volume of superconducting winding was reduced by 29% in comparison with the volume minimized coil with a rectangular winding cross-section.     

一种新型板束结构的板翅式换热器的数值模拟和优化设计

换热器是热力系少轴向导热的影响,设计出的换热器往往很长.为了使换热器的结构型式更为紧凑合理,提出了一种新型板束结构,将换热器沿长度方向折叠.对物理模型用有限元的方法进行了数值模拟,对新型板束结构换热器的传热性能和压力损失进行分析.计算发现,合理选择板束间隔热片的材料和厚度,可以使新型板束结构的板翅式换热器在保持优良传热性能的基础上,实现高效紧凑、布局合理的良好效果.