Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. 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 problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

On two formulations of an optimal insulation problem

Two formulations for the design of the optimal insulation of a domain are investigated by computational means. The results illustrate the similarities and differences that result from the two approaches. One method is in the format of a topology design problem of distributing insulating material in a domain surrounding a non-design domain that is heated by a given heat source; this problem is treated in both a relaxed format and a penalized material format. The other approach deals with the optimal distribution of a thin layer of insulation on the boundary of the non-design domain; this problem is more in the realm of shape design, or rather, it is similar to optimal design of support conditions for structures. In both cases, mathematical programming is used, but for the shape design case, it is applied to the non-linear analysis problems that arise when the optimal design is explicitly solved for.     

A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory

The theory of optimal plastic design of structures via optimality criteria (W. Prager approach) transforms the optimal design problem into a certain nonlinear elastic structural analysis problem with appropriate stress-strain laws, which are derived by the adopted specific cost function for the members of the structure and which generally have complete vertical branches. Moreover, the concept of structural universe (introduced by G.I.N. Rozvany) permits us to tackle complicated optimal layout problems.On the other hand, a significant effort in the field of nonsmooth mechanics has recently been devoted to the solution of structural analysis problems with complete material and boundary laws, e.g. stress-strain laws or reaction-displacement laws with vertical branches.In this paper, the problem of optimal plastic design and layout of structures following the approach of Prager-Rozvany is revised within the framework of recent progress in the area of nonsmooth structural analysis and it is treated by means of techniques primarily developed for the solution of inequality mechanics problems. The problem of the optimal layout of trusses is used here as a model problem. The introduction of general convex, continuous and piecewise linear specific cost functions for the structural members leads to the formulation of linear variational inequalities or equivalent piecewise linear, convex but nonsmooth optimization problems. An algorithm exploiting the particular structure of the minimization problem is then described for the numerical solution. Thus, practical structural optimization problems of large size can be treated. Finally, numerical examples illustrate the applicability and the advantages of the method.On leave from the Institute of Applied Mechanics, Department of Engineering Sciences, Technical University of Crete, GR-73100 Chania, Greece     

Application of an energy-based model for the optimal design of structural materials and topology

This paper describes an implementation of recent developments in modelling for the design of continuum structures into a general program for computational solution of such problems. In the basic model, the unrestricted material tensor appears as the design variable. The algorithm for this program is presented and the method of solution is described. The approach is applicable to predict both the optimal unrestricted material design and as well for design with a specified material. In either case, the distributions (fields) of all designable components of the material tensor are predicted. Results are given for 2D and 3D examples, in the form of continuously varying material properties and for various values of volume fraction. The associated zero-one or topology designs are obtained by application of an additional procedure to these results. Comparison against results from earlier approaches indicates that optimization of the material may lead to considerable improvement in structural performance.     

Nonlinear analysis and optimal design of structures via force method and genetic algorithm

In this paper analysis, design and optimization of structures are performed considering material and geometric nonlinearity. For this purpose, the force method, energy concepts and genetic algorithm are employed. The first part of this paper contains the formulation of the problem based on the force method and energy principles using linear analysis. In this method, the material nonlinearity is also included. The formulations are examined by simple illustrative examples. Reduction of the complementary energy is efficiently incorporated in this approach. The second part of the article combines the process of the analysis and design to achieve specified stress ratios for the members of the structure. This problem is especially important in the seismic deign of structures. Geometric nonlinearity is then formulated, in the third part by employing two approaches. Considering the energy term next to the weight of the structure, optimal dimensions of the structures are selected. In each part, the efficiency of the methods is illustrated by means simple examples.     

Design of multi-component structural systems for optimal layout topology and joint locations

Present day topology optimization techniques for continuum structures consider the design of single structural components, while most real life engineering design problems involve multiple components or structures. It is therefore necessary to have a methodology that can address the design of multi-component systems and generate designs for the optimal layouts of individual structures and locations for interconnections. The interconnections include supports provided by the ground, joints and rigid connections like rivets, bolts and welds between components. While topology optimization of structures has been extensively researched, relatively little work has been done on optimizing the locations of the interconnections. In this research, a method to model and define domains for the interconnections has been developed. The optimization process redistributes material in the component design domains and locates the connections optimally based on an energy criterion. Some practical design examples are used to illustrate the capability of this method.     

Optimization of nonlinear trusses using a displacement-based approach

A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust. Received December 4, 2000     

A crossing sensitivity filter for structural topology optimization with chamfering,rounding, and checkerboard-free patterns

Chamfering and rounding are the most common features of structures and mechanisms. Introducing these features is another way to make material usage more efficient by lowering or avoiding stress con centrations and making the structure even stronger. In this brief note, a new crossing sensitivity filter is proposed for structural topology optimization with chamfering and rounding. This method also ensures the final optimal solution without checkerboard patterns or mesh-dependency. It can also partially prevent one-node hinges. Numerical examples that include both design of stiffest/strongest structures and synthesis of compliant mechanisms are provided.     

The generation of hierarchic structures via robust 3D topology optimisation

Commonly used building structures often show a hierarchic layout of structural elements. It can be questioned whether such a layout originates from practical considerations, e.g. related to its construction, or that it is (relatively) optimal from a structural point of view. This paper investigates this question by using topology optimisation in an attempt to generate hierarchical structures. As an arbitrarily standard design case, the principle of a traditional timber floor that spans in one direction is used. The optimisation problem is first solved using classical sensitivity and density filtering. This leads indeed to solutions with a hierarchic layout, but they are practically unusable as the floor boarding is absent. A Heaviside projection is therefore considered next, but this does not solve the problem. Finally, a robust approach is followed, and this does result in a design similar to floor boarding supported by timber joists. The robust approach is then followed to study a floor with an opening, two floors that span in two directions, and an eight-level concrete building. It can be concluded that a hierarchic layout of structural elements likely originates from being optimal from a structural point of view. Also clear is that this conclusion cannot be obtained by means of standard topology optimisation based on sensitivity or density filtering (as often found in commercial finite element codes); robust 3D optimisation is required to obtain a usable, constructible (or in the future: 3D printable) structural design, with a crisp black-and-white density distribution.     

Generalized topology design of structures with a buckling load criterion

Material based models for topology optimization of linear elastic solids with a low volume constraint generate very slender structures composed mainly of bars and beam elements. For this type of structure the value of the buckling critical load becomes one of the most important design criteria and so its control is very important for meaningful practical designs. This paper tries to address this problem, presenting an approach to introduce the possibility of critical load control into the topology optimization model.Using the material based formulation for topology design of structures, the problem of optimal structural reinforcement for a critical load criterion is formulated. The stability problem is conveniently reduced to a linearized eigenvalue problem assuming only material effective properties and macroscopic instability modes. The respective optimality criteria are presented by introducing the Lagrangian associated with the optimization problem. Based on this Lagrangian a first-order method is used as a basis for the numerical update scheme. Two numerical examples to validate the developments are presented and analysed.     

Constrained multi-objective optimization algorithms: Review and comparison with application in reinforced concrete structures

Engineering design problems are often multi-objective in nature, which means trade-offs are required between conflicting objectives. In this study, we examine the multi-objective algorithms for the optimal design of reinforced concrete structures. We begin with a review of multi-objective optimization approaches in general and then present a more focused review on multi-objective optimization of reinforced concrete structures. We note that the existing literature uses metaheuristic algorithms as the most common approaches to solve the multi-objective optimization problems. Other efficient approaches, such as derivative-free optimization and gradient-based methods, are often ignored in structural engineering discipline. This paper presents a multi-objective model for the optimal design of reinforced concrete beams where the optimal solution is interested in trade-off between cost and deflection. We then examine the efficiency of six established multi-objective optimization algorithms, including one method based on purely random point selection, on the design problem. Ranking and consistency of the result reveals a derivative-free optimization algorithm as the most efficient one.     

Optimization of passive vibration isolators mechanical characteristics

The contribution of optimization has been essential to the more recent developments in design of new mechanical structures and materials. The objective of this work is to apply the models of material and structural optimization to the design of passive vibration isolators. A computational tool to identify the optimal viscoelastic characteristics of a nonlinear one-dimensional isolator was developed. The cost functional involves the minimization of a weighted average of the maximum transient and steady state response amplitudes for a set of predefined dynamic loads. The optimal isolator behaviour is obtained by a simulated annealing method. The solutions obtained are analyzed and discussed concerning their dependence on the applied forces and objective function selection. The results obtained can facilitate the design of elastomeric materials with improved behaviour in terms of dynamic stiffness for passive vibration control.     

Topology optimization of couple-stress material structures

Conventional topology optimization is concerned with the structures modeled by classical theory of mechanics. Since it does not consider the effects of the microstructures of materials, the classical theory can not reveal the size effect due to material’s heterogeneity. Couple-stress theory, which takes account of the microscopic properties of the material, is capable of describing the size effect in deformations. The purpose of this paper is to investigate the formulation for topology optimization of couple-stress material structures. The artificial material density of each element is chosen as design variable. Based on the basic idea of SIMP (Solid Isotropic Material with Penalization) method, the effective material stiffness matrix of couple-stress material is related to the artificial density by power law with penalty. The structural analysis is implemented by finite element method for couple-stress materials, and a 4-noded quadrilateral couple-stress element is formulated in which C ₁ continuity requirement is relaxed. Some typical problems are solved and the optimal results based on the couple-stress theory are compared with the conventional ones. It is found that the optimal topologies of couple-stress continuum show remarkable size effect.     

Robust topology optimization accounting for misplacement of material

The use of topology optimization for structural design often leads to slender structures. Slender structures are sensitive to geometric imperfections such as the misplacement or misalignment of material. The present paper therefore proposes a robust approach to topology optimization taking into account this type of geometric imperfections. A density filter based approach is followed, and translations of material are obtained by adding a small perturbation to the center of the filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics during the optimization process. The proposed method is successfully applied to three example problems: the minimum compliance design of a slender column-like structure and a cantilever beam and a compliant mechanism design. An extensive Monte Carlo simulation is used to show that the obtained topologies are more robust with respect to geometric imperfections.     

Sculpturic design of structures using finite elements

A new philosphy of optimal structural design called ‘sculpturic design’ is introduced in this paper. A companion methodology is presented through which an optimal design is achieved using a systematic procedure. To illustrate this concept and methodology several examples are treated, and by using the finite-element method, certain structures which may be termed ‘sculpturic structures’ are obtained.     

Simultaneous optimization of the material properties and the topology of functionally graded structures

A level set based method is proposed for the simultaneous optimization of the material properties and the topology of functionally graded structures. The objective of the present study is to determine the optimal material properties (via the material volume fractions) and the structural topology to maximize the performance of the structure in a given application. In the proposed method, the volume fraction and the structural boundary are considered as the design variables, with the former being discretized as a scalar field and the latter being implicitly represented by the level set method. To perform simultaneous optimization, the two design variables are integrated into a common objective functional. Sensitivity analysis is conducted to obtain the descent directions. The optimization process is then expressed as the solution to a coupled Hamilton-Jacobi equation and diffusion partial differential equation. Numerical results are provided for the problem of mean compliance optimization in two dimensions.     

Addendum to: An energy model for the optimal design of linear continuum structures

The original paper of the above title presents an analytical model for problems in the optimal design of linearly elastic continuum structures, where the material modulus tensor has the role of design variable. Both internal (strain) energy and the expression of generalized cost are represented conveniently there, in a form where the modulus tensor is transformed into vector coordinates. The general design of linear continuum structures is stated as a max-min problem. Optimality conditions for the transformed design problem have particularly simple form.¶Both local properties, represented by the relative values of components of the modulus tensor, and the global distribution of structural resource (material) are variable in the design.With some modification to the original formulation, these separate aspects of design can be represented explicitly in the model. This modified form, which directly facilitates study of the role of local properties in the prediction of optimal design, and which ultimately serves as the basis for schemes to perform computational solution, is described and substantiated here.     

Optimal topology design of structures under dynamic loads

When elastic structures are subjected to dynamic loads, a propagation problem is considered to predict structural transient response. To achieve better dynamic performance, it is important to establish an optimum structural design method. Previous work focused on minimizing the structural weight subject to dynamic constraints on displacement, stress, frequency, and member size. Even though these methods made it possible to obtain the optimal size and shape of a structure, it is necessary to obtain an optimal topology for a truly optimal design. In this paper, the homogenization design method is utilized to generate the optimal topology for structures and an explicit direct integration scheme is employed to solve the linear transient problems. The optimization problem is formulated to find the best configuration of structures that minimizes the dynamic compliance within a specified time interval. Examples demonstrate that the homogenization design method can be extended to the optimal topology design method of structures under impact loads.Presented at WCSMO-2, held in Zakopane, Poland, 1997     

Topology optimization for minimum weight with compliance and stress constraints

The paper deals with a formulation for the topology optimization of elastic structures that aims at minimizing the structural weight subject to compliance and local stress constraints. The global constraint provides the expected stiffness to the optimal design while a selected set of local enforcements require feasibility with respect to the assigned strength of material. The Drucker?CPrager failure criterion is implemented to handle materials with either equal or unequal behavior in tension and compression. A suitable relaxation of the equivalent stress measure is implemented to overcome the difficulties related to the singularity problem. Numerical examples are presented to discuss the features of the achieved optimal designs along with performances of the adopted procedure. Comparisons with pure compliance?Cbased or pure stress?Cbased strategies are also provided to point out differences arising in the optimal design with respect to conventional approaches, depending on the assumed material behavior.     

On topology optimization of damping layer in shell structures under harmonic excitations

This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.