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A parameter-free free-form optimization method to solve multidisciplinary shape optimization problems of solid structures is presented in this paper. The design objective is both to minimize the mean compliance of linear elastic structures and to control the temperature at arbitrary locations and time periods for arbitrary loadings and under volume constraint, simultaneously. The multidisciplinary design problem is formulated as a distributed-parameter shape optimization problem based on the variational method, and the global objective function is defined using an original normalized weighted sum method. The shape gradient density functions with respect to the shape variation are derived based on the material derivative and the adjoint variable methods. The shape gradient density functions theoretically derived are applied to the H1 gradient method, a gradient method in the function space, to obtain the optimal shape while maintaining the smoothness of the shape. Numerical examples are considered to demonstrate the validity and the effectiveness of the developed optimization method.

Design of engineering structures may benefit from reduction in assembly complexity through use of periodic components, in which uniform sub-structures combine to form a relatively simple topology. The benefits of periodic structures include lower manufacture costs as well as ease of assembly. Recent developments in periodic topology optimization have shown its efficacy for addressing a range of design objectives. However, constraints such as assembly conditions and the connection configuration of periodic sub-components present limiting factors in the application of periodic optimization to real-world engineering problems. This study addresses the current knowledge gap in periodic optimization assembly through inclusion of common interfacing connections between periodic components, such as screws, welds, or rivets, thus accounting for real assembly conditions. A bi-directional evolutionary structural optimization (BESO) method and solid isotropic material with penalization (SIMP) method are presented, for stiffness and frequency criteria, which simultaneously optimizes the topology of the periodic components and the joint configuration connecting components. Elemental sensitivities are derived and utilized to drive the design of both the periodic component and the connection layouts. Iterative updating of the topological design, guided by elemental sensitivities, allows for optimization of the periodic topology for given objective functions. To demonstrate the effectiveness of the proposed method, optimized structures are explored through different periodicities. Application of the methodology presented in this study will assist in providing new design capabilities to reduce the costs of manufacturing, transport, and assembly through optimized periodic components.

In this study, we propose a parameter-free free-form optimization method to control the time-dependent response of shell structures. The design objective is to minimize the dynamic compliance or to control the dynamic responses at arbitrary domains and times for an arbitrary time-dependent loading under volume constraint. The unsteady optimum design problems are formulated as a distributed-parameter shape optimization problem based on the variational method, and the shape gradient functions with respect to the shape variation in the normal direction to the surface are derived based on the material derivative and the adjoint variable methods. The shape gradient functions theoretically derived are applied to the H1 gradient method for shells, a gradient method in the function space. With the proposed approach, the optimum free form of the shell structures for the time-dependent response problems can be obtained while minimizing the objective functional and maintaining the smoothness of the form. Several shape design examples under a continuous dynamic force or an impulse force are demonstrated to show the effectiveness of the proposed method, and the results are discussed.

The stiffened plates are of demonstrable advantages and potential in offering high resistance to such extreme loading scenarios as blast. Since the distribution of the stiffeners has considerable effect on their performance, its design signifies an important topic of research. However, existing research has mainly focused on empirical design, and the configurations were largely experience based, which limits structural explosion-proof capacity. In order to improve the performance of stiffened plates against blast loading, we introduced here two new structural configurations of stiffened plates. In this study, the modified ant colony optimization (MACO) algorithm which introduces the mass constraint factor to the pheromone update function and integrates the idea of crossover and mutation was used to design the subjected to given working conditions. Specifically, material distribution of stiffeners is taken to be the design variables, and minimization of the maximum deflection of the center point of the plate to be the design objective under predetermined mass constraints. Compared with the baseline structure, the optimal designs largely improved the explosion-proof performance through distributing stiffener topology on the plates. The results showed that the optimum designs all present the reinforcement stiffeners to link with the fixed boundaries against the deformation. Moreover, the optimum designs placed more reinforcement materials in the central regions instead of four angles, and with the increase of the mass fraction, the reinforcement placement gradually extends from the center to the edges. The proposed method and new topological configurations are expected to provide some insights into design for novel protective structures.

In this work, we propose to extend the bi-directional evolutionary structural optimization (BESO) method for compliance minimization design subject to both constraints on volume fraction and maximum von Mises stress. To this end, the aggregated p-norm global stress measure is first adopted to approximate the maximum stress. The conventional compliance design objective is augmented with p-norm stress measures by introducing one or multiple Lagrange multipliers. The Lagrange multipliers are employed to yield compromised designs of the compliance and the p-norm stress. An empirical scheme is developed for the determination of the Lagrange multipliers such that the maximum von Mises stress could be effectively constrained through the controlling of the aggregated p-norm stress. To further enforce the satisfaction of stress constraints, the stress norm parameter p is assigned to a higher value after attaining the objective volume. The update of the binary design variables lies in the computationally efficient sensitivity numbers derived using the adjoint method. A series of comparison studies has been conducted to validate the effectiveness of the method on several benchmark design problems.

A topology optimization approach is proposed for the optimal design of bi-material distribution on underwater shell structures. The coupled finite element method (FEM) / boundary element method (BEM) scheme is used for the system response analysis, where the strong interaction between the structural and the acoustic domain is considered. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The design variables are the artificial densities of design material elements in a bi-material model constructed by the solid isotropic material with penalization (SIMP) method, and the minimization of sound power level (SWL) is chosen to be the design objective. In this study, the adjoint operator method is employed to calculate the sensitivity of the objective function with respect to the design variables. Based on the sensitivity information, the gradient-based optimization solver is finally applied for updating the design variables during the optimization process. Numerical tests are provided to illustrate the correctness of the sensitivity analysis approach and the validity of the proposed optimization procedure. Results show that the heavy fluid feedback has a big impact on the final design, and thus it is necessary to conduct a strong coupling scheme between the fluid and structures. In addition, the optimal design is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.

Naturally evolved biological structures exhibit the optimal characteristics of light weight, high stiffness, and high strength. Based on the growth mechanism of biological branch systems in nature, an optimization method for internal stiffener plate distribution in box structures is suggested. Under the given load and support conditions, the internal stiffener plates of machine pedestal structures grow, bifurcate, and degenerate towards the direction of maximum overall structural stiffness in accordance with the adaptive growth law. The optimal and distinct distribution of internal stiffener plates with the most effective load path is thus obtained. Based on this, a size optimization for lightweight design is conducted, in which the self-weight of the structures is taken as the design objective, and the natural vibration frequency and static stiffness in the direction that is sensitive to machining accuracy are set as constraints. Finally, an optimized structure is obtained. The effectiveness of the proposed method is verified by using a precision grinder bed as an example. The results of numerical simulation and 3D–printed model experiment indicate that both the dynamic and the static performance of the optimized structure are improved, while the structural weight is reduced by compared with the initial structure. The suggested design method provides a new solution approach for the design optimization of machine pedestal structures.

© 2017, Springer-Verlag GmbH Germany. Multi-cell thin-walled structures exhibit significant advantages in maximizing energy absorption and minimizing mass during vehicle crashes. Since the topological distribution of wall members has an appreciable effect on the crashworthiness, their design signifies an important area of research. As a major energy absorber, multi-cell tubes are more commonly encounter oblique loading in real life. Thus, this study aimed to optimize multi-cell cross-sectional configuration of tubal structures for multiple oblique loading cases. An integer coded genetic algorithm (ICGA) is introduced here to optimize topological distribution of multi-celled web members for single/multiple oblique impacting conditions. Specifically, material distribution in a form of allocating web wall thickness, starting from zero, is considered as design variables and maximization of energy absorption (EA) as the design objective under the predefined peak crushing force and structural mass constraints. The optimization allows generating uniform or non-uniform thickness distribution in different web wall configurations to maximize usage efficiency of material. Compared with the baseline structure, the optimized configurations largely improved the energy absorption in both single and multiple load cases. The examples demonstrate that the proposed ICGA-based design method not only provides a useful approach to searching for novel crashworthy structures in a systematic fashion, but also develops a series of novel multi-cell topologies for multiple oblique loading cases.

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.

Topology optimization for coupled acoustic-structural systems subjected to stationary random excitations is investigated. Bi-material elastic continuum structures without damping are considered. The finite element method is utilized to deal with coupled acoustic-structural problems. An accurate and highly efficient algorithm series for stationary random analysis techniques, pseudo excitation method, is extended to calculate the acoustic random response generated by the vibrating structures. Minimization of the auto power spectral density of sound pressure is taken as design objective and its sensitivities with respect to topological variables are derived by adjoint method. A penalty term is proposed to suppress the intermediate volumetric densities. Numerical examples are given to demonstrate the validity of the presented methods.

With the rapid development of the vehicle industry, crashworthiness has become a crucial aspect in vehicle body design. In fact, crashworthiness is a multivariable optimization design problem for a vehicle body, regardless of structure or material. However, when crashworthiness involves a large number of design variables, including both material and structure variables, it is more difficult to deal with. In this paper, an integrated design technique for materials and structures of vehicle body under crash safety consideration is suggested. First, a finite element model of the vehicle body is established according to relevant vehicle safety standards. Then, the material parameters of the vehicle body are set as analytical factors for factor screening. Next, significant factors are obtained using a three-level saturated design integrated with multi-index comprehensive balance analysis and the MaxUr(3) method, with an improved evaluation method. These screened material parameters along with the corresponding continuous variables of the structure, are considered as the design variables of the integrated design of the vehicle body. Both the weight and the crashworthiness properties are set as the design objectives. Optimal Latin hypercube sampling and radius basis functions are utilized to construct highly accurate surrogate models. Furthermore, the non-dominated sorting genetic algorithm II is implemented to seek the optimal solutions. Finally, two cases considering the roof module and the frontal module of a vehicle body are analyzed to verify the proposed method.

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.

This paper presents a new approach for the topological design of materials with extreme properties. The method is based on hybrid cellular automaton (HCA), which is an implicit optimization technique that uses local rules to update design variables iteratively until meeting the described optimality conditions. By means of an energy-based homogenization approach, the effective properties of the considered material are calculated in terms of element mutual energies. By this method, no sensitivity information is required to find the optimal topology for the considered design objectives: bulk modulus, shear modulus, and negative Poisson’s ratio. The proposed method is validated by a series of numerical examples.

A reliability-based vibro-acoustic microstructural topology optimization (RBVAMTO) model taking into consideration the uncertainty of several design-independent parameters, such as the direction of the load, the excitation frequency, or their combinations is presented. The design objective is to minimize the sound power of the vibrating composite macrostructure that is assumed to be constructed by periodic micro unit cell filled up by two kinds of prescribed isotropic materials. The SIMP based bi-material interpolation model is employed at the micro-scale and the design variable is the relative material volume density of the micro unit cell. A design process consisting of the uncertainty analysis and vibro-acoustic microstructural topology optimization is proposed and implemented. The influences of different combination values of the normalized variables corresponding to the uncertainty parameters on the design results are investigated. Numerical examples show that in the vibro-acoustic microstructural design the uncertainty of the excitation frequency may play a very important role. It is also shown that when the normalized variable corresponding to the random excitation frequency takes the higher value, the optimum microstructural topology may not be so sensitive to perturbation of the loading direction. Monte Carlo simulation results demonstrate that the RBVAMTO designs normally lead to the optimum results that are more robust for perturbations of both the excitation frequency and the loading direction than the deterministic design.

This paper deals with the problem of parameter optimization of the motion control mechanism of the hatch door of ARJ21-700, a regional airliner of China. Motion improvement of the hatch door is implemented by two kinds of passive designs. Firstly, a single-layer optimization model for trajectory modification is developed to find the optimum size of the key parts of the control mechanism. Secondly, a novel nested bi-level optimization model is presented for the design of the size tolerance limits of the selected parts. The design objective is minimization of the total extremum deviation of the motion trajectory of the objective point of the hatch door, where the extremum deviation is obtained by solution of the inner-level size optimization problem for the fixed size tolerance limits. The optimization models for motion control of the hatch door mechanism are solved using the response surface method. Numerical examples show that the precision of the real running trajectory of the objective point of the hatch door mechanism may be improved effectively by using the methods presented. A home-made multi-body dynamics solver (THUSOLVER) and the corresponding optimization software have been developed to implement the above tasks.

Faced with the constantly increasing requirements for machine efficiency and reliability, designers have to pay attention to every detail, including machine foundation. Possible outage costs due to foundation damage can easily exceed the costs of design and construction of the foundation. This paper presents a novel and systematic procedure for finding an optimal design of reinforced concrete columns of the spring mounted machine foundation subjected to dynamic loads. The proposed approach assumes considerably decoupled natural modes of the critical structural members which can be optimized as simplified analytical models. It includes the following steps: FE modal analysis of the complete structure, identification of expected dynamic excitation, determination of the critical structural members, optimization of the corresponding analytical models and verification by FE modal analysis of the complete optimized structure. Turbine-generator-foundation structure was employed as an illustrative example. Operating speeds of the turbine and the generator and their higher order harmonics are taken as a critical dynamic excitation. Therefore, avoiding possible resonance region became the design objective, while the design variables were dimensions of the foundation columns cross-section. Lower and upper bounds of the design variables are determined on the basis of structural stability and space limitations regarding condenser and other auxiliary components of the turbine-generator system. The results of the analyzed example confirm that the optimization was successfully performed, since the optimized dimensions of the columns yielded natural frequencies outside the critical frequency range.

Design optimization is presented for the crashworthiness improvement of an automotive body structure. The optimization objective was to improve automotive crashworthiness conditions according to the defined criterion (occupant chest deceleration) during a full frontal impact. The controllable factors used in this study consisted of six internal parts of the vehicle's frontal structure in a condition that their thickness was the "design parameter". First using the Taguchi method, this study analyzed the optimum conditions in discontinuous design area and impact factors and their optimal levels of design objectives were obtained by analyzing the experimental results. Next to model a precise understanding of the explicit mathematical input---output relationship, fuzzy logic is utilized which make use of full factorial design set of experimental test cases resulted from Taguchi predicting formulations. Interestingly, the optimum conditions for automotive crashworthiness occurred with 2.72 % improvement in the defined crashworthiness criterion in comparison with the baseline design while selected structural parts experienced mass reduction by 8.23 %.

The Energy based topology optimization method has been used in the design of compliant mechanisms for many years. Although many successful examples from the energy based topology optimization method have been presented, optimized configurations of these designs are often very similar to their rigid linkage counterparts; except using compliant joints in place of rigid links. These complaint joints will endure large strain under the applied forces in order to perform the specified motions which are very undesirable in a compliant mechanism design. In this paper, a strain based topology optimization method is proposed to avoid a localized high strain of the compliant mechanism design, which is one of the drawbacks using strain energy formulation. Therefore, instead of minimizing the strain energy for structural rigidity, a global effective strain function is minimized. This is done in order to distribute the strain within the entire mechanism while maximizing the structural rigidity. Furthermore, the physical programming method is adopted to accommodate both flexibility and rigidity design objectives. Design examples from both the strain energy based topology optimization and the strain based method are presented and discussed.

The optimization of thermal-composite structural designs for heat flux shielding, focusing, and reversal is explained in this paper. Heat conduction in anisotropic solids is reviewed, and a detailed two-phase material microstructure description for non-symmetric inclusions embedded in a matrix medium is provided. Objective functions related to heat flux shielding and focusing are derived from a general energy formulation of thermal compliance by using information related to the magnitude of the thermal gradients. Additionally, a multi-term objective function for heat flux reversal is developed based on the directionality of the thermal gradients. A computational optimization technique is then employed to examine the validity of these three design objectives in the context of an established benchmark example. The results of these numerical experiments are shown to be in good agreement with experimental results from the literature. Additionally, the optimization method is shown to be capable of handling structures with arbitrary geometry. Logical extensions of the method include the development of novel structural layouts for thermal circuits, thermal cloaks, and negative differential thermal resistance materials.

Formulation space exploration is a new strategy for multiobjective optimization that facilitates both divergent exploration and convergent optimization during the early stages of design. The formulation space is the union of all variable and design objective spaces identified by the designer as being valid and pragmatic problem formulations. By extending a computational search into the formulation space, the solution to an optimization problem is no longer predefined by any single problem formulation, as it is with traditional optimization methods. Instead, a designer is free to change, modify, and update design objectives, variables, and constraints and explore design alternatives without requiring a concrete understanding of the design problem a priori. To facilitate this process, we introduce a new vector/matrix-based definition for multiobjective optimization problems, which is dynamic in nature and easily modified. Additionally, we provide a set of exploration metrics to help guide designers while exploring the formulation space. Finally, we provide an example to illustrate the use of this new, dynamic approach to multiobjective optimization.

A technique is proposed for determining the material distribution of a structure to obtain desired eigenmode shapes for problems of maximizing the fundamental eigenfrequency. The design objective is achieved using the solid isotropic method with penalization (SIMP) for topology optimization. Weighted constraints added in bound formulation are proposed to maximize the fundamental natural frequency, which provides an easy and straightforward way to prevent mode switching in the optimization process. Aside from maximizing the fundamental frequency, a method to modify existing eigenmodes to continuously evolve and assume the same shapes as the desired modes within the optimization process is proposed. The topology layout of a structure with desired eigenmodes is obtained by adding the modal assurance criterion (MAC) as additional constraints in the bound formulation optimization. Examples are presented to illustrate the proposed method, and a potential application of the proposed technique in decoupling a mechanical system is demonstrated.

When non-zero prescribed support displacements are involved in addition to design independent loads for a continuum/structure, then the objectives of minimum compliance (total elastic energy) and of maximum strength lead to different designs. This is verified by the presented sensitivities. Designs from neither of the two objectives are characterized by uniformly distributed energy density. However, simple iterations with the goal of obtaining uniform energy density show that the strength is favored by this approach. These observations leads to a rejection of the objectives of compliance minimization as well as that of direct strength maximization; we choose the objective of obtaining uniform energy density and show by examples that the obtained solutions are close to fulfilling also strength maximization, with the price of increased compliance. Optimal design examples are presented and discussed in detail for different combinations of non-zero prescribed support displacements and design independent loads.

A multi-objective topology optimization formulation for the design of dynamically tunable fluidic devices is presented. The flow is manipulated via external and internal mechanical actuation, leading to elastic deformations of flow channels. The design objectives characterize the performance in the undeformed and deformed configurations. The layout of fluid channels is determined by material topology optimization. In addition, the thickness distribution, the distribution of active material for internal actuation, and the support conditions are optimized. The coupled fluid-structure response is predicted by a non-linear finite element model and a hydrodynamic lattice Boltzmann method. Focusing on applications with low flow velocities and pressures, structural deformations due to fluid-forces are neglected. A mapping scheme is presented that couples the material distributions in the structural and fluid mesh. The governing and the adjoint equations of the resulting fluid-structure interaction problem are derived. The proposed method is illustrated with the design of tunable manifolds.

In this paper, a linear passive vehicle suspension model with eight DOF, including six vertical and two rotation motions is given. Considering road surface damage, and ride comfort, taking stiffness and damping of seat and suspension system as design variables; taking seat acceleration RMS, tire’s relative dynamic load and suspension’s maximum dynamic stroke as objective functions, it established a tri-objective optimization model; and it also gave a multi-objective method based on game theory, takes three design objectives as three players, by calculating the affecting factors of the design variables to objective functions and fuzzy clustering, the design variables are divided into different strategic spaces owned by each player. Bionic research on three kinds of male side-blotched lizard’s behavior and survival of multiply mechanism, it defines three lizards as opportunism, egoism and collectivism; sets the three behaviors as corresponding player, and takes self-adaptive function as objective as optimal mono-objective its own strategic spaces and obtains the best strategy to deal with the others. All the best strategies are combined as a game strategy set. By multi-gaming, the final solution meeting given convergence of criterion is obtained. Calculation results show the effectiveness and practicability of the proposed method.

This paper presents an innovative approach to shape optimisation of three-dimensional, damage-tolerant structures. In this approach, a new and simple method, which we termed Failure Analysis of Structures (FAST), is used to estimate the stress-intensity factor for cracks at a notch. The methodology and software used to automate damage-tolerance calculations are developed using computer-aided design and FAST codes. The worst crack locations are found by modeling many cracks along fractured critical edges of the structure by using FAST. This software is then used to evaluate damage-tolerance objective functions for optimisation algorithms. A particular stress-based biological growth method is employed to study the problem of optimisation with fatigue life as the design objective. This work confirms that a stress-optimised structure does not necessarily give the longest fatigue life by numerical examples.

In automotive industry, structural optimization for crashworthiness criteria is of special importance. Due to the high nonlinearities, however, there exists substantial difficulty to obtain accurate continuum or discrete sensitivities. For this reason, metamodel or surrogate model methods have been extensively employed in vehicle design with industry interest. This paper presents a multiobjective optimization procedure for the vehicle design, where the weight, acceleration characteristics and toe-board intrusion are considered as the design objectives. The response surface method with linear and quadratic basis functions is employed to formulate these objectives, in which optimal Latin hypercube sampling and stepwise regression techniques are implemented. In this study, a nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes. The results demonstrate the capability and potential of this procedure in solving the crashworthiness design of vehicles.

In industrial applications, several objectives are often managed simultaneously (e.g., minimizing the cost and the weight of a mechanical structure satisfying some constraints). Although lots of optimization studies deal with only one objective, this approach is often not realistic for engineering optimization. Therefore, improvements in multiobjective optimization methods are required. This paper presents the formulation of a new utopia hyperplane that improves the proposal of the original normalized normal constraint method using two approaches: a redefinition of the anchor points and an exact linear transformation between the design objectives space and the normalized space. Both approaches always produce a normalized space with equal scales that improves the even distribution of the solutions over the Pareto frontier. Examples of the method proposed are presented related with mechanical engineering and structure design including a challenging non-convex Pareto frontier.

This paper addresses optimal design of simply supported symmetrically laminated composite plates with central circular holes. The design objective is the maximization of the buckling load, and the design variable is considered as the fiber orientation. The first-order shear deformation theory is used for the finite element analysis. The study is complicated because the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used to solve the optimization problems. Finally, the effect of different number of layers, boundary conditions, width-to-thickness ratio, plate aspect ratios, hole daimeter-to-width ratio, and load ratios on the results is investigated.

This paper presents a structural optimisation method using the genetic programming (GP) technique. This method applied linear GP to derive optimum geometry and sizing of discrete structure from an arbitrary initial design space. The linear GP was used to find out the optimum nodal locations and member sizing of the structure through a linear sequence of programming instructions. The nodal locations and member cross-sectional areas of the structure were used as the design variable for these instructions, with the optimal geometry and sizing obtained by evolving a population of GP individuals satisfying the optimisation design objective. The approach was applied to the benchmark example of ten-bar planar truss for verification. Other truss examples, including 18-bar planar truss and 25-bar space truss, were also used to demonstrate the effectiveness of this method. The optimum results obtained demonstrate the practicability and generality of using the proposed method in geometry and sizing optimisation problems.

The reinforcement with a thin composite plate of a 3D linear elastic medium on its external boundary or inside is considered. A linear analysis of the 3D problem leads to a variational formulation in which the reinforcement is modelled by a Kirchhoff–Love plate. Considering the sum of the compliance and a cost as the design objective, a numerical example of the optimization of this reinforcement is performed taking into account the in-plane membrane rigidity only (i.e. the bending aspects are not treated numerically).

Topology optimization of regional strain energy is studied in this paper. Unlike the conventional mean compliance formulation, this paper considers two main functions of structure: rigidity and compliance. For normal usages, rigidity is chosen as the design objective. For compliant design, a portion of the structure absorbs energy, while another part maintains the structural integrity. Therefore, we implemented a regional strain energy formulation for topology optimization. Sensitivity to regional strain energy is derived from the adjoint method. Numerical results from the proposed formulation are presented.

The authors recently proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier – for multiobjective optimization problems. Since few methods offer this desirable characteristic, the new method can be of significant practical use in the choice of an optimal solution in a multiobjective setting. This paper’s specific contribution is two-fold. First, it presents a new formulation of the NC method that incorporates a critical linear mapping of the design objectives. This mapping has the desirable property that the resulting performance of the method is entirely independent of the design objectives scales. We address here the fact that scaling issues can pose formidable difficulties. Secondly, the notion of a Pareto filter is presented and an algorithm thereof is developed. As its name suggests, a Pareto filter is an algorithm that retains only the global Pareto points, given a set of points in objective space. As is explained in the paper, the Pareto filter is useful in the application of the NC and other methods. Numerical examples are provided.

Structural optimization problems have been traditionally formulated in terms of crisply defined objective and constraint functions. With a shift in application focus towards more practical problems, there is a need to incorporate fuzzy or noncrisp information into an optimization problem statement. Such practical design problems often deal with the allocation of resources to satisfy multiple, and frequently conflicting design objectives. The present paper deals with a genetic algorithm based optimization procedure for solving multicriterion design problems where the objective or constraint functions may not be crisply defined. The approach uses a genetic algorithm based simulation of the biological immune system to solve the multicriterion design problem; fuzzy set theory is adopted to incorporate imprecisely defined information into the problem statement. A notable strength of the proposed approach is its ability to generate a Pareto-Edgeworth front of compromise solutions in a single execution of the GA.