Shape optimization of structures under earthquake loadings

The optimum design of structures under static loads is well-known in the design world; however, structural optimization under dynamic loading faces many challenges in real applications. Issues such as the time-dependent behavior of constraints, changing the design space in the time domain, and the cost of sensitivities could be mentioned. Therefore, optimum design under dynamic loadings is a challenging task. In order to perform efficient structural shape optimization under earthquake loadings, the finite element-based approximation method for the transformation of earthquake loading into the equivalent static loads (ESLs) is proposed. The loads calculated using this method are more accurate and reliable than those obtained using the building regulations. The shape optimization of the structures is then carried out using the obtained ESLs. The proposed methodologies are transformed into user-friendly computer code, and their capabilities are demonstrated using numerical examples.     

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.     

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     

Technical overview of the equivalent static loads method for non-linear static response structural optimization

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.     

Design of structure/control systems with transient response constraints exhibiting relative minima

Structural optimization problems involving dynamic behaviour constraints often exhibit nonconvex design spaces. The direct application of a global optimization algorithm requires a large number of function evaluations which in turn require a large number of dynamic structural analyses. This work presents a strategy aimed at finding the global optimum for problems with transient dynamic behaviour constraints based on approximation concepts. The method consists of generating and solving a sequence of approximate problems using a global optimizer. The approximations are explicit and capture most of the inherent nonconvexity of the exact functions. A simple example. problem is presented to illustrate the procedure set forth.     

Buckling design optimization of complex built-up structures with shape and size variables

The design optimization of buckling behaviour is studied for complex built-up structures composed of various kinds of elements and implemented within JIFEX95, a general-purpose software for finite element analysis and design optimization. The direct and adjoint methods of sensitivity analysis for critical buckling loads are presented with detailed computational procedures. Particularly, the variations of prebuckling stresses and external loads have been accounted for. The design model and solution methods presented in this paper are available for both shape and size optimization, and buckling optimization can also be combined with static, frequency and dynamic response optimization. The numerical examples show the applications of the buckling optimization method and the effectiveness of the methods and the program of this paper. Received February 23, 1999     

Design Optimization of Multibody Systems by Sequential Approximation

Design optimization of multibody systems is usually established by a direct coupling of multibody system analysis and mathematical programming algorithms. However, a direct coupling is hindered by the transient and computationally complex behavior of many multibody systems. In structural optimization often approximation concepts are used instead to interface numerical analysis and optimization. This paper shows that such an approach is valuable for the optimization of multibody systems as well. A design optimization tool has been developed for multibody systems that generates a sequence of approximate optimization problems. The approach is illustrated by three examples: an impact absorber, a slider-crank mechanism, and a stress-constrained four-bar mechanism. Furthermore, the consequences for an accurate and efficient accompanying design sensitivity analysis are discussed.     

Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load

This paper presents an efficient structural design optimization strategy that combines the reduced basis method (RBM) with the equivalent static load (ESL). In dynamic response optimization using ESL, the computation of a static system optimization is repeatedly executed under multiple static loads. In this process, we propose parametrizing the static system and employing the RBM with global proper orthogonal decomposition (POD). In general, the snapshots for the sampling procedure under multiple loads increase proportionally to the number of loads, which results in an inefficient computational procedure. Thus, we propose taking snapshots with a proper orthogonal mode (POM) of the multiple loads rather than for the multiple loads themselves. The number of snapshots then decreases, and the original system can be efficiently reduced. We directly employ the framework of the proposed RBM with the POM of multiple loads to ESL-based design optimization, and the results indicate that the proposed method is more efficient than conventional ESL-based optimization as well as a full order model. Various numerical examples, including comparisons of relative errors and the dynamic response optimizations, support the strength of the proposed strategy.     

Optimization of flexible components of multibody systems via equivalent static loads

An optimization methodology that iteratively links the results of multibody dynamics and structural analysis software to an optimization method is presented to design flexible multibody systems under dynamic loading conditions. In particular, rigid multibody dynamic analysis is utilized to calculate dynamic loads of a multibody system and a structural optimization algorithm using equivalent static loads transformed from the dynamic loads are used to design the flexible components in the multibody dynamic system. The equivalent static loads, which are derived from equations of motion, are used as multiple loading conditions of linear structural optimization. A simple example is solved to verify the proposed methodology and the pelvis part of the biped humanoid, a complex multibody system which consists of many bodies and joints, is redesigned using the proposed methodology.     

Structural dynamic topology optimization based on dynamic reliability using equivalent static loads

An approach for reliability-based topology optimization of interval parameters structures under dynamic loads is proposed. We modify the equivalent static loads method for non linear static response structural optimization (ESLSO) to solve the dynamic reliability optimization problem. In our modified ESLSO, the equivalent static loads (ESLs) are redefined to consider the uncertainties. The new ESLs including all the uncertainties from geometric dimensions, material properties and loading conditions generate the same interval response field as dynamic loads. Based on the definition of the interval non-probabilistic reliability index, we construct the static reliability topology optimization model using ESLs. The method of moving asymptotes (MMA) is employed as the optimization problem solver. The applicability and validity of the proposed model and numerical techniques are demonstrated with three numerical examples.     

Application of Micro-GA for optimal cost base isolation design of bridges subject to transient earthquake loads

Seismic isolation and energy dissipation systems are innovative strategies for seismic design and upgrade or retrofit of bridges. In a retrofit design, base isolation devices can be easily incorporated into existing bridges to replace conventional bearings and to improve the overall structural performance. In this paper, an optimal cost base isolation design or retrofit design method for bridges subject to transient earthquake loads is studied. The goal of this study is to push forward the concept of retrofit design optimization of structures using this isolation design as an example. This is achieved by combining nonlinear time history analyses with an optimization procedure to select base isolators that minimize the cost of the isolation system while satisfying certain design requirements. An improved genetic algorithm (GA), Micro-GA, is employed to search for the optimal solutions for such discrete optimization problems. An example of the optimal design of a highway bridge is presented and the minimum cost expense of the isolation system is achieved with improved structural response under multiple transient earthquake loads.     

Simulated annealing for the design of structures with time-varying constraints

The optimal design of structural systems with conventional members or systems with conventional as well as passive or active members is presented. The optimal sizes of the conventional members of structural systems are obtained for dynamic loads. A modified simulated annealing algorithm is presented which is used to solve the optimization problem with dynamic constraints. The present algorithm differs from existing simulated annealing algorithms in two respects; first, an automatic reduction of the search range is performed, and second, a sensitivity analysis of the design variables is utilized. The present method converges to the minimum in less iterations when compared to existing simulated annealing algorithms. The algorithm is advantageous over classical methods for optimization of structural systems with constraints arising from dynamic loads. For certain initial values of the design variables, classical optimization methods either fail to converge or produce designs which are local minima; the present algorithm seems to be successful in yielding the global minimum design.     

A mixed GA-PSO-based approach for performance-based design optimization of 2D reinforced concrete special moment-resisting frames

A mixed genetic algorithm and particle swarm optimization in conjunction with nonlinear static and dynamic analyses as a smart and simple approach is introduced for performance-based design optimization of two-dimensional (2D) reinforced concrete special moment-resisting frames. The objective function of the problem is considered to be total cost of required steel and concrete in design of the frame. Dimensions and longitudinal reinforcement of the structural elements are considered to be design variables and serviceability, special moment-resisting and performance conditions of the frame are constraints of the problem. First, lower feasible bond of the design variables are obtained via analyzing the frame under service gravity loads. Then, the joint shear constraint has been considered to modify the obtained minimum design variables from the previous step. Based on these constraints, the initial population of the genetic algorithm (GA) is generated and by using the nonlinear static analysis, values of each population are calculated. Then, the particle swarm optimization (PSO) technique is employed to improve keeping percent of the badly fitted populations. This procedure is repeated until the optimum result that satisfies all constraints is obtained. Then, the nonlinear static analysis is replaced with the nonlinear dynamic analysis and optimization problem is solved again between obtained lower and upper bounds, which is considered to be optimum result of optimization solution with nonlinear static analysis. It has been found that by mixing the analyses and considering the hybrid GA-PSO method, the optimum result can be achieved with less computational efforts and lower usage of materials.     

A design methodology for structural and control systems of a prosthetic leg

A design methodology is proposed to optimize a prosthetic leg considering the structural and control aspects. Previous studies mainly focused on each of a structural design problem or a control problem. The structural design variables of a prosthetic leg are determined in a structural design problem while the trajectory tracking based on the human gait cycle is found to be a control problem. The two problems have been separately solved. However, they should be simultaneously considered to obtain a better design. Then the problem would be nonlinear dynamic response optimization of structural and control systems. An optimization method was recently developed for a linear system of structural and control design problems using the equivalent static loads (ESLs). In this study, a design methodology for the prosthetic leg is presented by expanding the equivalent static loads method (ESLM) to nonlinear dynamic systems. A simple example is solved to validate the proposed methodology, and then a prosthetic leg is optimized to reduce the weight and energy consumption.     

Modified iterated simulated annealing algorithm for structural synthesis

A new hybrid simulated annealing method is presented for the optimization of structural systems subjected to dynamic loads. The optimization problem is formulated as a structural weight minimization, with time-varying constraints on floor displacements, velocities, accelerations, or floor drifts, and structural member combined stresses. In addition, time-invariant constraints on structural frequencies and member sizes that will satisfy the strong column–weak beam philosophy of the building codes can be imposed. The method uses elements of existing simulated annealing algorithms and introduces certain new procedures. Firstly, the search range is automatically reduced, by using the updated information of the current design, at each iteration. Secondly, the inner and outer iteration loops are implemented. Thirdly, sensitivity analysis of the time-varying global displacements is performed with respect to the design variables that are the structural member cross-sectional areas. The results of the sensitivity analysis identify which design variables must be modified to decrease the global displacements in the most effective manner. However, once the variables are identified from the sensitivity analysis, the new values of these variables are determined in a random manner. The possibility of attaining a global minimum is thus maintained. The method is suited for structural optimization problems with time-varying constraints because the annealing is a random search technique and can locate global rather than local minima.     

Shape optimization of buckling sensitive structures

The optimal design of structures with distinct geometrically non-linear behavior has attracted a great deal of interest in the last years mainly with respect to sizing for prescribed external loads. In the present contribution a method is proposed to maximize the critical load under certain constraints, e.g. for a given volume, allowing varying shape as well as cross-sections. The combination of direct computation of the critical load and path-following methods is integrated into a general optimization procedure consisting of mathematical programming techniques, sensitivity analysis and computer aided geometric design methods. The formulation includes imperfection sensitivity as an important part within the optimization process.     

A hybrid formulation for treatment of point-wise state variable constraints in dynamic response optimization

This paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems. The point-wise state variable constraint function is divided into time sub-domains such that each sub-domain contains only one local maximum point. Then, the original constraint is replaced by a number of equivalent functional constraints. Each functional constraint is the integration of the positive value of the original constraint over its own time sub-domain. A direct differentiation method and three adjoint variable methods of design sensitivity analysis are presented. All of these methods are discussed and compared. It turns out that two of the adjoint variable methods are more efficient than others. A hybrid optimization algorithm based on these methods is proposed in detail. Two problems are solved for optimal design. Comparisons of results with those available in the literature are made. Numerical experience with the proposed method is discussed in detail. It is concluded that the new formulation is extremely efficient and converges to either optimal or near optimal solutions without any difficulty.     

Review of options for structural design sensitivity analysis. Part 1: Linear systems

The paper reviews options for structural design sensitivity analysis, including global finite differences, continuum derivatives, discrete derivatives, and computational or automated differentiation. The objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency, computational cost, and implementation options and effort. Linear static analysis and transient dynamic analysis are reviewed. In a separate appendix, special attention is paid to the semi-analytical method. A future paper will address design sensitivity analysis in nonlinear structural problems.     

Stress-based design of thermal structures via topology optimization

The design of thermal structures in the aerospace industry, including exhaust structures on embedded engine aircraft and hypersonic thermal protection systems, poses a number of complex design challenges. These challenges are particularly well addressed by the material layout capabilities of structural topology optimization; however, no topology optimization methods are readily available with the necessary thermoelastic considerations for these problems. This is due in large part to the emphasis on cases of maximum stiffness design for structures subjected to externally applied mechanical loads in the majority of topology optimization applications. In addition, while limited work in the literature has investigated thermoelastic topology optimization, a direct treatment of thermal stresses is not well documented. Such a treatment is critical in the design of thermal structures where excessive thermal stresses are a primary failure mode. In this paper, we present a method for the topology optimization of structures with combined mechanical and thermoelastic (temperature) loads that are subject to stress constraints. We present the necessary steps needed to address both the design-dependent thermal loads and accommodate the challenges of stress-based design criteria. A relaxation technique is utilized to remove the singularity phenomenon in stresses and the large number of stress constraints is handled using a scaled aggregation technique that has been shown previously to satisfy prescribed stress limits in mechanical problems. Finally, the stress-based thermoelastic formulation is applied to two numerical example problems to demonstrate its effectiveness.     

Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm

Natural frequencies offer useful knowledge on the dynamic response of the structures. It is possible to avoid from the destructive effects of dynamic loads on the structures by optimizing layout and size of their subject to constraints on natural frequencies. Since optimization problems including frequency constraints are highly nonlinear, this kind of problems forms a challenging area to test the performance of the different optimization techniques. This study tests the performance of an integrated particle swarm optimization algorithm (iPSO), a new particle swarm optimizer integrated with the improved fly-back mechanism and the weighted particle concept, in four weight minimization of truss structures with sizing and layout variables under multiple frequency constraints. Optimization results demonstrate that the new algorithm is competitive with other state-of-the-art metaheuristic algorithms in dynamic and static structural optimization problems.