Minimum Weight Design Research Articles

Numerical Solution of Elliptic Shape Optimization Problems using wavelet-based BEM

In this article we study the numerical solution of elliptic shape optimization problems with additional constraints, given by domain or boundary integral functionals. A special boundary variational approach combined with a boundary integral formulation of the state equation yields shape gradients and functionals which are expressed only in terms of boundary integrals. Hence, the efficiency of (standard) descent optimization algorithms is considerably increased, especially for the line search. We demonstrate our method for a class of problems from planar elasticity, where the stationary domains are given analytically by Banichuk and Karihaloo in [N.V. Banichuk and B.L. Karihaloo (1976). Minimum-weight design of multi-purpose cylindrical bars. International Journal of Solids and Structures, 12, 267–273.]. In particular, the boundary integral equation is solved by a wavelet Galerkin scheme which offers a powerful tool. For optimization we apply gradient and Quasi–Newton type methods for the penalty as well as for the augmented Lagrangian functional.

Optimum Structural Design of High-Speed Surface Effect Ships Built of Composite Materials

A surface effect ship is known to be comparable to a high-speed ship. For the structural design of surface effect ships, advanced design methods are needed which can reflect the various loading conditions different from those of conventional ships. Also, minimum weight design is essential because hull weight significantly affects the lift, thrust powering and high-speed performance. This paper presents the procedure of optimum structural design and a computer program to minimize the hull weight of surface effect ships built of composite materials. By using the developed computer program, the optimum structural designs for three types of surface effect ships—built of sandwich plate only, stiffened single skin plate only, and both plates—are carried out and the efficiency of each type is investigated in terms of weight. The computer program, developed herein, successfully reduced the hull weight of surface effect ships by 15–30% compared with the original design. Numerical results of optimum structural designs are presented and discussed.

2302 空力弾性特性を考慮した複合材板翼の最適設計について

In the present paper, the optimum design method of composite plate wings for aeroelastic characteristics is examined. First, the minimum weight design using real-coded genetic algorithm is treated. Lamination parameters are used as design variables in this optimization procedure. Next, the active flutter suppression using piezoelectric actuators/sensors is treated. PVDF films are used as a modal transducer for identifying specific modes in vibration measurement, and PZT actuators are used as actuators of flutter suppression. The validity of these design approaches is examined through the numerical examples.

積層パラメータを用いた複合材板翼の空力弾性特性最適化に関する研究（第２報）―複合材板翼のフラッタ・ダイバージェンス特性最適化―

The present paper treats the flutter and divergence characteristics of composite plate wings with various sweep angles. In the second report of this paper, a minimum weight design of composite plate wings under the flutter and divergence velocity constraints is conducted by using a genetic algorithm in which lamination parameters are used as design variables. Aeroelastic analysis of composite plates is based on the finite element method and the subsonic unsteady lifting surface theory. The effectiveness of aeroelastic tailoring, which improves the aeroelastic characteristics of composite wings by the layup optimization, is demonstrated through the optimal results.

Optimisation of Structures by the Optimality Criteria Method

A robust computer procedure for weight minimisation of a high–rise and portal frames under the deflection constraint at various locations is presented. It replaces the conventional tedious and unreliable manual assessment for minimum weight design. Several practical considerations such as the Invariance or lower limit of wall thickness for fire resistance or building services requirements are Included in the optimisation procedure. The method is readily used in practical optimisation design of steel and reinforced concrete structures against the deflection limit.

Minimum weight design of non‐linear elastic structures with multimodal buckling constraints

AbstractIt well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non‐linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non‐differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account. Copyright © 2002 John Wiley & Sons, Ltd.

A matrix displacement formulation for minimum weight design of frames

A static linear programming formulation for minimum weight design of frames that is based on a matrix displacement method is presented in this paper. According to elementary theory of plasticity, minimum weight design of frames can be carried out by using only the equilibrium equations, because the system is statically determinate when at an incipient collapse state. In the present formulation, a statically determinate released frame is defined by introducing hinges into the real frame and the bending moments in yield constraints are expressed in terms of unit hinge rotations and the external loads respectively, by utilizing the matrix displacement method. Conventional Simplex algorithm with some modifications is utilized for the solution of linear programming problem. As the formulation is based on matrix displacement method, it may be easily adopted to the weight optimization of frames with displacement and deformation limitations. Four illustrative examples are also given for comparing the results to those obtained in previous studies.

Flexible Composite Wing with Internal Actuation for Roll Maneuver

Abstract : This paper is concerned with designing an optimum composite flexible wing structure to enhance roll maneuver capability at high dynamic pressures using an embedded actuating system without external control surfaces. A minimum weight design, with constraints on strength for three different flight conditions, on the frequency distribution and lift effectiveness was used for this study. The elastic twist and camber is achieved by providing a system of actuating elements distributed within the internal substructure of the wing to provide control forces. The modal approach is used to develop the dynamic equilibrium equations which culminates in the steady roll maneuver of a wing subjected to aerodynamic loads and the actuating forces. The distribution of actuating forces to achieve the specified steady roll rate and roll angle of the flexible vehicle within a specified time was determined by using Independent Modal-Space Control (IMSC) design approach. Here, a full-scale realistic wing is considered for the assessment of the strain energy and distribution of actuator forces required to produce the antisymmetric twist and camber deformation to achieve the specified roll performance.

Weight and crash optimization of foam-filled three-dimensional ?S? frame

The optimization process with the target of minimum weight design of a foam-filled three-dimensional thin-walled S frame under the prescribed values of stiffness and energy absorption was formulated and solved. A distinctive feature of the present approach is that the response function is given as an analytical expression. Three design parameters were introduced in this study, which are the width of the column wall, gauge thickness, and the relative foam density. The sequential quadratic programming (SQP) was employed to find the optimum design variables and the necessary calculation can be completed within a few minutes. Comparing with the optimized empty S frames, about 75% increase in the specific energy absorption was achieved for the aluminum foam-filled S frames. All the analytical solution and the optimization results were verified with the confirmation runs made by commercial FE code PAM-CRASH. The error was within 6%. It was observed that least weight is achieved with a low relative foam density (of an order of 5%) and a thin gauge outer shell. Also, the cross-sectional dimensions b×b are realistic from the packaging point of view (for example, 90 mm × 90 mm). The present methodology can be used as a valuable tool in the early stage of crash-oriented car body design.

Optimization of steel structures using distributed simulated annealing algorithm on a cluster of personal computers

This paper presents a distributed simulated annealing (SA) algorithm for optimal structural design of steel structures under stress, maximum displacement, and inter-story drift constraints. To effectively harness the distributed computing capabilities of a cluster of PCs, a two-phased SA algorithm consisting of simulated quenching (SQ) and SA is developed and used for development of the distributed algorithm. The distributed algorithm is based on two different levels of parallelism, design variable level for a distributed SQ algorithm and candidate design level for a distributed SA algorithm. The distributed SA algorithm is applied to the discrete minimum weight design of a verifying example and two steel braced frame structures. The results show that the distributed SA algorithm implemented on a network of PCs can reduce the computational requirement significantly, and yield more stable convergence histories.

A parallel nodal-based evolutionary structural optimization algorithm

This paper is concerned with the minimum weight design of structures using Finite Element Analysis (FEA). A new evolutionary structural optimization (ESO) algorithm is presented. This method departs from previous studies of ESO in that it exploits the movements of the nodes in an unstructured finite element mesh in an appropriate way. An attractive feature of the scheme presented is that it carries out topology optimization in the interior of the domain concurrently with shape optimization of the exterior of the domain. Circular cavities are inserted into the interior of the domain from which the internal topology is then revealed by migration of the cavity edge nodes. Due to the complexity of the resulting cavity geometry the FE mesh tends to be refined internally. A scheme for maintaining a roughly uniform density unstructured finite element mesh throughout the optimization in a two-dimensional domain is presented. The designs produced posses smooth internal and external boundaries. The method uses iterative finite element analysis and re-meshing to correct for any element distortion. The benchmark “Michell Arch" problem is used to demonstrate the approach.

Minimum-weight seismic design of a moment-resisting frame accounting for incremental collapse

It was shown in the previous study (Lee and Bertero 1993) that incremental collapse can lead to the exhaustion of the plastic rotation capacity at critical regions in a structure when subjected to the number of load cycles and load intensities as expected during maximum credible earthquakes and that this type of collapse can be predicted using the shakedown analysis technique. In this study, a minimum-weight design methodology, which takes into account not only the prevention of this incremental collapse but also the requirements of the serviceability limit states, is proposed by using the shakedown analysis technique and a nonlinear programming algorithm (gradient projection method).

Optimum design of cold-formed steel space structures using neural dynamics model

Optimization of space structures made of cold-formed steel is complicated because an effective reduced area must be calculated for members in compression to take into account the non-uniform distribution of stresses in thin cold-formed members due to torsional/flexural buckling. The effective area varies not only with the level of the applied compressive stress but also with its width-to-thickness ratio. For statically indeterminate structures a new effective area has to be calculated for each member in every iteration of the optimization process. As such, the constraints are implicit, non-smooth, and discontinuous functions of design variables. The patented robust neural dynamics model of Adeli And Park (U.S. patent 5,815,394 issued on September 29, 1998) has been adapted for optimum design of space trusses made of commercially available cold-formed shapes in accordance with AISI specification. A CPN network was developed to learn the relationship between the cross-sectional area and dimensions of cold-formed channels. The model has been used to find the minimum weight design for several space trusses commonly used as roof structures in long-span commercial buildings and canopies, including a large structure with 1548 members with excellent convergence results.

End compression of sandwich columns

Sandwich columns, comprising woven glass fibre reinforced epoxy face sheets and PVC polymer foam cores, have been tested under edgewise compressive loading. Failure is by Euler macrobuckling, shear macrobuckling or by face sheet microbuckling, depending upon the material combination and geometry of column. Simple analytical models are developed for the axial strength, and these are in good agreement with the experimental values for each failure mode. Collapse mechanism maps are constructed to illustrate the dependence of failure mode upon the geometry and relative density of the core; and minimum weight designs are determined as a function of the appropriate structural load index.

Optimal structural design for flexible space structure with control system based on LMI

A simultaneous optimal design problem of structural and control systems is discussed by taking a 3-D truss structure as an object. We use descriptor forms for a controlled object and a generalized plant because the structural parameters appear naturally in these forms. We consider a minimum weight design problem for structural system and disturbance suppression problem for the control system. The structural objective function is the structural weight and the control objective function isH∞ norm from the disturbance input to the controlled output in the closed-loop system. The design variables are cross sectional areas of the truss members. The conditions for the existence of controller are expressed in terms of linear matrix inequalities (LMI). By minimizing the linear sum of the normalized structural objective function and control objective function, it is possible to make optimal design by which the balance of the structural weight and the control performance is taken. We showed in this paper the validity of simultaneous optimal design of structural and control systems.

Displacement-Based Optimization for Truss Structures Subjected to Static and Dynamic Constraints

The use of displacements as design variables for truss structure optimization is considered as an alternative to the conventional finite-element-analysis-based design approach. A two-level nested optimization has been developed; in an inner level, cross-sectional areas of truss members are designed for a given displacement field, and, at an outer level, optimal displacements corresponding to the minimum weight designs are searched through the use of sequential quadratic programming. The computational efficiency of the method is demonstrated through three examples, and the evolutions of the cross-sectional areas and optimal weight as a function of the displacements are studied. Static, dynamic, and topology problems are considered. It is shown that the method is highly efficient compared to the conventional design approaches. It is also demonstrated that the weight is always continuous throughout the design history, although areas might exhibit large discontinuities.

Study on Structural Optimization Technique Using Evolutionary Cellular Automata. Local Direct Rule Based on New Transition Function.

In this paper, a local direct rule based on a new transition function of the Cellular Automata (CA) is presented. It is the structural optimization method by the Evolutionary Cellular Automata (ECA). The local direct rule uses the neighborhood states as the CA's input. The ECA is the CA optimized by the Genetic Algorithm (GA) for the certain problems. This new transition function makes to decrease drastically the degrees of local direct rule freedom depending on the number of the neighborhood cells. The design conditions of the structure are introduced as virtual cells to apply the same rule in all the design space. A local direct rule here is decided by the evolutionary calculation of GA. The method is applied to the minimum weight design problems and the adaptive solutions are obtained. From the results, it is observe that the local direct rule based on a new transition function is effective in the self-organization problem of the complex system.

Recursive Optimization of Self-Similar Structural System.

In this paper, we discuss the recursive optimal design approach: an application of self-similarity to optimal structural design. The approach is formulated in a recursive manner based on a resourcebased optimization concept. An actually self-similar topology of the initial truss structure is introduced in order to realize a completely self-similar optimization process. Computational cost and suitability to parallel processing of the recursive optimal design approach are discussed from the view-point of self-similarity. Minimum-weight design of two-dimensional truss structure is conducted; the availability of the actually self-similar topology of the truss to the recursive optimization process is also discussed.

Optimality-criteria-based probabilistic structural design

The possibility of extending absolute minimum weight designs for sandwich beam systems or fibre-reinforced beams and plates is considered when both the extreme loading applied to the system and the beam bending resistance properties are described in probabilistic rather than deterministic terms. It is shown that such an extension is possible provided certain plausible assumptions are made about the probabilistic modelling of the system. Examples considered include simple and continuous beams under loading of uncertain magnitude, direction or location. It is shown that the classical (deterministic) optimal solutions remain valid under reasonable assumptions. The extension of the probabilistic solutions to a wider class of examples is discussed.

Optimal discrete truss design using improved sequential and genetic algorithm

The potential of two distinct approaches applied to the truss discrete optimization problem is presented in the paper. The sequential discrete optimization method SDO (which is a deterministic procedure, using heuristics based on the idea of fully stressed truss design) and the genetic algorithm GA (a stochastic search method, inspired by the natural evolution model) are compared. The minimum weight design of truss structures subjected to stress and displacement constraints is investigated, including the case of multiple load conditions. The discrete design variables are areas of members, selected from a finite catalogue of available sections. Benchmark 2D and 3D problems are presented in numerical examples. The effectiveness of two approaches is discussed. The improvements of both algorithms and GA integrating the results of SDO method are proposed. They enable us to accelerate the convergence, diminish the number of structural analyses and guide to refined “near optimal” solutions.