Minimum Weight Design Of Structures Research Articles

Size effect of lattice material and minimum weight design

The size effects of microstructure of lattice materials on structural analysis and minimum weight design are studied with extented multiscale finite element method (EMsFEM) in the paper. With the same volume of base material and configuration, the structural displacement and maximum axial stress of micro-rod of lattice structures with different sizes of microstructure are analyzed and compared. It is pointed out that different from the traditional mathematical homogenization method, EMsFEM is suitable for analyzing the structures which is constituted with lattice materials and composed of quantities of finite-sized micro-rods. The minimum weight design of structures composed of lattice material is studied with downscaling calculation of EMs-FEM under stress constraints of micro-rods. The optimal design results show that the weight of the structure increases with the decrease of the size of basic sub-unit cells. The paper presents a new approach for analysis and optimization of lattice materials in complex engineering constructions.

Functional morphology of jaw trabeculation in the lesser electric rayNarcine brasiliensis, with comments on the evolution of structural support in the Batoidea

The design of minimum-weight structures that retain their integrity under dynamic loading regimes has long challenged engineers. One solution to this problem found in both human and biological design is the optimization of weight and strength by hollowing a structure and replacing its inner core with supportive struts. In animals, this design is observed in sand dollar test, avian beak, and the cancellous bone of tetrapod limbs. Additionally, within the elasmobranch fishes, mineralized trabeculae (struts) have been reported in the jaws of durophagous myliobatid stingrays (Elasmobranchii: Batoidea), but were believed to be absent in basal members of the batoid clade. This study, however, presents an additional case of batoid trabeculation in the lesser electric ray, Narcine brasiliensis (Torpediniformes). The trabeculae in these species likely play different functional roles. Stingrays use their reinforced jaws to crush bivalves, yet N. brasiliensis feeds by ballistically protruding its jaws into the sediment to capture polychaetes. In N. brasiliensis, trabeculae are localized to areas likely to experience the highest load: the quadratomandibular jaw joints, hyomandibular-cranial joint, and the thinnest sections of the jaws immediately lateral to the symphyses. However, the supports perform different functions dependent on location. In regions where the jaws are loaded transversely (as in durophagous rays), "load leading" trabeculae distribute compressive forces from the cortex through the lumen of the jaws. In the parasymphyseal regions of the jaws, "truss" trabeculae form cross-braces perpendicular to the long axes of the jaws. At peak protrusion, the jaw arch is medially compressed and the jaw loaded axially such that these trabeculae are positioned to resist buckling associated with excavation forces. "Truss" trabeculae function to maintain the second moment of area in the thinnest regions of the jaws, illustrating a novel function for batoid trabeculation. Thus, this method of structural support appears to have arisen twice independently in batoids and performs strikingly different ecological functions associated with the distribution of extreme loading environments.

Optimization in aircraft engineering including approximate structural evaluations by support vector machines

AbstractThe minimum weight design of structures made of fiber reinforced composite materials leads to a class of mixed‐integer optimization problems for which evolutionary algorithms (EA) are well suited. Based on these algorithms the optimization tool package GEOPS has been developed at TU Dresden.For each design generated by an EA the structural response has to be evaluated. This is often based on a finite element analysis which results in a high computational complexity for each single design. Typical runs of EA require the evaluation of thousands of designs. Thus, an efficient approximation of the structural response could improve the performance considerably. To achieve this aim the constraints on the structural response are approximated by means of support vector machines (SVM). It is trained by means of exact structural evaluations for selected design alternatives only. Several ways to enhance the efficiency of such an optimization procedure are presented.As an example for a typical aircraft structure, a stiffened composite panel under compressive and shear loading is considered. The SVM is trained on geometrical and material data. Representing the design space of composite panels by ABD matrices turned out to be a valuable means for obtaining well trained SVMs. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Comparisons of optimality criteria for minimum-weight dual material structures

A criterion established by Prager for minimum-weight design of structures, using different materials for tension and compression members, referred to here as bi-material structures, appears to have been completely overlooked in the literature. Cantilever and simply supported beam designs are analyzed to compare the result of applying this criterion with the conditions established by Michell in his groundbreaking paper on structural optimization. Surprisingly, for statically determinate bi-material structures, Michell's original conditions for minimum volume, also ensure minimum weight. However, for statically indeterminate structures, Michell's criterion does not give least weight unless the ratio, of yield stress to density, is the same for tension and compression members. For statically indeterminate cantilever design cases analyzed in this paper, the weight differences between those satisfying Michell's and Prager's conditions are shown to be very small, even when the ratio of yield stress to density tension differs by a factor of 7 between the tension and compression members. This allows the selection of alternative structural layouts for other attributes than minimum weight.

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.

Structral–control optimization withH2- andH∞-norm bounds

An integrated approach to minimum weight structural design and robust control system design is presented. The controller is designed to tolerate both the structured real parameter uncertainty in the structural natural frequencies and the unstructured unmodelled dynamics due to model truncation. The control approach utilizes a mixed H2–H∞ LQG compensator with an H∞-norm bound on the disturbance to the output transfer matrix in the closed-loop system and an upper bound on the H2-norm signifying the maximum value of the quadratic control performance index for the uncertain system. In structural–control optimization problems, constraints are imposed on the fundamental structural frequency and on the dB gain separation between the open-loop and closed-loop singular values at prescribed frequencies. The method is demonstrated on a 48-state structural truth model of a three-dimensional tapered box truss with collocated actuators and sensors in which the control design model is eighth-order including the compensator dynamics. © 1997 John Wiley & Sons, Ltd.

Minimum weight design of structures under frequency and frequency response constraints

A technique for the determination of the least weight design which satisfies the desired frequency and frequency response constraints, plus upper and lower bounds on the design variables, is introduced. The design algorithm is based on Zoutendijk's method of feasible directions [1]. This method requires the analytical gradients of the objective function and the constraint functions which are active at a given stage in the design process. A considerable improvement in convergence has been achieved by considering each pushoff factor as a linear function of the corresponding active constraint. An initial step length based on a pre-selected decrement of the objective function is used. The results, obtained using an IBM-compatible microcomputer, assert that the method leads to good results even when a very limited number of modes is used in calculating the sensitivities of the frequency response function. This makes the method very efficient with regard to both accuracy, computation time and suitability for use on microcomputers.

Merits and limitations of optimality criteria method for structural optimization

AbstractThe merits and limitations of the Optimality Criteria (OC) method for the minimum weight design of structures subjected to multiple load conditions under stress, displacement and frequency constraints were investigated by examining several numerical examples. The examples were solved utilizing the OC design code that was developed for this purpose at the NASA Lewis Research Center. This OC code incorporates OC methods available in the literature with generalizations for stress constraints, fully utilized design concepts, and hybrid methods that combine both techniques. It includes multiple choices for Lagrange multiplier and design variable update methods, design strategies for several constraint types, variable linking, displacement and integrated force method analysers, and analytical and numerical sensitivities. On the basis of the examples solved, the optimality criteria for general application were found to be satisfactory for problems with few active constraints or with small numbers of design variables. However, the OC method without stress constraints converged to optimum even for large structural systems. For problems with large numbers of behaviour constraints and design variables, the method appears to follow a subset of active constraints that can result in a heavier design. The computational efficiency of OC methods appears to be similar to some mathematical programming techniques.

Robust Optimization of Large-Scale Systems

Mathematical programming models with noisy, erroneous, or incomplete data are common in operations research applications. Difficulties with such data are typically dealt with reactively—through sensitivity analysis—or proactively—through stochastic programming formulations. In this paper, we characterize the desirable properties of a solution to models, when the problem data are described by a set of scenarios for their value, instead of using point estimates. A solution to an optimization model is defined as: solution robust if it remains “close” to optimal for all scenarios of the input data, and model robust if it remains “almost” feasible for all data scenarios. We then develop a general model formulation, called robust optimization (RO), that explicitly incorporates the conflicting objectives of solution and model robustness. Robust optimization is compared with the traditional approaches of sensitivity analysis and stochastic linear programming. The classical diet problem illustrates the issues. Robust optimization models are then developed for several real-world applications: power capacity expansion; matrix balancing and image reconstruction; air-force airline scheduling; scenario immunization for financial planning; and minimum weight structural design. We also comment on the suitability of parallel and distributed computer architectures for the solution of robust optimization models.

OPTIMAL PLACEMENT OF ACTUATORS IN ACTIVELY CONTROLLED STRUCTURES

The paper deals with the influence of actuator/sensor locations and feedback gains on the optimum design of actively controlled structures. Two related problems are addressed. The first is a parametric study dealing with the effect of number and location of actuators on the minimum weight structural design, while satisfying constraints on the closed loop eigenvalues and damping ratios. The second problem addresses the optimal placement of actuators with the damping augmentation provided by the control action being used as the performance criterion. A solution methodology which allows for an integrated determination of feedback gains and actuator/sensor locations while maximizing the energy dissipated by the controller is presented. Since the actuator locations are spatially discrete whereas the feedback gains are continuous, the resulting optimization problem has mixed discrete-continuous design variables. This problem is solved using a hybrid optimization method which is a synergistic blend of artificial genetic search and gradient-based search techniques. The effectiveness of the proposed approach is illustrated via an application to an ACOSS-FOUR structure. The optimum results obtained using the hybrid optimization method are shown lo outperform the optimum results obtained using gradient-based optimization techniques by factors ranging from 2.5 to 7.2.

Optimal Design of Structures with Kinematic Nonlinear Behavior

This paper suggests an optimization-based methodology for the design of minimum weight structures with kinematic nonlinear behavior. Attention is focused on three-dimensional reticulated structures idealized with beam elements under proportional static loadings. The algorithm used for optimization is based on a classical optimality criterion approach using an active-set strategy for extreme limit constraints on the design variables. A first-order necessary condition is derived and used as the basis of a fixed-point iteration method to search for the optimal design. A fixed-point iteration algorithm is used based on the criterion that at optimum design the nonlinear strain energy is equal in all members. A nonlinear analysis procedure for three-dimensional structures is discussed and used in developing the optimization algorithm. Several examples are given to evaluate the validity of the underlying assumptions and to demonstrate some of the characteristics of the proposed procedures. The procedure is verified using two well-known examples.

Optimum design of multiple configuration structures for frequency constraints

Introduction D the service of an aerospace structure, its may vary for different operating conditions. When we restrict the word configuration to structural geometry, the structures with multiple configurations that come to mind are, for example, a tilt-rotor aircraft in airplane or helicopter mode and a stowed vs deployed solar array system for a spacecraft. If we generalize the word to encompass boundary conditions and mass distribution, then all aerospace vehicles can be considered to be structures with multiple configurations. In application, design specifications may impose specific natural frequency constraints for different configurations. For a structure with a single configuration, the minimum weight design with frequency constraints can be found readily.' However, the resulting optimum design may not satisfy frequency constraints for the structure in other configurations. This often leads to design iterations among various configurations. After many design iterations, the final design may be feasible but not optimum. The purpose of this paper is to develop an approach for minimum weight design of structures with several configurations under natural frequency constraints. In the optimum design problem, frequency constraints for all configurations are considered simultaneously. In each design cycle, the constrained minimum weight design problem is solved iteratively by a combined analysis/optimization procedure. To increase the computational efficiency, the iterative analysis is approximated by using reduced-order modal space models.' Sequential linear programming (SLP)'' for optimization procedure is considered. In SLP, the optimum design problem is linearized at each design iteration. Together with a strategy to compute move limits, the resulting linear optimization problem is solved as a linear programming problem. The combination of finite element analysis together with optimization method (SLP) constitutes an effective and reliable approach for solving practical optimum design problems.

Minimum-weight design of structures with natural-frequency constraints

The problem of minimum-weight design of structures with several natural-frequency constraints is considered in this paper. The problem is solved by using a combined finite element method and sequential linear programming (FEM-SLP) formulation. The unique features of the present approach include the use of the assumed mode reanalysis formulation for the repeated eigensolution and the associated sensitivity analysis. The present approach has been implemented using the general-purpose finite element program msc/nastran. Two examples are included to illustrate the effectiveness of the present approach.

Sensitivity and optimization of composite structures in MSC/NASTRAN

The main purpose of this paper is to present the considerations and the resultant approach used to implement design sensitivity capability for composites into a large-scale, general-purpose finite element system ( msc/nastran). The design variables for composites can be lamina thicknesses, orientation angles, material properties, or a combination of all three. As a secondary goal, the sensitivity analysis has been coupled with a general-purpose optimizer to validate sensitivity analysis results and also demonstrate how easy it is couple to an optimizer once a general senssitivity capability is available. This preliminary version of the optimizer is capable of dealing with minimum weight structural design with a rather general design variable linking capability at the element level or the system level. Only sizing type of design variables (i.e., lamina thicknesses) can be handled by the optimizer. Test cases have been run and validated by comparison witj independent finite element packages. The linking of a design sensitivity capability for composites in msc/ nastran with an optimizer would give designsrs a powerful, automated tool to carry out practical optimization design of real-life, complicated composite structures.

OPTIMIZATION OF STRUCTURES WITH NONLINEAR RESPONSE

Minimum weight design of structures with geometrically nonlinear behaviour is treated. The problem is formulated in two different ways. In the first formulation all the equilibrium equations are relaxed and treated as constraints. Both design variables and displacements are independent variables. In the second approach the displacements are transformed such that only a few of the equilibrium equations need to be treated as constraints in the optimization problem. The design variables and only a few transformed displacements are independent variables. The optimization problems associated with both formulations are solved numerically using a sequential quadratic programming (SQP) method. Truss structures are used to illustrate the performance of the SQP method when used to solve the two types of problems.

Finite element method and optimality criterion based structural optimization

In this paper several approaches to the design of minimum weight structures through the application of the Optimality Criterion are presented, using structures analyzed by means of the Finite Element Method. Two different recurrence relations to modify the design variables and two methods of calculating the Lagrange multipliers are studied. In order to show the efficiency of the different strategies, several examples of application are presented.

Minimum weight design of structures with random parameters

A general formulation of the minimum-weight optimization problem is presented in the paper. Formulation of the problem is based on the concept of the expected value. Solution of the corresponding mathematical programming problem has been obtained by means of indirect method. To evaluate the magnitude of the influence of random character of the structure parameters two numerical examples concerning plane truss and free vibrating plane strain structure are presented.

A MIXED METHOD FOR SHAPE OPTIMIZATION OF SKELETAL STRUCTURES

This paper presents a new efficient mixed method for minimum weight structural design (geometrical and member sizing) of structures. The objective function is approximated by a second-order Taylor series expansion and the constraints by first-order Taylor series expansions. In order to eliminate the tedious work in evaluating the Hessian matrix and performing its inversion, an approximating matrix employed by the variable metric method is used to approach the inverse of the Hessian matrix. Applying the Kuhn-Tucker conditions to the problem, the K-T multipliers can be determinated by quadratic programming, and finally, the problem is transformed to one of linear programming with intercomplementarity conditions. The examples show that the method is generally applicable, convenient to use and has rapid convergence.

Minimum weight design of truss structures with geometric nonlinear behavior

The paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior. The nonlinear critical load is determined by finding the load level at which the Hessian of the potential energy ceases to be positive definite. A recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm. Sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables. NUMBER of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements, element stresses, system stability, etc.1'2 The optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables. The optimality criterion was derived by, differentiating the Lagrangian with respect to the design variables. The displacements were assumed to be small and, in the finite element analysis, linear equilibrium equations were solved to determine the response of the structure to the ap- plied loads. In the case of system stability the constraints were defined (see Refs. 3-6) by the associated linear eigenvalue problem. This definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections. The correct procedure for these structures would be to analyze the structure* by using nonlinear equilibrium equations. This will be particularly true for large space structures (LSS). In this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior. In the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode. This design tends to become im- perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub- stantially. The imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal (see Ref. 7). A real structure has geometric im- perfections and these structures tend to have nonlinear

An optimality criterion method for structures with stress, displacement and frequency constraints

Presented in this paper is a procedure, based on optimality criterion methods, for the minimum weight design of structures subjected to stress, displacement and natural frequency constraints. The technique presented is a combination of a previously developed method for stress and displacement constraints alone and one for frequency limited structures. The method is applied to a delta-wing example, and the optimal designs obtained are compared to previously published results. The new method is capable of obtaining the optimal design in a small number of iterations, without significant calculations beyond a standard analysis. No approximate analyses or determination of large numbers of Lagrange multipliers are involved.