Minimum mass design of elastic frames subjected to multiple load cases

For frames with stress- and displacement constraints subjected to multiple load cases the formulation is given that enables use of the unified optimization approach which combines finite element and linear programming. Sensitivity analysis is shown analytically for Timoshenko beam models with transformations for eccentricities. For a specific case of a portal frame for a crane a study is made of the influence of the given portal columns (boundary conditions), the relations to fully stressed designs, and the influence of slenderness, when displacement constraints are involved.     

Structural optimization of product families subjected to multiple crash load cases

This paper discusses the problem of structural optimization of product families subjected to multiple load cases, evaluated by computationally costly finite element analysis. Product families generally have a complex composition of shared components that makes individual product optimization difficult as the relation between the shared variables is not always intuitive. More optimal is to treat the problem as a product family optimization problem. Though, for product families subjected to multiple and computationally costly crash loads, the optimization problem takes too long time to solve with traditional methods. Therefore, a new optimization algorithm is presented that decomposes the family problem into sub-problems and iteratively reduces the number of sub-problems, decouple and solve them. The algorithm is applicable for module based product families with predefined composition of generalized commonality, subjected to multiple load cases that can be analyzed separately. The problem reduction is performed by only considering the constraints that are critical in the optimal solution. Therefore the optimization algorithm is called the Critical Constraint Method, CCM. Finally the CCM algorithm is evaluated by two product family optimization problems.     

A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions

The paper deals with minimization of the weighted sum of compliances related to the load cases applied non-simultaneously. The design variables are all components of the Hooke tensor, subject to the isoperimetric condition bounding the integral of the sum of the Kelvin moduli. This free material design problem is reduced to an equilibrium problem – in two formulations – of an effective body with locking. The stress-based formulation reduces to minimization of an integral of a certain norm of stress fields over the stress fields which equilibrate the given loads. The equivalent displacement-based formulation involves a locking locus defined by using a norm being dual to the previous one. The optimal Hooke tensor is determined by using the stress fields solving the auxiliary locking problem. To make the optimal Hooke tensor positive definite one should consider at least 3 load conditions in the 2D case and not less than 6 load conditions in the 3D case.     

A compliance based design problem of structures under multiple load cases

There are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading (the layouts would be the same if in the compliance problem the volume is set equal to the result of the first problem). In this paper, we introduce the “envelope strain energy” problem where we minimize the volume integral of the worst case strain energy of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate (the term non-indeterminate includes both statically determinate trusses and mechanisms) trusses, this compliance method gives the same optimal design as the stress based design method.     

Generalized shape optimization using stress constraints under multiple load cases

Optimal shape design using numerical techniques is an increasingly useful engineering tool. Generalized or layout optimal design where the topology of the object is not fixed is one of the emerging applications. These problems are numerically difficult to solve due to the large number of design variables and equality/inequality constraints. Solutions have focused primarily on compliance based minimization under a fixed volume. A more usual engineering approach would be one of minimizing the volume under a stress or deflection constraint. This, however, can lead to problems as stress is a local quantity and volume minimization of multiple load cases under stress constraints may not result in the stiffest design for the remaining material. The approach adopted here is based on a differential rate equation governed by a local operator that defines the state of each element at each time step. This algorithm forms the optimality criteria for the problem. To satisfy the global stress constraints, a feedback derivative is used, analogous to a Lagrange multiplier. The original method for a single load case developed by these authors is extended to deal with multiple load cases. Additionally, a discussion of the global behaviour is included.     

Optimum dynamic design for beam shapes subjected to a moving concentrated load

We are concerned with an optimum dynamic design of beams subjected to a moving concentrated load with constant speed. The influence of the dynamic behaviour of the beams is considered in a proposed optimum design problem. The optimum shapes of beams are determined by the minimization of two kinds of performance indices. The optimization procedure is performed by non-linear programming on the basis of the exterior penalty function and BFGS methods. Optimization is calculated by the modal coordinates transformation and the numerical integration method     

Computerized modular ratio design of reinforced concrete members subjected to axial load and biaxial bending

A computerized method of analysis and design of reinforced concrete members of arbitrary cross-sections subjected to axial load and biaxial bending is proposed. The design process has been computerized to full automation—in the sense that given the concrete section and the applied loading, the program directly evaluates the amount of reinforcement required and the corresponding stress envelope. The strength of concrete in tension is neglected in the analysis. An iterative process which successively adjusts the section properties according to the stress state is employed. The design process is basically an iterative process of gradually increasing the amount of reinforcement till the permissible stresses are not exceeded. Reinforcement is added at locations of highest mean sequare stress in order to utilize fully the reinforcement added. Multiple loading cases are considered.     

An evolutionary shape optimization for elastic contact problems subject to multiple load cases

Most structures in the real life are subject to multiple load cases. This paper aims at extending the evolutionary structural optimization (ESO) algorithm to optimal contact shape design for elastic bodies under the multiple load cases. To evaluate the reference stresses of each contact node in a finite element framework, an extreme stress criterion (the worst case design) and a weighted average criterion (Pareto design) are presented. In the extreme stress method, the highest nodal contact stress under all load cases is adopted as the reference level. In the weighted average method, the weighted sum of nodal contact stresses over all the load cases is regarded as the reference. It is found that these two criteria can produce different results. In this paper, the examples are presented to demonstrate some new features of contact shape optimization in the presence of the multiple load cases.     

Stability of columns subjected to an intermediate concentrated load

In column stability studies where columns are subjected to intermediate concentrated loads, effects of shear deformation will be predominant even though the columns are slender when the loaded length of the column is appreciably small compared with its overall length. This aspect is studied in detail in this paper and realistic estimates for the stability parameters of columns subjected to intermediate concentrated loads are presented.     

Layout optimization for multi-bi-modulus materials system under multiple load cases

An optimization model is presented for obtaining optimal layout of multiple bi-modulus materials systems under multiple load cases (MLC). In the optimization model, the objective function is the linearly weighted structural compliance under MLC. The bi-modulus materials in a finite element are replaced by isotropic materials according to the stress state of that element. The equivalent mechanical properties of an element are expressed as the power–law function of the volume fractions (design variables) and moduli of the solid phases. Numerical experiments are presented to verify the validity and efficiency of the present algorithm. The effects of factors including the bi-modulus behavior of materials, the load directions and the weighting schemes of MLC are also investigated numerically.     

Nonlinear deformations of spherical panels subjected to transversal load action

Nonlinear behaviour of flexible shallow rectangular spherical panels subjected to a uniformly distributed transversal load is analysed. Detection of bifurcation points and construction of bifurcation branches on a ‘load–deflection’ characteristic are mainly addressed. The proposed algorithm is based on a set-up method. Geometrical parameters of shallow shells bifurcations are estimated numerically, and an evolution of bifurcation diagrams vs geometrical parameters k_x, k_y is traced. An increase of geometrical parameters yields an increase of the set of solutions. Besides symmetrical ones, unsymmetrical solutions are detected, illustrated, and discussed as well.     

Density-based topological design of structures subjected to water pressure using a parametric loading surface

Topological design of structures has enjoyed a period of steady growth since the publication of a seminal paper by Bendsøe and Kikuchi. Nowadays, topology design can be recognized as a discipline in structural and multidisciplinary optimization with its idiosyncrasies, specific formulations, and solution techniques. A class of problems, known as topological optimization with design-dependent loading (see Rozvany and Prager (1979) for first closed-form solutions and Fuchs and Moses (2000) for semianalytical results) present often some intrinsic obstacles. Such is the case, in particular, in the presence of pressure loads. Indeed, when water pressure is the applied load, the density-based technique seems to run into trouble because not until the structure has converged is there a clear water/material interface where the pressure can be applied. Several authors have proposed solutions, and this paper adds its contribution. We introduce, independent of the material distribution, a parametric loading surface, the shape of which is an additional unknown. Water pressure is applied at this surface as if it were the water/material interface. By doing so, one factually severs the physical link between the smooth pressure surface and the indistinct material distribution. In order to remove any lingering material from the water region, a small elastic modulus E is assumed there. Additional care was exercised for computing the sensitivities as the loading surface meanders through the fixed grid of the finite element model. For this purpose, a smooth transition for E from the water to the material zone is imposed to alleviate any numerical instabilities that may occur in computing the sensitivities of E over the abrupt transition at the loading surface. Several cases, primarily the design of optimal dams, were tested. The method proved very robust and produced crisp water/material interfaces, which coalesced with the loading surfaces.     

具有承载能力约束的集装箱装入问题的求解方法

对有承载能力约束的三维集装箱装入问题进行了描述,定义了货物承载约束的表示形式,并在全局寻优能力很强的集装箱装入的禁忌搜索算法中加入承载能力的计算和检查方法,设计了相关的处理策略.在优化空间利用率的同时减少或杜绝货物损坏现象的发生.实验结果表明,该算法对有承载能力约束的三维集装箱装入问题的有效性和实用性.     

Cover geometry design using multiple convex hulls

We present a design method to create close-fitting customized covers for given three-dimensional (3D) objects such as cameras, toys and figurines. The system first computes clustering of the input vertices using multiple convex hulls, then generates multiple convex hulls using the results. It then outputs a cover geometry to set union operation of these hulls, and the resulting intersection curves are set as seam lines. However, as some of the regions created are not necessarily suitable for flattening, the user can design seam lines by drawing and erasing. The system flattens the patches of the target cover geometry after segmentation, allowing the user to obtain a corresponding 2D pattern and sew the shapes in actual fabric. This paper’s contribution lies in its proposal of a clustering method to generate multiple convex hulls, i.e., a set of convex hulls that individually cover part of the input mesh and together cover all of it. The method is based on vertex clustering to allow handling of mesh models with poor vertex connectivity such as those obtained by 3D scanning, and accommodates conventional meshes with multiple connected components and point-based models with no connectivity information. Use of the system to design actual covers confirmed that it functions as intended.     

Digital filter design using multiple pareto fronts

Evolutionary approaches have been used in a large variety of design domains, from aircraft engineering to the designs of analog filters. Many of these approaches use measures to improve the variety of solutions in the population. One such measure is clustering. In this paper, clustering and Pareto optimisation are combined into a single evolutionary design algorithm. The population is split into a number of clusters, and parent and offspring selection, as well as fitness calculation, are performed on a per-cluster basis. The objective of this is to prevent the system from converging prematurely to a local minimum and to encourage a number of different designs that fulfil the design criteria. Our approach is demonstrated in the domain of digital filter design. Using a polar coordinate based pole-zero representation, two different lowpass filter design problems are explored. The results are compared to designs created by a human expert. They demonstrate that the evolutionary process is able to create designs that are competitive with those created using a conventional design process by a human expert. They also demonstrate that each evolutionary run can produce a number of different designs with similar fitness values, but very different characteristics.Part of the material presented in this paper was published in Third NASA Workshop on Evolvable Hardware (EH 2001), 12–14 July 2001, Long Beach, California, pp. 136–145This research is generously supported by a grant from Marconi, plc. The leadership and support of John Evans is gratefully acknowledged.     

Dynamic response of double-FG porous beam system subjected to moving load

In this paper, the vibration response of the double-FG porous beam system (DFGPBS) acted by a moving load is investigated. The DFGPBS composed of two parallel FG porous beams with their material properties varying along both the axial and transverse directions, i.e., bi-directional FG material distribution, is taken into account. The porous imperfection is simulated by distributing the porosity along the beam thickness with even and uneven patterns. The governing equations of this bi-directional DFGPBS under a moving load are established with the aid of the Hamilton principle associated with the Timoshenko beam theory. The Ritz method is adopted to discrete the differential governing equations, which are solved by the Newmark-β approach. The validation of the present model is performed by comparing the numerical results with two previous works. Then, the parametric study is carried out to investigate the influences of bi-directional gradient indices, porosity volume fraction, boundary conditions, stiffness of elastic layer, and velocity of the moving load on the vibration response of bi-directional DFGPBSs excited by a moving load. It is demonstrated that the vibration response of the double-beam system subjected to moving loads can be governed by tailoring the distribution of the bi-directional FG materials. The present work can be used to guide the multi-functional design of a double-beam system under dynamic loadings.

     

Probabilistic free vibration analysis of beams subjected to axial loads

A stochastic finite-element-based algorithm for the probabilistic free vibration analysis of beams subjected to axial forces is proposed in this paper through combination of the advantages of the response surface method, finite element method and Monte Carlo simulation. Uncertainties in the structural parameters can be taken into account in this algorithm. Three response surface models are proposed. Model I: star experiment design using a quadratic polynomial without cross-terms; Model II: minimum experiment design using a quadratic polynomial with cross-terms; Model III: composite experiment design using a quadratic polynomial with cross-terms.A separate set of finite element data is generated to verify the models. The results show that the Model II is the most promising one in view of its accuracy and efficiency. Probabilistic free vibration analysis of a simply supported beam is performed to investigate the effects of various parameters on the statistical moments of the frequency response of beams. It is found that the geometric properties of beams have significant effects on the variation of frequency response.     

Topology design of material layout in structured composites of high stiffness and strength

General continuous topology design formulations based on families of classical Voigt and Reuss mixing assumptions are developed and applied to solve the multiple material layout problem for the design of high stiffness/high strength composites. In the novel design framework, computational homogenization is employed to compute stiffness and strength characteristics of individual composite designs. Alternative design formulations for both high stiffness and high strength are investigated along with design sensitivity analysis algorithms. Demonstrative material design problems for boron-epoxy and graphite-epoxy composites are solved with robust sequential quadratic programming (SQP) techniques.     

Optimal design of rigid-plastic beams subjected to dynamical loading

In the present paper the optimization problem of dynamically loaded simply supported beams is handled. The concept of a rigid-plastic body is used. The shape of the beam is sought, for which the integral residual deflection for a given time-instant and load is minimal. Two numerical methods for solving the problem are proposed. The numerical results are compared with those obtained by Lepik (1982).