SHAPE OPTIMIZATION AND MINIMUM WEIGHT LIMIT DESIGN OF ARCHES

The paper is concerned with the optimization of arches, using classical beam finite elements, for the minimum elastic displacements and the minimum weight designs under ultimate loading conditions.

The concept of separate but dependent design spaces for node coordinates and member plastic capacities is introduced. The shape optimization problem, whereby a norm of the elastic displacement vector is minimized, is formulated in the space of node coordinate variables. Then the minimum weight limit design problem in the space of member plastic capacities is considered using the static theorem of limit analysis. An iterative procedure alternating these two approaches is presented in the paper. The nonlinear unconstrained optimization and linear programming techniques are used to solve the corresponding numerical problems. The proposed method is illustrated by numerical examples.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

轴心受压钢柱抗火设计的优化分析

基于"双重优化设计"概念给出了轴心受压钢柱抗火优化设计方案,利用有限元软件ANSYS的APDL语言及其优化设计模块对其进行了截面尺寸和抗火保护优化设计。分析中采用了两种优化设计方案,即在常温和一定临界温度条件下对轴心受压钢柱进行截面尺寸及抗火保护优化设计。通过对两种设计方案的优化结果进行比较和分析,得到了高温下轴心受压钢柱的抗火优化设计结果随温度变化的规律,给出了考虑经济成本条件下如何选择更加经济合理的优化设计方案的建议。分析结果表明,给出的优化方案可以得到安全、经济、可靠的设计结果,能够节省钢材用量和防火保护材料用量。     

Panel flutter optimization—a refined finite element approach

This paper presents the results of an aeroelastic optimization study. In this study the weight of a large aspect ratio panel immersed in high Mach number supersonic flow is minimized subject to the requirement that a critical aeroelastic parameter for flutter be held within specified limits. A refined finite element technique is used to model the panel flutter equilibrium equations, equations which act as constraints on the design search. The optimization mechanism itself is studied and discussed to provide qualitative results which may be useful to the solution of other aeroelastic optimization problems. The numerical results of this investigation are compared to converged numerical results to illustrate finite element accuracy. The results of the study show that the weight savings and material distribution found with an effective finite element model are comparable to those found by more complicated numerical methods.     

Design of ultrasonic array elements for acoustic power considerations

In sound-transmitting applications such as therapeutic ultrasound, the acoustic power at a particular operating frequency is a critical figure of merit for transducer/array design. A design methodology for enhancing the acoustic power radiated from fluid-loaded piezoelectric array elements at a fixed frequency is developed in this paper. A gradient-based optimization algorithm is integrated within the finite element framework to guide the determination of the two design variables, the piezoelectric element thickness and the matching layer thickness, to optimize the acoustic power output. A method for avoiding explicit remeshing in the optimization iteration is presented. Optimized designs are determined numerically, and the effectiveness of the design method is confirmed by experimental measurements. The validated numerical analysis also shows that conventional design strategies using one-dimensional transducer analysis and rule-of-thumb matching layer or protection layer sizing rules may not give the best design for array elements in acoustic power applications     

Design optimization using multiple dominance relations

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization.     

Optimum Structural and Manufacturing Design of a Braided Hollow Composite Part

Simultaneous material consolidation and shaping, as performed in manufacturing of composite materials, causes a strong interconnection between structural and manufacturing parameters which makes the design process complicated. In this paper, the design of a carbon fiber bicycle stem is examined through the application of a multi-objective optimization method to illustrate the interconnection between structural and manufacturing objectives. To demonstrate the proposed method, a test case dealing with the design of composite part with complex geometry, small size and hollow structure is described. Bladder-assisted Resin Transfer Molding is chosen as the manufacturing method. A finite element model of the stem is created to evaluate the objectives of the structural design, while a simplified 2D model is used to simulate the flow inside the preform during the injection process. Both models are formulated to take into account the variation of fiber orientation, thickness and fiber volume fraction as a function of braid diameters, injection pressure and bladder pressure. Finally, a multiobjective optimization method, called Normalized Normal Constraint Method, is used to find a set of solutions that simultaneously optimizes weight, filling time and strength. The solution to the problem is a set of optimum designs which represent the Pareto frontier of the problem. Pareto frontier helps to gain insight into the trade-off among objectives, whose presence and importance is confirmed by the numerical results presented in this paper.     

Using multicriterion optimization for strength design of composite laminates

Using multicriterion optimization methods for weight minimization of failure strength controlled composite structures is described. A composite lay-up design problem for minimizing the number of layers under strength constraints with respect to multiple loading conditions is formulated. The constrained problem is transferred into a sequence of unconstrained problems and solved with an interactive descent method. A typical design cycle that comprises the finite element mesh generator and solver as well as the laminate analysis and optimization modules is illustrated with a numerical example.     

Metamaterial topology optimization of nonpneumatic tires with stress and buckling constraints

This article presents the design of a metamaterial for the shear layer of a nonpneumatic tire using topology optimization, under stress and buckling constraints. These constraints are implemented for a smooth maximum function using global aggregation. A linear elastic finite element model is used, implementing solid isotropic material with penalization. Design sensitivities are determined by the adjoint method. The method of moving asymptotes is used to solve the numerical optimization problem. Two different optimization statements are used. Each requires a compliance limit and some aspect of continuation. The buckling analysis is linear, considering the generalized eigenvalue problem of the conventional and stress stiffness matrices. Various symmetries, base materials, and starting geometries are considered. This leads to novel topologies that all achieve the target effective shear modulus of 10 MPa, while staying within the stress constraint. The stress-only designs generally were susceptible to buckling failure. A family of designs (columnar, noninterconnected representative unit cells) that emerge in this study appears to exhibit favorable properties for this application.     

Horsetail matching: a flexible approach to optimization under uncertainty*

It is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated.     

Optimal design of gating systems by gradient search methods

A numerical optimization technique based on gradient-search is applied to obtain an optimal design of a typical gating system used for the gravity process to produce aluminum parts. This represents a novel application of coupling nonlinear optimization techniques with a foundry process simulator, and it is motivated by the fact that a scientifically guided search for better designs based on techniques that take into account the mathematical structure of the problem is preferred to commonly found trial-and-error approaches. The simulator applies the finite volume method and the VOF algorithm for CFD analysis. The direct gradient optimization algorithm, sequential quadratic programming (SQP), was used to solve both a 2D and a 3D gating system design problems using two design variables. The results clearly show the effectiveness of the proposed approach for finding high quality castings when compared with current industry practices.     

A PROCEDURE TO SELECT THE BEST MATERIAL COMBINATIONS AND OPTIMALLY DESIGN HYBRID COMPOSITE PLATES FOR MINIMUM WEIGHT AND COST

The optimal layup with least weight or cost for a symmetrically laminated plate subject to a buckling load is determined using a hybrid composite construction. A hybrid construction provides further tailoring capabilities and can meet the weight, cost and strength constraints while a non-hybrid construction may fail to satisfy the design requirements. The objective of the optimization is to minimize either the weight or cost of the plate using the ply angles, layer thicknesses and material combinations as design variables. As the optimization problem contains a large number of continuous (ply angles and thicknesses) and discrete (material combinations) design variables, a -sequential solution procedure is devised in which the optimal variables are computed in different stages. The proposed design method is illustrated using graphite, kevlar and glass epoxy combinations and the efficiencies of the hybrid designs over the non-hybrid ones are computed.     

OPTIMUM COST DESIGN OF A STEEL-FRAMED BUILDING

The paper describes an investigation into the minimum cost design of a steel portal framed building. Computer programmes have been developed which permit:1)The simple “plastic” design of this type of structure in accordance with the relevant B.S. Codes and common industrial practice.

2)The costing of any “feasible” design in accordance with a cost model devised in conjunction with several members of the steelwork fabrication industry.3)The economical determination of the “least cost” design from the hyperspace of “feasible” designs.It has been assumed that the optimum shape of a structure (i.e. the specific combination of problem variables which produces the design of least cost) is a function of material and fabrication costs only. Accordingly, for any feasible design, the cost is defined as the sum of the individual material and fabrication costs for the structural steelwork, purlins and cladding.

A model has been devised in which the problem has been formulated as a constrained non-linear optimization problem. The independent variables included in the model are: number of bays, frame spacing, eaves height, roof pitch, purlin spacing, building length and overall building width. Constraints are introduced by a consideration of site limitations, floor area and/or building volume requirements. Individual member selection is treated as a sub-optimization problem using simple “plastic” design. Using appropriate combinations of the independent variables any number of “feasible” designs may be generated.

The “search” through the feasible design space for the least cost structure is accomplished using the non-linear variable metric simplex method. Methods by which the initial simplex is generated and subsequent “improved” designs are found, is discussed.

Results illustrating the behaviour of the significant problem variables are presented in the form of cost response surfaces.     

Structural optimization to include material selection

The mini-max dual method ia applied for solving the material selection structural optimization problems. A somewhat general objective function is introduced to consider structural weight and material cost simultaneously. It is expressed by the sum of scaled structural weight and material cost. The present formulation can yield minimum cost and minimum weight designs as two distinct special cases. It is shown that practically useful intermediate optimal designs can exist between the two extremes. Only truss structures are considered as a representative example. One numerical example is provided to illustrate the change in optimal material distribution when various objective functions are employed. A particular advantage in using the dual method for the material selection problem is also described.     

Topology synthesis of large‐displacement compliant mechanisms

This paper describes the use of topology optimization as a synthesis tool for the design of large‐displacement compliant mechanisms. An objective function for the synthesis of large‐displacement mechanisms is proposed together with a formulation for synthesis of path‐generating compliant mechanisms. The responses of the compliant mechanisms are modelled using a total Lagrangian finite element formulation, the sensitivity analysis is performed using the adjoint method and the optimization problem is solved using the method of moving asymptotes. Procedures to circumvent some numerical problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Multimaterial topology design for optimal elastic and thermal response with material-specific temperature constraints

We present an original method for multimaterial topology optimization with elastic and thermal response considerations. The material distribution is represented parametrically using a formulation in which finite element–style shape functions are used to determine the local material properties within each finite element. We optimize a multifunctional structure that is designed for a combination of structural stiffness and thermal insulation. We conduct parallel uncoupled finite element analyses to simulate the elastic and thermal response of the structure by solving the two-dimensional Poisson problem. We explore multiple optimization problem formulations, including structural design for minimum compliance subject to local temperature constraints so that the optimized design serves as both a support structure and a thermal insulator. We also derive and implement an original multimaterial aggregation function that allows the designer to simultaneously enforce separate maximum temperature thresholds based upon the melting point of the various design materials. The nonlinear programming problem is solved using gradient-based optimization with adjoint sensitivity analysis. We present results for a series of two-dimensional example problems. The results demonstrate that the proposed algorithm consistently converges to feasible multimaterial designs with the desired elastic and thermal performance.     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

Design of dissipative multimaterial viscoelastic-hyperelastic systems at finite strains via topology optimization

This study focuses on the topology optimization framework for the design of multimaterial dissipative systems at finite strains. The overall goal is to combine a soft viscoelastic material with a stiff hyperelastic material for realizing optimal structural designs with tailored damping and stiffness characteristics. To this end, several challenges associated with incorporating finite-deformation viscoelastic-hyperelastic materials in a multimaterial design framework are addressed. This includes consideration of a thermodynamically consistent finite-strain viscoelasticity model for simulating energy dissipation together with F-bar finite elements for handling material incompressibility. Moreover, an effective multimaterial interpolation scheme is proposed, which preserves the physics of material mixtures in the context of density-based topology optimization. A numerically accurate analytical design sensitivity calculation is also presented using a path-dependent adjoint method. Furthermore, both prescribed-load and prescribed-displacement boundary conditions are considered in the optimization formulations, together with various strategies for controlling stiffness. As demonstrated by the numerical examples, the use of the stiffer hyperelastic material phase in a design not only improves stiffness but also increases energy dissipation capacity. Moreover, with the finite-deformation theory, the effect of the loading magnitude on the optimized designs can be observed.     

Optimum detailed design of reinforced concrete frames using genetic algorithms

This article presents the application of the genetic algorithm to the optimum detailed design of reinforced concrete frames based on Indian Standard specifications. The objective function is the total cost of the frame which includes the cost of concrete, formwork and reinforcing steel for individual members of the frame. In order for the optimum design to be directly constructible without any further modifications, aspects such as available standard reinforcement bar diameters, spacing requirements of reinforcing bars, modular sizes of members, architectural requirements on member sizes and other practical requirements in addition to relevant codal provisions are incorporated into the optimum design model. The produced optimum design satisfies the strength, serviceability, ductility, durability and other constraints related to good design and detailing practice. The detailing of reinforcements in the beam members is carried out as a sub-level optimization problem. This strategy helps to reduce the size of the optimization problem and saves computational time. The proposed method is demonstrated through several example problems and the optimum results obtained are compared with those in the available literature. It is concluded that the proposed optimum design model can be adopted in design offices as it yields rational, reliable, economical, time-saving and practical designs.     

Response surface-based robust parameter design optimization with both qualitative and quantitative variables

The response surface-based robust parameter design, with its extensive use of optimization techniques and statistical tools, is known as an effective engineering design methodology for improving production processes, when input variables are quantitative on a continuous scale. In many engineering settings, however, there are situations where both qualitative and quantitative variables are considered. In such situations, traditional response surface designs may not be effective. To rectify this problem, this article lays out a foundation by embedding those input variables into a factorial design with pseudo-centre points. A 0–1 mixed-integer nonlinear programming model is then developed and the solutions found using three optimization tools, namely the outer approximation method, the branch-and-bound technique and the hybrid branch-and-cut algorithm, are compared with traditional counterparts. The numerical example shows that the proposed models result in better robust parameter design solutions than the traditional models.