Sensitivity analysis and shape optimization for preform design in thermo‐mechanical coupled analysis

In complex forging processes, it is essential to find the optimal deformation path and the optimal preform shape that will lead to the desired final shape and service properties. A sensitivity analysis and optimization for preform billet shape in thermo‐mechanical coupled simulation is developed in this work. Non‐linear sensitivity analysis of temperatures, flow‐stresses, strains and strain‐rates are presented with respect to design variables. Both analytical and finite‐difference gradients are employed to validate the effectiveness of sensitivity analysis developed in this work. Numerous iterations of coupled thermo‐mechanical analysis are performed to determine an optimum preform shape based on a given criterion of minimizing the objective function on effective strain variance within the final forging. The design constraints are imposed on die underfill, material scrap, folding defects and temperatures. In addition, a method for material data processing is given in order to determine the flow stress and its derivatives. The shape optimization scheme is demonstrated with the preform designs of an axisymmetric disk and a plane strain problem. Copyright © 1999 John Wiley & Sons, Ltd.     

结构-声场耦合系统声压响应优化设计研究

为了减小结构振动产生的噪声,基于结构-声场耦合有限元模型,对其在简谐力激励下的声响应进行了计算。在此基础上,以结构厚度为设计变量,保持结构重量相对不变并以设计变量的上、下限为约束,以设计域点在各频率响应声压平方和的平方根为目标函数,对其进行了优化设计研究。用软件N astran计算声响应及声敏度,基于iS IGHT对N astran的集成用可行方向法实施了优化。以一矩形板结构-立方体声场耦合系统为实例,得到了优化的结构厚度分布。优化后,设计域点的峰值声压级均有不同程度的降低,声压级降低最大值达32.8dB。结果表明声压响应优化设计,在保持结构重量相对不变的情况下,通过对结构重量的重新分布能达到较明显的降噪效果。     

A unified parametric design approach to structural shape optimization

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given:     

Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

Sensitivity analysis of viscoplastic deformation process with application to metal preform design optimization

The sensitivity analysis of rigid viscoplastic deformation processes with application to metal preform design optimization is investigated. For viscoplastic constitutive models, the deformation process is path-dependent in nature and thus the sensitivity analysis of the deformation history is formulated in an incremental procedure. To this end, an algorithm is derived on the basis of the time integration scheme used in the primary finite element analysis, where the contact conditions are treated with the penalty method. The discretized equilibrium equations, as well as the time integration equations, are directly differentiated with respect to the design variables. The discrete form of the sensitivity equations is then solved with procedures similar to those used in the direct analysis, where the secant matrix decomposed in the direct analysis can also be utilized at each time instant. Thus the sensitivity of the deformation history is evaluated in a step-wise procedure. The present algorithm can be employed for the optimization of metal forming processes. The accuracy of the proposed sensitivity analysis as well as its applicability are demonstrated by numerical examples with reference to preform design optimization problems, where the aggregate function method is employed for converting the non-smooth Min–max type objective function into a numerically tractable one.     

飞机总体设计中的一种多目标鲁棒优化方法

 提出一种基于灵敏度的多目标鲁棒优化方法。针对各维设计变量存在扰动的情况,在原约束多目标优化模型上,附加偏差目标函数,并采用最差估计法对约束条件进行鲁棒可行性调整。采用全局敏度方程方法来计算目标函数和约束函数对设计变量的敏度,进而采用Pareto遗传算法搜索约束多目标优化问题的非劣解集,设计者可以根据不同的设计准则从中选择合适的设计点。将上述方法用于飞机总体参数优化设计,并与采用常规优化方法所得的优化结果进行了分析和比较。     

Variational design sensitivity analysis in the context of structural optimization and configurational mechanics

Variational design sensitivity analysis is a branch of structural optimization. We consider variations of the material configuration and we are interested in the change of the state variables and the objective functional due to these variations. In the same manner in configurational mechanics we are interested in changes of the material body. In this paper, we derive the physical and material residual problem by using standard optimization procedures and we investigate sensitivity relations for the physical and material problem. These sensitivity relations are used in order to solve the coupled physical and material problem. Both problems are coupled by the pseudo load operator, which play an important role for the solution of structural optimization problems. Furthermore, we derive explicit formulations for the variations of the physical and material problem and propose different solution algorithms for the coupled problem.     

A continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design problems

A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the differentiation of the governing field equations of the direct problem and development of weak forms for the corresponding field sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non‐differentiability of the contact conditions, appropriate regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity fields with respect to various process parameters including the die surface. The accuracy and effectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a Bézier representation of the die curve is introduced. The control points of the Bézier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a finite difference approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the final product or minimize the required extrusion force for a given reduction ratio. An open‐forging die is also designed which for a specified stroke and initial workpiece produces a final product of desired shape. Copyright © 2000 John Wiley & Sons, Ltd.     

Multidisciplinary design optimization based on independent subspaces

Optimization has been successfully applied to systems with a single discipline. Since many disciplines are involved in a coupled fashion in modern engineering, multidisciplinary design optimization (MDO) technology has been developed. MDO algorithms are designed to solve the coupled aspects generated from the interdisciplinary relationship. In a general MDO algorithm, a large design problem is decomposed into smaller ones which can be easily solved. Although various methods have been proposed for MDO, research is still in the early stage. This study proposes a new MDO method which is named MDO based on independent subspaces (MDOIS). Many real engineering problems consist of physically separate components and they can be independently designed. The inter‐relationship occurs through coupled physics. MDOIS is developed for such problems. In MDOIS, a large system is decomposed into small subsystems. The coupled aspects are solved via system analysis which solves the coupled physics. The algorithm is mathematically validated by showing that the solution satisfies the Karush–Kuhn–Tucker condition. Copyright © 2005 John Wiley & Sons, Ltd.     

Geometrical design of thermoelectric generators based on topology optimization

This paper discusses an application of the topology optimization method for the design of thermoelectric generators. The proposed methodology provides the optimized geometry in accordance with various arbitrary conditions such as the types of materials, the volume of materials, and the temperature and shape of the installation position. By considering the coupled equations of state for the thermoelectric problem, we introduce an analytical model subject to these equations, which mimics the closed circuit composed of thermoelectric materials, electrodes, and a resistor. The total electric power applied to the resistor and the conversion efficiency are formulated as objective functions to be optimized. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization method used in topology optimizations. Simple relationships are formulated between the density function of the solid isotropic material with penalization method and the physical properties of the thermoelectric material. A sensitivity analysis for the objective functions is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes. Finally, numerical examples are provided to demonstrate the validity of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Sensitivity analysis and design optimization of three‐dimensional non‐linear aeroelastic systems by the adjoint method

We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.     

Evolutionary topology optimization using design space adjustment based on fixed grid

Design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown. In the conventional topology optimization, the design domain is fixed. It is, however, desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables, which we call design space optimization. A breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background, but due to coupling effect and others, sensitivity results have not been satisfactory. Three innovative implementations are developed in this paper. Firstly, the proper characteristics of artificial material are defined. The second one is to decouple neighbouring elements for exact design space sensitivities. The previous design space optimization has been tedious because only one layer can be added. So, the third innovation is a new expansion strategy with multi‐layers based on design space sensitivities. As a result, the proposed evolutionary method can get an optimum much faster than ever before especially for large‐scale problems. It is also conjectured that this gives higher probability of getting the global optimum, as confirmed by numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Truss layout design under nonprobabilistic reliability-based topology optimization framework with interval uncertainties

A nonprobabilistic reliability-based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown-but-bounded uncertainties) is proposed in this paper. The cross-sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single-loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.     

Integrated multi-objective robust optimization and sensitivity analysis with irreducible and reducible interval uncertainty

Uncertainty considered in robust optimization is usually treated as irreducible since it is not reduced during an optimization procedure. In contrast, uncertainty considered in sensitivity analysis is treated as partially or fully reducible in order to quantify the effect of input uncertainty on the outputs of the system. Considering this, and the usual existence of both reducible and irreducible uncertainty, an approach that can perform robust optimization and sensitivity analysis simultaneously is of much interest. This article presents such an integrated optimization model that can be used for both robust optimization and sensitivity analysis for problems that have irreducible and reducible interval uncertainty, multiple objective functions and mixed continuous-discrete design variables. The proposed model is demonstrated by two engineering examples with differing complexity to demonstrate its applicability.     

Design space optimization using a numerical design continuation method

A generalized optimization problem in which design space is also a design to be found is defined and a numerical implementation method is proposed. In conventional optimization, only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted. A design space is specified by the number of design variables, and the layout or configuration. To solve this type of design space problems, a simple initial design space is selected and gradually improved while the usual design variables are being optimized. To make the design space evolve into a better one, one may increase the number of design variables, but, in this transition, there are discontinuities in the objective and constraint functions. Accordingly, the sensitivity analysis methods based on continuity will not apply to this discontinuous stage. To overcome the difficulties, a numerical continuation scheme is proposed based on a new concept of a pivot phase design space. Two new categories of structural optimization problems are formulated and concrete examples shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

Die shape design optimization of sheet metal stamping process using meshfree method

A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity‐based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum‐based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation‐dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring‐back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.     

The continuous adjoint approach to the k–ω SST turbulence model with applications in shape optimization

In this article, the gradient of aerodynamic objective functions with respect to design variables, in problems governed by the incompressible Navier–Stokes equations coupled with the k–ω SST turbulence model, is computed using the continuous adjoint method, for the first time. Shape optimization problems for minimizing drag, in external aerodynamics (flows around isolated airfoils), or viscous losses in internal aerodynamics (duct flows) are considered. Sensitivity derivatives computed with the proposed adjoint method are compared to those computed with finite differences or a continuous adjoint variant based on the frequently used assumption of frozen turbulence; the latter proves the need for differentiating the turbulence model. Geometries produced by optimization runs performed with sensitivities computed by the proposed method and the ‘frozen turbulence’ assumption are also compared to quantify the gain from formulating and solving the adjoint to the turbulence model equations.     

Application of multiple objective optimization theory to a preliminary missile design study

In this large-scale project, the design tasks were divided into two parts which allowed for increased efficiency. The modelling and generation of optimal candiatte missile designs were separated from the preference-laden design selection process. A set of performance indices including range, cost, trajectory error, system susceptibility, and reliability were established as the system measures with which all the subsystems were subsequently optimized. A team of technical designers used computer-based submodels in the areas of propulsion, lifecycle cost, and system susceptibility along with analytical models in the areas of trajectory analysis and system reliability to construct a statespace model. The system statespace included a set of established constants, and it mathematically linked the various subsystems to the performance measures. This model was then nested inside a vector optimization algorithm on a digital computer. The output of this interactive computer program is a set of efficient or nondominated missile designs. Each design is defined by its set of state variables, and accompanied by a set of performance index scores and control variables which define its particular trajectory. These nondominated designs show explicitly the tradeoffs among the performance indices for the missile systems as one moves along the efficient frontier of designs. Additional sensitivity analysis is provided by the optimization software for each efficient design. The second phase of the design process consisted of identifying one missile design from the efficient set for further development. This identification process results from ranking of the efficient designs according to a scalar scoring function which relies on the decision maker's preferences.