Multi-disciplinary design optimization of a combustor

A technique for design optimization of a combustor is presented. This technique entails the use of computational fluid dynamics (CFD) and mathematical optimization to minimize the combustor exit temperature profile. The empirical and semi-empirical correlations commonly used for optimizing combustor exit temperature profile do not guarantee optimum. As an experimental approach is time consuming and costly, use is made of numerical techniques. However, using CFD without mathematical optimization on a trial and error basis does not guarantee optimal solutions. A better approach, which is often viewed as too expensive, is a combination of the two approaches, thus incorporating the influence of the variables automatically. In this study the combustor exit temperature profile is optimized. The optimum (uniform) combustor exit temperature profile mainly depends on the geometric parameters. Combustor parameters have been used as optimization variables. The combustor investigated is an experimental liquid-fuelled atmospheric combustor with a turbulent diffusion flame. The CFD simulations use the Fluent code with a standard k–? model. The optimization is carried out using the Dynamic-Q algorithm, which is specifically designed to handle constrained problems where the objective and constraint functions are expensive to evaluate. The optimization leads to a more uniform combustor exit temperature profile compared with the original.     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

Design, optimization and manufacturing of wood–plastic composite pallet

This paper presents the application of an innovative method of optimization to the design of an I-shape profile used in a wood–plastic composite (WPC) pallet. The pallet was made via assembling three WPC extruded profiles manufactured in the extrusion process. The middle profile was considered to be I-shaped, a design which known to have a high load bearing capability. However, due to the characteristics of WPC products, a delicate design and thus optimization is highly required. A multi-objective-optimization program of micro-genetic algorithm was developed in Visual Basic environment to accomplish the optimization task. By specifying the dimensional variables of the profile section and applying finite elements analysis on the profile and then using the optimization program, an optimal profile section was obtained. The objective was to withstand the maximum load while yielding the minimum deflection and mass. The optimized design was used to manufacture a die and then the product was produced to validate the design. The comparison of simulations and experimental results indicted that the given design method is reasonably reliable. The final mass of the produced pallet was less than 20 kg whereas its strength against bending and distributed smooth restraint loading were greater than 500 kg and 2000 kg, respectively.     

Constrained minimization of vibrational magnitudes using a modal reduction method

A numerically effective method is suggested and applied for evaluating objective and constraint functions when so-called vibrational magnitudes of a mechanical structure are minimized. General damped linear structures under external harmonic loading are considered. The magnitude functions studied can relate to displacements, velocities and accelerations and also to sectional and reactive forces. Both magnitudes at a specific frequency and peak magnitudes and averaged magnitudes over a frequency range are investigated. An arbitrary set of magnitude functions can be used in the constraints. Design variables are masses, dampings and stiffnesses of discrete and discretized continuous elements contained in the structure. The objective and constraint functions are expressed by use of the modal parameters (generally complex-valued) of the structural system. A reduced modal model is established and updated during the optimization process. Approximate derivatives (sensitivities) of the objective and constraint functions with respect to changes in design variables are calculated employing perturbed modal parameters. The optimization problem is solved by use of a primal method. Numerical examples demonstrate applications to the classical damped vibration absorber with two design variables and to a beam system used in a light-weight machine foundation with 14 design variables.     

概率及非概率不确定性条件下结构鲁棒设计方法

提出了在概率不确定性和非概率不确定性同时存在时的约束函数鲁棒性和目标函数鲁棒性的实现策略及结构鲁棒设计方法。将传统优化设计问题的约束条件改造成为能同时反映两类不确定性量波动变化影响的约束条件,以实现约束函数的鲁棒性;在传统优化设计问题目标函数中增加若干个关于目标函数灵敏度的新目标函数,构成一个多目标函数设计问题,以实现目标函数的鲁棒性。所提方法应用于一个10杆桁架结构设计,采用宽容排序法求解。计算结果表明,在相同的结构总质量限制条件下,目标函数鲁棒性程度随着变量不确定性程度的增加而降低;在相同的变量不确定性程度条件下,增加结构总质量能提高目标函数鲁棒性的程度。     

Shape optimization of flow guides in three‐dimensional extrusion processes by an approximation scheme based on state variable linearization

A new scheme of shape optimization is applied to the design of a flow guide in flat‐die extrusion processes. In general, tremendous time is inevitably required for the optimization of large‐scale three‐dimensional extrusion processes. This is because the finite element analysis requires large computation time owing to the complexity of the die geometry and flow behaviour. The proposed scheme effectively reduces the computation time for the optimization process by approximating the objective function. This is achieved by introducing a transformed equation of the state variables. The scheme is then applied to the practical extrusion processes to produce ‘l’, ‘H’ and ‘L’ sections. The objective of the optimization is to make a balanced flow of the material in the exit region. Control points of a Bezier curve describing the outline of the flow guide are regarded as the design variables. Through application to large‐scale problems, the effectiveness and usefulness of the proposed scheme is demonstrated. Copyright © 2005 John Wiley & Sons, Ltd.     

Process and die design for rod extrusion of γ iron

This study is related to materials modeling and die and process design of rod extrusion of γ iron. Strain dependent rate power law is used for materials modeling whose coefficients are arrived at through genetic algorithm (GA). Die profile of the rod extrusion process is optimized to produce products of desirable microstructure at maximum production speed and minimum left out material in the die. The design problem is formulated as a nonlinear programming problem which is solved using GA. Selection of the processing parameters is carried out using dynamic materials modeling (DMM). Using this approach rod extrusion process of γ iron is successfully designed. FE simulation on the optimum profile is also attempted to study deformation behaviour and load requirement.     

Multi-objective Optimization of Sheet Metal Forming Die Using Genetic Algorithm Coupled with RSM and FEA

Present study describes the approach of applying response surface methodology (RSM) with a Pareto-based multi-objective genetic algorithm to assist engineers in optimization of sheet metal forming. In many studies, finite element analysis and optimization technique have been integrated to solve the optimal process parameters of sheet metal forming by transforming multi-objective problem into a single-objective problem. This paper aims to minimize objective functions of fracture and wrinkle simultaneously. Design variables are blank-holding force and draw-bead geometrical parameters (length and diameter). RSM has been used for design of experiment and finding relationship between variables and objective functions. Forming limit diagram has been used to define objective functions. Finite element analysis applied for simulating the process. Proposed approach has been investigated on a fuel tank drawing part and it has been observed that it is more effective and accurate than traditional finite element analysis method and the “trial and error” procedure.     

Developing a fuzzy proportional–derivative controller optimization engine for engineering design optimization problems

In real world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as purely mathematical. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all. In this article, the optimization process is analogous to a closed-loop control system, and a fuzzy proportional–derivative (PD) controller optimization engine is developed for engineering design optimization problems with monotonicity and implicit constraints. Monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. In this research, monotonicity of the design variables and activities of the constraints determined by the theory of monotonicity analysis are modelled in the fuzzy PD controller optimization engine using generic fuzzy rules. The designer only needs to define the initial values and move limits of the design variables to determine the parameters in the fuzzy PD controller optimization engine. In the optimization process using the fuzzy PD controller optimization engine, the function value of each constraint is evaluated once in each iteration. No sensitivity information is required. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested.     

Reliability-based optimal structural design by the decoupling approach

A decoupling approach for solving optimal structural design problems involving reliability terms in the objective function, the constraint set or both is discussed and extended. The approach employs a reformulation of each problem, in which reliability terms are replaced by deterministic functions. The reformulated problems can be solved by existing semi-infinite optimization algorithms and computational reliability methods. It is shown that the reformulated problems produce solutions that are identical to those of the original problems when the limit-state functions defining the reliability problem are affine. For nonaffine limit-state functions, approximate solutions are obtained by solving series of reformulated problems. An important advantage of the approach is that the required reliability and optimization calculations are completely decoupled, thus allowing flexibility in the choice of the optimization algorithm and the reliability computation method.     

Locally weighted regression models for surrogate-assisted design optimization

We consider engineering design optimization problems where the objective and/or constraint functions are evaluated by means of computationally expensive blackboxes. Our practical optimization strategy consists of solving surrogate optimization problems in the search step of the mesh adaptive direct search algorithm. In this paper, we consider locally weighted regression models to build the necessary surrogates, and present three ideas for appropriate and effective use of locally weighted scatterplot smoothing (LOWESS) models for surrogate optimization. First, a method is proposed to reduce the computational cost of LOWESS models. Second, a local scaling coefficient is introduced to adapt LOWESS models to the density of neighboring points while retaining smoothness. Finally, an appropriate order error metric is used to select the optimal shape coefficient of the LOWESS model. Our surrogate-assisted optimization approach utilizes LOWESS models to both generate and rank promising candidates found in the search and poll steps. The “real” blackbox functions that govern the original optimization problem are then evaluated at these ranked candidates with an opportunistic strategy, reducing CPU time significantly. Computational results are reported for four engineering design problems with up to six variables and six constraints. The results demonstrate the effectiveness of the LOWESS models as well as the order error metric for surrogate optimization.     

PREFORM DIE SHAPE DESIGN IN METAL FORMING USING AN OPTIMIZATION METHOD

An optimization algorithm for preform die shape design in metal-forming processes is developed in this paper. The preform die shapes are represented by cubic B-spline curves. The control points of the B-spline are used as the design variables. The optimization objective is to reduce the difference between the realized and desired final forging shapes. The sensitivities of the objective function with respect to the design variables are developed in detail. The numerical examples show that the optimization method and the sensitivity analysis developed in this paper are very useful and the design results are satisfactory. Importantly, the preform die shapes designed by this method are easily manufacturable and can be implemented in practical metal-forming operations. This optimization method and the sensitivity analysis can also be applied in the preform design of complex industrial metal-forming problems. © 1997 by John Wiley & Sons, Ltd.     

Sensitivity Analysis Based Multiple Objective Preform Die Shape Optimal Design in Metal Forging

The multiple objective preform design optimization was put forward. The final forging＇s shape and deformation uniformity were considered in the multiple objective. The objective is to optimize the shape and the deformation uniformity of the final forging at the same time so that a more high integrate quality of the final forging can be obtained. The total objective was assembled by the shape and uniformity objective using the weight adding method. The preform die shape is presented by cubic B-spline curves. The control points of B-spline curves are used as the design variables. The forms of the total objective function, shape and uniformity sub-objective function are given. The sensitivities of the total objective function and the sub-objective functions with respect to the design variables are developed. Using this method, the preform die shape of an H-shaped forging process is optimally designed. The optimization results are very satisfactory.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

An uncertain multidisciplinary design optimization method using interval convex models

This article proposes an uncertain multi-objective multidisciplinary design optimization methodology, which employs the interval model to represent the uncertainties of uncertain-but-bounded parameters. The interval number programming method is applied to transform each uncertain objective function into two deterministic objective functions, and a satisfaction degree of intervals is used to convert both the uncertain inequality and equality constraints to deterministic inequality constraints. In doing so, an unconstrained deterministic optimization problem will be constructed in association with the penalty function method. The design will be finally formulated as a nested three-loop optimization, a class of highly challenging problems in the area of engineering design optimization. An advanced hierarchical optimization scheme is developed to solve the proposed optimization problem based on the multidisciplinary feasible strategy, which is a well-studied method able to reduce the dimensions of multidisciplinary design optimization problems by using the design variables as independent optimization variables. In the hierarchical optimization system, the non-dominated sorting genetic algorithm II, sequential quadratic programming method and Gauss–Seidel iterative approach are applied to the outer, middle and inner loops of the optimization problem, respectively. Typical numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

Reducing dimensionality in topology optimization using adaptive design variable fields

Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright © 2009 John Wiley & Sons, Ltd.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

Redundancy allocation for multi-state systems using physical programming and genetic algorithms

This paper proposes a multi-objective optimization model for redundancy allocation for multi-state series–parallel systems. This model seeks to maximize system performance utility while minimizing system cost and system weight simultaneously. We use physical programming as an effective approach to optimize the system structure within this multi-objective optimization framework. The physical programming approach offers a flexible and effective way to address the conflicting nature of these different objectives. Genetic algorithm (GA) is used to solve the proposed physical programming-based optimization model due to the following three reasons: (1) the design variables, the number of components of each subsystems, are integer variables; (2) the objective functions in the physical programming-based optimization model do not have nice mathematical properties, and thus traditional optimization approaches are not suitable in this case; (3) GA has good global optimization performance. An example is used to illustrate the flexibility and effectiveness of the proposed physical programming approach over the single-objective method and the fuzzy optimization method.     

Die shape optimal design in three-dimensional shape metal extrusion by the finite element method

A new approach to die shape optimal design in shape extrusion is presented. In this approach, the design problem is formulated as an optimization problem incorporating the three-dimensional finite element analysis model, and optimization of the die shape is conducted on the basis of the design sensitivities. The approach is applied to the determination of the die shapes for extrusion of parts with various cross sections including polygons and T sections. © 1998 John Wiley & Sons, Ltd.     

基于遗传算法汽车动力总成悬置系统解耦优化

为避免传统优化算法在对汽车动力总成悬置系统优化中陷入局部最优解,采用遗传算法对其进行优化。在深入分析设计变量选取、约束函数的提取及目标函数的选取原则基础上,以悬置刚度为优化变量、固有频率的范围和固有频率之差为约束函数、六自由度方向的解耦率为目标函数,利用MATLAB平台的遗传算法进行优化。开发基于遗传算法汽车动力总成悬置系统解耦优化系统,并对某型号汽车动力总成系统优化。优化结果表明:系统的固有频率的分配和解耦率得到极大的改善,效率和精度都得到很大的提升。