Optimal design of composite shells based on minimum weight and maximum feasibility robustness

A robust design optimization (RDO) approach for minimum weight and safe shell composite structures with minimal variability into design constraints under uncertainties is proposed. A new concept of feasibility robustness associated to the variability of design constraints is considered. So, the feasibility robustness is defined through the determinant of variance–covariance matrix of constraint functions introducing in this way the joint effects of the uncertainty propagations on structural response. A new framework considering aleatory uncertainty into RDO of composite structures is proposed. So, three classes of variables and parameters are identified: deterministic design variables, random design variables and random parameters. The bi-objective optimization search is performed using on a new approach based on two levels of dominance denoted by Co-Dominance-based Genetic Algorithm (CoDGA). The use of evolutionary concepts together sensitivity analysis based on adjoint variable method is a new proposal. The examples with different sources of uncertainty show that the Pareto front definition depends on random design variables and/or random parameters considered in RDO. Furthermore, the importance to control the uncertainties on the feasibility of constraints is demonstrated. CoDGA approach is a powerfully tool to help designers to make decision establishing the priorities between performance and robustness.     

Worst‐case robust design optimization under distributional assumptions

Presented in this paper is a novel robust design optimization (RDO) methodology. The problem is reformulated in order to relax, when required, the assumption of normality of objectives and constraints, which often underlies RDO. In the second place, taking into account engineering considerations concerning the risk associated with constraint violation, suitable estimates of tail conditional expectations are introduced in the set of robustness metrics. A computationally affordable yet accurate implementation of the proposed formulation is guaranteed by the adoption of a reduced quadrature technique to perform the uncertainty propagation. The methodology is successfully demonstrated with the aid of an industrial test case performing the sizing of a mid‐range passenger aircraft. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Structural shape and topology optimization of cast parts using level set method

This paper presents a level set‐based shape and topology optimization method for conceptual design of cast parts. In order to be successfully manufactured by the casting process, the geometry of cast parts should satisfy certain moldability conditions, which poses additional constraints in the shape and topology optimization of cast parts. Instead of using the originally point‐wise constraint statement, we propose a casting constraint in the form of domain integration over a narrowband near the material boundaries. This constraint is expressed in terms of the gradient of the level set function defining the structural shape and topology. Its explicit and analytical form facilitates the sensitivity analysis and numerical implementation. As compared with the standard implementation of the level set method based on the steepest descent algorithm, the proposed method uses velocity field design variables and combines the level set method with the gradient‐based mathematical programming algorithm on the basis of the derived sensitivity scheme of the objective function and the constraints. This approach is able to simultaneously account for the casting constraint and the conventional material volume constraint in a convenient way. In this method, the optimization process can be started from an arbitrary initial design, without the need for an initial design satisfying the cast constraint. Numerical examples in both 2D and 3D design domain are given to demonstrate the validity and effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Gradient and parameter sensitivity estimation for systems evaluated using Monte Carlo analysis

The performance evaluation of many practical systems can be handled only through computationally intensive Monte Carlo simulation. Although a number of specialist techniques have been proposed, in general, estimation of the sensitivity of the outcome to changes in parameters involves duplicate simulations and finite differences for each parameter of interest. An approximate technique for gradient sensitivity estimation was outlined previously. It is appropriate when the performance function is uni-modal and relatively smooth in the region of interest. It generates all gradients simultaneously by converting Monte Carlo simulation run outcomes to an approximate analytic problem defined by a simplified response surface. The gradients then follow immediately. No extra simulation runs are required. Herein that approach is extended to non-Normal random variables and to the estimation of parameter sensitivities for random variable means and standard deviations. Some illustrative examples are given with comparisons to sensitivities computed by conventional Monte Carlo. The influence of constraint function(s) defining the admissible solution region is also considered.     

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.     

3D adaptive RKPM method for contact problems with elastic–plastic dynamic large deformation

The adaptive procedure of reproducing kernel particle method (RKPM) for 3D contact problems with elastic–plastic dynamic large deformation is presented. In this study, a modified cell energy error (MCEE) estimate model is constructed to capture the high gradients of stresses behavior in large deformation. Refinement particles with a new proper refinement function are inserted into the high error distribution regions. A domain decomposition method is proposed to determine the support domain size for nodes. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a particle-to-segment contact algorithm, the contact constraints are imposed directly on the new added contact nodes, consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. For verification of the simulation results, a general benchmark test is applied to justify the accuracy and efficiency of the adaptive RKPM method. Several numerical examples are provided to illustrate the effectiveness and robustness of the suggested approach.     

Development of a simple and efficient method for robust optimization

Robust optimization problems are newly formulated and an efficient computational scheme is proposed. Both design variables and design parameters are considered as random variables about their nominal values. To ensure the robustness of objective performance, we introduce a new performance index bounding the performance together with a constraint limiting the performance variation. The constraint variations are regulated by considering the probability of feasibility. Each probability constraint is transformed into a sub‐optimization problem by the advanced first‐order second moment (AFOSM) method for computational efficiency. The proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second‐order sensitivity information is not required even in case of gradient‐based optimization process. The suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Design of efficient stiffened shells of revolution

A method to produce efficient piecewise uniform stiffened shells of revolution is presented. The approach uses a first order differential equation formulation for the shell prebuckling and buckling analyses and the necessary conditions for an optimum design are derived by a variational approach. A variety of local yielding and buckling constraints and the general buckling constraint are included in the design process. The local constraints are treated by means of an interior penalty function and the general buckling load is treated by means of an exterior penalty function. This allows the general buckling constraint to be included in the design process only when it is violated. The self adjoint nature of the prebuckling and buckling formulations is used to reduce the computational effort. Results for four conical shells and one spherical shell are given.     

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。     

Optimization of Tablet Formulations Containing

An optimized direct compression tablet formulation of a conventional theophylline tablet was developed using the technique of response surface methodology and successive quadratic programming (SQP). The response surfaces were obtained from fitting data generated from a secondorder uniformprecision rotatable hexagonal experimental design. The tablet formulation was optimized for mean in vitro dissolution time using disintegration time, ejection force, friability and hardness, as constraints within the experimental region by the SQP technique. The response surface model was validated by preparing and evaluating the predicted formulation. The characteristics of the tablet formulation were analyzed by principal component analysis. Sensitivity analysis for the optimal solution was performed for each constraint, while all remaining constraints were held constant. The robustness of the response surface model was evaluated by simulation for error in the compression force values.     

Optimization of Tablet Formulations Containing

Abstract

An optimized direct compression tablet formulation of a conventional theophylline tablet was developed using the technique of response surface methodology and successive quadratic programming (SQP). The response surfaces were obtained from fitting data generated from a secondorder uniformprecision rotatable hexagonal experimental design. The tablet formulation was optimized for mean in vitro dissolution time using disintegration time, ejection force, friability and hardness, as constraints within the experimental region by the SQP technique. The response surface model was validated by preparing and evaluating the predicted formulation. The characteristics of the tablet formulation were analyzed by principal component analysis. Sensitivity analysis for the optimal solution was performed for each constraint, while all remaining constraints were held constant. The robustness of the response surface model was evaluated by simulation for error in the compression force values.     

Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

With the increasing complexity of engineering systems, ensuring high system reliability and system performance robustness throughout a product life cycle is of vital importance in practical engineering design. Dynamic reliability analysis, which is generally encountered due to time-variant system random inputs, becomes a primary challenge in reliability-based robust design optimization (RBRDO). This article presents a new approach to efficiently carry out dynamic reliability analysis for RBRDO. The key idea of the proposed approach is to convert time-variant probabilistic constraints to time-invariant ones by efficiently constructing a nested extreme response surface (NERS) and then carry out dynamic reliability analysis using NERS in an iterative RBRDO process. The NERS employs an efficient global optimization technique to identify the extreme time responses that correspond to the worst case scenario of system time-variant limit state functions. With these extreme time samples, a kriging-based time prediction model is built and used to estimate extreme responses for any given arbitrary design in the design space. An adaptive response prediction and model maturation mechanism is developed to guarantee the accuracy and efficiency of the proposed NERS approach. The NERS is integrated with RBRDO with time-variant probabilistic constraints to achieve optimum designs of engineered systems with desired reliability and performance robustness. Two case studies are used to demonstrate the efficacy of the proposed approach.     

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.     

Noise-resilient estimation of optical flow by use of overlapped basis functions

Conventional techniques for the computation of optical flow from image gradients are used to formulate the problem as a nonlinear optimization that comprises a gradient constraint term and a field smoothness factor. The results of these techniques are often erroneous, highly sensitive to noise and numerical precision, determined sparsely, and computationally expensive. We regularize the gradient constraint equation by modeling optical flow as a linear combination of a set of overlapped basis functions. We develop a theory for estimating model parameters robustly and reliably. We prove that the extended-least-squares solution proposed here is unbiased and robust to small perturbations in the gradient estimates and to mild deviations from the gradient constraint. Our solution is obtained with a numerically stable sparse matrix inversion, which gives a reliable flow-field estimate over the entire frame. To validate our claims, we perform a series of experiments on standard benchmark data sets at a range of noise levels. Overall, our algorithm outperforms by a wide margin the others considered in the comparison. We demonstrate the applicability of our algorithm to image mosaicking and to motion superresolution through experiments on noisy compressed sequences. We conclude that our flow-field model offers greater accuracy and robustness than conventional optical flow techniques in a variety of situations and permits real-time operation.     

A novel isoparametric finite element displacement formulation for axisymmetric analysis of nearly incompressible materials

It is well accepted that severe numerical difficulties arise when using the conventional displacement method to analyse incompressible or nearly incompressible solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic-plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation functions to minimize the constraints. The theoretical formulation of a new six-noded isoparametric displacement finite element, which is well suited for elastic-plastic analysis of axisymmetric constrained solids by using a rational displacement interpolation function, is presented in this paper. The proposed displacement interpolation function implies that the displacement in the axial direction and the product of the displacement in the radial direction and the radius should be treated as two independent basic variables. Alternatively, the proposed displacement interpolation function can also be implemented in a conventional displacement formulation simply by using a modified shape function matrix. The suitability of the proposed formulations is first studied theoretically by assessing the number of degrees of freedom per constraint and then verified by performing numerical experiments on typical boundary value problems which involve incompressible behaviour.     

多工况下杆系结构的概率优化设计

研究了基于概率的杆系结构多工况优化设计问题。建立了以杆系截面积为设计变量,结构位移、单元应力可靠度和尺寸限为约束条件,使结构重量极小化设计的数学模型。通过将概率约束等价化处理使之转变为常规约束形式,以紧约束处理策略确定有效约束。最后利用齿行法求解。算例表明文中的模型和方法是合理的和有效的。     

Soft constraint handling for a real-world multiobjective energy distribution problem

Real-world optimisation problems usually involve some conflicting objectives and a number of constraints. In such cases, finding a feasible, Pareto-optimal solution poses a demanding challenge. In reality, constraints bear different importance levels to these conflicting objectives. If some constraints are relaxed within an acceptable degree, quality infeasible solutions could be found on the boundary from the infeasible side of the searching region. This paper formulates an energy distribution problem arising from a real-world iron and steel production as a multiobjective optimisation problem. During the course of the optimisation search, this paper attempts to handle certain constraints in a soft manner to find solutions with good balance among objective and constraints violation. Based on the analysis of constraints from the real-world perspective, different tolerance values are defined. The proposed constraint violation degree-based soft handling approach is incorporated into the advanced version of non-dominated sorting genetic algorithm framework, as a case study, to examine the efficiency of the proposed soft constraint handling approach for a real-world energy distribution problem. The proposed approach is also implemented in different ways of constraint handling and tested on some benchmark functions to further demonstrate the performance of soft constraint handling for multiobjective optimisation problems.     

Shape from shading using wavelets and weighted smoothness constraints

A new method that allows capturing shapes from an input image using an optimisation-based approach is presented. An objective function is designed by introducing two terms: the first term is used to minimise the difference between the shading of the reconstructed shape and the input image, and the second term is to apply smoothness constraints to the reconstructed shape. To achieve shape reconstruction in high quality, the authors propose weighted smoothness constraint, which is designed to be anti-proportional to the intensity gradients in the input image. Under this constraint, flat image areas make more contribution towards the smoothness of the reconstructed shape, while the fine details from the image areas with large intensity gradients are preserved in the reconstructed result. Given the objective function, wavelets are used to obtain the solution effectively. Since wavelets accurately preserve high-frequency data, they can be used to solve the objective function with the advantage of allowing for a good recovery of fine details from the input image. The authors have chosen to use the Daubechies wavelets, which are orthonormal and compactly supported. Here the formulation of the algorithm based on the mathematical details is provided. Finally, the authors present experimental results on a number of different images and compare them against some well-known methods and ground truth (where available). The comparison shows that the method is effective and offers good results.     

Wind farm layout design optimization through multi-scenario decomposition with complementarity constraints

This article presents a multi-scenario decomposition with complementarity constraints approach to wind farm layout design to maximize wind energy production under region boundary and inter-turbine distance constraints. A complementarity formulation technique is introduced such that the wind farm layout design can be described with a continuously differentiable optimization model, and a multi-scenario decomposition approach is proposed to ensure efficient solution with local optimality. To combine global exploration and local optimization, a hybrid solution algorithm is presented, which combines the multi-scenario approach with a bi-objective genetic algorithm that maximizes energy production and minimizes constraint violations simultaneously. A numerical case study demonstrates the effectiveness of the proposed approach.