The inverse electromagnetic shaping problem

The inverse problem concerning electromagnetic casting of molten metals consists of looking for an electric current density distribution such that the induced electromagnetic field makes a given mass of liquid metal acquire a predefined shape. This problem is formulated here as an optimization problem where the positions of a finite set of inductors are the design variables. Two different formulations for this optimization problem for the two-dimensional case are proposed. The first one minimizes the difference between the target and the equilibrium shapes while the second approach minimizes the L ² norm of a fictitious surface pressure that makes the target shape to be in mechanical equilibrium. The optimization problems are solved using Feasible Arc Interior Point Algorithm, a line search interior-point algorithm for nonlinear optimization. Some examples are presented to show the effectiveness of the proposed approaches.     

Inverse and direct magnetic shaping problems

In this work, we study the direct and inverse problems arising from electromagnetic shaping in applications such as continuous casting processes. The magnetic field produces a surface pressure which forces a molten metal to change its shape until it reaches an equilibrium state between the magnetic pressure and the surface tension. The arising direct problem is a free boundary problem which is to determine the shape of molten metals for a given magnetic field. On the other hand, the inverse problem is to seek a configuration of electromagnetic field generators (i.e. inductors) in order that molten metals have prescribed shapes. A level set method will be presented to determine an equilibrium shape formed by a given configuration of magnetic fields. Also a computational method as well as uniqueness results for the inverse problem will be introduced and illustrated with examples.     

Optimization of a sheet metal forming process using successive multipoint approximations

An automated optimization method based on multipoint approximations and applied to the design of a sheet metal forming process is presented. Due to the highly complex nature of the design functions, it was decided to focus on the characterization of the final product thickness distribution as a function of the preforming die shape variables. This was achieved by constructing linear approximations to the noisy responses usingresponse surface methodology (RSM). These approximations are used to obtain an approximate solution to an optimization problem. Successive approximations are constructed, which improve the solution. An automated panning-zooming scheme is used to resize and position the successive regions of approximation. The methodology is applied to optimally design the preforming die shape used in the manufacture of an automotive wheel centre pressing from a sheet metal blank. The die shape is based on a cubic spline interpolation and the objective is to minimize the blank weight, subject to minimum thickness constraints. A weight saving of up to 9.4% could be realized for four shape variables. Restart is introduced to escape local minima due to the presence of noise and to accelerate the progress of the optimization process.     

Synthesis of tensegrity structures of desired shape using constrained minimization

There are two types of problems in tensegrity design: (i) form-finding when the tensegrity shape is not specified and (ii) synthesis when the tensegrity shape is specified. We address synthesis problems in this paper. We first formulated and solved an optimization problem to synthesize tensegrity structures of specified shape when the connectivity of the elements (bars and cables) is known a priori. We minimize the error in force-balance at the vertices in the desired equilibrium configuration by using force densities as the design variables. This constrained minimization problem enabled us to synthesize a known asymmetric tensegrity arch and a hitherto unknown tensegrity of biconcave shape similar to that of a healthy human red blood cell. We also extend the above method to a reduced order optimization problem for synthesizing complex symmetric tensegrity structures. Using this approach, we synthesized a truncated dodecahedron inside another truncated dodecahedron to emulate a nucleus inside a cell. We use a restricted global structure on an already available two-step mixed integer linear programming (MILP) topology optimization formulation to synthesize a non-convex tensegrity structure when only the coordinates are provided. We further improve this two-step MILP to a single-step MILP. We also present static analysis of a tensegrity structure by minimizing the potential energy with unilateral constraints on the lengths of the cables that cannot take compressive loads. Furthermore, we use this method to synthesize a tensegrity table of desired height and area under a predefined load. The prototypes of three synthesized tensegrities were made and validated.     

Constrained global optimization for wine blending

Assemblage consists in blending base wines in order to create target wines. Recent developments in aroma analysis allow us to measure chemical compounds impacting the taste of wines. This chemical analysis makes it possible to design a decision tool for the following problem: given a set of target wines, determine which volumes must be extracted from each base wine to produce wines that satisfy constraints on aroma concentration, volumes, alcohol contents and price. This paper describes the modeling of wine assemblage as a mixed constrained optimization problem, where the main goal is to minimize the gap to the desired concentrations for every aromatic criterion. The deterministic branch and bound solvers Couenne and IbexOpt behave well on the wine blending problem thanks to their interval constraint propagation/programming and polyhedral relaxation methods. We also study the performance of other optimization goals that could be embedded in a configuration tool, where the different possible interactions amount to solving the same constraints with different objective functions. We finally show on a recent generic wine blending instance that the proposed optimization process scales up well with the number of base wines.     

Shape and topology optimization for closed liquid cell materials using extended multiscale finite element method

A new multiscale shape and topology optimization method is presented to design closed liquid cell materials based on the extended multiscale finite element method, which directly captures the small scale features to the large scale computation. The multiscale optimization method firstly focuses on seeking the optimum geometrical parameters and volume expansion of the fluid in the closed liquid cells in the microscale level in terms of maximizing the macroscale mechanical response of the structure. Furthermore, a new hierarchical multiscale optimization method is developed to optimize the macroscale distributions of closed liquid cells and the microscale shape of the fluid inclusion in the cells. In the macroscale level of the multiscale optimization method, the macroscale design domain is discretized by the multiscale coarse elements, while the shape of the fluid inclusions is set to be the design parameters in the microscale level. This method is firstly utilized to minimize the system compliance of the closed liquid cell structure. Moreover, due to the fact that non-uniform volume expansions of the fluid in cells can induce the elastic action, the multiscale optimization method is further extended to design biomimetic compliant actuators of the closed liquid cell materials. The multiscale optimization methods developed are implemented in the FE-package SiPESC, and the numerical examples are carried out to validate the accuracy of the methods proposed.     

Optimum material design of minimum structural compliance under seepage constraint

In filtration and chemical engineering industry the load carrying capacity and seepage performances are very important for a successful filter design. We study a two-scale structural design optimization problem to minimize structural compliance under given seepage flow rate and material porosity constraints. Structural size, shape and topology are given because of other functional requirements. Structural material used is macro homogeneous porous material with periodic microstructure and is to be designed. Since structural compliance and seepage performances in macro-scale are implicit functions of material microstructural topology, it becomes a two-scale design optimization problem. The cross scale sensitivities are derived by the adjoint method. A new volume preserving nonlinear density filter is proposed which makes the process of optimization iteration more stable. The optimization problem is solved by GCMMA. Examples under the equality constraints of different seepage flow rate are presented to illustrate the effectiveness of two-scale design optimization formulation and solution approach.     

An interactive method for the selection of design criteria and the formulation of optimization problems in computer aided optimal design

This paper presents an interactive method for the selection of design criteria and the formulation of optimization problems within a computer aided optimization process of engineering systems. The key component of the proposed method is the formulation of an inverse optimization problem for the purpose of determining the design preferences of the engineer. These preferences are identified based on an interactive modification of a preliminary optimization result that is the solution of an initial problem statement. A formulation of the inverse optimization problem is presented, which is based on a weighted-sum multi-objective approach and leads to an explicit optimization problem that is computationally inexpensive to solve. Numerical studies on structural shape optimization problems show that the proposed method is able to identify the optimization criteria and the formulation of the optimization problem which drive the interactive user modifications.     

Shape optimization of heavy load carrying components for structural performance and manufacturing cost

This paper addresses the trade-off between structural performance and manufacturing cost of heavy load carrying components by incorporating virtual machining (VM) technique in computer-aided design (CAD)-based shape optimization problem. A structural shape optimization problem is set up to minimize total cost, subject to the limits on structural performance measures. For every design iteration, finite element analysis (FEA) is conducted to evaluate structural performance, and VM is employed to ascertain machinability and estimate machining time. Design sensitivity coefficients of objective function and constraints are computed and supplied to the optimization algorithm. Based on the gradients, the algorithm determines design changes, which are used to update FEA and VM models. The process is repeated until specified convergence criterion is satisfied. Application programs developed to integrate commercially available CAD/CAM/FEA/Design optimization tools enable implementation in virtual environment and facilitate automation. The application programs can be reused for similar design problems provided that the same set of tools is used.     

Covering a set of points in a plane using two parallel rectangles

In this paper we consider the problem of finding two parallel rectangles in arbitrary orientation for covering a given set of n points in a plane, such that the area of the larger rectangle is minimized. We propose an algorithm that solves the problem in O(n³) time using O(n²) space. Without altering the complexity, our approach can be used to solve another optimization problem namely, minimize the sum of the areas of two arbitrarily oriented parallel rectangles covering a given set of points in a plane.     

A Pareto archive particle swarm optimization for multi-objective job shop scheduling

In this paper, we present a particle swarm optimization for multi-objective job shop scheduling problem. The objective is to simultaneously minimize makespan and total tardiness of jobs. By constructing the corresponding relation between real vector and the chromosome obtained by using priority rule-based representation method, job shop scheduling is converted into a continuous optimization problem. We then design a Pareto archive particle swarm optimization, in which the global best position selection is combined with the crowding measure-based archive maintenance. The proposed algorithm is evaluated on a set of benchmark problems and the computational results show that the proposed particle swarm optimization is capable of producing a number of high-quality Pareto optimal scheduling plans.     

Non-parametric free-form optimization method for frame structures

In this paper, we propose a parameter-free shape optimization method based on the variational method for designing the smooth optimal free-form of a spatial frame structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of shape design problems of frame structures. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and that the cross section is prismatic. The shape gradient function and the optimality conditions are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a fictitious distributed force both to vary the member in the optimum direction and to minimize the objective functional without shape parametrization, while maintaining the members’ smoothness. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures were obtained by this method.     

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.     

Single- and multi-objective shape design of Y-noise barriers using evolutionary computation and boundary elements

The optimum shape design of Y-noise barriers is carried out using single and multi-objective evolutionary algorithms and the Boundary Element Method (BEM). Reduction of noise impact efficiency (using the insertion loss-IL-magnitude) and cost of the barrier (using its total length magnitude) are considered. A two-dimensional problem of sound propagation in the frequency domain is handled, defined by a fixed position emitting source, which pulses in a frequency range, and receptor. A noise barrier (limiting its maximum effective height) is situated between both. Its shape is modified to minimize the receptor measured sound level, which is calculated using BEM. Results of an inverse problem using the IL barrier curve as reference are successfully performed to validate the methodology. The proposed methodology is then used to obtain Y-barriers with 15% and 30% improved IL spectrum. Finally, six non-dominated solutions of the multi-objective optimum design problem are presented in detail.     

A combined shape control procedure of cable mesh reflector antennas with optimality criterion and integrated structural electromagnetic concept

A combined shape control procedure with optimality criterion and integrated structural electromagnetic concept for cable mesh reflector antennas is presented in this study. Using the optimality criterion, the shape control algorithm drives the distorted surface towards the ideal shape. The optimality criterion is implemented by pseudo inverse of sensitivity matrix of surface nodal displacements with respect to cable member dimensions to accelerate the iterative convergence. The following integrated structural electromagnetic design is performed to make good electromagnetic performance by a sequential quadratic programming optimization model. A distorted offset cable mesh reflector antenna is employed to show its effectiveness.     

Stress isolation through topology optimization

This paper introduces a problem of stress isolation in structural design and presents an approach to the problem through topology optimization. We model the stress isolation problem as a topology optimization problem with multiple stress constraints in different regions. The shape equilibrium constraint approach is employed to effectively control the local stress constraints. The level set based structural optimization is implemented with the extended finite element method (X-FEM) for providing an adequately accurate stress analysis. Numerical examples of stress isolation design in two dimensions are investigated as a benchmark test of the proposed method. The results, from the force transmittance point of view, suggest that the guard “grooves” obtained can change the force path to successfully realize the stress isolation in the structure.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

基于优化法的金属材料应力-应变曲线反演

采用优化法反演金属材料的应力-应变曲线.建立与试验边界条件一致的有限元仿真模型,并用Abaqus进行有限元仿真.设计变量为给定应变下的应力参数,优化目标是使模拟的应力-位移曲线与试验曲线一致,采用自适应响应面法和序列二次规划法开展优化迭代.结果表明,反演参数能很好地表现材料行为特征,为下一步仿真分析提供材料数据基础.     

A hybrid genetic algorithm for constrained multi-objective optimization under uncertainty and target matching problems

This work presents a new approach for interval-based uncertainty analysis. The proposed approach integrates a local search strategy as the worst-case-scenario technique of anti-optimization with a constrained multi-objective genetic algorithm. Anti-optimization is a term for an approach to safety factors in engineering structures which is described as pessimistic and searching for least favorable responses, in combination with optimization techniques but in contrast to probabilistic approaches. The algorithm is applied and evaluated to be efficient and effective in producing good results via target matching problems: a simulated topology and shape optimization problem where a ‘target’ geometry set is predefined as the Pareto optimal solution and a constrained multiobjective optimization problem formulated such that the design solutions will evolve and converge towards the target geometry set.