Intelligent design of induction motors by multiobjective fuzzy genetic algorithm

In this paper an approach using multi-objective fuzzy genetic algorithm (MFGA) for optimum design of induction motors is presented. Single-objective genetic algorithm optimization is compared with the MFGA optimization. The efficiency of those algorithms is investigated on motor’s performance. The comparison results show that MFGA is able to find more compromise solutions and is promising for providing the optimum design. Besides, a design tool is developed to evaluate and analysis the steady-state characteristics of induction motors.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Hybrid multi-objective shape design optimization using Taguchi’s method and genetic algorithm

This research is based on a new hybrid approach, which deals with the improvement of shape optimization process. The objective is to contribute to the development of more efficient shape optimization approaches in an integrated optimal topology and shape optimization area with the help of genetic algorithms and robustness issues. An improved genetic algorithm is introduced to solve multi-objective shape design optimization problems. The specific issue of this research is to overcome the limitations caused by larger population of solutions in the pure multi-objective genetic algorithm. The combination of genetic algorithm with robust parameter design through a smaller population of individuals results in a solution that leads to better parameter values for design optimization problems. The effectiveness of the proposed hybrid approach is illustrated and evaluated with test problems taken from literature. It is also shown that the proposed approach can be used as first stage in other multi-objective genetic algorithms to enhance the performance of genetic algorithms. Finally, the shape optimization of a vehicle component is presented to illustrate how the present approach can be applied for solving multi-objective shape design optimization problems.     

A new design optimization framework based on immune algorithm and Taguchi's method

This paper describes an innovative optimization approach that offers significant improvements in performance over existing methods to solve shape optimization problems. The new approach is based on two-stages which are (1) Taguchi's robust design approach to find appropriate interval levels of design parameters (2) Immune algorithm to generate optimal solutions using refined intervals from the previous stage. A benchmark test problem is first used to illustrate the effectiveness and efficiency of the approach. Finally, it is applied to the shape design optimization of a vehicle component to illustrate how the present approach can be applied for solving shape design optimization problems. The results show that the proposed approach not only can find optimal but also can obtain both better and more robust results than the existing algorithm reported recently in the literature.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

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.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

Numerical method for shape optimization using meshfree method

A numerical method for continuum-based shape design sensitivity analysis and optimization using the meshfree method is proposed. The reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method. Special features of the meshfree method from a sensitivity analysis viewpoint are discussed, including the treatment of essential boundary conditions, and the dependence of the shape function on the design variation. It is shown that the mesh distortion that exists in the finite element-based design approach is effectively resolved for large shape changing design problems through 2-D and 3-D numerical examples. The number of design iterations is reduced because of the accurate sensitivity information.     

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.     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.     

Simultaneous optimization of shape parameters and weight factors in ensemble of radial basis functions

Radial basis functions (RBFs) are approximate mathematical models that can mimic the behavior of fast changing responses. Different formulations of RBFs can be combined in the form of an ensemble model to improve prediction accuracy. The conventional approach in constructing an RBF ensemble is based on a two-step procedure. In the first step, the optimal values of the shape parameters of each stand-alone RBF model are determined. In the second step, the shape parameters are fixed to these optimal values and the weight factors of each stand-alone RBF model in the ensemble are optimized. In this paper, simultaneous optimization of shape parameters and weight factors is proposed as an alternative to this two-step procedure for further improvement of prediction accuracy. Gaussian, multiquadric and inverse multiquadric RBF formulations are combined in the ensemble model. The efficiency of the proposed method is evaluated through example problems of varying dimensions from two to twelve. It is found that the proposed method improves the prediction accuracy of the ensemble compared to the conventional two-step procedure for the example problems considered.     

Efficient shape parameterization method for multidisciplinary global optimization and application to integrated ship hull shape optimization workflow

Multidisciplinary global shape optimization requires a geometric parameterization method that keeps the shape generality while lowering the number of free variables. This paper presents a reduced parameter set parameterization method based on integral B-spline surface capable of both shape and topology variations and suitable for global multidisciplinary optimization. The objective of the paper is to illustrate the advantages of the proposed method in comparison to standard parameterization and to prove that the proposed method can be used in an integrated multidisciplinary workflow. Non-linear fitting is used to test the proposed parameterization performance before the actual optimization. The parameterization method can in this way be tested and pre-selected based on previously existing geometries. Fitting tests were conducted on three shapes with dissimilar geometrical features, and great improvement in shape generality while reducing the number of shape parameters was achieved. The best results are obtained for a small number (up to 50) of optimization variables, where a classical applying of parameterization method requires about two times as many optimization variables to obtain the same fitting capacity.The proposed shape parameterization method was tested in a multidisciplinary ship hull optimization workflow to confirm that it can actually be used in multiobjective optimization problems. The workflow integrates shape parameterization with hydrodynamic, structural and geometry analysis tools. In comparison to classical local and global optimization methods, the evolutionary algorithm allows for fully autonomous design with an ability to generate a wide Pareto front without a need for an initial solution.     

New approximate-iterative technique in shape optimization of continuum structures

An efficient technique that examines the optimum shape of elastic continuum structures is presented. The total number of stiffness matrix inversions required to obtain the optimal shape was significantly reduced as a result of using a newly developed iterative and approximate algorithm. The new technique is capable of approximating both displacements and stresses, and allows the user to have control over the degree of approximation. The efficiency of the proposed iterative approach was verified through various numerical examples and proved to be a viable alternative to the direct approach (which requires matrix inversion), especially in large-scale problems. The proposed approach is not limited, however, to shape optimization, but can be applied to other branches of structural optimization.     

Parameter-free optimum design method of stiffeners on thin-walled structures

In this paper, we present a shape optimization method for designing stiffeners on thin-walled or shell structures. Solutions are proposed to deal with a stiffness maximization problem and a volume minimization problem, which are subject to a volume constraint and a compliance constraint, respectively. The boundary shapes of the stiffeners are determined under a condition where the stiffeners are movable in the in-plane direction to the surface. Both problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using a material derivative method and an adjoint variable method. The optimal free-boundary shapes of the stiffeners are obtained by applying the derived shape gradient function to the $H^{1}$ gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several stiffener design examples are presented to validate the proposed method and demonstrate its practical utility.     

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.     

Truss structure optimization using adaptive multi-population differential evolution

This paper applies multi-population differential evolution (MPDE) with a penalty-based, self-adaptive strategy—the adaptive multi-population differential evolution (AMPDE)—to solve truss optimization problems with design constraints. The self-adaptive strategy developed in this study is a new adaptive approach that adjusts the control parameters of MPDE by monitoring the number of infeasible solutions generated during the evolution process. Multiple different minimum weight optimization problems of the truss structure subjected to allowable stress, deflection, and kinematic stability constraints are used to demonstrate that the proposed algorithm is an efficient approach to finding the best solution for truss optimization problems. The optimum designs obtained by AMPDE are better than those found in the current literature for problems that do not violate the design constraints. We also show that self-adaptive strategy can improve the performance of MPDE in constrained truss optimization problems, especially in the case of simultaneous optimization of the size, topology, and shape of truss structures.     

An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems

This paper presents an approach to shape and topology optimization of fluid-structure interaction (FSI) problems at steady state. The overall approach builds on an immersed boundary method that couples a Lagrangian formulation of the structure to an Eulerian fluid model, discretized on a deforming mesh. The geometry of the fluid-structure boundary is manipulated by varying the nodal parameters of a discretized level set field. This approach allows for topological changes of the fluid-structure interface, but free-floating volumes of solid material can emerge in the course of the optimization process. The free-floating volumes are tracked and modeled as fluid in the FSI analysis. To sense the isolated solid volumes, an indicator field described by linear, isotropic diffusion is computed prior to analyzing the FSI response of a design. The fluid is modeled with the incompressible Navier-Stokes equations, and the structure is assumed linear elastic. The FSI model is discretized by an extended finite element method, and the fluid-structure coupling conditions are enforced weakly. The resulting nonlinear system of equations is solved monolithically with Newton’s method. The design sensitivities are computed by the adjoint method and the optimization problem is solved by a gradient-based algorithm. The characteristics of this optimization framework are studied with two-dimensional problems at steady state. Numerical results indicate that the proposed treatment of free-floating volumes introduces a discontinuity in the design evolution, yet the method is still successful in converging to meaningful designs.     

Level set based robust shape and topology optimization under random field uncertainties

A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in loading and material properties is developed in this work. The proposed approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen–Loeve expansion is employed. The univariate dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is then employed for calculating statistical moments of the design response. The combination of the above techniques greatly reduces the computational cost in evaluating the statistical moments and enables a semi-analytical approach that evaluates the shape sensitivity of the statistical moments using shape sensitivity at each quadrature node. The applications of our approach to structure and compliant mechanism designs show that the proposed RSTO method can lead to designs with completely different topologies and superior robustness.