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This paper presents a topology optimization method for two dimensional turbulent flow based on the Reynolds-averaged Navier-Stokes (RANS) equations using a level set boundary expression and the immersed boundary method (IBM). In this study, two-equation turbulence models, the k-ϵ and the k-ω, are considered. In our proposed method, the level set method is used for capturing the exact fluid-solid interfaces. Additionally, the no-slip boundary condition along the fluid-solid interfaces is imposed explicitly using the IBM during the optimization process. Based on the information of the exact boundary position, the interpolated velocity and pressure values within the viscous sublayer are estimated using the standard wall function. From the above formulations, we construct a topology optimization method for the total pressure drop minimization problems considering two dimensional turbulent flows, under the frozen turbulence assumption. We provide numerical examples to confirm the validity and utility of the proposed method.

This papers proposes a level set-based topology optimization method for the simultaneous design of acoustic and structural material distributions. In this study, we develop a two-phase material model that is a mixture of an elastic material and acoustic medium, to represent an elastic structure and an acoustic cavity by controlling a volume fraction parameter. In the proposed model, boundary conditions at the two-phase material boundaries are satisfied naturally, avoiding the need to express these boundaries explicitly. We formulate a topology optimization problem to minimize the sound pressure level using this two-phase material model and a level set-based method that obtains topologies free from grayscales. The topological derivative of the objective functional is approximately derived using a variational approach and the adjoint variable method and is utilized to update the level set function via a time evolutionary reaction-diffusion equation. Several numerical examples present optimal acoustic and structural topologies that minimize the sound pressure generated from a vibrating elastic structure.

This paper presents a multi-objective topology optimization method for a magnetic actuator design considering both magnetic force and actuator volume. We use a gradient-based optimization algorithm and the multi-objective optimization problem is formulated using a weighted sum of the objective functions. An adaptive weight determination scheme is iteratively employed to update the weighting coefficients of the objective functions to obtain non-dominated solutions. To ensure that the optimal solutions are evenly distributed in the objective domain, e-equality constraints are applied in the empty regions of the non-dominated solutions. The values of the constraints are determined by considering the distance ratios of the objective functions. The optimization problem is formulated to maximize the magnetic force and minimize the volume of the actuator. A level set function is used as a topological design variable to obtain optimal designs that have clear structural boundaries.

This paper presents a local-in-time (LT) discrete adjoint-based topology optimization method for unsteady incompressible viscous flows incorporating the lattice Boltzmann method (LBM). For the optimization of unsteady flows, straightforward global implementations of the time-dependent optimization are usually adopted. However, such global implementations require that the entire flow solution history be available to calculate the solution of the adjoint equation reversed in time. For 3-D design optimization problems, the storage requirements can become prohibitively large. In this paper, the LT discrete adjoint-based method is applied to a LBM-based topology optimization to reduce the storage requirement. The basic idea of the LT method is to divide the entire time interval into several subintervals and to approximate the global sensitivity derivative as a combination of local sensitivity derivatives computed for each time subinterval. In this approach, flow solutions for only a single subinterval need to be stored. Since each time subinterval includes only a few (possibly one) time steps, the data storage requirements can be tremendously reduced. This method is applied in a pressure drop minimization problem considering unsteady viscous fluid. Two- and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

This paper proposes an optimal design method for plate-type microchannel reactor manifolds, using a level set-based topology optimization method that targets flow uniformity among the microchannels. For plate-type microchannel reactors, manifold designs must consider both the uniformity of flow among the microchannels and minimization of pressure drop in the device. To address these design requirements, the pressure drop in the device is defined as the objective functional to be minimized under a flow rate inequality constraint, defined as the deviation in flow rate among the microchannels during the optimization. In addition, we propose a comparably simple and stable augmented Lagrangian method using an exponential function, to determine the Lagrange multiplier that acts to satisfy the flow rate inequality constraint. To demonstrate the utility of our proposed method, two-dimensional Z-type and U-type manifold design problems are presented in the numerical examples.

Topology optimization has been used in many industries and applied to a variety of design problems. In real-world engineering design problems, topology optimization problems often include a number of conflicting objective functions, such to achieve maximum stiffness and minimum mass of a design target. The existence of conflicting objective functions causes the results of the topology optimization problem to appear as a set of non-dominated solutions, called a Pareto-optimal solution set. Within such a solution set, a design engineer can easily choose the particular solution that best meets the needs of the design problem at hand. Pareto-optimal solution sets can provide useful insights that enable the structural features corresponding to a certain objective function to be isolated and explored. This paper proposes a new Pareto frontier exploration methodology for multiobjective topology optimization problems. In our methodology, a level set-based topology optimization method for a single-objective function is extended for use in multiobjective problems, using a population-based approach in which multiple points in the objective space are updated and moved to the Pareto frontier. The following two schemes are introduced so that Pareto-optimal solution sets can be efficiently obtained. First, weighting coefficients are adaptively determined considering the relative position of each point. Second, points in sparsely populated areas are selected and their neighborhoods are explored. Several numerical examples are provided to illustrate the effectiveness of the proposed method.

This paper proposes a topology optimization method for thermal-fluid flow problems using the lattice Boltzmann method (LBM). The design sensitivities are derived based on the adjoint lattice Boltzmann method (ALBM), whose basic idea is that the adjoint problem is first formulated using a continuous adjoint approach, and the adjoint problem is then solved using the LBM. In this paper, the discrete velocity Boltzmann equation, in which only the particle velocities are discretized, is introduced to the ALBM to deal with the various boundary conditions in the LBM. The novel sensitivity analysis is applied in two flow channel topology optimization problems: 1) a pressure drop minimization problem, and 2) a heat exchange maximization problem. Several numerical examples are provided to confirm the utility of the proposed method.

This paper proposes techniques to improve the diversity of the searching points during the optimization process in an Aggregative Gradient-based Multiobjective Optimization (AGMO) method, so that well-distributed Pareto solutions are obtained. First to be discussed is a distance constraint technique, applied among searching points in the objective space when updating design variables, that maintains a minimum distance between the points. Next, a scheme is introduced that deals with updated points that violate the distance constraint, by deleting the offending points and introducing new points in areas of the objective space where searching points are sparsely distributed. Finally, the proposed method is applied to example problems to illustrate its effectiveness.

Damping material is usually applied to steel panels of vehicles to reduce vibration levels. On the other hand, the weight of a vehicle must be reduced to improve the rate of fuel consumption. Therefore, the modal loss factors caused by the treatment of damping material on the steel panels of a vehicle body structure must be maximized within a given volume. In this paper, we propose a practical design method to maximize modal loss factors by optimizing the layout of damping material under a volume constraint. The modal loss factor for an eigenmode can be obtained conventionally by the modal strain energy method as the material loss factor multiplied by the ratio of the strain energy stored in the damping material over the total strain energy in the system under consideration. In the proposed method, we assume that the eigenvectors with damping material are almost identical with the eigenvectors without damping material. The modal loss factor can then be expressed approximately by using a corresponding real eigenvalue, for which the stiffness of the damping material is taken into account but its mass density is set to zero and ignored. Several numerical examples are provided to demonstrate that the proposed method obtains optimal layouts of damping material applied to a flat rectangular panel. Our results indicate that the damping material is mainly distributed in areas where strain energy is stored, which agrees well with the results obtained using conventional design methodologies. Moreover, by applying a design sensitivity filter that was improved recently, the layout of damping material can be unified into a single domain to meet practical requirements for manufacturing.

This paper presents a topology optimization method for fluid dynamics problems, based on the level set method and using the lattice Boltzmann method (LBM). In this optimization method, the optimization problems are formulated based on the original Boltzmann equation, and the design sensitivities are precisely obtained without the time-consuming numerical operations encountered when dealing with a large-scale asymmetric matrix, in contrast to previous research in which the LBM uses the lattice Boltzmann equation (LBE) for the formulations of optimization problems and the derivation of their adjoint equations. That is, we newly derive sensitivity formulations from the original Boltzmann equation, not the LBE that can be said to be an approximated equation, and these formulations yield strictly correct sensitivities that are error free. Based on the above formulations, we construct a level set-based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction. Furthermore, two- and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

A process of compromise that addresses conflicting objective functions such as performance and cost is often involved in real-world engineering design activities. If such conflicting relationships among objective functions exist in a multiobjective design optimization problem, no single solution can simultaneously minimize all objective functions, and the solutions of the optimization problem are obtained as a set of design alternatives called a Pareto optimal solution set. This paper proposes a new gradient-based multiobjective c that incorporates a population-based aggregative strategy for obtaining a Pareto optimal solution set. In this method, the objective functions and constraints are evaluated at multiple points in the objective function space, and design variables at each point are updated using information aggregatively obtained from all other points. In the proposed method, a multiobjective optimization problem is converted to a single objective optimization problem using a weighting method, with weighting coefficients adaptively determined by solving a linear programming problem. A sequential approximate optimization-based technique is used to update the design variables, since it allows effective use of design sensitivities that can be easily obtained in many engineering optimization problems. Several numerical examples, including a structural optimization problem, are provided to illustrate the effectiveness and utility of the proposed method.

This study presents a multi-level optimization (MLO) method applied in particle swarm optimization (PSO) to solve a 3-objective optimization problem for an antenna array design consisting of two L-shaped folded monopole antennas (LFMAs) operating in the IEEE 802.11 b/g WLAN band (2400-2484 MHz). The method incorporates a simple concept whereby three sub-objective functions, representing the reflection coefficient S 11, the mutual coupling S 21, and the entire antenna volume of the LFMA array, are minimized one by one in the PSO to obtain an optimal LFMA array configuration that provides reduced size and higher antenna performance. The advantages of the proposed method, both in terms of deriving appropriate optimization results and reducing computational cost for the optimization problem, are demonstrated by comparing its performance with that of a conventional weighted aggregation (CWA) method, and a method using the concept of Pareto dominance (PD). The optimal LFMA array configuration obtained by the proposed method has a 10% resonant bandwidth ( S 11 ≤ -10 dB for 2.32-2.57 GHz), S 21 reduced to less than -15 dB in this band, and an antenna volume that was markedly decreased, to 28.5% of that of the original design.

International audience; The aim of this research is to construct a shape optimization method based on the convected level set method, in which the level set function is defined as a truncated smooth function obtained by using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent function dramatically reduces the generation of red the error between the specified profile of the hyperbolic tangent function and the level set function that is updated using a time evolution equation. In addition, the small size of the error facilitates the use of convective reinitialization, whose basic idea is that the reinitialization is embedded in the time evolution equation, whereas such treatment is typically conducted in a separate calculation in conventional level set methods. The convected level set method can completely avoid the need for additional calculations when performing reinitialization. The validity and effectiveness of our presented method are tested with a mean compliance minimization problem and a problem for the design of a compliant mechanism.

This paper presents a level set-based topology optimization method for the design of negative permeability dielectric metamaterials. Metamaterials are artificial materials that display extraordinary physical properties that are unavailable with natural materials. The aim of the formulated optimization problem is to find optimized layouts of a dielectric material that achieve negative permeability. The presence of grayscale areas in the optimized configurations critically affects the performance of metamaterials, positively as well as negatively, but configurations that contain grayscale areas are highly impractical from an engineering and manufacturing point of view. Therefore, a topology optimization method that can obtain clear optimized configurations is desirable. Here, a level set-based topology optimization method incorporating a fictitious interface energy is applied to a negative permeability dielectric metamaterial design problem. The optimization algorithm uses the Finite Element Method (FEM) for solving the equilibrium and adjoint equations, and design problems are formulated for both two- and three-dimensional cases. First, the level set-based topology optimization method is explained, and the optimization problems for the design of metamaterials are then discussed. Several optimum design examples for the design of dielectric metamaterials that demonstrate negative effective permeability at prescribed frequencies are provided to confirm the utility and validity of the presented method.

This study presents optimization methods for the design of rotor poles in interior permanent magnet machines, which aim to effectively obtain a sinusoidal distribution of the air-gap flux density. A sinusoidal field in the air-gap offers several advantages, such as reduction of torque ripple when using sinusoidal phase currents, and reduction in cogging torque and higher harmonic components of back electromotive force. In this study, the optimization process is achieved based on a phase field method using an Allen-Cahn equation. For the optimization problem we address, regions with meaningful design sensitivity values may exist outside the interfacial layer, which makes design variable updating impossible. To resolve this problem, the design sensitivities are diffused in the radial direction, using a diffusion equation. The time evolution of the design variables is based on the implicit finite element method, and the augmented Lagrangian method is used to deal with the volume constraint.

A robust topology optimization approach in consideration with random field uncertainty in loading condition is proposed in this paper. The proposed method integrates the level set-based topology optimization and the robust design method using stochastic process model. The stochastic process model is applied to describe the uncertainty of design parameters with nonuniform distribution in space with a reduced set of random variables. The robust optimization is formulated to minimize the robust compliance that is defined as a weighted sum of the expected value and the standard deviation of mean compliance under volume constraint. The standard deviation of mean compliance is approximated by the first-order sensitivity with respect to random variables that are described through the assumed spectral distributions from Wiener-Khintchine theorem. As a topology optimization, a level set-based approach incorporating a fictitious interface energy proposed by parts of the authors is applied. The method has several advantages to makes it possible to change structural topology as well as boundary shapes and to control the geometrical complexity of the optimum configuration qualitatively. Through numerical examples, the efficiency of the proposed robust topology optimization approach is demonstrated by comparing between the deterministic and robust optimum configurations.

In this paper, we propose a level set-based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high-frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary-tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso-contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso-contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso-contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz-type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

This paper proposes a new macroscopic model for sound-absorbing poroelastic media which is derived by using the homogenization theory based on the method of asymptotic expansions. The derivation of the macroscopic properties and governing equations takes into account the multiphysics occurring in poroelastic media for sound absorption, including elastic motions of the solid phase, compressible viscous fluid flow, and the distributions of pressure and temperature in the fluid phase. The coupled effects between the elastic solid and the fluid pressure, and the temperature and the fluid pressure are also considered. In contrast to the conventional Biot’s model, which includes heuristic formulae, the proposed method yields a rigorous model that is consistent with the principal governing equations on the microscopic scale. Utilizing several models that have simple microscopic geometry and comparing the numerical solutions obtained using the proposed method with corresponding analytical solutions, we demonstrate that the derived macroscopic governing equations can provide accurate and effective predictions.

Topology optimization for structures has been applied to nonlinear structural problems, however conventional topology optimization methods for structures with geometrical nonlinearity encounter difficulties during nonlinear analysis using the FEM (Finite Element Method), due to the use of a mesh. In this study, we propose a new level set-based topology optimization method considering geometrical nonlinearity using a mesh-free/particle technique, for optimizing elastic structures that undergo large deformation. In the proposed method, the MPS (Moving Particle Semiimplicit) method, a particle method, is used for the response analysis, since it does not rely on a mesh for geometrically nonlinear analysis. In this paper, first, a topology optimization problem is formulated based on the level set method and a method for regularizing the optimization problem by the Tikhonov regularization method is explained. The reactiondiffusion equation that updates the level set function is then derived and an optimization algorithm, which uses the FEM to solve the equilibrium equations and the reaction-diffusion equation when updating the level set function, is constructed. Next, the particle interaction model and the treatment of geometrical nonlinearity in the MPS method are shown, and the implementation of combining the level set-based topology optimization and the MPS methods is explained. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed method of topology optimization for geometrically nonlinear problems.

This paper proposes a new topology optimization method, which can adjust the geometrical complexity of optimal configurations, using the level set method and incorporating a fictitious interface energy derived from the phase field method. First, a topology optimization problem is formulated based on the level set method, and the method of regularizing the optimization problem by introducing fictitious interface energy is explained. Next, the reaction–diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed, which uses the finite element method to solve the equilibrium equations and the reaction–diffusion equation when updating the level set function. Finally, several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method.

This paper discusses a structural optimization method that optimizes shape and topology based on the phase field method. The proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions. The advantage of the method is the simplicity of computation, since extra operations such as re-initialization of functions are not required. Structural shapes are represented by the phase field function defined in the design domain, and optimization of this function is performed by solving a time-dependent reaction diffusion equation. The artificial double well potential function used in the equation is derived from sensitivity analysis. The proposed method is applied to two-dimensional linear elastic and vibration optimization problems such as the minimum compliance problem, a compliant mechanism design problem and the eigenfrequency maximization problem. The numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations.

Structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function, and updated by solving the so-called Hamilton–Jacobi equation based on a Eulerian coordinate system. These new methods can allow topological alterations, such as the number of holes, during the optimization process whereas the boundaries of the target structure are clearly defined. However, the re-initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures. In this paper, we propose a new structural optimization method based on the level set method using a new geometry-based re-initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the Finite Element Method. The stiffness maximization, eigenfrequency maximization, and eigenfrequency matching problems are considered as optimization problems. Several design examples are presented to confirm the usefulness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

The SLSV (single-loop-single-vector) method is modified for the reliability-based optimization problem with multiple reliability constraints. The design problem is formulated to minimize the structural volume of frame structure in terms of cross-sectional area of each frame element subjected to the two mode reliability constraints. The two mode reliability criteria consist of the mean compliance and mean eigenfrequency. The limit state functions are formulated as normalized form to achieve numerical stability of the SLSV method, because the functions are directly adopted as constraint conditions. That is a large difference from the conventional double-loop method, where the limit state functions do not appear in the optimization loop. Through numerical examples of 2-D and 3-D frame design problems, higher computational efficiency and sufficient reliability approximation accuracy by the SLSV method are demonstrated in comparison with the conventional double loop method that the mode reliabilities are evaluated by the first order reliability method (FORM) in each optimization step. Additionally, the importance of normalization of the limit state functions in the SLSV method is also demonstrated.

Multilevel redundancy allocation optimization problems (MRAOPs) occur frequently when attempting to maximize the system reliability of a hierarchical system, and almost all complex engineering systems are hierarchical. Despite their practical significance, limited research has been done concerning the solving of simple MRAOPs. These problems are not only NP hard but also involve hierarchical design variables. Genetic algorithms (GAs) have been applied in solving MRAOPs, since they are computationally efficient in solving such problems, unlike exact methods, but their applications has been confined to single-objective formulation of MRAOPs. This paper proposes a multi-objective formulation of MRAOPs and a methodology for solving such problems. In this methodology, a hierarchical GA framework for multi-objective optimization is proposed by introducing hierarchical genotype encoding for design variables. In addition, we implement the proposed approach by integrating the hierarchical genotype encoding scheme with two popular multi-objective genetic algorithms (MOGAs)—the strength Pareto evolutionary genetic algorithm (SPEA2) and the non-dominated sorting genetic algorithm (NSGA-II). In the provided numerical examples, the proposed multi-objective hierarchical approach is applied to solve two hierarchical MRAOPs, a 4- and a 3-level problems. The proposed method is compared with a single-objective optimization method that uses a hierarchical genetic algorithm (HGA), also applied to solve the 3- and 4-level problems. The results show that a multi-objective hierarchical GA (MOHGA) that includes elitism and mechanism for diversity preserving performed better than a single-objective GA that only uses elitism, when solving large-scale MRAOPs. Additionally, the experimental results show that the proposed method with NSGA-II outperformed the proposed method with SPEA2 in finding useful Pareto optimal solution sets.

Electromagnetic structures that incorporate certain structural periodicities are known to display special behavior when subjected to electromagnetic waves, and can be designed to have specific functions such as inhibiting the intrusion of electromagnetic waves of certain frequencies into the periodic structure. This paper proposes a novel topology optimization method for periodic microstructures of electromagnetic materials using the concept of propagation behavior to implement designs that inhibit electromagnetic wave propagation. First, a way to apply topology optimization to the design of electromagnetic structures is briefly discussed. Next, the design specifications are clarified, and a new objective function is proposed to satisfy these specification. The optimization algorithm is developed using sequential linear programming (SLP) and the adjoint variable method (AVM). Several numerical examples are provided to confirm that the proposed method is capable of automatically generating physically reasonable periodic structures that have desired specified functions without relying on a predefined basic lattice.

This paper describes a new design method to optimize thickness distribution of a multilayered structure which is located on the coupling surface between a structure and an acoustic cavity. The design method is based on the concept of the density approach in topology optimization incorporating a transfer matrix for a multilayered structure that includes a poroelastic media layer. The one-dimensional transfer matrix adopted here is an approximate representation addressing vibro-acoustic effects inherent in a multilayered structure, and balances calculation resources and desired accuracy. Applying the transfer matrix representation as boundary conditions on the coupling surface between a structure and an acoustic cavity, the modified equilibrium equation of the vibro-acoustic system is derived which is approximately but efficiently solved by the modal approach. In this study, the problem of minimizing the acoustic pressure within the cavity over the prescribed frequency range is formulated under the volume constraint of the poroelastic media layer. The continuous approximation of thickness distribution is assumed, and the thickness of the poroelastic media layer at each nodal point is chosen as design variables. Numerical results show that an acoustic response is significantly reduced by the optimal thickness distribution having a total weight equal to or less than that in the initial uniform thickness. These demonstrate that the proposed method is effective to design the optimal thickness distribution of a multilayered structure.

Multi-axis force sensors are extensively utilized in engineering fields, for example in automotive development, where 6-axis force transducers are used to measure force components applied to wheels, and as sensors in robotic manipulators. This paper describes a new structural optimization method for multi-axis force sensor structures considering the accuracy of force detection. First, we formulate homogenization design methods (HDM) in which continuous material distributions are assumed using a continuous interpolation function at each node. Next, we clarify the mechanical requirements and design specifications of the sensor structure, using a methodology based on singular value decomposition, and construct objective functions that aim to satisfy given design specifications. The sensitivities of the objective function with respect to the design variables are formulated using eigen values. Based on these formulations, an optimization algorithm is constructed using sequential linear programming (SLP). Finally, we examine the characteristics of the optimization formulations and the generated optimal configurations and confirm the usefulness of our proposed methodology for the optimization of multi-axis force sensor structures.

In this paper, a topology optimization method is constructed for thermal problems with generic heat transfer boundaries in a fixed design domain that include design-dependent effects. First, the topology optimization method for thermal problems is briefly explained using a homogenization method for the relaxation of the design domain, where a continuous material distribution is assumed, to suppress numerical instabilities and checkerboards. Next, a method is developed for handling heat transfer boundaries between material and void regions that appear in the fixed design domain and move during the optimization process, using the Heaviside function as a function of node-based material density to extract the boundary of the target structure being optimized so that the heat transfer boundary conditions can be set. Shape dependencies concerning heat transfer coefficients are also considered in the topology optimization scheme. The optimization problem is formulated using the concept of total potential energy and an optimization algorithm is constructed using the Finite Element Method and Sequential Linear Programming. Finally, several numerical examples are presented in order to confirm the usefulness of the proposed method.

In this study, a robust topology optimization method is proposed for compliant mechanisms, where the effect that variation of the input load direction has on the output displacement is considered. Variations are evaluated through a sensitivity-based robust optimization approach, with the variance evaluated using first-order derivatives. The robust objective function is defined as a combination of maximizing the output deformation under the mean input load and minimizing variation in the output deformation as the input load is varied, where variance due to changes in load can be obtained through mutual compliance and the presence of a pseudo load. For the topology optimization, a modified homogenization design method using the continuous approximation assumption of material distribution is adopted. The validity of the proposed method is confirmed with two compliant mechanism design problems. The effect that varying the input load direction has upon the obtained configurations is investigated by comparing these with deterministic optimum topology design results.

This article proposes a new multiobjective optimization method for structural problems based on multiobjective particle swarm optimization (MOPSO). A gradient-based optimization method is combined with MOPSO to alleviate constraint-handling difficulties. In this method, a group of particles is divided into two groups—a dominated solution group and a non-dominated solution group. The gradient-based method, utilizing a weighting coefficient method, is applied to the latter to conduct local searching that yields superior non-dominated solutions. In order to enhance the efficiency of exploration in a multiple objective function space, the weighting coefficients are adaptively assigned considering the distribution of non-dominated solutions. A linear optimization problem is solved to determine the optimal weighting coefficients for each non-dominated solution at each iteration. Finally, numerical and structural optimization problems are solved by the proposed method to verify the optimization efficiency.

The designs of on-board electronic devices used in automobiles need to satisfy not only electronic requirements but also physical requirements such maintaining sufficient stiffness under loads. The development of multi-disciplinary optimization methods that also handle multi-physics issues can facilitate a systematic approach to the design of such devices. In this study, we propose a new multi-disciplinary topology optimization method that handles multi-physics phenomena, namely, the coupling of electromagnetic wave propagation problems with static structural mechanics problems. The Finite Difference-Time Domain method is used to solve the electromagnetic wave propagation problem, while the static structural mechanics problem is solved using the Finite Element Method. New relaxation schemes for the design domain and design variable settings are proposed to deal with the use of the different numerical methods. The integral reflection electric energy and the mean compliance are formulated as objective functions to be minimized in an optimization problem. A multi-objective function is also formulated based on a weighted sum formulation to deal with the two objective functions. The optimization algorithm is constructed based on sequential linear program-ming. A design example of a multiple-band dielectric resonator antenna is provided to show the usefulness of the proposed method.

This paper proposes a structural optimization method for obtaining mechanical structures that maximize structural stiffness, based on the level set method. First, the basic details of a structural optimization method using the level set method are briefly explained, and an optimization problem that addresses the minimum mean compliance problem is formulated. Next, an optimization algorithm is constructed based on this formulation. In this optimization scheme, a new re-initialization method is proposed, in which a level set function is re-initialized, based on a zero level set surface. This re-initialization method can easily deal with level set function singularities, and can be applied to non-structural meshes. A second new method is also proposed in which the level set function is modified to satisfy a volume constraint while preserving topology of the structure being optimized. By using these two new methods, structural optimization problems can be solved using the finite element method rather than the finite difference method. Finally, several examples are provided to confirm the usefulness of the proposed structural optimization method.

This paper describes a new design method for optimizing the thickness distribution of a multilayered structure located on the coupling surface between a structure and an acoustic cavity. The design method is based on the concept of a topology optimization method incorporating a transfer matrix for the multilayered structure that includes a poroelastic media layer. The one-dimensional transfer matrix adopted here is an approximate representation addressing vibro-acoustic effects inherent in a multilayered structure over a range of low frequencies, and balances calculation times and desired accuracy. In this study, the problem of minimizing the acoustic pressure within the cavity over the prescribed frequency range is formulated under the constraint of the total weight of the design domain. The thickness of the poroelastic media layer at each nodal point is chosen as the design variables. Numerical results show that acoustic response is significantly reduced by an optimal thickness distribution having a total weight equal to or less than that of the initial thickness.

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization...

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.

To prevent the numerical instabilities in topology optimization, continuous approximation of material distribution (CAMD) is proposed to the homogenization design method (HDM) and the simple isotropic material with penalization (SIMP) method. The continuous FE approximation of design variables including high order elements is applied to the formulation of SIMP method. Numerical examples are presented to compare the efficiency of CAMD both in HDM and SIMP.

In vibration optimization problems, eigenfrequencies are usually maximized in the optimization since resonance phenomena in a mechanical structure must be avoided, and maximizing eigenfrequencies can provide a high probability of dynamic stability. However, vibrating mechanical structures can provide additional useful dynamic functions or performance if desired eigenfrequencies and eigenmode shapes in the structures can be implemented. In this research, we propose a new topology optimization method for designing vibrating structures that targets desired eigenfrequencies and eigenmode shapes. Several numerical examples are presented to confirm that the method presented here can provide optimized vibrating structures applicable to the design of mechanical resonators and actuators.

This paper presents a new topology optimization method that maximizes multiple eigen-frequencies to obtain, at the conceptual design phase, the most stable designs of mechanical structures for dynamic problems. This method is developed based on frame elements that can consider the principal direction of the second moment of inertia in the cross-section to obtain optimal results that illustrate the relationship between the objective function and desired cross-sectional properties. For each frame element, an ellipsoidal cross-section is assumed as the simplest case. The normalized cross-sectional area, the rotational angle denoting the principal direction, and the ratio of the lengths of the major and minor axes, are considered as design variables. A method for obtaining the optimal angle is derived, based on the minimization of complementary energy. Finally, several numerical examples confirm the validity and utility of the proposed method, which should be useful for mechanical design engineers at the conceptual design phase.

A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.