Additive manufacturing (AM) is employed for fabricating industrial products with complex geometries. As topological optimization is suitable for designing complex geometries, studies have combined AM and topological optimization, evaluating the density optimization of lattice structures as a variant of topological optimization. The lattice structures of components fabricated via AM comprise voids. Models designed using topological optimization should be modified to ensure structures suitable for AM. As the lattice unit can be easily fabricated using AM with fewer design modifications, this study uses lattice density optimization for an industrial AM product. We propose a method of optimizing the lattice distribution for controlling the surface temperature uniformity of industrial products, such as molds. The effective thermal conductivity of the lattice is calculated using the homogenization and finite element methods. The effective thermal conductivity changes depending on the internal pore sizes. The proposed methodology is validated using a 3D example; the minimization problem of surface temperature variations in the target domain is considered. The variable density of the embedded lattice in the target domain is optimized, and we experimentally validated the performance of the lattice unit cell and optimal 3D structure using metal powder bed fusion AM.

<p>This paper addresses shock absorption behavior of 3-D additive manufactured truncated octahedron for the landing gear of lunar and planetary explores. The deformation modes of truncated octahedron were predicted by the form finding analyses. The collapse load of each deformation load was calculated by the plastic hinge theory. The predicted load--displacement curve agreed with the experimental results, and thus, the proposed prediction method was verified. </p>

© 2018 Elsevier B.V. Controlling the orientation of crystal grains in metal additive manufacturing is an active field of research. Assuming that three-dimensional control of orthotropic metal crystal grain can be achieved, we numerically studied the effective performance of a topology-optimized lattice made of crystal grains with optimal directions in orientation. The objective function is maximized for an effective isotropic stiffness. The effective properties of the lattice are calculated using the homogenization method. Each discretized finite element is regarded as a crystal grain and its artificial density and orientation are optimized. Through several numerical studies using a single orthotropic crystal of Ni exhibiting large anisotropy in its single crystal stiffness, the optimization and the usefulness of the lattice structure composed of optimal orientation crystal grains is confirmed. In the design of a lattice with effective isotropic stiffness, an improvement in stiffness was observed compared with that using equivalent isotropic Ni as a material.

Designing sophisticate ship structures that satisfy several design criteria simultaneously with minimum weight and cost is an important engineering issue. For a ship structure composed of a shell and stiffeners, this issue is more serious because their mutual effect has to be addressed. In this study, a two-stage optimization method is proposed for the conceptual design of stiffeners in a ship's prow. In the first stage, a topology optimization method is used to determine a potential stiffener distribution based on the optimal results, whereupon stiffeners are constructed according to stiffener generative theory and the material distribution. In the second stage, size optimization is conducted to optimize the plate and stiffener sections simultaneously based on a parametric model. A final analysis model of the ship-prow structure is presented to assess the validity of this method. The analysis results show that the two-stage optimization method is effective for stiffener conceptual design, which provides a reference for designing actual stiffeners for ship hulls.

© 2018 Elsevier B.V. Electron-beam melting (EBM) exhibits advantages over other metal-additive manufacturing techniques owing to its low residual stress, rapid fabrication speed, and high energy efficiency. However, in EBM, metal powder is preheated and sintered to stabilize the temperature gradient and powder position during melting with a high-power electron beam. When making a lattice structure by EBM, a certain size of the powder-removing hole is required to remove the sintered remaining metal powder from the lattice. However, a large powder-removing hole can reduce the lattice mechanical performance. We conducted topology optimization to derive an optimal lattice structure shape with high isotropic stiffness assuming fabrication by EBM and minimizing the performance reduction owing to fixed large powder-removing holes. The optimized structure was fabricated via the EBM of a Ti–6Al–4V alloy. The optimal lattice structure achieved 83% of the performance of the Hashin–Shtrikman upper bound in numerical simulations, but an approximate 20% stiffness reduction was observed in the experiments. The isotropy was high with an error in Young's modulus and a strength of less than 9% and 6%, respectively. These results are discussed based on numerical and experimental results.

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.

Manufacturing-oriented topology optimization has been extensively studied the past two decades, in particular for the conventional manufacturing methods, for example, machining and injection molding or casting. Both design and manufacturing engineers have benefited from these efforts because of the close-to-optimal and friendly-to-manufacture design solutions. Recently, additive manufacturing (AM) has received significant attention from both academia and industry. AM is characterized by producing geometrically complex components layer-by-layer, and greatly reduces the geometric complexity restrictions imposed on topology optimization by conventional manufacturing. In other words, AM can make near-full use of the freeform structural evolution of topology optimization. Even so, new rules and restrictions emerge due to the diverse and intricate AM processes, which should be carefully addressed when developing the AM-specific topology optimization algorithms. Therefore, the motivation of this perspective paper is to summarize the state-of-art topology optimization methods for a variety of AM topics. At the same time, this paper also expresses the authors’ perspectives on the challenges and opportunities in these topics. The hope is to inspire both researchers and engineers to meet these challenges with innovative solutions.

Ca2RuO4%y ceramics exhibit a large volumetric negative thermal expansions (NTE), although the crystallographic volume contraction on heating is much smaller than the NTE. Therefore, we examine the differences in the mechanisms underlying the volumetric thermal expansion for ruthenate ceramics and crystals in the context of the elasticity. We identify the possible microstructure of ruthenate ceramics composed of crystal grains and cavities using structural topology optimization. We conclude that the measured large volumetric NTE of ruthenate ceramics is certainly possible via anisotropic crystallographic thermal expansion through an elastic mechanism.

Magneto-mechanical resonators—magnetically-driven vibration devices—are used in many mechanical and electrical devices. We develop topology optimization (TO) to configure the magnetic fields of such resonators to enable large vibrations under specified current input to be attained. A dynamic magneto-mechanical analysis in the frequency domain is considered where we introduce the surface magnetic force calculated from the Maxwell stress tensor. The optimization problem is then formulated involving specifically the maximization of the dynamic compliance. This formulation is implemented using the solid-isotropic-material-with-penalization method for TO by taking into account the relative permeability, Young’s modulus, and the mass density of the magnetic material as functions of the density function. Through the 2D numerical studies, we confirm that this TO method works well in designing magnetic field patterns and providing matching between the external current frequency and eigenfrequency of the vibrating structure.

<p>Structure can get various mechanical characteristics by applying periodic structures as typified by lattice structures. Lattice structures are generally used inside the structural member in order to reduce the weight. One advantage of lattice structures is that we do not need to change the whole structural shape when we replace the solid part of a component with the lattice structures. Another advantage is the lightness of the weight, and hence it is important to design a high performance lattice shape with low weight. However, a framework for development of micro lattice structures considering both stiffness and weight has not been established. Thus, we propose a method for designing and producing micro lattice structures. We use a topology optimization method for a designing methodology. Topology optimization is an effective method in designing high performance lattice structure since topology optimization allows us to change the topology and to design a complicated shape. We use a metal additive manufacturing (AM) machine for producing the optimal lattice structures. AM allows us to produce a complicated structure which removal and forming manufacturing cannot produce. We use a bulk modulus as the objective function since it is one of the important mechanical characteristics in design. In this research, we use a homogenization method to compute the bulk modulus. Objective function was modified so that isotropy of the optimal shape is retained when the solution is updated. In addition, structures produced by AM need holes so that internal metal powder can be removed. Hence, we defined the design domain so that the optimal structure becomes open cell structure. Then, high bulk modulus shapes were derived using topology optimization. The lattice structures were produced by metal AM machine after being modified for production.</p>

To realize negative thermal expansion (NTE), porous composites made of two materials with different coefficients of thermal expansion are being actively researched. NTE can be realized by taking advantage of the thermal deformation mechanisms of a composite material's internal geometry. However, in addition to negative thermal expansion, materials with anisotropic and large positive thermal expansion are also desirable for various applications. Also, additive manufacturing provides new ways to fabricate composites by layering multiple materials at arbitrary points in three-dimensional space. In this study, we developed a design methodology for porous composites, which showed defined thermal expansion characteristics, including negative and positive thermal expansion as well as isotropic and anisotropic thermal expansion. Our approach was tested based on the fabrication of a multi-material photopolymer by additive manufacturing. The internal geometries required to produce such characteristics were designed by topology optimization, which is the most effective structural optimization method for realizing macroscopic inward deformation and for maintaining stiffness. The designed structures were converted to three-dimensional models and fabricated by multi-material photopolymer additive manufacturing. Using laser scanning dilatometry, we measured the thermal expansion of these specimens, revealing well-ordered thermal expansion, from anisotropic positive thermal expansion to anisotropic negative thermal expansion, over a wide range of about -3 x 10(-4) K-1 to 1 x 10(-3) K-1. (C) 2017 Elsevier Ltd. All rights reserved.

© 2017 Elsevier B.V. Recent additive manufacturing technologies can be used to fabricate porous metals with precise internal pore structures and effective performance. We use topology optimization to derive an optimal pore structure shape with high stiffness that is verified experimentally. The design maximizes the effective bulk modulus and isotropic stiffness, and the performance is compared with Hashin–Shtrikman (HS) bounds. The optimized structure is fabricated via selective laser melting of maraging steel, which is a high-strength, iron-nickel steel that cannot easily be made porous with conventional methods. The optimal porous structure achieved 85% of the performance of the HS upper bound in numerical simulations, and at least 90% of them were realized in compressive testing. Finally, the performance is discussed relative to that of other metals.

Non-destructive testing (NDT) detects damage according to a difference in a physical phenomenon between a normal structure and damaged structure. As a solution avoiding human errors in NDT, a numerical method based on a dynamical numerical analysis model and a structural optimization algorithm was proposed. This method automatically derives a structure with a response that is equal to that of a damaged structure through an optimization procedure. Among structural optimization methods, topology optimization can optimize the structure fundamentally by changing the topology and not just the shape of a structure. Thus, topology optimization is employed together with eigenfrequency analysis, which is the most fundamental methodology of NDT. The proposed method derives a structure that has the same eigenfrequencies as a damaged structure employing topology optimization. The shape and location of damage can be identified through the optimal shape obtained.

Additive manufacturing may be a novel method for fabricating porous materials. These materials can achieve effective performance because of their internal geometries. Metal-additive manufacturing is expected to utilize thermal conduction materials and devices. We have developed a porous metal with effective isotropic thermal conductivity by using metal-selective laser melting additive manufacturing. The internal pore structure was designed by topology optimization, which is the most effective structural optimization technique to maximize effective thermal conductivity. The designed structure was converted to a three-dimensional STL model, which is a native digital format of additive manufacturing, and assembled as a test piece. Effective thermal conductivity was measured by a steady-state method in which the effective thermal conductivity was calculated from a one-dimensional temperature gradient and the heat flux of the test pieces. The test pieces showed an effective thermal conductivity close to the Hashin-Shtrikman or Maxwell-Eucken bound, which is the theoretical limit of effective performance with an error less than 10%. (C) 2016 Elsevier Ltd. All rights reserved.

© 2016 Elsevier B.V. With a view to developing a highly biocompatible and highly reliable material for artificial hip joints, cellular lattice structures with high strength and low Young's modulus (E) were designed using computational shape optimization. These structures were fabricated from a biomedical Co-Cr-Mo alloy via electron beam melting. As a starting point for shape optimization, inverse body-centered-cubic (iBCC)-based structures with different porosities and aspects were fabricated. The strength tended to increase with increasing E. Then, the structures were re-designed using shape optimization based on the traction method, targeting a simultaneous increase in yield strength with retention of the low E. The shapes were optimized through minimization of the maximum local von Mises stress and control of E to 3/2 or 2/3 of the original value, while maintaining constant porosity. The re-designed cellular structures were fabricated and subjected to mechanical testing. The E values of the porous structures were comparable to the design values, but the strength of the cellular lattice with E = 2/3 (design value) was lower than expected. This discrepancy was attributed to inhomogeneities in the microstructures and their impact on the lattice mechanical properties. Thus, shape optimization considering crystal orientation is a significant challenge for future research, but this approach has considerable potential.

In this research, we propose a new objective function for optimizing damping materials to reduce the resonance peak response in the frequency response problem, which cannot be achieved using existing criteria. The dynamic compliance in the frequency response problem is formulated as the scalar product of the conjugate transpose of the amplitude vector and the force vector of the loading nodes. The proposed objective function methodology is implemented using the common solid isotropic material with penalization (SIMP) method for topology optimization. The optimization problem is formulated as maximizing the complex part of the proposed complex dynamic compliance under a volume constraint. 2D and 3D numerical examples of optimizing the distribution of the damping material on the host structure are provided to illustrate the validity and utility of the proposed methodology. In these numerical studies, the proposed objective function worked well for reducing the response peak in both lower and upper excitation frequencies around the resonance. By adjusting the excitation frequency, multi-resonance peak reduction may be achieved with a single frequency excitation optimization. (C) 2015 Elsevier Ltd. All rights reserved.

Structural optimization for reinforcing the anti-vibration characteristics of the generators in the engine room of a ship is presented. To improve the vibration characteristics of the structures, topology optimization methods can be effective because they can optimize the fundamental characteristics of the structure with their ability to change the topology of the target structure. Topology optimization is used to improve the characteristics of the anti-vibration reinforcement of the generators in the engine room. First, an experimentally observed vibration phenomenon is simulated using the finite element method for frequency response problems. Next, the objective function used in topology optimization is set as the dynamic work done by the load based on the energy equilibrium of the structural vibration. The optimization problem is then constructed by adding the volume constraint. Finally, based on finite element analysis and the optimization problem, topology optimization is performed on several vibration cases to improve their performance and reduce weight.

Additive manufacturing (AM) could be a novel method of fabricating composite and porous materials having various effective performances based on mechanisms of their internal geometries. Materials fabricated by AM could rapidly be used in industrial application since they could easily be embedded in the target part employing the same AM process used for the bulk material. Furthermore, multi-material AM has greater potential than usual single-material AM in producing materials with effective properties. Negative thermal expansion is a representative effective material property realized by designing a composite made of two materials with different coefficients of thermal expansion. In this study, we developed a porous composite having planar negative thermal expansion by employing multi-material photopolymer AM. After measurement of the physical properties of bulk photopolymers, the internal geometry was designed by topology optimization, which is the most effective structural optimization in terms of both minimizing thermal stress and maximizing stiffness. The designed structure was converted to a three-dimensional stereolithography (STL) model, which is a native digital format of AM, and assembled as a test piece. The thermal expansions of the specimens were measured using a laser scanning dilatometer. Negative thermal expansion corresponding to less than -1 x 10(-4) K-1 was observed for each test piece of the N = 3 experiment. (C) 2015 Author(s).

Optimal cross-sectional shapes for whispering-gallery mode sensors with prescribed emission wavelengths and resonance modes are generated through topology optimization based on the finite element method. The sensor is assumed to detect the state of the domain surrounded by the sensor. We identified the integral of the square of the electric field intensity over the detection domain and the quality factor (Q factor), which should be maximized, as key values for the sensor sensitivity, representing the detection limit for the relative permittivity change of the test object. Based on this, the integral of the square of the electric field intensity over the detection domain and the Q factor are studied as the optimization targets. In our numerical study, their optimal configuration characteristics are identified and analyzed. The resulting device has a small radius, a small detection domain and a concave shape with a center located next to the detection domain. We also succeeded in performing simultaneous optimization of the integral of the square of the electric field intensity over the detection domain and the Q factor.

In this research, a topology optimization with constraints of structural strength and thermal conductivity is proposed. The coupled static linear elastic and heat conduction equations of state are considered. The optimization problem was formulated; viz., minimizing the volume under the constraints of p-norm stress and thermal compliance introducing the qp-relaxation method to avoid the singularity of stress-constraint topology optimization. The proposed optimization methodology is implemented employing the commonly used solid isotropic material with penalization (SIMP) method of topology optimization. The density function is updated using sequential linear programming (SLP) in the early stage of optimization. In the latter stage of optimization, the phase field method is employed to update the density function and obtain clear optimal shapes without intermediate densities. Numerical examples are provided to illustrate the validity and utility of the proposed methodology. Through these numerical studies, the dependency of the optima to the target temperature range due to the thermal expansion is confirmed. The issue of stress concentration due to the thermal expansion problem in the use of the structure in a wide temperature range is also clarified, and resolved by introducing a multi-stress constraint corresponding to several thermal conditions. (C) 2014 Elsevier B.V. All rights reserved.

This research develops a stress-based topology optimization method (STOM) using the phase-field method representing topological changes. This research shows that to apply the phase field method, regional and localized stress constraints should be addressed. Thus, we use an Augmented Lagrange multiplier approach for the stress constraints and present a new numerical solution for the Lagrange multipliers inside the Allen-Cahn equation with the topological derivatives. Through several two dimensional illustrative problems, the results of the phase-field method have larger objective values, but are robust from a stress point of view compared with the results of the STOM by the density method. (C) 2013 Elsevier Ltd. All rights reserved.

This paper describes a design methodology for piezoelectric energy harvesters that thinly encapsulate the mechanical devices and exploit resonances from higher-order vibrational modes. The direction of polarization determines the sign of the piezoelectric tensor to avoid cancellations of electric fields from opposite polarizations in the same circuit. The resultant modified equations of state are solved by finite element method (FEM). Combining this method with the solid isotropic material with penalization (SIMP) method for piezoelectric material, we have developed an optimization methodology that optimizes the piezoelectric material layout and polarization direction. Updating the density function of the SIMP method is performed based on sensitivity analysis, the sequential linear programming on the early stage of the optimization, and the phase field method on the latter stage of the optimization to obtain clear optimal shapes without intermediate density. Numerical examples are provided that illustrate the validity and utility of the proposed method.

An optimization methodology is proposed for the piezoelectric transducer (PZT) layout of an energy-recycling semi-active vibration control (ERSAVC) system for a space structure composed of trusses. Based on numerical optimization techniques, we intend to generate optimal location of PZTs under the constraint for the total length of PZTs. The design variables are set as the length of the PZT on each truss based on the concept of the ground structure approach. The transient problems of the mechanical and electrical vibrations based on the ERSAVC theory are considered as the equations of state. The objective is to minimize the integration of the square of all displacement over the whole analysis time domain. The sensitivity of the objective function is derived based on the adjoint variable method. Based on these formulations, an optimization algorithm is constructed using the fourth-order Runge-Kutta method and the method of moving asymptotes. Numerical examples are provided to illustrate the validity and utility of the proposed methodology. Using the proposed methodology, the optimal location of PZTs for the vibration suppression for multi-modal vibration is studied, which can be benchmark results of further study in the context of ERSAVC systems. (C) 2013 Elsevier Ltd. All rights reserved.

We discuss a phase field method for shape optimization in the context of electromagnetic wave propagation. The proposed method has the same functional capabilities as the level set method for shape optimization. The first advantage of the method is the simplicity of computation, since extra operations such as re-initialization of functions are not required. The second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis. Structural shapes are represented by the phase field function defined in the design domain, and this function is optimized by solving a time-dependent reaction diffusion equation. The artificial double-well potential function used in the equation is derived from sensitivity analysis. We study four types of 2D or 2.5D (axisymmetric) optimization problems. Two are the classical problems of photonic crystal design based on the Bloch theory and photonic crystal wave guide design, and two are the recent topics of designing dielectric left-handed metamaterials and dielectric ring resonators. (C) 2013 Elsevier Inc. All rights reserved.

Cloaking techniques conceal objects by controlling the flow of electromagnetic waves to minimize scattering. Herein, the effectiveness of homogenized anisotropic materials in non-resonant dielectric multilayer cloaking is studied. Because existing multilayer cloaking by isotropic materials can be regarded as homogenous anisotropic cloaking from a macroscopic view, anisotropic materials can be efficiently designed through optimization of their physical properties. Anisotropic properties can be realized in two-phase composites if the physical properties of the material are within appropriate bounds. The optimized anisotropic physical properties are identified by a numerical optimization technique based on a full-wave simulation using the finite element method. The cloaking performance measured by the total scattering width is improved by about 2.8% and 25% in eight-and three-layer cylindrical cloaking materials, respectively, compared with multilayer cloaking by isotropic materials. In all cloaking examples, the optimized microstructures of the two-phase composites are identified as the simple lamination of two materials, which maximizes the anisotropy. The same performance as published for eight-layer cloaking by isotropic materials is achieved by three-layer cloaking using the anisotropic material. Cloaking with an approximately 50% reduction of total scattering width is achieved even in an octagonal object. Since the cloaking effect can be realized using just a few layers of the laminated anisotropic dielectric composite, this may have an advantage in the mass production of cloaking devices. (C) 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

The non-destructive testing for detecting structural damage at the earliest possible stage is significant in the safe and long life operation of large marine structure such as ships. In the non-destructive testing, the variation in structural characteristics from the original one to the damaged one is used to identify the damage. Vibration characteristic is one of the most popular criteria used for it due to its sensitivity for the structural damage. In this paper, a damage identification method used in non-destructive testing is proposed based on topology optimization focusing on the difference of eigenfrequency between the original structure and the damaged one. Eigenfrequency analysis of a structure is considered. A inimization of the least squared error between the eigenfrequency of the damaged structure and the target eigenfrequency is set as an objective function. An optimization algorithm is constructed based on the topology optimization and the method of moving asymptotes (MMA). The validity and the usefulness of the proposed method is confirmed by several numerical examples.

The sensitivity analysis of objective functions including the eigenmodes of continuum systems governed by scalar Helmholtz equations is carried out in continuum form. In addition, based on the sensitivity, the mode shapes are specified through numerical optimization. Using the continuum sensitivity and adjoint equation, the physical nature of them can be analyzed, which helps to explain the nature of the target optimization problem. Moreover, the continuum sensitivity and adjoint equation contribute to the quick numerical implementation of sensitivity analysis using software that can solve an arbitrary partial differential equation directly. A scalar Helmholtz equation in 1D or 2D domain is considered. The sensitivity analysis is performed, for the general objective function formulated as a function of the eigenmode in continuum form. A minimization problem using the least squared error (i.e., difference) between the eigenvector and target mode shape is set as a sample objective function for both the first and second eigenmodes. The sensitivity and the adjoint equation are derived for this objective function. 1D and 2D numerical sensitivity analysis and optimization examples are studied to illustrate the validity of the derived sensitivity. (C) 2012 Elsevier Ltd. All rights reserved.

Optimal cross-sectional shapes of whispering-gallery ring resonators with prescribed emission wavelength and resonance mode are generated using topology optimization based on the finite element method. The two critical performance indices, the quality factor (Q factor) and mode volume of a resonator, are treated as the objective functions in the optimization. In our numerical study, characteristics of optimal configurations are identified and analyzed. Since the Q factor and mode volume have a tradeoff relationship, i.e., an increasing Q factor increases mode volume, a Pareto-optimal set of solutions can be identified under certain device specifications. These configurations achieve better performances than existing shapes in producing both a high Q factor and low mode volume.

Piezoresistive sensors are commonly made of a piezoresistive membrane attached to a flexible substrate, a plate. They have been widely studied and used in several applications. It has been found that the size, position and geometry of the piezoresistive membrane may affect the performance of the sensors. Based on this remark, in this work, a topology optimization methodology for the design of piezoresistive plate-based sensors, for which both the piezoresistive membrane and the flexible substrate disposition can be optimized, is evaluated. Perfect coupling conditions between the substrate and the membrane based on the `layerwise' theory for laminated plates, and a material model for the piezoresistive membrane based on the solid isotropic material with penalization model, are employed. The design goal is to obtain the configuration of material that maximizes the sensor sensitivity to external loading, as well as the stiffness of the sensor to particular loads, which depend on the case (application) studied. The proposed approach is evaluated by studying two distinct examples: the optimization of an atomic force microscope probe and a pressure sensor. The results suggest that the performance of the sensors can be improved by using the proposed approach.

This paper discusses an application of the topology optimization method for the design of thermoelectric generators. The proposed methodology provides the optimized geometry in accordance with various arbitrary conditions such as the types of materials, the volume of materials, and the temperature and shape of the installation position. By considering the coupled equations of state for the thermoelectric problem, we introduce an analytical model subject to these equations, which mimics the closed circuit composed of thermoelectric materials, electrodes, and a resistor. The total electric power applied to the resistor and the conversion efficiency are formulated as objective functions to be optimized. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization method used in topology optimizations. Simple relationships are formulated between the density function of the solid isotropic material with penalization method and the physical properties of the thermoelectric material. A sensitivity analysis for the objective functions is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes. Finally, numerical examples are provided to demonstrate the validity of the proposed method. Copyright (C) 2012 John Wiley & Sons, Ltd.

This paper proposes a topology optimization method for the design of compliant circular path mechanisms, or compliant mechanisms having a set of output displacement vectors with a constant norm, which is induced by a given set of input forces. To perform the optimization, a simple linear system composed of an input force vector, an output displacement vector and a matrix connecting them is constructed in the context of a discretized linear elasticity problem using FEM. By adding two constraints: 1, the dimensions of the input and the output vectors are equal; 2, the Euclidean norms of all local input force vectors are constant; from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector, the optimization problem, which specifies and equalizes the norms of all output vectors, is formulated. It is a minimization problem of the weighted summation of the condition number of the matrix and the least square error of the second singular value and the specified value. This methodology is implemented as a topology optimization problem using the solid isotropic material with penalization method, sensitivity analysis and method of moving asymptotes. The numerical examples illustrate mechanically reasonable compliant circular path mechanisms and other mechanisms having multiple outputs with a constant norm. Copyright (C) 2011 John Wiley & Sons, Ltd.

A structural optimization system using the Finite Element Method (FEM) for the initial design stage of a ship is presented in this paper. A general bulk carrier is selected as the object for the optimization. Some dimensions determining the shapes of the ship&apos;s bottom structure are taken as design variables. Since the design variables affect the shape of the structure, the FEM model needs to be updated during the optimization. Further, the structure of the ship is so large and complicated that optimizing the shape of the ship&apos;s structure is very difficult. The individual mesh subdivision technique and the multi-point constraint method are introduced to make this optimization possible. However, creating the FEM datasets for the ship&apos;s structure requires a lot of manpower. To remove this drawback, Prime Ship-Hull is used because it has a function for recognizing the ship&apos;s structural members. The FEM dataset which can be applied to the individual mesh subdivision technique and the multi-point constraint method is made automatically from the structural members categorized by Prime Ship-Hull. Five key design variables for shape optimization of the ship&apos;s bottom structure-height and width of the double bottom, height of the bilge hopper tank, and the 2 widths of the lower stool-are considered here. A numerical example shows that the proposed method makes it possible to optimize the shape of the ship&apos;s bottom structure.

This paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load. The design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control is introduced to assess the value of the robust compliance. The aggregation solution technique provides the direct relationship between the uncertain input load and output displacement, as a small linear system composed of these vectors and the reduced size of a symmetric matrix, in the context of a discretized linear elasticity problem, using the finite element method. Introducing the constraint that the Euclidean norm of the uncertain load set is fixed, the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the Rayleigh-Ritz theorem for symmetric matrices. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures. (C) 2011 Elsevier B.V. All rights reserved.

Topology optimization has been successfully used in many industries, especially those engaged in the design and manufacturing of mechanical devices, but numerical problems are often encountered, such as grayscale representations of obtained composites. A type of structural optimization method using the level set theory for boundary expressions has been proposed, in which the outlines of target structures are implicitly represented using the level set function, and optimal configurations are obtained by updating this function based on the shape sensitivities. Level set-based methods typically have a drawback, however, in that topological changes that increase the number of holes in the material domain are not allowed. To overcome the above numerical and topological problems, this paper proposes a new topology optimization method incorporating level set boundary expressions based on the concept of the phase field method, which we apply to a minimum mean compliance problem. First, a structural optimization problem is formulated based on a boundary expression, using the level set function. Next, a time evolutionary equation for updating the level set function is formulated based on the concept of the phase field method, and the minimum mean compliance problem is formulated using a level set boundary expression. An optimization algorithm for the topology optimization incorporating the level set boundary expression based on the concept of the phase field method is then derived. Several examples are provided to confirm the usefulness of the proposed structural topology optimization method.

This paper proposes a robust topology optimization method for a linear elasticity design problem subjected to an uncertain load. The robust design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control research is introduced to evaluate the value of the robust compliance. The aggregation is applied to provide the direct relationship between the uncertain input load and output displacement using a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method. The robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix subject to the constraint that the Euclidean norm of the uncertain load set is fixed. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed robust structural optimization method is implemented using the topology optimization method, sensitivity analysis and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.

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. (C) 2010 Elsevier B.V. All rights reserved.

Load cells are used extensively in engineering fields. This paper describes a novel structural optimization method for single- and multi-axis load cell structures. First, we briefly explain the topology optimization method that uses the solid isotropic material with penalization (SIMP) method. Next, we clarify the mechanical requirements and design specifications of the single- and multi-axis load cell structures, which are formulated as an objective function. In the case of multi-axis load cell structures, a methodology based on singular value decomposition is used. The sensitivities of the objective function with respect to the design variables are then formulated. On the basis of these formulations, an optimization algorithm is constructed using finite element methods and the method of moving asymptotes (MMA). Finally, we examine the characteristics of the optimization formulations and the resultant optimal configurations. We confirm the usefulness of our proposed methodology for the optimization of single- and multi-axis load cell structures.

This paper discusses a structural optimization method that optimizes shape and topology based oil the phase field method. The proposed method has the same functional capabilities as a structural optimization method based oil 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 oil the optimal configurations (C) 2009 Elsevier Inc All rights reserved.

This paper proposes a robust topology optimization method for a linear elasticity design problem subjected to an uncertain load. The robust design problem is formulated to minimize a robust compliance that is defined as the maximum compliance for the worst load case of an uncertain load set. Since the robust compliance can be formulated the scalar product of the uncertain input load and output displacement, the idea of "aggregation" used in the field of control research is introduced to evaluate the value of the robust compliance under the uncertainty of input load. Aggregation is to represent the direct relationship between the uncertain input load and output displacement by a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method. According to the Rayleigh-Ritz theorem for symmetric matrices, the robust compliance minimization problem is formulated as the minimization problem of the maximum eigenvalue of the aggregated symmetric matrix when Euclidean norm of the uncertain load set is fixed.

This paper discusses a structural optimization method that achieves optimum 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. Since the proposed method does not require extra operations such as re-initialization of the level set function or smoothing of sensitivities, the computational cost is lower than that of typical level set methods. 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 toa two-dimensional linear elastic problem. The provided numerical examples illustrate the convergence of the compliance minimization problem.

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. Since the proposed method does not require extra operations such as re-initialization of the level set function or smoothing of sensitivities, the computational cost is lower than that of typical level set methods. 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 function and the effect that perimeter control has on the optimal configurations.

Topology optimization has been successfully used in many industries, such as mechanical industries, but it often encounters numerical problems such as grayscale representations of obtained composites. A type of structural optimization method using the level set theory for boundary expressions has been proposed, in which the outlines of target structures are implicitly represented using the level set function, and optimal configurations are obtained by updating this function based on the optimal criteria. However, this method has a drawback that it does not allow topological changes that either introducing a hole in the material domain. To overcome the above two problems, this paper proposes a new topology optimization method incorporating level set boundary expressions based on the concept of the phase field method, and we apply it to the minimum mean compliance problem. First, a structural optimization problem is formulated based on a boundary expression using the level set function. Next, a time evolution equation for updating the level set function is formulated based on the concept of the phase field method, and the minimum mean compliance problem is formulated using the level set boundary expression. An optimization algorithm for the topology optimization incorporating the level set boundary expression based on the concept of the phase field method is derived. Finally, several examples are provided to confirm the usefulness of the proposed structural optimization method.

Band-gap structures can inhibit the propagation of waves in certain frequency ranges, or bands. Phononic band-gap structures that inhibit certain acoustic and elastic waves from propagating have received a great deal of attention recently, and have great potential in engineering applications, as it is hoped that they can be utilized as sound or vibration protection devices, acoustic lasers, acoustic mirrors, frequency filters and waveguides in mechanical structures and MEMS. This paper presents a new structural optimization method for the design of more generic and extendible phononic band-gap structures using discrete structural elements, based on topology optimization techniques. The use of a variety of discrete structural elements has the advantage of enabling the design of higher performance or more sophisticated structures. In addition, in this research, a new objective function is proposed that enables several bands to be simultaneously considered as optimization targets, in contrast to the objective function typically employed in existing design methods for phononic band-gap structures that considers only a single band where attenuation is desired. Using this new objective function, structures incorporating band-gap effects across multiple eigen-frequency bands can be created. The resulting optimal configurations have the potential to dramatically reduce the propagation of waves in specified frequency ranges. Furthermore, several numerical examples confirm the validity and utility of the proposed method.

This paper discusses a new structural optimization method, based on topology optimization techniques, using frame elements where the cross-sectional properties can be treated as design variables. For each of the frame elements, the rotational angle denoting the principal direction of the second moment of inertia is included as a design variable, and a procedure to obtain the optimal angle is derived from Karush-Kuhn-Tucker (KKT) conditions and a complementary strain energy-based approach. Based on the above, the optimal rotational angle of each frame element is obtained as a function of the balance of the internal moments. The above methodologies are applied to problems of minimizing the mean compliance and maximizing the eigen frequencies. Several examples are provided to show the utility of the presented methodology.

This paper presents a new structural optimization method that supports decision-making processes to obtain innovative designs at the conceptual design phase. This method is developed based on structural and function-oriented elements, such as frame and panel elements that have specific functions. For each of the frame elements, the rotational angle denoting the principal direction of the second moment of inertia is included as a design variable, and a procedure to obtain the optimal angle is derived from Karush-Kuhn-Tucker conditions. For the panel elements, two types of panel elements are introduced based the assumed stress method. Several examples are provided to show the utility of the methodology presented here for mechanical design engineers.

Computer-aided engineering (CAE) has been successfully used in mechanical industries such as automotive industries. CAE enables us to quantitatively evaluate the mechanical performances of products and to propose an effective way to improve their performances using optimization techniques without building physical prototypes. However, CAE tools are usually utilized not at the conceptual design phase, but at the evaluation phase following the detailed design phase. This is because current CAE tools require detailed design data that does not yet exist at the conceptual design phase, and such tools also inhibit the provision of useful design suggestions that, ideally, match the way of thinking and insight of design engineers. Thus, at present, no CAE tools exist that can assist the conceptual design decision making process of design engineers. On the other hand, conceptual design processes are of great significance when seeking to create innovative and high-performance products and to shorten their development time. In order to fulfill the designer's needs during the conceptual design phase, a new type of CAE method must be constructed, one that enables concurrent design support and evaluation, and fits the way design engineers think and explore design insights.
This article presents a new structural optimization method that supports concurrent decision making so that design engineers can work to obtain innovative designs and evaluate the mechanical design details of mechanical structures at the conceptual design phase. This method is developed based on the concept of product-oriented analysis and discrete, function-oriented elements, such as beam and panel elements, since these can provide design suggestions concerning the structural evaluation of reasons as to why certain design ideas obtained are reasonable or optimal in the design sense.
The basic ideas and specifications needed to construct the method are explained and the construction of the structural optimization design method is discussed. The optimization algorithm is developed using the ground structure approach and CONLIN sequential convex programming. The examples provided demonstrate the utility of the proposed methodology for supporting design engineers' concurrent decision making, so that innovative mechanical designs can be evaluated at the conceptual design phase.

This paper discusses a structural topology optimization method based on frame elements. The purpose of this method is to support decision-making as engineers work to obtain innovative designs and evaluate mechanical design details at the conceptual design phase for mechanical structures such as automotive bodies and machine tools. When developing mechanical products, enrichment of the conceptual phase is an important issue. At the conceptual design phase, basic structural and mechanical design features that directly affect product performance are decided. Especially, vibration characteristic is concerned with product performances such as durability, and one of the most important features. So, in this paper, we present an eigen value maximize topology optimization method based on discrete and function-oriented elements such as frame, panel and spring elements, and sequential convex programming. In addition, examples are provided to show the utility of the methodology presented here for mechanical design engineers.

Additive manufacturing may be a novel method for fabricating porous materials. These materials can achieve effective performance because of their internal geometries. Metal-additive manufacturing is expected to utilize thermal conduction materials and devices. We have developed a porous metal with effective isotropic thermal conductivity by using metal-selective laser melting additive manufacturing. The internal pore-structure was designed by topology optimization, which is the most effective structural optimization technique to maximize effective thermal conductivity. The designed structure was converted to a three-dimensional STL model, which is a native digital format of additive manufacturing, and assembled as a test piece. Effective thermal conductivity was measured by a steady-state method in which the effective thermal conductivity was calculated from a one-dimensional temperature gradient and the heat flux of the test pieces.

This paper discusses a geometrical optimization method for thermoelectric generators. Our method provides the optimized geometry in accord with various arbitrary conditions such as types of materials, volume of materials, temperatures of installation position, and shape of installation position. It has the potential to contribute to improving thermoelectric devices in supporting design stages. By establishing the coupled equations of state for the thermoelectric problem, an analytical model subject to these equations is introduced that mimics the closed circuit composed of the thermoelectric materials, electrodes and a resistor. The total electric power lost through the resistor is formulated as an objective function for optimization. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization (SIMP) method used in topology optimizations. Simple relations are formulated between the density function of SIMP method and physical properties of thermoelectric material. Sensitivity analysis for the objective function is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes, which has numerous benefits in various optimization problems by virtue of its combination with topology optimization. Finally, the numerical examples are provided to demonstrate the validity of our method.

構造最適化とは,設計者の勘と経験に頼らずに,物理的,数学的根拠に基づいて,構造物の最適な形状や形態を決定する方法論である.近年,構造最適化の新しい方法として,レベルセットモデリングに基づいた形状最適化が世界的に注目を集めている.この方法は,一次元高位の関数である,レベルセット関数を用いて,外形形状を陰的に表現する.そして,最適化問題の定式化に基づいて,レベルセット関数を更新させることにより,最適形状を示すレベルセット関数を得る方法である.しかし,この方法は基本的には形状最適化の方法であるため,物体領域に穴がそう出されるようなトポロジー変化を許容しない問題を持つ.さらには,最適構造は無数の局所解を持つが,設計者が意図した最適解を得るのが非常に難しい問題を持つ.そこで,本研究では,著者らが提案した,Topological Phase-Field法に基づく構造最適化を用いることにより,上述の問題を抜本的に解決する方法を構築する.そして,機械構造物の最適化問題に適用することにより,その方法論の有効性を検証する.

Faculty of Engineering at Hiroshima University participates in JAPAN INTERNATIONAL BIRDMAN RALLY every year since 2002. The human-powered aircraft consists of various members, but main wing is the most important part which produces necessary lifting power most effectively so that human-powered aircraft flies in the sky in that. A ready-made thing has been used for the air foil of the main wing till now. Therefore optimize the whole main wing as a new trial, and finding the air foil of the most suitable solution of this study is aimed. Air foil is expressed with a NURBS curve and XFOIL is used for two-dimensional fluid analysis in this study. As an optimization technique, GA is used. Lift produced by the whole main wing and a moment hanging to the wing root are constraint conditions. The objective functions are that the drag in installation attack angle of 3 degrees is minimized to have high performance in a design speed and that the lift drag ratio from an attack angle 0 degrees to 5 degrees is maximized to have more stable wing performance. And then, the most suitable solutions for two cases are derived. In this study, fluid analysis is limited to a two-dimensional problem and the structural consideration is ignored. In addition, in consideration of a fluid characteristic, two kinds of objective functions are introduced and each solution is found. The comparison and the examination of obtained solutions are performed, and the characteristic of each solution is observed to get. Previously, evaluation of the performance was performed so far after having decided air foil. However, air foil which matched a designed purpose was able to be found. As a result of the optimization in this study, a characteristic was found in each solution.

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 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. Based on these formulations, an optimization algorithm is constructed using sequential linear programming.

A structure optimization for the initial design of ship is under consideration in this paper. A bulk career is selected for the object of the optimization. Both the sizes and the plate thicknesses of a bottom structure of ship are taken as design variable simultaneously. Individual mesh sub-division technique and multi-point constrain method are introduced for making this optimization possible. In order for reducing computational effort, a sensitivity analysis is used instead of FEM, and the validity of using it is examined. Computational results are compared with a real ship data and a great performance is observed for the optimal design with considering the size and the plate thickness of a ship structure. Genetic algorithm is used for this optimization in this study.

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 higher order function. But these methods have the problem which does not allow the topological changes increasing the number of holes in the material domain. In this research, we proposed a new topology optimization method based on the level set model and concept of the phase-field theory. In addition, the method is expanded to minimizing the mean compliance problem. Simple example is provided to confirm the usefulness of the proposed structural optimization method.

This paper presents an advanced multi-step optimal design method using function oriented elements for the development of the development of the leaning spring for artificial satellite. For the sake of the efficient design, the appropriate methodology must be selected in each design phase. In this research, the topology optimization method and the shape optimization method are used in the conceptual design phase and the detailed design phase respectively. Frame elements are used for both phases as the function oriented element. In addition, the numerical example of based on the actual analysis model of leaning spring part are provided to show the utility of the methodology presented here.

This paper discusses a method to determine the optimal direction of the principal moment of inertia in frames element cross-sections for the design of mechanical structures at the conceptual design phase. The direction in each frame element is determined based on the KKT-conditions and the complementary energy concept. This method is implemented as an application in a structural topology optimization that maximizes multiple eigenfrequencies to obtain the most stable designs of mechanical structures for dynamic problems. 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.

This paper presents a new topology optimization method that supports decision-making to obtain innovative designs at the conceptual design phase for mechanical structures. This method is developed based on structural and function-oriented elements such as frame and panel elements. For each of the frame elements, the rotational angle denoting the principal direction of the second moment of inertia is included as a design variable, and a way to obtain the optimal angle is derived from Karush-Kuhn-Tucker(KKT) conditions. For the panel elements, two types of panel element are introduced based the assumed stress method. 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.

This paper presents the outcome of research concerning the development of a structural topology optimization methodology that can be used at the conceptual design phase of mechanical products. The objective function is formulated as the minimization of the mean compliance. The optimization method is constructed based on frame elements, which are a typical discrete structural element. 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 length 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.

This paper presents the,outcome of research concerning the development of a structural topology optimization methodology which can be used at the conceptual design phase of mechanical products. This optimization method is constructed based on beam elements with ellipsoidal cross-section and its cross-sectional area and rotational angle denoting the principal direction of second moment of inertia. First, the optimization problem is formulated for the eivenvalue maximization problem, and its algorithm is constructed based on the ground structure approach and the convex approximation method. The update procedure of the rotational angle denoting the principal direction of second moment of inertia in ellipsoidal cross-section is derived form KKT-conditions. In addition, several examples are provided to show the utility of the methodology presented here.

This paper presents the outcome of research concerning the development of a structural topology optimization methodology which can be used at the conceptual design phase of mechanical products. This optimization method is constructed based on beam elements with ellipsoidal cross-section. First, the optimization problem is formulated for the stiffness maximization problem, and its algorithm is constructed based on the ground structure approach and the convex approximation method. The update procedure of the rotational angle regarding the principal direction of second moment of inertia in ellipsoidal cross-section is derived form KKT-conditions. In addition, several examples are provided to show the utility of the methodology presented here.