Free-form Optimization Method Research Articles

Concurrent optimization method of shape and topology for natural vibration problem of shell structures

In this paper, a concurrent optimization method of shape and topology for natural vibration design of shell structures is presented. The fundamental vibration eigenvalue is maximized under volume constraints for shape and topology optimization. The free-form optimization method for shells and the solid isotropic material with penalization (SIMP) method are respectively employed and combined effectively for shape and topology design, in which shape and fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H1 gradient method, where the sensitivity functions are respectively applied as the Robin condition to the design surface in order to determine the optimal shape and topology. We also show an improved method for the spurious (or localized) mode problem of shell structures, which often occurs in the lower density region in topology optimization for vibration design. With the proposed method, shell structure for light weight and improved low-order vibration characteristics can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag-shape problems.

Multidisciplinary free-form optimization of solid structures for mean compliance minimization and time-dependent temperature control

A parameter-free free-form optimization method to solve multidisciplinary shape optimization problems of solid structures is presented in this paper. The design objective is both to minimize the mean compliance of linear elastic structures and to control the temperature at arbitrary locations and time periods for arbitrary loadings and under volume constraint, simultaneously. The multidisciplinary design problem is formulated as a distributed-parameter shape optimization problem based on the variational method, and the global objective function is defined using an original normalized weighted sum method. The shape gradient density functions with respect to the shape variation are derived based on the material derivative and the adjoint variable methods. The shape gradient density functions theoretically derived are applied to the H1 gradient method, a gradient method in the function space, to obtain the optimal shape while maintaining the smoothness of the shape. Numerical examples are considered to demonstrate the validity and the effectiveness of the developed optimization method.

Free-form optimization method for frame structures aiming at controlling time-dependent responses

In this paper, a parameter-free optimization method for controlling the time-dependent response of frame structures is proposed. The proposing method deals with three optimization problems, including a vibration displacement minimization problem, a vibration displacement control problem, and a dynamic compliance minimization problem, which is the special case of the vibration minimization problems. These time-dependent optimum design problems are formulated as distributed-parameter optimization problems and the sensitivity functions for the shape variation to obtain the optimal shape are derived based on the material derivative method, Lagrange multiplier method, and the adjoint method. The derived sensitivity function is applied to the H1 gradient method for frames, which is a gradient method in the function space to determine the optimal shape variation. With the proposed method, the optimal shape for time-dependent response problems such as a forced vibration, a free vibration, or a transient response is obtained while minimizing the objective functional and maintaining the smoothness of a frame structure. Several numerical design examples including a continuous dynamic force or an impulse force as an input force are shown to verify and validate the effectiveness of the proposed method and the results are discussed.

Fundamental investigation of designing vibration lure using the free-form optimization method

Fundamental investigation of designing vibration lure using the free-form optimization method

Free-form optimization for controlling time-dependent responses of shell structures

In this study, we propose a parameter-free free-form optimization method to control the time-dependent response of shell structures. The design objective is to minimize the dynamic compliance or to control the dynamic responses at arbitrary domains and times for an arbitrary time-dependent loading under volume constraint. The unsteady optimum design problems are formulated as a distributed-parameter shape optimization problem based on the variational method, and the shape gradient functions with respect to the shape variation in the normal direction to the surface are derived based on the material derivative and the adjoint variable methods. The shape gradient functions theoretically derived are applied to the H1 gradient method for shells, a gradient method in the function space. With the proposed approach, the optimum free form of the shell structures for the time-dependent response problems can be obtained while minimizing the objective functional and maintaining the smoothness of the form. Several shape design examples under a continuous dynamic force or an impulse force are demonstrated to show the effectiveness of the proposed method, and the results are discussed.

Free-form optimization method for controlling time-dependent response of 3D linear elastic structure

Free-form optimization method for controlling time-dependent response of 3D linear elastic structure

Design optimization of cable–membrane structures for form-finding and stiffness maximization

Membrane structures are light weight and have a unique characteristic that their surfaces can be curved optionally. On the other hand, the stiffness of membrane structures can be significantly improved by cables. However, cable–membrane structures have mechanical characteristics of that they can maintain their shape only with in-plane tensile stress, so it is important to design their optimal shapes. In the present work, we propose a free-form optimization method for designing cable–membrane structures for form-finding and stiffness maximization. The proposed free-form optimization method is a kind of non-parametric optimization method based on the H1 gradient method. In the form-finding problem, to find the minimum surface areas of membrane structures with specified arbitrary boundaries, the area minimization problem is formulated as a distributed-parameter shape optimization problem considering total length of cables as a constraint condition. Then, we perform shape optimization of cable–membrane structures in the stiffness maximization problem under the constraint conditions of total length of cables, membrane area, and membrane perimeter. The numerical results show the effectiveness of the proposed optimization method, and confirm that the optimal shape of the cable–membrane structures can be designed better than membrane structures without cables and has higher stiffness.

Simultaneous shape and topology optimization method based on the H<sup>1</sup> gradient method for creating light weight plate and shell structures

In this paper, we present a simultaneous optimization method of shape and topology for designing a light-weight plate and shell structure. The free-form optimization method for shells and SIMP method are respectively employed for shape and topology optimization, and combined effectively. Shape and fictitious homogenized-density variations are used as the design variables, and simultaneously determined in one iteration of the convergence process. With this method, the optimal topology is determined in the variable design surface optimized by shape optimization. Compliance is used as the objective functional, and minimized under the volume constraint. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined by the H1 gradient method, where the sensitivity functions are applied as the Robin condition to the design surface. With the proposed method, the optimal lighter and stiffer shell structure with smooth surface can be obtained without any design parameterization and numerical instabilities such as checkerboard and zigzag-shape problems.

Reaction forces control design of shell structures in plastic range based on free-form optimization method

In this study, we propose an optimization method for the reaction forces control problem of shell structures in plastic range. The sum of squared error norms between reaction forces in the plastic range and the target forces are minimized under a volume constraint. The total strain theory is employed for simplicity under the assumption that the considered force or enforced displacement increases monotonically, so that the final deformation is path-independent. The optimum shape design problem is formulated as a distributed-parameter system, and we assume that a shell is varied in the out-of-plane direction to the surface whereas the thickness is not varied during the shape change. The shape gradient function and the optimality conditions for this problem are derived theoretically using the material derivative method and the adjoint variable method. The shape gradient function calculated from a non-linear finite-element analysis is applied to the H1 gradient method for shells proposed by one of the authors to determine optimum shape variation. The optimal shape of a shell structure can be obtained without shape parameterization while maintaining surface smoothness. Numerical examples are presented to demonstrate the validity and practical utility of the proposed free-form optimization method.

Fundamental frequency maximization of orthotropic shells using a free-form optimization method

In this study, a free-form optimization method is proposed that maximizes the fundamental frequencies of the orthotropic shells to avoid vibrational resonance. The negative fundamental vibrational eigenvalue is employed as the objective function, which is minimized by subjecting to the governing equation of the natural frequency analysis and area constraint. In the free-form optimization process, the natural frequency analysis of the orthotropic shells is performed to calculate the shape gradient function. The shape gradient function is then applied to the velocity analysis for determining the optimal shape variation. The repeated eigenvalues are considered by converting the fundamental eigenvalue to a summation form of the repeated eigenvalues. The proposed optimization method is validated using three examples of the orthotropic shells. The numerical results show that the optimized shapes of the orthotropic shells are smooth, and their fundamental frequencies are significantly enhanced using the proposed free-form optimization method.

欠陥導入によるナノ炭素材料の構造最適設計法

Because of their superior mechanical, structural, and electronic properties, carbon nanomaterials (CNs) (e.g. graphene sheets and carbon nanotubes) are supposed to be base materials for nanoelectromechanical systems (NEMS). In the present work, we propose a structural optimization method of carbon nanomaterials by introducing topological defects, which consists of the molecular mechanics method, the free-form optimization method, the Phase-Field-Crystal (PFC) method, Voronoi tessellation, and molecular dynamics (MD) simulation. The C-C bonds of CNs are simulated as equivalent continuum beams by a combination of molecular and continuum mechanics, so the atomic structures of CNs can be treated as frame structures. We adopt the free-form optimization method for frames to determine the optimal shapes of CNs in stiffness maximization problem. For obtaining the stable atomic structures of the optimal shapes of CNs, topological defects are introduced in the optimal shapes of CNs using a combination of PFC method, Voronoi tessellation, and MD simulation. The numerical results show that the compliance of CNs can be significantly reduced according to the structural optimization method, which is helpful for designing CNs components in NEMS.

Node-based free-form optimization method for vibration problems of shell structures

We have previously proposed a numerical node-based parameter-free shape optimization method for designing the optimal free-form surface of shell structures. In this paper, this method is extended to deal with two vibration problems including a vibration eigenvalue maximization problem and a frequency response minimization problem. To avoid the repeated eigenvalue problem when a specified vibration eigenvalue is maximized, we provide two optional approaches, i.e., tracking the specified natural mode or increasing all the repeated eigenvalues. Each vibration problem is formulated as a distributed-parameter shape optimization problem, and the derived shape gradient function is applied to the H1 gradient method for the shells proposed by the authors, where the shape gradient function is used as a distributed force function to vary the surface. With this method, the optimal and smooth free-form shape including a natural bead pattern can be obtained. Several calculated examples are presented to demonstrate the effectiveness of the proposed method for the free-form design of shell structures involving vibration problems.

Non-parametric free-form optimal design of frame structures in natural frequency problem

Optimal design of structures with respect to their mechanical behavior is essential and basically required in structural engineering. In this study, we propose a non-parametric free-form optimization method based on the variational method to design frame structures composed of arbitrarily curved linear elastic members. The natural frequency maximization problem of frame structures is formulated as a non-parametric shape optimization problem under the volume constraint. Under the assumption that each member varies in the out-of-plane direction to its centroidal axis, the shape gradient functions and the optimality conditions are theoretically derived by the Lagrange multiplier method and the formulae of the material derivative. Then, the derived shape gradient functions are applied to a gradient method in the Hilbert space with a P.D.E (Partial Differential Equation) smoother, which is referred as the H1 gradient method for frame structures. Moreover, a simple switching technique of the objective functional is presented for overcoming the discontinuity problem of repeated eigenvalues, which often appears in natural frequency maximization problem. With this combination of the three techniques, the optimal free-form frame structures owning smoothly curved members can be obtained without any preliminary shape parameterization, and the effectiveness and validity of the proposed method are verified through three design examples.

Free-form optimization of sandwich structures for controlling thermal displacement

Considering the different thermal expansion coefficients of skins and cores, sandwich structures whose design optimization is performed have the potential for application in thermal actuators. In the present work, we propose a free-form optimization method for the design optimization of sandwich structures composed of two thermoelastic materials (A and B) for controlling their thermal displacements. We use the square displacements error norm between the thermal displacement and target displacement at the designated boundary as the objective function, and minimize it by optimizing the interface and the surface shapes of the sandwich structures under thermal load. We derive the shape gradient function by using the material derivative method and apply it to the free-form optimization method for determining the optimal shape. Two volume constraints, the total volume and volume of material A, are considered in the problem formulation. The numerical results show that the thermal deformation of the optimal sandwich structure can be well identified; this is helpful for applying sandwich structures in thermal actuators.

Design optimization of composite structures composed of dissimilar materials based on a free-form optimization method

Design optimization of composite structures has been a significant topic of research over the recent years. In the present work, we propose a developed free-form optimization method for designing the shape of composite structures composed of dissimilar materials (A and B). In this shape design optimization, we use compliance as the objective function and minimize it under two volume constraints (total volume and volume of material A). According to the proposed free-form optimization method, we theoretically derive the shape gradient function used in the compliance minimization problem, and determine the optimal interface and surface shapes of composite structures under external loading or body force without any shape parameterization. The optimal results of numerical examples show that the compliance of composite structures composed of dissimilar materials can be significantly reduced by using the proposed free-form optimization method.

Free-form optimization method for buckling of shell structures under out-of-plane and in-plane shape variations

In the present work, in order to tackle the linear elastic buckling problem, we develop a shape optimization process based on a free-form optimization method to optimize a shell structure under out-of-plane and in-plane shape variations. The free-form optimization method is a node-based method in which mesh regularity can be maintained and shape design parameterization is not required. It has the advantages of sufficient efficiency for treating large-scale problems and the ability to realize a smooth shape. The buckling coefficient in buckling mode 1 is set as the objective function in the shape optimization problem. We consider repeated eigenvalues and volume constraint for the whole optimization process. Three numerical examples are presented in this work to illustrate the shape optimization algorithm and to show that it can increase the buckling strength substantially, especially under shape variation in the out-of-plane direction.

異種材料からなる複合構造体の形状最適設計

Designing composite structures to satisfy specific requirements has been an important topic for a long time. In this study, we review a free-form optimization method for the optimal shape design of composite structures composed of dissimilar materials in terms of the stiffness problem, the shape identification problem, and the vibration control problem. The free-form optimization method is based on the H1 gradient method, in which the shape gradient functions are theoretically derived using the material derivative method, and applied to the design boundaries as traction forces. We set the interface and surface of composite structures as the design boundaries, and consider the volume constraints of the dissimilar materials in each design problem. At last, we give some numerical examples to verify the validity of the proposed free-form optimization method.

多層シェルの多目的設計問題に対する形状－トポロジー同時最適化

This paper describes a new approach for creating the optimal shape and topology of a multi-layered shell considering multi-objective design. By implementing topology optimization in the variable design domain which is optimized by shape optimization at every iteration, the optimal topology and shape is simultaneously determined. The free-form optimization method and the SIMP method for multi-layered shells are applied to shape and topology optimization, respectively. The compliance vector minimization problem is formulated and its sensitivity functions for shape and density variations are theoretically derived. Both the optimal shape and density variations are determined by using the H1 gradient method, where the sensitivity functions are applied to vary the shape and density. The results show that the proposed simultaneous optimization method provides stiffer and lighter structures with less numerical instabilities while maintaining smooth surface and density distribution.

強度設計問題に対する骨組構造体の形状最適設計法

強度設計問題に対する骨組構造体の形状最適設計法

弾性座屈荷重最大化を目的とする骨組構造体のフリーフォルム最適化

弾性座屈荷重最大化を目的とする骨組構造体のフリーフォルム最適化