We propose a novel shape optimization for designing a multiscale structure with desired static deformation. The square error norm between actual and target displacements of the macrostructure is minimized as an objective function. The design variables are the shape variation field of the outer shape of the macrostructure, the interface shape of the macrostructure and the shapes of pores of the microstructures. In this study, the macrostructure is divided into some arbitrary domains, which have independent periodic microstructures. The homogenized elastic tensors are calculated using the asymptotic homogenization method, and apply to the correspondent domains of the macrostructure. The shape gradient functions with the state and the adjoint variables are derived for the shape variation of the macrostructure and the microstructures are introduced, and apply to the H1 gradient method to determine the optimum shapes. The proposed method is applied to a both ends fixed beam and it is confirmed that the objective function decrease to zero, or the desired static deformation can be achieved as expected while obtaining the clear and smooth boundaries.

In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed; minimization of the maximum stress of the microstructure and minimization of the maximum stress of the macrostructure. The homogenization method is used to bridge the macrostructure and the microstructures, and also to calculate the local stress in the microstructures. By replacing the maximum value of the stress with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability on the maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem with the area constraint including the whole microstructures. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H¹ gradient method is used to determine the unit cell shapes of the microstructures, while reducing the objective function and maintaining the smooth design boundaries. In the numerical examples, the optimal shapes obtained for the maximum local stress minimization of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.

In this study, we present a 3D shape optimization method for designing micro- and macro-structures concurrently. We assume the macrostructure consists of several arbitrary subdomains, which have different periodic microstructures. The macro- and microstructures are bridged by the homogenized elastic tensors, which are calculated by applying the homogenization method to the unit cells of the microstructures. Defining the boundary shapes of the macro-, microstructures and the interface shapes between the subdomains as design variable, the compliance of the macrostructure is minimized. The volume of the macrostructure considering the whole holes in the microstructures is used as the constraint. The homogenization equations for the microstructures and the equilibrium equation for the macrostructure are also used as the constraint. This design problem is formulated as a distributed-parameter optimization problem, and the shape sensitivity functions are theoretically derived. The optimum boundaries of the macro- and microstructures are determined by the H¹ gradient method. The proposed method is applied to a numerical example to confirm the effectiveness of the proposed method.

In this paper, we present a multi-scale shape optimization method for shape design of a 3D periodic microstructure. The method consists of two-step optimization. As the first step, the material properties of a macrostructure are optimized using a size-optimization technique, where the components of the liner elastic tensor are set as the design variables, in which the objective function and the constraints are set according to design purpose. As the second step, the shapes of 3D periodic-microstructures are determined to identity the material properties of the macrostructures obtained in the first step, where the inverse homogenization method and the H¹ gradient method are employed. The shape optimization problem of a microstructure with hetero materials is formulated as a distributed-parameter system, in which the mass is set as an objective functional and the shape variation field is set as a design variable, the target macroscopic material properties and the homogenization equation are used as the constraints. Through numerical calculations, we confirmed the effectiveness of the proposed method based on the inverse homogenization method and the H¹ gradient method.

In this study, we propose a concurrent shape optimization method for stiffness design of porous multiscale structures. The macrostructure is divided into several arbitrary domains, which have independent periodic microstructures. We optimize the shape of the macrostructure, the shapes of the voids distributed in the microstructures periodically and the boundary shapes of the interfaces between the domains. In order to bridge the macrostructure and the microstructures, the homogenized elastic moduli are calculated using the homogenization method, and applied to the elastic modulus of each domain in the macrostructure. The compliance of the macrostructure is minimized under the constraint conditions of the whole area of the macrostructure including the microstructures, the area of domains, the state equation of the macrostructure, and the homogenization equation of the microstructures. The sensitivity functions distributed on the design boundaries are derived using the material derivative of the Lagrange function. We use the H¹ gradient method to obtain the clear and smooth boundaries.

In this study, we propose a novel multiscale shape optimization method for designing the shapes of periodic microstructures using the homogenization method and the H1 gradient method. The compliance of a macrostructure is minimized under the constraint conditions of the total area of the unit cells of the microstructures distributed in the macrostructure. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function is then theoretically derived. The shape gradient function is calculated with the two state and two adjoint equations related to the micro- and macro structures. Clear and smooth boundary shapes of the unit cells can be determined with the H¹ gradient method. The homogenized elastic moduli of the updated unit cells are calculated and applied to the macrostructure. The proposed method is applied to a multiscale structure, in which the numbers of domains with the microstructures are varied and the optimal shapes of the unit cells and the compliances obtained are compared. The numerical results confirm the effectiveness of the proposed method for creating the optimal shapes of microstructures distributed in macrostructures.

In this study, we propose a novel shape optimization method for designing micro- and macro-structures concurrently. We assume the macro-structure consists of several arbitrary domains, which have different periodic micro-structures. The macro-structure and the micro-structures are connected by the homogenized elastic tensors, which are calculated by applying the homogenization method to the unit cells of the micro-structures. Defining the boundary shapes of the macro-, the micro-structures and the interface shapes between the domains as design variable, the compliance of the macrostructure is minimized. The volume of the macro-structure considering the whole holes in the micro-structures is used as the constraint. The homogenization equations for the micro-structures and the equilibrium equation for the macro-structure are also used as the constraint. This design problem is formulated as a distributed-parameter optimization problem, and the shape sensitivity functions are theoretically derived. The optimum boundary and the interface shapes of the macro- and the micro-structures are determined by applying the shape sensitivity functions to the H¹ gradient method. The proposed concurrent shape optimization method is applied to several numerical examples to confirm the effectiveness of the proposed method for designing the shapes of multi-scale structures. Also, the compliance and the shapes optimized are, compared and discussed for the different domains.

In this paper, we present a shape optimization method for periodic microstructures to maximize a specified vibration eigenvalue of a porous macrostructure. The homogenized elastic moduli calculated by the homogenization method are applied to the macrostructure to connect the microstructures with the macro structure. The KS function is introduced to solve the repeated eigenvalue problem hidden in vibration eigenvalue optimization. The shape optimization problem subject to the volume constraint considering the microstructures is formulated as a distributed-parameter optimization problem, and the shape gradient function is derived by the Lagrange multiplier method and the adjoint variable method. The shape gradient function is applied as a distributed force to update the design boundaries of the unit cells of the microstructures by the H¹ gradient method. The smooth boundary shapes obtained by the H¹ gradient method are suitable for manufacturing with a 3D printer. In the numerical examples, the eigenvalues and the optimum shapes were compared changing the number of the domains of the microstructures in the macrostructure. As a result, the effectiveness of shape optimization method for microstructures aimed at maximizing the vibration eigenvalue of a macrostructure was confirmed.