H1 Gradient Method Research Articles

Concurrent multiscale and multi‐material optimization method for natural vibration design of porous structures

AbstractThis paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aimed at maximizing a specified natural frequency. The multi‐material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization, respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi‐material topology optimization using the generalized solid isotropic material with penalization (GSIMP) method, which is a method to obtain multi‐material topology by implementing the successive binarization. The KS (Kreisselmeier–Steinhauser) function is introduced to solve the repeated natural frequencies issue that lies in the maximization of a specified natural frequency. An area‐constrained multiscale optimization problem is formulated as a distributed‐parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method (i.e., a nonparametric shape/topology optimization method). The effectiveness of the proposed method for maximizing a specified natural frequency is confirmed through 2D and 3D numerical examples, in which the influences of sub‐regions of the macrostructure and anisotropic material of the microstructures are also investigated and the results are discussed.

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

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.

Shape optimization method for strength design problem of microstructures in a multiscale structure

AbstractIn 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 maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. 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 microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress 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.

Numerical and experimental study on optimization of distributed material-orientation for frequency response of CFRP shell structures

In this paper, we present a non-parametric material-orientation optimization method for frequency response, a dynamic design problem of a laminated shell structure composed of orthotropic carbon-fiber-reinforced plastic (CFRP) materials. The squared vibration displacements in the arbitrarily prescribed evaluation area are minimized at a given frequency or in a given frequency band. This optimal design problem to determine the optimal material-orientation distribution is formulated as a distributed-parameter system optimization problem based on the variational method, and the sensitivity function with respect to the variation of the material-orientation is theoretically derived. The optimal material-orientation distribution is determined by the H1 gradient method with Poisson’s equation. With this method, the objective function is minimized while controlling the smoothness and the curvature distribution of the material-orientation. Numerical results show the effectiveness of the proposed method for obtaining the optimal material-orientation distribution of each layer that minimizes the squared vibration displacements at the given frequency or in the given frequency band. Also, the prototypes for the initial and the optimized material-orientation distribution are fabricated by the tailored fiber placement technique (TFP), and the static stiffness and the frequency response are experimentally evaluated. The results are compared with the calculated ones and discussed.

Shape optimization for strain energy minimization problem with damper element

Dampers are often used for vibration suppression of a civil or mechanical structure. It is considered that vibration can be further suppressed by optimizing the shape of the structure after the installation of dampers. In this paper, we proposed a shape optimization method for minimizing strain energy for a structure with a viscous damper. The shape gradient function was derived using the Lagrange multiplier method. Both the frequency response and adjoint variable were solved using the modal analysis. It is expected that these calculations will be simplified. To obtain a smooth shape of the solution, H1 gradient method was used as the shape modification method. The numerical analysis was performed by combining the FEM software, HyperWorks, and a self-made program. A solid model excided by a load over a wide frequency range was used as a numerical example. From the analysis result, we confirmed the effectiveness of the shape optimization method considering the damper by H1 gradient method.

Concurrent shape optimization for macro- and micro-structure of 3D multiscale porous structure

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.

Micropore shape optimization of porous laminated shell structures

In this study, we propose a multiscale shape optimization method for designing micropores in porous laminated shell structures. The shapes of the unit cells of the periodic micropores distributed in each layer of the laminated shell structure are optimized. The homogenization method is used to bridge the macrostructure and the periodic microstructures. A squared error norm is minimized for controlling the displacements at arbitrary points of the laminated shell structure to the target values under the total volume constraint including the microstructures. The equilibrium equation of the macrostructure and the homogenization equations of the unit cells are also used as the constraints. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function is theoretically derived. The H1 gradient method is used for shape optimization of the unit cells of the micropores. The validity of the proposed method is confirmed by a numerical example for designing the optimal shapes of the micropores distributed in a laminated shell structure. With the proposed method, arbitrary stiff and compliant porous laminated shell structures can be created.

Shape optimization method for strength design problem of microstructures in a multiscale structure

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.

粘性ダンパーを有する構造の広帯域振幅規定問題における形状最適設計

It is efficient to use the modal method for shape design of structures considering vibrations in a wide frequency range. However, when there are viscous dampers in the structure, it is difficult to use the modal method because the equation of motion cannot be decoupled. In this paper, we propose an efficient shape optimization analysis method using the modal method for the displacement regulation problem in a wide frequency range with a viscous damper. The shape gradient function is derived analytically, and is given as a modal solution. In addition, the H1 gradient method was used for shape modification analysis. Using a simple numerical model, it was shown that this method can be applied not only to the shape-optimal design of structures, but also to the shape-optimal design of the materials for eardrum reconstruction.

Macropore shape optimization of porous laminated shell structures

In this study, we propose a multi-scale shape optimization method to find the optimal macrostructure and microstructures that achieves the target displacement for a porous laminated shell structure. The displacement at any point of the laminated shell structure is controlled to the target value under the total volume constraint by minimizing the squared error. The equilibrium equation for the macrostructure and the homogenization equations for the unit cells are used as constraints. The shape optimization problem is formulated, and the shape gradient functions theoretically derived are applied to the H1 gradient method. The homogenization method is used for connecting the macrostructure and the microstructures. With the H1 gradient method for shape optimization, a smooth macrostructure and microstructure can be obtained.

Concurrent multiscale optimization method for natural vibration design of porous structures

This paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aiming at maximizing the vibration eigenvalue. The multi-material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi-material topology optimization using the GSIMP method. The KS function is introduced to solve the repeated-root problem that lies in the maximization of vibrational eigenvalues. The area-constrained multiscale optimization problem is formulated as a distributed parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method. The effectiveness of the proposed method for maximizing vibration eigenvalues is confirmed through numerical examples.

Simultaneous optimization of material orientation angle and shape/topology of CFRP shell structure

The objective of this study is to develop a method for simultaneous optimization of shape, topology and orientation angle to maximize the stiffness of CFRP shell structures under volume constraints. In the optimization of orientation angle, the material directions of orthotropic anisotropic material in finite elements are used as design variable. The optimal shape is determined by the H1 gradient method for vector variables, and the optimal solutions for topology optimization and orientation angle are determined by the H1 gradient method for scalar variables. The effectiveness of the proposed method is confirmed through a numerical example.

Simultaneous optimization of shape and topology for stiffener creation

In this paper, we propose a method for designing stiffeners on thin plate structures by shape and topology optimization. The compliance is minimized under the constraints of volume and equilibrium equations. The optimal stiffeners’ shape and placement are simultaneously determined. Shape optimization and topology optimization are formulated as distributed-parameter optimization problem, in which the shape gradient function and density gradient function are derived by using the adjoint equations and the equilibrium equation. The shape optimization is performed by applying the shape gradient function to the H1 gradient method for vector variable, and the topology optimization is performed by applying the density gradient function to the H1 gradient method for scalar deign variable using the SIMP method. Several numerical examples are presented to confirm the effectiveness of the proposed method for the design of stiffeners.

Identification of muscle activity in tongue motion during swallowing through medical image data

This paper presents a formulation to identify muscle activities from the variation in shapes of organs during swallowing. We assume that each organ consists of a three-dimensional hyperelastic body, and the contraction movement of the muscle is caused by a contractive inelastic stress in the organ. A function distributed in the organ domain to control the magnitude of the inelastic stress is chosen as a design variable in the same manner as the density in the topology optimization problem of density variation type. The identification problem is formulated as a problem of determining the design variable that minimizes an objective cost function defined by the squared L2 norm of the reaction force in the normal direction on the boundary when an enforced displacement to fit the varied boundary of the organ and the inelastic stress modeling the muscle activity are applied. The finite deformation problem of the hyperelastic body is analyzed using the finite element method. The direction of the muscle fiber is assumed to be the direction of the minimum principal stress obtained as the solution to the finite deformation problem. The solution to the identification problem is presented based on a scheme using the H1 gradient method for the topology optimization problem of density variation type. A numerical example using a previously developed model of the tongue is introduced to demonstrate the effectiveness of the proposed approach.

Lagrangian approach for optimizing fiber bundle placements of CFRP shell structures

We present a non-parametric shape and topology optimization method for fiber bundle placement and orientation design aimed at controlling the rigidity of CFRP shell structures. Since the fiber bundles are modeled by the beam element, the placement and the orientation of the fiber bundles can be explicitly obtained while moving the positions. In order to achieve the target displacement at an arbitrarily specified position, we considered the problem of minimizing the squared error under the length constraint of the fiber bundles, and formulated it as a distributed-parameter optimization problem. Using the derived shape gradient function and the density gradient function, the optimum placement and the orientation of the fiber bundles are determined simultaneously by the H1 gradient method for vector and scalar variables. The effectiveness of the proposed method is confirmed by numerical examples.

Simultaneous shape and topology optimization method for frame structures with multi-materials

In this study, we present a novel non-parametric shape-topology optimization method for frame structures with multi-materials, aiming at designing more light and stiff frame structures. The sum of squared error norm for achieving the target displacements on the specified members is minimized under the volume constraints of multi-materials as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the material gradient functions for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization. The gradient functions derived are applied to the unified H1 gradient method for frame structures to determine the optimal shape and material variations. With this method, the optimal free-form and topology for arbitrary large-degree of design freedom frame structures with multi-materials can be obtained without any shape and topology parameterization. Numerical examples with various materials and different boundary conditions are demonstrated and the results are discussed.

Shape optimization of periodic-microstructures for stiffness maximization of a macrostructure

In this study, we present a shape optimization approach 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 microstructures distributed in the macrostructure, the elastic equation of the macrostructure and the homogenization equation of the unit cells. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function involving the state and adjoint variables for both the macro- and micro-structures is theoretically derived. Clear and smooth boundary shapes of the unit cells can be determined with the H1 gradient method. The proposed method is applied to multiscale structures, in which the numbers of domains with the microstructures are varied and the optimized 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

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.