Numerical Solution for Min-Max Shape Optimization Problems. (Minimum Design of Maximum Stress and Displacement).

Shimoda, Masatoshi; Azegami, Hideyuki; Sakurai, Toshiaki

doi:10.1299/jsmea.41.1

Cited by 14 publications

(5 citation statements)

References 6 publications

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“…The objective functions in Equations (11) and (12) have problems of singularity and nondifferentiability with respect to shape variation. Since the KS function, a power function of exponential has sufficiently differentiable smoothness, the authors have used it for various min‐max problems, 40,41 and confirmed its effectiveness. Besides the KS function, a function called l p ‐norm

{\left[{\sum}_{d=1}^{DN}{\int}_{\Omega^d}\left\{{\int}_{\Omega_y^d}{\left(\frac{\sigma_{Mises}^{(d)}\left(\boldsymbol{x},\boldsymbol{y}\right)}{\sigma_a}\right)}^pd{\Omega}_y\right\}d\Omega \right]}^{1/p}

can also be used for the same purpose.…”

Section: Formulation Of a Multiscale Shape Optimization Problemmentioning

confidence: 97%

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

TORISAKI

Shimoda

Ali

2022

Numerical Meth Engineering

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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 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 H 1 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.

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{\left[{\sum}_{d=1}^{DN}{\int}_{\Omega^d}\left\{{\int}_{\Omega_y^d}{\left(\frac{\sigma_{Mises}^{(d)}\left(\boldsymbol{x},\boldsymbol{y}\right)}{\sigma_a}\right)}^pd{\Omega}_y\right\}d\Omega \right]}^{1/p}

can also be used for the same purpose.…”

Section: Formulation Of a Multiscale Shape Optimization Problemmentioning

confidence: 97%

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

TORISAKI

Shimoda

Ali

2022

Numerical Meth Engineering

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“…In (5), p ∈ (0, ∞) is assumed to be a constant in the previous papers [1,2]. However, in the present paper, we assume that…”

Section: Shape Optimization Problemmentioning

confidence: 99%

“…We can define the shape derivative of the KS function and obtain a numerical solution by the finite element method, as shown in [2].…”

Section: Introductionmentioning

confidence: 99%

Construction method of the cost function for the minimax shape optimization problem

Shintani

Azegami

2013

JSIAM Letters

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The present paper describes a method by which to formulate a shape optimization problem of a linear elastic continuum for minimizing the maximum value of a strength measure, such as the von Mises stress. In order to avoid the irregularity of the shape derivative of the maximum value, the Kreisselmeier-Steinhauser function of the strength measure is used as the cost function. In the cost function, a parameter is used to control the regularity of the shape derivative. In the present paper, we propose a rule by which to appropriately determine the parameter. The effectiveness of the proposed rule is confirmed through a numerical example of a cantilever problem.

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“…The concept underlying the method is the gradient method in Hilbert space [3,4]. By the traction method, many different types of shape optimization problems involving elastic continua [5][6][7][8][9] and heat-transfer fields [10,11] have been solved. However, our studies conducted to shape optimization problem with coupling boundary value problems have not applied.…”

Section: Introductionmentioning

confidence: 99%

Solution to Shape Optimization Problems of Continua on Thermal Elastic Deformation

Azegami

Yokoyama

Katamine

2002

Inverse Problems in Engineering Mechanics III

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This paper presents a numerical solution to shape optimization problems of continua minimizing steady-state thermal elastic deformation. The potential energy by thermal elastic strain was used as an objective functional. Coupling boundary value problems of partial differential equations for steady-state heat-transfer and linear elastic deformation were employed as state equations. The shape gradient function of the shape optimization problem was theoretically derived using the Lagrange multiplier method, or the adjoint variable method, and the formulae of the material derivative. Reshaping was carried out by the traction method that was proposed by one of the authors as an approach to solving domain optimization problems. The validity of the presented method was confirmed by results of numerical analyses.

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Numerical Solution for Min-Max Shape Optimization Problems. (Minimum Design of Maximum Stress and Displacement).

Cited by 14 publications

References 6 publications

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

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

Construction method of the cost function for the minimax shape optimization problem

Solution to Shape Optimization Problems of Continua on Thermal Elastic Deformation

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