Material Derivative Method Research Articles

J1201-3-2 材料境界面の形状最適化のための一解法 : 剛性設計問題への適用(解析・設計の高度化・最適化III)

In this paper we propose a shape optimization method for the interface design of solid structures consist of two different materials. As the design boundaries to determine the shape of the solid structure, we consider the outer boundaries and the interface boundaries. The compliance is minimized subjected to the volume constraints for two materials. The shape gradient function derived using the material derivative method is applied to the Neumann-type traction method. By this method, the optimal outer and interface boundary shapes with smoothness can be determined without shape design parameterization. The validity of our method for the interface design problem of solid structures was verified through the obtained results for the design problems.

407 構造体の断面形状最適化のための一解法

In this paper, we present a numerical method to optimize a cross section shape of a solid structure, which is often demanded at the early stage in structural designs. The cross section area is minimized subject to the constraints of sectional properties including the torsional constant, the moments of inertia of area, the center of gravity and the boundary length. The problem is formulated as a distributed shape optimization problem. The shape gradient function is derived using the Lagrange multipliers and the material derivative method. The Robin-type traction method, or a gradient method in a Hilbert space, is applied to determine the smooth optimal shape. The validity of this method is verified through design examples for obtaining the optimal shape of the cross section under the constraints of sectional properties.

Shape and topology optimization of contact problems by level set methods

AbstractThis paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Formulation of cost functionals for different measurement principles in nonlinear capacitance tomography

A model-based, nonlinear reconstruction for electrical capacitance tomography (ECT) is considered as an inverse problem to find the spatial distributed permittivities in a pipe. The corresponding forward problem is defined by the Laplace equation and depending on the measurement principle two different sets of boundary data can be obtained for the inverse problem. Dirichlet boundary data correspond to voltage measurements and Neumann boundary data to the measurement of charges on the electrodes. For each type of data different cost functionals are defined. In this paper, the influence of the functionals on the inverse problem is examined. Therefore, two different, nonlinear reconstruction methods validate the functionals. One is based on a fixed grid and the forward problem is solved by the finite-element method. Second, a moving contour described by a level set formulation is used to reconstruct the boundaries of different phases. In this case, the field problem is solved by a boundary element formulation. The reconstructions are based on a Gauss–Newton scheme and the gradient is calculated analytically by the material derivative method. Reconstruction results for measurements with an ECT prototype sensor are presented and good results are reported independent of the implemented cost functional.

3203 予備設計における断面形状を制約条件にした断面積の最小設計(OS8 設計と最適化(I),未来社会を支えるものづくりとひとづくり(設計・システムから))

In this paper, we present a numerical method to optimize the cross section shape of solid structures, which is often needed at the preliminary stage in the structural designs of automotives and airplanes and so on. The cross section area is minimized subject to sectional properties such as moments of inertia of area, center of gravity, boundary length. The problem is formulated as a distributed optimization problem, or a non-parametric optimization problem. The shape gradient function is derived using the Lagrange multipliers and the material derivative method. The traction method, or a gradient method in Hilbert space is employed to determine the smooth optimal shape using the derived shape gradient function. The validity of this method is verified to obtain the optimal shape of cross section under the constraints of sectional properties through an example.

Minimum Weight Shape Design For The NaturalVibration Problem Of Plate And Shell Structures

In this paper we present a solution to a shape optimization problem involving plate and shell structures subject to natural vibration. The volume is chosen as the response to be minimized under a specified eigenvalue constraint with mode tracking. The designed boundaries are assumed to be movable in the in-plane direction so as to maintain the initial curvatures. The surfaces are discretized by plane elements based on the Mindlin-Reissner plate theory. A non-parametric or a distributed shape optimization problem is formulated and the shape gradient function is theoretically derived using the material derivative method and the Lagrange multiplier method. The traction method, a shape optimization method developed by the authors, is applied to obtain the optimal shape in this problem. The validity of the numerical solution for minimizing the weight of the plate and shell structures is verified through application to fundamental design problems and an actual automotive suspension component.

3308 ノンパラメトリック形状最適化手法による自動車部品の多目的剛性設計(OS14 曲面構造の解析と設計III)

In this paper, we present a solution to the multi-objective shape optimization problem for the rigidity design of sheet metal structures. The compliances for multi-loadings are used as indexes of rigidity. The lp norm method is employed to scalarize the vector objective functional of compliances. The volume is set as the constraint. The boundaries to be optimized are assumed to be movable only in-plane direction to maintain the curvatures of the initial shape. The shape gradient function and the optimality conditions are theoretically derived using the Lagrangian approach and the material derivative method.. The traction method is applied to determine the smooth shape variations that minimizes the objective functional, while constraining the shape variations in the normal direction on the sheet metal surfaces. This method is applied to a simple sheet metal structure and a practical automotive chassis component to verify the effectiveness and practical utility for improving the rigidity of sheet metal structures under multi-loading conditions.

Optimal Shape Design of Inductor Coils for Surface Hardening

We study a mathematical model for induction hardening of steel. It consists of a vector potential formulation of Maxwell's equations coupled with a heat equation and an evolution equation for the volume fraction of the high temperature phase in steel called austenite. An important task for practical applications of induction hardening is to find the optimal coupling distance between inductor and workpiece. To this end we control the volume fraction of austenite with respect to perturbations of the coupling distance. The coil is modeled as a tube and is defined by a regular curve. We formulate the shape optimization problem over the set of admissible curves and prove the existence of an optimal curve. We apply the material derivative method for the shape sensitivity analysis of the state system. Finally, the shape gradient is specified for an optimal curve and the first order necessary optimality conditions are established.

Optimization for boiling heat transfer determination and enhancement in pressure die casting

Boiling in cooling channels has recently been demonstrated to be an effective mechanism for heat extraction in pressure die casting. Boiling heat transfer can be enhanced by cooling channel shape optimization. The occurrence of boiling presents a non-linear thermal problem which, when combined with shape optimization, necessitates the solving of non-linear equations for each channel configuration. In this paper a methodology is presented that involves the use of optimization for the combined determination of channel shapes and heat transfer coefficients. It is shown in the paper how this approach results in the accurate determination of boiling heat transfer coefficients on the final optimized cooling channel configuration. The non-linear thermal problem is calculated at very little computational cost over that required for a comparable linear problem. Focus in the paper is on the application of the methodology to the pressure die casting process. The approach adopted is founded on a design sensitivity analysis using the material derivative adjoint variable method. The thermal model for the pressure die casting process is founded on the boundary element method and the optimization is performed using a conjugate gradient scheme. Geometrical constraints are enforced using buffer elements superimposed on to the boundary element mesh. Numerical and experimental trials are performed to demonstrate the potential of the new optimization methodology.

Sensitivity and optimization for shape and non‐linear boundary conditions in thermal boundary elements

AbstractThis paper is concerned with the minimization of functionals of the form ∫Γ(b) f(h,T(b, h)) dΓ(b) where variation of the vector b modifies the shape of the domain Ω on which the potential problem, ∇2T=0, is defined. The vector h is dependent on non‐linear boundary conditions that are defined on the boundary Γ. The method proposed is founded on the material derivative adjoint variable method traditionally used for shape optimization. Attention is restricted to problems where the shape of Γ is described by a boundary element mesh, where nodal co‐ordinates are used in the definition of b.Propositions are presented to show how design sensitivities for the modified functional ∫Γ(b) f(h,T (b, h)) dΓ(b) +∫Ω(b) λ(b, h) ∇2T(b, h) dΩ(b) can be derived more readily with knowledge of the form of the adjoint function λ determined via non‐shape variations. The methods developed in the paper are applied to a problem in pressure die casting, where the objective is the determination of cooling channel shapes for optimum cooling. The results of the method are shown to be highly convergent. Copyright © 2002 John Wiley & Sons, Ltd.

Configuration design sensitivity analysis and optimization of beam structures

A general method for configuration design sensitivity analysis over a three-dimensional beam structure is developed based on a variational formulation of the classical beam in linear elasticity. A sensitivity formula is derived based on a variational equation for the beam structure using the material derivative concept and adjoint variable method. The formulation considers not only the shape variation in a three dimensional direction, which includes translational as well as rotational change of the beam but also the orientation angle variation of the beam's cross section. The sensitivity formula can be evaluated with generality and ease even by employing a piecewise linear design velocity field despite the fact that the bending model is a fourth order differential equation. The design sensitivity analysis is implemented using the post-processing data of a commercial code ANSYS. Several numerical examples are given to show the excellent accuracy of the method. Optimization is carried out for a tilted arch bridge and an archgrid structure to show the method's applicability.

The method of arbitrary lines in optimal shape design: problems with an elliptic state equation

Optimal shape design problems with elliptic state equations are defined. The shape of an optimised domain is controlled by a vector of design variables uniquely denning a Bézier curve. The state equations are semi-discretised by the h- p method of arbitrary lines (MAL) and approximate shape design problems are derived. The existence and convergence of both semi-discrete state solutions and approximate optimal shapes is proved. In shape design sensitivity analysis, the material derivative method is used. The resulting formulae are expressed by boundary integrals which contain the normal derivative of the state solution. The derivative is approximated by means of the MAL semi-discrete state solution and the convergence properties of the approximation are analysed. Finally, numerical examples are given which demonstrate the performance of MAL solutions as compared with other approaches based on (conforming and nonconforming) finite element methods.

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

This paper presents a numerical shape optimization method for continua that minimizes some maximum local measure such as stress or displacement. A method of solving such min-max problems subject to a volume constraint is proposed. This method uses the Kreisselmeier-Steinhauser function to transpose local functionals to global integral functionals so as to avoid non-differentiability. With this function, a multiple loading problem is recast as a single loading problem. The shape gradient functions used in the proposed traction method are derived theoretically using Lagrange multipliers and the material derivative method. Using the traction method, the optimum domain variation that reduces the objective functional is numerically and iteratively determined while maintaining boundary smoothness. Calculated results for two- and three-dimensional problems are presented to show the effectiveness and practical utility of the proposed method for min-max shape design problems.

The material derivative method for shape optimization problem in electro-chemical machining

The main idea of this paper is the application of the material derivative (or speed) method (Zolesio [8]) for a shape optimization problem in the electrochemical machining. We develop the material derivative method for the optimal shape design for systems described by a variational inequality. We introduce a penalty method for the problem and discuss a numerical method using the finite element method.

Optimal shape design of systems governed by variational inequalities

In this paper, we present some results concerning the optimal shape design problem governed by the variational inequalities of the fourth order. This problem can be considered as a model example for the design of the shape for elastic-plastic problem. The performance criterion is minimized by the material derivative method.

Three-Dimensional shape optimization approach based on natural design variables and the boundary element method

This paper deals with shape optimization of three-dimensional elastic bodies. The proposed approach is based on natural design variables and shape functions and utilizes the boundary element method. The design variables are the magnitudes of a set of fictitious loads applied on the structure. Shape optimal design problems based on minimum mean compliance, peak stress minimization and stress constraints are considered. A general method for shape design sensitivity analysis using the material derivative concept and adjoint variable method is used. The sensitivity formula for a general stress constraint imposed over the optimized boundary is derived. Optimal shapes for three-dimensional problems are presented to show numerical applications.

Boundary Shape Determination of Continua with Desired Streess Distribution.

In this paper we present a numerical analysis method for shape determination of continua based on a strength criterion to control the stress distribution to a desired one. As an objective functional, we introduce a squared stress error norm on the prescribed domains. Using the Lagrange multipliers, an optimization problem subjected to a volume constraint is formulated. The shape gradient function and the optimality condition are derived using the material derivative method and the adjoint method. With the traction method, the domain variation that minimizes the objective functional is numerically and iteratively determined. The shape determination system is developed using a general-purpose FEM code, which is used to solve the state equation and the adjoint equation. The calculated results show the effectiveness of the proposed method in controlling the stress distribution.

Green's function BEM for 2-D optimal shape design

A new approach has been proposed (Melnikov and Titarenko, in Dynamics and Strength of Heavy Machines, Dniepropetrovsk State University, 1987 (in Russian); Proc. IABEM-92 Symp., University of Colorado, 1992; Int. J. Numer. Meth. Engng, 1993, 36, 2017–2030) for the solution of two-dimensional problems in optimal eigenvalue shape design. The idea behind the approach is based on an appropriate combination of the material derivative method (Zalezio, in Optimization of Distributed Parameter Structures, Sijthoff and Nordhoff, 1981) used for the shape sensitivity analysis of natural vibrations, and a specially adapted Green's function formulation of the boundary element method used for the eigenvalue problem itself. Emphasis (Melnikov and Titarenko, in Dynamics and Strength of Heavy Machines, Dniepropetrovsk State University, 1987 (in Russian); Proc. IABEM-92 Symp., University of Colorado, 1992; Int. J. Numer. Meth. Engng, 1993, 36, 2017–2030) was made mainly on the mathematical basis of the approach, with numerical examples exhibited just to illustrate its computational capability. The present paper, an extension of those studies, covers a variety of computational aspects, such as the accuracy of the sensitivities obtained, the practical convergence of the algorithm, and so forth. It also contains a detailed description of the computer programming system developed to realize the approach. Additionally, some of the theory behind the approach is formulated and proven.

A systematic approach to shape sensitivity analysis

A fully detailed development of the domain parameterization method is presented for shape sensitivity analysis. It is shown that equivalent results are obtained from the material derivative method. In fact, the material derivative method may be viewed as a special case of the domain parameterization method which occurs when the reference configuration coincides with the body configuration. The method is illustrated for the Laplace problem in which explicit shape sensitivities are derived by the adjoint and direct differentiation methods. Both finite element and boundary element applications are discussed. The similarities between this approach and the isoparametric finite/boundary element method are transparent. In the finite element approach, it is shown that the sensitivity integrals may be transformed to the boundary (as is commonly done in the material derivative method) for the adjoint method, however, this does not seem possible for the direct differentiation method. Finally, in the boundary element approach, the sensitivities do not require the differentiation of the fundamental solutions.

Optimal shape design for a frictionless contact problem

In this paper we develop the material derivative method for the optimal shape design of contact problems described by variational inequalities. Since the method can be used for solving the optimal shape problem for systems described by partial differential equations, here we shall use it to solve it for differential inequalities by taking limits of the equations resulting from a penalized approximation. The computations are done by the finite-element method; the gradient of the criteria as a function of coordinates moving nodes is computed, and the performance criterion is then minimized by a material derivative (or speed) method (Zolesio (1981)).