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This paper presents a numerical solution to a non-parametric shape optimization problem for design of suspension arm in which strength of suspension arm is evaluated by reaction force to plastic buckling load due to compulsory displacement. To deal with buckling phenomena, the geometrical non-linearity and material non-linearity are considered. Hyper-elastic theory is applied to calculate the deformation of suspension arm, under assumption of monotonous loading. Mass and the reaction force integral to the buckling phenomena are chosen as an objective function and a constraint function, respectively. The shape derivatives of these functions are evaluated by the shape optimization theory. A numerical scheme based on a sequential quadratic approximation method is applied to reshape by using the shape gradients. In this scheme, the traction method is used to find the decent directions of the cost functions. The scheme is implemented by using a commercial shape optimization program. In this program, the shape gradients are calculated by a user sub-program which is developed by using the result of non-linear FEM analysis of a commercial solver. The numerical example for a suspension arm model shows 12% of mass reduction while keeping the reaction force integral constant.

This paper presents a numerical solution to boundary shape optimization problems taking into account material non-linearity. The goal aimed in this paper is to minimize external work by varying a boundary shape under a volume constraint. Shape variation is described by using a one-parameter family of mappings defined in a domain where a continuum lies initially. The shape sensitivity is derived using the Lagrange multiplier method and the formula of the material derivative. A procedure to solve this problem using the traction method is presented, which one of the authors has proposed as an approach to solving domain optimization problems. The varidity of proposed method is verified by applying basic numerical examples and by comparing with results considering only linearity.

A shape optimization method for linear buckling load maximization problems of continua is presented. Using a one-parameter family of mappings defined in a domain where a continuum lies initially, a buckling load maximization problem considering multimodal conditions was formulated. The shape gradient density function for the problem was derived using the Lagrange multiplier method and the formula of the material derivative. The shape gradient density function was then used in a shape optimization analysis conducted with the traction method. The validity of the proposed method is demonstrated by numerical analyses of column-like and arch-like two-dimensional continua in a plane strain condition.

The purpose of this study is to develop a topology optimization method to get a given detormation mode, such as a homologous structure. A minimization problem of a square error in relation to a given deformation and an active deformation is formulated for a linear elastic structure. To solve this problem the homogenized elastic constants are employed and a topology optimization is replaced with an optimal density distribution problem. using the similar method proposed by Bendsoe and Kikuchi. The homogenized elastic constants are calculated by the homogenization method for a microstructure having a rectangular hole. Using this method, a structural deformation is controlled to a given deformation. Simple examples are presented to confirm the validity of this optimization method.

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.

The purpose of this study is to present an optimization method to create multiobjective structures by integrating shape optimization with topology optimization. The topology optimization method used here is based on the homogenization method proposed by Bendsoe and Kikuchi. As the shape optimization method, we used the traction method proposed by Azegami. By integrating these two methods, the optimal shape with smooth boundaries considering optimal topology can be obtained. A multiobjective mean compliance minimization problem of a linear elastic structure is formulated for both topology and shape optimization using the Lagrangian multipliers and the weighted Tchebycheff norm. The integrated optimization system is developed using a general-purpose FEM code to create multiply-connected Pareto optimal shapes. An application example of an automotive component in addition to a simple example is presented to confirm the validity and the practical utility of this integration method and the system.

This paper presents a numerical analysis technique for shape optimization of linear elastic structures subject to multiple loading conditions. The technique presented here is based on the traction method for determining the domain variation that reduces the objective functional. Using the speed method, a volume minimization problem is formulated in which the mean compliances associated with individual load cases act as constraints. The shape gradient function, which is equivalent to the shape sensitivity, for this problem is obtained using the Lagrange multiplier method. With the traction method, the domain variation is numerically analyzed using a commercial FEM code. A method is also proposed for finding the domain variation so as to satisfy equality and inequality constraints. The analytical results show the effectiveness and practical utility of the proposed method in solving shape optimization problems involving multiple loading conditions.

This paper describes the shape optimization analysis of solid structures, where the shape optimization problems of linear elastic structures are treated to improve the strength or to reduce the weight of solid structures. The shape optimization system is developed based on the growth-strain method. The growth-strain method, which modifies a shape by generating bulk strain, was previously proposed for analysis of the uniform-strength shape. The generation law of the bulk strain is given as a function of a distributed parameter to be made uniform such as von Mises stress. In this paper, improved generation laws are proposed for the volume and maximum stress constraints. The laws make the stress uniform while controlling the volume and maximum stress to target values. PID control is introduced as the control method. A simple example using a cantilever beam and examples applied to suspension arms are presented. The results show sufficient validity and practicality of this method and developed system in terms of strength improvement and weight reduction of solid structures.

The growth-strain method was previously proposed as a shape optimization method which is the deforming of shapes by generating bulk strain. Hitherto, we have shown that the uniform-strength shapes can be analyzed by the method where the bulk strain is generated according to the magnitude of a strength parameter. The objective of the present paper is to devise the formulation of the method in order to apply it to maximum-stiffness shape analysis. First, by reforming the inverse variational principle of the principle of minimum potential energy, we obtain a uniform condition of the potential energy density as a necessary condition to obtain maximum stiffness. Based on the theory, the proposed formulation is to set the potential energy density on the generation measure of the bulk strain. The effectiveness of this formulation is examined on beam and column problems. The notable advantage is confirmed with the latter problem.

Optimum cross sections of cylindrically coiled spring are analyzed by the growth-strain method. This method was previously proposed as a shape optimization method which deforms shapes by generating bulk strain. In the present analyses, the helix angle is neglected. The stress distributions are analyzed by the finite element method based on the stress function. The growth deformation is analyzed by the finite element method on the assumption of a roll of coil as a rotationally symmetric ring under free restriction. The bulk strain is generated based on the strain energy density in a manner in which volume change in the cross section is absorbed in the circuit direction. In consequence, a convergent curve of mass improvement ratio and corresponding cross sections are obtained. The cross sections are egg-shaped tapering to the outside of the coil.

A simple method for shape optimization of a static elastic body in a sense of uniform strength is newly proposed. The idea from which the method originated came from the growth behavior of living things, such as trees or bones, responding to a stress accompanying a load. On the use of the method, the optimum shape is formed by iterating a usual elastic analysis and incremental growth analysis by using a constitutive equation of growth with the result of the elastic analysis, without a particular technique for mathematical optimization programming or a steep extension of memory for calculation of the sensitivity. The growth law is given as a relation between incremental growth strain and a deviation of objective stress indicating the strength from basic stress. Two examples of a cantilever beam loaded at its tip and a column loaded at its tip under the influence of gravity are analyzed to show the effectiveness of the proposed method.

In the previous reports, an experimental method to evaluate the fracture resistance of a stably growing crack by crack energy density was proposed, and its applicability and superiority were shown through the applications of the method to the stable crack growth problems of thin plates and the comparison with the fracture resistances evaluated by the J-integral. In this report, experiments on the stable growth fractures of thin plates are carried out, the fracture resistances evaluated by CTOD and CTOA are obtained, and they are compared with prior fracture resistances evaluated by crack energy density. The results are as follows : (1) Both show a similar tendency in their variations with crack extensions as was expected theoretically. (2) In comparison with the latter, the former are scattered because of the difficulty of the measurement of local deformation around a crack tip. Therefore, the proposed method is superior to the method based on CTOD and CTOA in the ease of the measurement of these quantities necessary for evaluation. The validity of the instability criterion on crack extension proposed before is also shown ion the Appendix.

き裂の安定 ; 不安定に関して小規模降伏を伴うき裂から大規模降伏を伴うき裂にまで有効なクライテリオンとしてき裂エネルギ密度の観点からTe(T*ee)クライテリオンを新たに提案した. またこのクライテリオンと従来有効とされて来たくらいてりおんの関係を考察することにより, 従来のクライテリオンの物理的意味を明らかにするとともにTeクライテリオンを軸としてそれらが統一的に把握できることを示した.

In the previous 2nd and 3rd Reports, the method proposed in the 1st Report to evaluate the Fracture resistance of a stable growing crack by crack energy density was applied to the stable crack growth problems of thin plates, and its applicability was confirmed. In this report, the fracture resistances based on the J-integral are evaluated using the experimental data obtained in the 2nd and 3rd reports. They are compared with the fracture resistances evaluated before by crack energy density and the superiority of the proposed method is shown through the comparison. The results are as follows: (1) The fracture resistances by the J-integral at the initiation of crack growth almost accord with those by crack energy density, as was predicted before. (2) The change of the fracture resistance of a growing crack evaluated by the J-integral does not correspond to the change of the fracture mode, although the fracture resistance by crack energy density varies corresponding to the change of the fracture mode.

The method proposed in the 1st Report for evaluating the fracture resistance of a stably growing crack by crack energy density is applied to the stable crack growth problems of thin plates, and its applicability and validity are confirmed. That is, the experiments of stable growth fractures of thin single-edge cracked specimens with different initial crack lengths under bending moment, and thin center cracked specimens also with different initial crack lengths under tensile force are carried out. The fracture resistances expressed by additional rates of crack energy density and crack energy density are evaluated, based on the results. When fracture modes are almost the same, fracture resistances have almost the same values regardless of initial crack lengths, and their values vary only corresponding to the change of fracture mode caused by crack extension. Also, the difference of specimen types has no remarkable influence on their values.