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In this paper, we present a parameter-free, or a node-based optimization method for finding the smooth optimal free-form of automotive shell structures, including global and local curvature distributions such as beads or embossed ribs. The design problems dealt with in this paper involve a stiffness problem. Stiffness is maximized using the compliance as an objective functional. The optimum design problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem under the assumptions that the shell is varied in the normal direction to the surface and the thickness is constant. The shape gradient function and the optimality conditions are then theoretically derived. The optimum free-form, or optimal curvature distribution, is determined by applying the derived shape gradient function in the normal direction to the shell surface as pseudo external forces to vary the surface and to minimize the objective functional.

In this paper, we present a numerical solution to shape optimization problems in automotive shell structural designs subject to a strength constraint. Using the proposed method, the optimal shape can be obtained without any parameterization of design variables. With the aim of reducing the weight, a volume minimization problem subject to a von Mises stress constraint is formulated as a distributed-parameter shape optimization problem, or a non-parametric shape optimization problem. It is assumed that the design domain is varied in the tangential direction to the surface to maintain the curvatures of the initial shape. The shape gradient function and the optimality conditions are theoretically derived for this problem using the material derivative method, Lagrange multiplier method and the adjoint variable method. The traction method we have proposed earlier is applied to determine the smooth domain variation that minimizes the objective functional.

This paper presents a free-form optimization method for achieving a desired stiffness in the shape design of automotive sheet metal structures. A squared error norm of displacements at loaded points is introduced as an objective functional in the formulation of a distributed-parameter shape identification problem. The shape gradient function theoretically derived for this problem is applied to the non-parametric free-form optimization method for shells that was developed by one of the authors. With this method, an optimal arbitrarily formed shell, or a shell with optimal curvature distribution can be obtained without any shape parameterization. The calculated results show the effectiveness and the practical utility of the proposed method for controlling stiffness when designing sheet metal structures.

This paper presents a numerical analysis technique for application to shape optimization problems of linear elastic structures subject to multiple loading conditions. The problems dealt with here are a mean compliance minimization problem in relation to individual load cases and a fully stressed design problem. The proposed technique is based on the traction method which analyzes the domain variation. A shape optimization system was developed and applied to fundamental problems in two and three dimensions. The computed results confirmed the validity and usefulness of the proposed technique.

In this paper we present a numerical shape optimization method of continua for solving min-max problems and identification problems. The min-max shape optimization problems involve minimization of maximum stress or maximum displacement; the shape identification problems involve the determination of shapes that achieve a given desired stress distribution or displacement distribution. Each problem is formulated and sensitivity functions are derived using the Lagrangian multiplier method and the material derivative method. The traction method, which is a shape optimization method, is employed to find the optimal domain variation that reduces the objective functional. The proposed numerical analysis method makes it possible to design optimal structures for maximizing strength and rigidity and for controlling stress and displacement distributions. Examples of computed results are presented to show the validity and practical utility of the proposed method.

This paper presents a shape optimization method for the rigidity design of sheet metal structures under multiple loading conditions with the aim of weight reduction. In order to maintain the curvatures of the given initial shape, it is assumed that the design domain is varied in the in-plane direction. Using compliance as an index of the rigidity, a volume minimization problem subjected to multiple rigidity constraints is formulated as a non-parametric shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived for this problem. The traction method is applied to determine the smooth in-plane domain variation that minimizes the objective functional. The calculated results of fundamental design examples and actual automotive chassis components will show the effectiveness and practical utility of the proposed method in solving shape optimization problems of sheet metal structures.