Shape optimization of thin-walled beam-like structures

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Abstract

This article presents a methodology for optimizing the shape of thin-walled structures having a beam-like dynamic behavior. The equivalent nominal beam characteristics (quadratic moments of inertia, torsional rigidity, etc.) of a refined finite element model are determined from a direct calculation based on explicit equations which are functions of the nodal coordinates defining the cross-sectional geometry. A subset of these coordinates is then taken as the design variables for a nonlinear optimization problem where new target physical properties are sought. A complementary model reduction procedure is introduced to improve the precision of the proposed method for beams with variable cross-section or topological accidents.

Introduction

A knowledge of the dynamic behavior of mechanical structures is essential for their design and optimization. In the context of the present article, we are interested in optimizing the response behavior of complex automobile assemblies with respect to a set of physical design variables. While the application of finite element methods to obtain and optimize discrete elastodynamic models of mechanical systems is now commonplace, the cost of analyzing the resulting large order models is often prohibitive. It is thus necessary to develop a strategy to reduce the size of the model in question while preserving its fidelity. The literature is replete with different methodologies for reducing the order of these discrete models via a generalized transformation matrix as exemplified by the component mode synthesis approaches (e.g. [1], [2], [3]). However, these reduction methods, which have the disadvantage of not preserving the topological structure of the initial model, are severely limiting due to the inaccessibility of the physical design parameters (local stiffness and mass modifications). An alternative reduction strategy consists of simplifying the refined model by replacing certain zones by dynamically equivalent but less finely meshed topologies. For example, a methodology has recently been developed for the automobile industry [4], which is particularly well adapted to structures having a beam-like dynamic behavior. A direct consequence of this strategy is the possibility of performing a parametric optimization on the resulting simplified model of the complete body-in-white. However, once the optimal parameters have been determined based on the simplified model, it is necessary to transform back to the real geometrical shapes that satisfy a number of relatively complex topological constraints.

This article addresses the shape optimization problem which determines the actual refined geometry of the thin-walled beam cross-section satisfying the complete set of optimized design constraints, both mechanical and topological. The presentation is organized as follows. Firstly, a brief review of the basic equivalent beam formulation will be given based on [4]. The principle consists of replacing the three-dimensional mesh by an equivalent model composed only of beam finite elements. The method identifies the set of physical beam parameters to be introduced in the equivalent model based on a straight beam of constant cross-section. The equivalent beam results are taken as reference values and are used to establish correction coefficients for the direct approach that follows.

Secondly, the beam properties are calculated using an explicit formula based on the position of the nodes defining the beam cross-section. The potential design parameters include the coordinates of the nodes in the plane, the wall thickness, and the orientation of the principal inertial directions.

Thirdly, the shape optimization is performed by minimizing a cost function computed from target and direct values. The design parameters in this final optimization consist of the nodal coordinates defining the cross-sectional geometry of the thin-walled beam. The proposed method establishes a simple relation between the two design spaces [5], the first containing the properties computed with the equivalent beam method and the second containing properties determined explicitly based on purely geometric arguments.

Section snippets

Background

A methodology for simplifying thin-walled members having a beam-like behavior has been developed for the automobile industry [4]. The following sections provide a brief review of the equivalent beam approach.

Formulation of the direct method

The methodology consists of expressing the different geometric properties A, J, IY and IZ as functions of the nodal coordinates of the end sections of the beam using an explicit formulation. We then optimize the shape of these two sections with respect to the target beam properties. This section is devoted to a description of the analytical functions relating these properties to the nodal coordinates of the beam end sections.

Two cross-sectional topologies will be considered independently,

Shape optimization

Two cases of shape optimization will be considered, namely for constant and variable section beams. The following paragraphs provide the definitions of the cost function which is minimized for the shape optimization. For a non-constant section beam, the use of correction coefficients allows the values obtained by the direct method to be calibrated with respect to the reference results provided by the equivalent beam method. For variable section beams, the presence of sharp discontinuities in

Numerical applications

The shape optimization method is applied to a straight 660 mm long steel beam whose section is defined in Fig. 2. The optimization is performed using the proposed procedure. The initial and attained beam properties, calculated with the equivalent beam method, are reported in Table 1, Table 2, Table 3 (values in bold represent the subset of beam characteristics whose target values are different from the nominal values).

In the first example, geometric constraints are imposed so as to preserve the

Conclusion

A method for optimizing the shape of thin-walled beam-like structures has been presented (Fig. 16). The strategy can be applied to structures having both constant and variable cross-sections and containing local topological accidents. A direct calculation procedure is formulated and correction coefficients used to calibrate the beam property estimations on the basis of reference properties obtained by a finite element method based on equivalent beam procedure. The proposed procedure can take

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