Keywords: shape optimization, boundary element, minimization methods.

In this work, the Boundary Element Method (BEM) is applied to simple problems of geometric optimization of two-dimensional continuous structural systems. A mathematical formulation is proposed based on the BEM and on deterministic optimization techniques of zero and first order. The optimal characterization of the geometric configuration of the structure is obtained with the minimization of an objective function. This function is written in terms of referential stresses (maybe strain or deformation) located at some referential points inside or at the boundary of the structure.

The process of geometric optimization of the structure is accomplished through a gradual reduction of the difference between the referential stresses and the stresses calculated by the BEM at referential points. In the proposed formulation, the geometry is expressed in terms of design variables which values are iteratively modified until a configuration that minimizes the objective function approximates the calculated stress values to the reference values. The design variables have to comply with certain limits and, therefore, always be inside a feasible domain. Such restrictions are enforced by the use of a penalty function augmented to the objective function. In addition, some heuristic rules are used so that the searching step during the minimization process will limit the design variables to feasible domains.

The use of the BEM for problems of geometric optimization of structures is quite attractive in relation to other methods that inevitably discretize the whole continuous, i.e., the Finite Elements and the Finite Difference Methods. When the body forces are not significant, the BEM discretizes just the boundary of the structure. During the process of objective function minimization, the geometry of the structure undergoes a sequence of topological modifications that, accordingly, should be accompanied with the appropriate discretization. With the BEM such modifications are easy to do while in formulations using the Finite Element or the Finite Difference methods such mesh modifications turn out to be a cumbersome process.

The formulation was based on quadratic boundary elements and on three optimization methods. One method is of zero order: Powell's method; and the other two are of first order: Conjugate Gradient and BFGS methods. For the latter methods, the derivatives of the BEM quantities (stress, strain, displacements) contained in the objective function are calculated by finite difference.

Four examples are presented in the article. The examples are based on the minimization of stresses available at few referential points. These points are located inside or at the border of the structural member.

One example is the determination of the height of a cantilever beam such that the maximum stress at a specific point is at yield stress. A second example deals with the determination of the minimum allowable thickness of a pipe section, for a fixed internal radius of the pipe. Another example deals with the determination of the position of a hole of radius inside a 2D panel such that the stresses at the boundary of the hole are minimized. The last example shows how to find a "smoother" configuration for a chamfer or notch such that the principal stresses at specific internal points are at minimized.

The number of design variables used in the examples varies from 1 to 6. Convergence was soon attained soon in all applications studied. The BEM showed good numerical stability, precision in the stress results, ease of mesh updates during the shape changes, and reduced matrices. The formulation proposed in this paper, for the shape optimization of 2D continuous structures, shows potential for simple applications.

1

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