تجزیه و تحلیل حساسیت برای بهینه سازی شکلی شبکه ثابت با استفاده از مایل تقریب منحنی مرز

The remesh-free property is the most attractive feature of the various versions of fixed-grid-based shape optimization methods. When the design boundary curves do not pass through the predetermined analysis grids, however, the element stiffness as well as the stress along the curves may be computed inaccurately. Even with the popular area-fraction-based stiffness evaluation approach, the whole optimization process may become quite inefficient in such a case. As an efficient alternative approach, we considered a stiffness matrix evaluation method based on the boundary curve approximation by piecewise oblique curves which can cross several elements. The main contribution of this work is the analytic derivation of the shape sensitivity for the discretized system by the fixed-grid method. Since the force term in the sensitivity equation is associated only with the elements crossed by the design boundary curve, we only need the design velocities of the intersecting points between the curve and the fixed mesh. The present results obtained for two-dimensional elasticity and Poisson’s problems are valid for both the single-scale standard fixed-grid method and the multiscale fictitious domain-based interpolation wavelet-Galerkin method.

In the standard shape optimization based on the finite element approach, remeshing cannot be avoided during the optimization process if accurate analysis is to be guaranteed, especially for design problems requiring large shape changes (Bennett and Botkin, 1985; Yao and Choi, 1989). Researchers have shown recent interest on the shape optimization based on the fixed-grid analysis or the Eulerian-type analysis because the analysis offers way to avoid cumbersome remeshing processes. Another advantage of the fixed-grid-based shape optimization method is that it requires only the boundary velocity field for design updates while the standard finite-element-based shape optimization method generally requires design velocities for all nodes, the so-called domain and boundary velocity fields (Choi and Chang, 1994). The fixed-grid based method shows this feature because its analysis nodes are independent of the shape changes. Though the fixed-grid method is equipped with the excellent remesh-free property, this method has some difficulties in accurately evaluating the stiffness matrices of the elements adjacent to curved boundaries. It has this difficulty mainly because the analysis grids or nodes are always predetermined, and the design boundary does not necessarily pass through these analysis grid points. Since the present sensitivity analysis is mainly for a method to overcome such a difficulty, it is worth stating the implementation technique of the fixed-grid method for shape optimization. In implementing the remesh-free fixed-grid analysis method, the most popular approach is to embed the original design domain ω encircled by curved boundaries into a fictitious domain Ω usually having a simple geometry. Then, the fixed-grid-based analysis is carried out for Ω. In Fig. 1, we illustrate a rectangular fictitious domain for a generally-shaped ω. The single-scale fixed-grid method usually uses uniformly distributed rectangular finite elements for two-dimensional cases. The stiffness of the elements inside ω is set to be the stiffness of the original material, but the elements inside Ω⧹ω are assigned to have a very weak material. The question is: How does one evaluate the stiffness of the boundary elements lying on the boundary ∂ω. In the fixed-grid method, the stiffness of the boundary elements changes when the boundary curve changes. Therefore, the boundary element stiffness must be estimated accurately for efficient shape optimization. Full-size image (10 K) Fig. 1. A two-dimensional problem with the domain of interest ω embedded in a fictitious domain Ω (View the MathML sourceΓωg: boundary under kinematic constraint, View the MathML sourceΓωh: boundary under natural condition, DΓω: design boundary). Figure options Until recently, the common approach has been the area-fraction-based stiffness evaluation method, as was used in Garcia and Steven (1998) and Kim and Chang, 2003 and Kim and Chang, submitted for publication. The concept of this approach is to evaluate the boundary element stiffness proportionally to the area fraction of the part belonging to ω within the boundary element. The boundary design velocity is thus related to the rate of the change of the area fraction of the boundary element. However, the boundary curve in this method needs to be approximated by zigzags that consist only of vertical and horizontal lines, so this area-fraction-based approach is not effective for curved boundaries. The only way to obtain accurate solutions near the boundary is to work with highly-dense grid distributions. In order to obtain accurate solutions near the curved boundary without excessive grid densities, Jang et al., 2002 and Jang et al., 2003 proposed a more direct approach; a curved boundary is approximated by piecewise oblique lines formed by the connections of the points between the curved boundary and the fixed-grid lines. In this case, the piecewise oblique line, the approximated boundary curve, does not usually pass through the analysis nodes. Therefore, the stiffness matrix of a boundary element should be integrated separately by considering the oblique line on the element, but this integration can be easily performed by the Gauss quadrature. Jang et al., 2002 and Jang et al., 2003 used this idea for the shape optimization method based on the adaptive multiscale interpolation wavelet-Galerkin method; the standard fixed-grid method is a non-adaptive, single-scale version of the wavelet-Galerkin method. Thus the piecewise oblique boundary curve approximation scheme works equally for the standard fixed-grid method. In Jang et al., 2002 and Jang et al., 2003, however, the resulting sensitivity analysis was carried out by the direct finite difference scheme. In this work, we present the semi-analytic sensitivity analysis for the fixed-grid shape optimization based on the oblique boundary curve approximation. By the semi-analytic analysis, we mean the analytic sensitivity analysis for a discretized structural system, or the continuum-discrete sensitivity analysis (Choi and Kim, in press). In the first part of this work, some results derived by Hansen et al. (2001) are utilized for the present analysis; the analysis grids not interacting with the boundary curves are stationary or fixed during the whole design process both in the method by Hansen et al. (2001) and in the oblique boundary curve approximation method. The sensitivity equations and the boundary conditions are derived for two-dimensional Poisson problems and elasticity problems. The shape change will be represented by the movement of the intersection points. The force vector for the sensitivity equations comes only from the boundary velocity fields of the intersection points. Once the semi-analytic sensitivity is calculated, it may be used to check the accuracy of the sensitivity by the finite difference scheme. To this end, we considered a domain having a simple geometry parameterized by a Bezier curve and compared the numerical and analytical sensitivities. The shape sensitivity for a microgripper whose boundary is parameterized by a B-spline curve was also considered for verification. Finally, we also remark on how the present sensitivity analysis based on the single-scale fixed-grid method can be extended for the multiscale interpolation wavelet-Galerkin method.

The analytic shape sensitivity for the fixed-grid method using the boundary approximation by piecewise oblique lines was derived. The shape sensitivity analyses for two-dimensional Poisson problems and plane elasticity problems were explicitly carried out. In the proposed fixed-grid method, the force terms in the sensitivity equations appeared only at the boundary elements. Thus, the calculation of the force terms required the boundary velocity field on the boundary elements, and the field was approximated linearly in terms of the velocities at the intersection points of the original boundary curve and the edges of the fixed finite elements. The analytic sensitivity derived in this investigation was validated by the comparison with the numerical sensitivity. We also showed that the analytic sensitivity was applicable for the multiscale wavelet-Galerkin method and presented the procedure to implement the shape sensitivity within the multiscale method. When the multiscale fixed-grid method is extended to three-dimensional problems, the proposed method is expected to greatly speed up the whole shape optimization process.