Full analytical sensitivities in NURBS based isogeometric shape optimization

Abstract

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.

Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.

This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.

The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.

Introduction

This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS (non-uniform rational B-spline) control points in structural shape optimization.

Even since the work of Braibant and Fleury [1], B-spline and its generalized representation, NURBS, have been widely used in shape parameterization in structural optimization. It has become the method of choice for parameterizing freeform shape in structural optimization [2] for two important reasons: 1) With a few control points, NURBS can represent complex freeform shape. The alternative representation, the use of finite element nodes as design variables, would often lead to wiggly, irregular shape. Fig. 1 gives one such example. 2) The output of NURBS-based shape optimization can be directly linked to a computer-aided design (CAD) system since NURBS is the standard shape representation underlying all major CAD software.

The recent advent of NURBS based isogeometric analysis [3] has made it even more advantageous to use NURBS in shape parameterization for design optimization since NURBS can not only be used to represent the geometry, it can also be used as a basis for approximating the physical fields. The use of the NURBS basis in finite element analysis has exhibited superior numerical properties, e.g. in terms of per-degree-of-freedom accuracy, over traditional finite element analysis [3]. Further, the tri-variate B-spline representation has been extended to represent both geometry and material composition in functionally gradient materials (FGM) parts [4] and used in the B-spline basis based graded finite element analysis of FGM objects [5]. It thus allows closer integration with CAD since the exact geometry and even material composition can be used in both design and analysis through the NURBS representation.

NURBS represented shape is affected by both the positions and weights of its control points. Fig. 2 presents a 4 × 3 control net for a NURBS surface consisting of 2 × 2 knot spans with degree 2 in ξ₁ direction and degree 1 in ξ₂ direction. Fig. 2.b shows when a control point changes its position from Q_a to Q_b, the underlying surface and knot spans change. Fig. 2.c shows the surface change and the knot span change when the weight of control point Q changes from 1 to 0.5.

Although both positions and weights of control points affect the NURBS geometry, as demonstrated in Fig. 2, until very recently, usually only positions of control points are used as design variables [1], [6], [7], [8], [9], [10]. In these work, analytical sensitivities when used are only given for positions of control points, thus they are referred to as partial sensitivity in this paper. Isogeometric analysis has recently been successfully applied in structural shape optimization [11], [12] where, again, only analytical sensitivities for positions of control points are given.

Thus far, the use of both weights and positions of control points as design variables has only occasionally been explored, e.g. in [13]. In particular, there has been a lack of analytical formula for sensitivity calculation for shape optimization with both positions and weights of control points as design variables. A notable exception is very recent work in [14] where both control points and weights are used to optimize one-dimensional beam structures with sensitivities analytically evaluated.

Analytical formula for computing the sensitivity of physical quantities over both positions and weights is important for the following reasons:

• Analytical formulas lead to more accurate and efficient calculation of derivative information required in gradient-based optimization. The finite difference based method for gradient calculation suffers from the “step-size dilemma” due to the potential truncation error and round-off error [15]. It is also inefficient. If we need to find the derivatives of the structural response with respect to n design variables, the forward-difference approximation requires n analyses, while the central-difference approximation would require 2n analyses.

• The use of NURBS weights as design variables in structural optimization, in addition to positions of control points, lends more flexibility in shape representation and enlarges the design space, which can lead to better design. In particular, it makes it possible to recover a class of optimal shapes such as conic curves, e.g. ellipses and circles, and surfaces, e.g. cylinders, spheres and ellipsoids, which are otherwise not possible. Note, without weights, NURBS shape degenerates into B-spline shape and B-spline representation cannot exactly represent these conic curves and surfaces.

The absence of analytical sensitivities of physical quantities such as compliance, displacement and stress over shape parameters, i.e. both positions and weights, is due perhaps to the seeming complexity of such a derivation. A naïve way of obtaining these sensitivities would be to expand the integrand involved in calculating these physical quantities into explicit expressions of design variables, which would be of daunting complexity. In the context of isogeometric shape optimization, the derivation could be more involving since the NURBS basis function used in analysis also becomes affected by the design variables (weights). For example, although the usefulness of weights as design variables was recognized in recent work in [11], the derivation of analytical sensitivities for weights was not available and was deemed “more complex”.

In this paper, we give a set of compact formulas for computing analytical sensitivities for both control point positions and weights, thus referred to as full sensitivity. Our approach follows that of [16], [17], [18], an isoparametric based technique for differentiating stiffness matrix and force vectors with respect to discrete design variables. We use the chain rule of differentiation and Jacobi's formula for the derivative of a determination to derive these compact formulas. These formulas are applicable to both traditional finite element based NURBS shape optimization and isogeometric shape optimization.

The calculation of these sensitivities involve two terms: analysis terms that are encountered during the usual finite element analysis and isogeometric analysis and geometric sensitivities that are represented as the derivatives of positions and weights over design variables. Mesh refinement is often required for accurate finite element analysis. This is especially true in isogeometric shape optimization since the weights are now design variables which can induce distorted distribution of isoparametric curves (element boundaries). Thus, an analytical method is also given for propagating geometric sensitivities of the control points in the design model to those of the control points in the refined analysis model.

Our numerical implementation is based on the isogeometric analysis due to its numerical advantages. Numerical examples demonstrate the availability of such analytical formulas has both theoretical implication and practical significance. Theoretically, they can be used in interpreting the optimality conditions and understanding behaviors of physical systems which may not been seen directly from the problem. For example, they can be used in determining whether an exact circle is the optimal shape for a hole in a plate under bi-axial load. In practice, they enlarge the design space, allow flexibility in shape representation and lead to better designs.

In the remainder of this paper, Section 2 gives a brief introduction to the NURBS basis and NURBS geometry and introduces some key notations used in sensitivity derivation. Section 3 gives a general formulation of shape optimization and the role of sensitivity in shape optimization. Section 4 gives the analytical formulas for sensitivities over positions and weights of NURBS control points. Section 5 discusses how the geometric sensitivity of a design model can be analytically propagated to that in the refined analysis model. Section 6 discusses the result of our numerical implementation on some common shape optimization problems. This paper concludes in Section 7. The derivation of the analytical formulas is given in the Appendix.