C0 C1 and C2 Continuity of Curves

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Shape variable definition with $\text{C}{}^0\text{,}\text{C}{}^1$ and C ² continuity functions

Abstract

The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).

Introduction

The correct definition of shape variables is of the utmost importance in the layout of an optimisation problem. The identification of the most useful geometry changes is crucial if a satisfactory solution is required. The shape variable definition task has to overcome different difficulties like the selection of the admissible geometrical changes and the preservation of several geometrical properties as well as geometrical continuity between fixed areas and areas subject to deformation.

One of the first researchers to point out the complexity and the difficulties in the shape variable definition was Imam. In his work [1] he identifies four main techniques for variable definition: (1) the independent node movement technique, (2) the design element technique, (3) the supercurve technique and (4) the shape superposition technique.

The independent node movement technique is very simple; it uses as shape variables the co-ordinates of the nodes of the discrete model of the structure.

The design element technique has been extensively used. The design element is a macro finite element consisting of one or more finite elements. The isoparametric shape representation, as used for individual elements, is used here to determine the isoparametric co-ordinates of any point on the surface. This technique also allows for the determination of the co-ordinates of the node points inside the design element for mesh generation, which is required every time the shape is changed during optimisation.

The third technique mentioned by Imam is the supercurve one. In structured meshes the boundary surface is fully determined given a few curves laying on it. Only the points on these curves are required to construct the finite element model. In such cases, the parametric representation of curves with a polynomial expression may be used; the coefficients of the polynomial expression serve in this case as the shape variables.

Finally, the shape superposition technique concerns the possibility to superimpose two or more shapes specified in terms of model location of points on a curve or surface in varying proportions. The linear combination of these predefined shapes allows the generation of a variety of shapes, the coefficients of the combination being the shape variables [5]. A similar technique has been used also by Vanderplaats [2] for airfoil optimisation.

In 1984, Braibant and Fleury [3] proposed a new approach for two-dimensional shape variable definition. The shape description is quite similar to the one based on the design element introduced by Imam. The region of the structure to be modified during the optimisation process is also defined by one or more design elements that still contain a part of the mesh. Instead of using the shape functions of a two-dimensional finite element, blending functions typical of Bezier or B-spline curves are used to determine the co-ordinates of any point inside the design element or on its boundaries. Therefore the shape variables are no longer the position of the nodes of an isoparametric two-dimensional element, but the points that control two families of curves whose cartesian product defines the design element. Slope discontinuities at the design element interfaces can be easily managed. Each boundary of the structure is an automatic piecing of spline of degree k. The value of k can be selected by the designer and this formulation guarantees the continuity of the piecing up to the order k−1 [4].

The five techniques mentioned above have several limitations. Let us consider first the node movement and the supercurve techniques. The former is practically unusable due to two severe drawbacks: it results in too many design variables and discontinuity of slope in the shape at the element interfaces takes place in the first few iterations of the process. The latter requires the identification of the parametric equations of several curves. Several practical applications can be found in two-dimensional problems concerning, for example, thickness distribution over a shell structure. Moreover, in three-dimensional problems, this technique requires a non-trivial work that is not compensated by the higher order surfaces that can be obtained.

These two techniques are not so common and are usually substituted by the curve superposition and the design element ones. In the former, difficulties arise in the selection and evaluation of the `master' curves or surfaces whose linear combination will allow identifying the optimal solution. These are usually taken as the first modal shapes of the structure or the shapes resulting by the application of user-defined distributed loads. Moreover, in this case it is difficult to control the surface modification of the structure especially if only local variations are to be considered. The design domain technique is one of the most useful and practical techniques indeed. It avoids the presence of discontinuities at the element interfaces and allows for the reduction of the number of the design variables. The limitations of this technique come from the interpolation functions used to extrapolate the design element nodal movements to all nodes present inside the design element itself. Isoparametric shape functions are used, and because of their origin, they can guarantee only zero continuity between adjacent elements. So, there is not any possibility to choose the continuity degree between two adjacent design elements. Moreover, only the nodes inside the design element are held in consideration for nodal movement extrapolation. If the design element dimensions are not large enough, a discontinuity in the movement distribution could be introduced leading to element distortion and to inaccurate results.

The technique proposed by Braibant and Fleury solves the lack of continuity control at the interface of two adjacent domains by introducing Bezier or B-spline interpolation functions. This approach can be easily adopted for two-dimensional problems, but difficulties arise in its extrapolation to three-dimensional ones. In any case the limitation linked to the movement distribution discontinuity due to the design element concept as introduced by Imam keep on being present.

All the mentioned techniques are characterised by a common problem: they are based on the discrete model of the structure analysed.

The present paper proposes a new shape design variable technique. It allows for defining the shape variables directly on the geometry of the analysed structure without any previous knowledge of the analytical expression of the CAD surfaces. The design element concept has been introduced and modified leading to radical simplifications. In three-dimensional problems, the domains are defined by two-dimensional surfaces whereas, in two-dimensional problems, the domains are defined by one-dimensional curves. Only the `grouping' capability has been preserved. Also the interpolation functions for nodal movement extrapolation have been modified being defined over a lower degree domain. The proposed technique allows for an a priori definition of the surface continuity to be preserved during the geometry modification process. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process in structural analysis, fluid-dynamic analysis, crash analysis, and aerodynamic analysis, among others.

Section snippets

The method

The proposed method is able to manage two-dimensional as well as three-dimensional shape variable definition problems. The way to operate is exactly the same in the two situations, the only difference being the degree of the geometrical entities. Whereas in a three-dimensional problem the geometry of the analysed structure is described using two-dimensional surfaces, in two-dimensional problems this is obtained by using one-dimensional curves.

Implementation of the method

The proposed method operates by the attribution of well-defined properties to several geometrical entities like points, lines or surfaces. So, the geometrical definition of the structure to be analysed is needed only. There is no problem indeed to use the same method for displacement propagation starting from a discrete model. In this case several facilities as the transfer of the geometrical properties to the discrete model are lost preserving the main characteristics of the method.

The

Examples

Two examples will be shown next. The first example concerns the shape variable definition in an airfoil fluid-dynamic optimisation problem. The airfoil is defined by several points connected by spline curves. The fluid domain has been defined as a rectangular `box' around the airfoil itself.

There exist different possibilities for the definition of the shape variables: for example, local shape variables changing only a portion of the geometry of the profile can be used or global ones taking in

Conclusions

A new method for shape design variable definition has been proposed. It overcomes some of the pitfalls of the methods formerly proposed by several authors. It is able to manage two-dimensional as well as three-dimensional problems allowing the user to work directly on the geometrical model of the problem and simplifying the variable definition phase of an optimisation problem layout. C ⁰, C ¹ or C ² continuity properties of the boundary surface of the analysed structure can be preserved leading to

Acknowledgements

This work has been supported by the EU Marie Curie Grant BRMA-CT97-5761 held by the first author. The authors want to acknowledge the support of E. N. BAZAN (Madrid) and of the International Centre of Numerical Methods in Engineering (CIMNE, Barcelona).

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