Local Surface Approximation and its Application to Smoothing in Three-dimensional Indirect Mesh Generation Dorit Merhof a , Roberto Grosso a, Udo Tremel b G¨unther Greiner a, aComputer Graphics Group, Friedrich-Alexander University, Erlangen, Germany bEADS Military Aircraft, Munich, Germany Abstract Smoothing techniques are of major importance for the generation of surface meshes. For this reason a considerable amount of research has been spent on developing a large variety of sophisticated smoothing approaches. However, these methods ei- ther require direct access to the analytic surface description or are restricted to flat meshes in two dimensions. In particular, if no analytic surface data is available as in the case of indirect mesh generation, it is not possible to reposition nodes on a three-dimensional geometry. To cope with this problem, this paper presents an approach based on the local approximation of an analytic surface which is used for repositioning nodes when smoothing is performed. Furthermore, the paper shows how established two-dimensional smoothing techniques such as length smoothing, angle smoothing and isoparametric smoothing are adapted to three dimensions by combining them with the local surface approximation. A quadrilateral mesh gener- ator that implements the presented techniques finally demonstrates the benefits of this approach. Key words: local surface approximation, smoothing, three-dimensional surface mesh, indirect mesh generation, quadrilateral 1 Introduction Nowadays, the generation of surface meshes is a common engineering problem. The surface to be meshed may be represented by an analytic function (para- metric tensor-product spline surfaces) (Farin, 1996) or by discrete data, i.e. a triangulation (STL-data, subdivision surfaces). The resulting surface meshes are either directly used for simulation and analysis or serve as intermediate step for the generation of volume grids. However, the high quality criteria required for simulation (e.g. smooth variation of element size and shape all over the domain) are in most cases not satisfied by the direct outcome of surface mesh generation algorithms such as advancing front or Delaunay type methods. A common approach to provide the high mesh quality required for simulation is to perform surface enhancements which are therefore an essential prerequisite for the generation of surface meshes. Among the different kinds of surface enhancements, smoothing is one of the mostly applied techniques. Regarding two-dimensional domains, a large variety of smoothing techniques is available. An overview is given by Canann et al. (1998) and George and Borouchaki (1995). For a three-dimensional domain the generation of surface meshes is more complex. In the case of an analytic surface description, a classification into two major groups is possible, parameter space and direct three-dimensional mesh generation. The parameter space approach takes advantage of the underlying uv presen- tation of an analytic surface. The mesh is thus computed within this two- dimensional parameter space using conventional smoothing techniques for two-dimensional space. The generation of isotropic elements is ensured by arc-length parameterizations (Farouki, 1997) or special metrics derived from the first fundamental form of the surface enabling the measurement of lengths, angles and area of faces (Chen and Bishop, 1997; Cuilliere, 1998; George and Borouchaki, 1995; Tristano et al., 1998). Direct three-dimensional approaches on the other hand generate surface ele- ments directly on a pre-discretized geometry without taking into account the two-dimensional parameter space representation of the surface (Cass et al., 1996; Lau and Lo, 1996; Lau et al., 1997). Those algorithms may produce better results in case of highly distorted mappings between parametric and physical space. In both cases the analytic surface description facilitates smoothing as the surface description may be consulted for the placement of nodes and thus guarantees that the shape of an object is retained. In contrary, when dealing with discrete surfaces no analytic surface description is available which could be utilized for smoothing. Up to now indirect mesh generation approaches were thus restricted to two dimensions. The aim of this paper therefore is to provide a smoothing approach applicable for three-dimensional discrete surfaces to enable indirect mesh generation for arbitrary three-dimensional domains. This is achieved by locally approximat- 2 ing an analytic surface in the vicinity of the node intended for smoothing. Smoothing is then performed by moving the node on this approximated sur- face. However, the approach for local surface approximation presented in this paper is not only suitable for three-dimensional smoothing which is chiefly in- vestigated in this paper but also for related problems such as point placement during mesh adaption or regridding of triangulations. The article is organized as follows: In Section 2 the approach for local sur- face approximation is presented. The combination of established smoothing techniques with this approach is investigated in Section 3. The techniques developed in this context were integrated into an indirect quadrilateral mesh generator to demonstrate their robustness and practical relevance. The results derived from this approach are presented in Section 4. 2 Local Surface Approximation The input for indirect quadrilateral mesh generation are triangular meshes. An essential prerequisite for successfully generating a quadrilateral mesh from the given triangular background mesh are adequate smoothing techniques. Exist- ing smoothing techniques for the indirect generation of quadrilateral surface meshes are only applicable for two-dimensional domains. For this reason a novel approach for three-dimensional smoothing is presented providing the possibility to process arbitrary triangular meshes in three-dimensional space. In order to smooth a node Ns in three-dimensional space, a local surface approximation is computed. Smoothing is then performed by repositioning node Ns on the approximated surface. The steps that are necessary to locally approximate a surface for smoothing of a node are described in the current Section. 2.1 Taylor Polynomial The basic idea of three-dimensional smoothing is to move the node that has to be smoothed on an analytic surface that locally approximates the geometry. For each node Ns that has to be smoothed an analytic surface has to be determined that contains the node itself and that best fits all direct neighbor nodes Ni connected to node Ns by an edge (Figure 1). A biquadratic Taylor polynomial is used to locally approximate the analytic surface: u2 v2 F (u; v) = uF + vF + F + uvF + F (1) u v 2 uu uv 2 vv 3 The two-dimensional parameter space coordinates of a node are denoted by u; v. The computation of the two-dimensional parameter space coordinates of the center node Ns and its neighbor nodes Ni is described in Section 2.2. The polynomial coefficients Fu, Fv, Fuu, Fuv and Fvv are the first and second order partial derivatives of F (u; v). A method to obtain these derivatives is presented in Section 2.3. N0 N5 NS N1 N4 N2 N3 Fig. 1. Three-dimensional smoothing scenario. The node selected for smoothing is denoted with Ns. Smoothing is performed by repositioning Ns on an analytic surface that runs through Ns and best fits its neighbor nodes Ni. 2.2 Parameterization For the computation of an approximated surface represented by the biquadratic Taylor polynomial given in equation (1), an almost isometric parameterization F (ui; vi) = Ni of the neighborhood of Ns has to be determined. A param- eterization is thereby called almost isometric if kFuk ≈ 1, kFvk ≈ 1 and FuFv ≈ 0. For all further computations a local coordinate system is uti- lized with the property that the center node Ns is positioned at the origin in three-dimensional space as well as in two-dimensional parameter space, i.e. Ns = (0; 0; 0) = F (0; 0) = F (us; vs). As regards the choice of a parameteriza- tion there are several possibilities. In the following sections two alternatives are discussed, a projection into the tangential plane of Ns (2.2.1) and an ex- ponential projection (2.2.2). 2.2.1 Projection into Tangential Plane The most evident way to compute a parameterization is to project all neigh- bor nodes Ni into the tangential plane of Ns. The normal vector ~n of this tangential plane is derived by averaging the normals of the faces around Ns. A parameterization is thus given by the transformation of the projected nodes into an orthonormal basis with ~n being one of the basis vectors. The selection 4 of an orthonormal basis thereby ensures that the resulting parameterization is almost isometric. A drawback of this method is that the projection into the tangential plane does not necessarily maintain the order of the neighbor nodes Ni around Ns. This might occur if the mesh is not flat enough or if an improper tangential plane was chosen. Figure 2 shows such a situation. 0 4 5 1 2 3 4 5 0 2 3 1 Fig. 2. Projection into tangential plane does not necessarily maintain the order of neighbor nodes Ni around Ns. 2.2.2 Exponential Projection A second parameterization that takes the lengths and angles between neighbor edges into account is the exponential projection. In contrary to the projection into the tangential plane the order of neighbor nodes Ni around Ns = (0; 0; 0) is maintained. For every neighbor node Ni the coordinates (ui; vi) within two- dimensional parameter space are computed as follows: i 1 i 1 − − exp(N ) 7! kN k cos( α~ ); sin( α~ ) (2) i i 0 j j 1 j=1 j=1 X X @ A The angles α~i are thereby computed by applying a function flat with α~i = flat(\(Ni; Ni+1)) = flat(αi) (3) where \(Ni; Ni+1) is the angle between Ni and Ni+1. The function flat scales the angles αi between two neighbor nodes Ni and Ni+1 so that the sum of the resulting angles α~i amounts to 2π (Figure 3).