EQSM: An efficient high quality surface grid generation ... - SPMS

Comput. Methods Appl. Mech. Engrg. 195 (2006) 5621–5633 www.elsevier.com/locate/cma

EQSM: An efficient high quality surface grid generation method based on remeshing Desheng Wang, Oubay Hassan, Kenneth Morgan *, Nigel Weatherill Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 25 April 2005; received in revised form 26 October 2005; accepted 26 October 2005

Abstract The EQSM method for the generation of unstructured triangular surface grids is presented. The method is based on remeshing techniques and avoids the need for traditional mesh triangulation approaches, such as the advancing front and the Delaunay methods. Normalized edge lengths, based on a metric derived from curvature or from a user-specified spacing, are employed as the remeshing criterion. It is assumed that the geometry is input in the form of composite parametric surfaces, with analytic definition or Ferguson- or Nurbstype multiple patch representation. Examples, based upon typical aircraft geometries, are included to demonstrate how high quality grids can be efficiently generated on surfaces that exhibit a high degree of geometric complexity. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Surface grid generation; Delaunay triangulation; Remeshing

1. Introduction Surface mesh generation is still a challenging and difficult prerequisite for numerical simulations involving large scale volumetric mesh generation. Existing approaches can be essentially categorized as being either direct or indirect [1–10]. In the direct approach, the mesh is generated directly on the 3D surface, using an octree based or an advancing front method [9,11,12]. The indirect approach relies on generation in the parametric domain, using any 2D mesh generation procedure [4,5,8,10,13–17]. The mesh is then mapped onto the 3D surface. Direct methods have difficulty in checking the validity of the mesh, while indirect methods have difficulty in controlling the size and shape of the elements generated in the 2D domain. For engineering simulations, in areas such as computational fluid dynamics, most boundary definitions come directly from CAD systems. All CAD systems support different output formats, with the most common being IGES, STEP and STL. The representation of a three-dimensional geometry, based on CAD patches, such as Ferguson patches, NURBS patches or patches with analytic expressions, will consist of both geometrical and topological data. The geometrical data is defined in the physical space and includes the basic parameters defining the shape of the support surfaces and the intersection curves. The topological data defines the surface regions to be generated, their support surfaces and bounding curves. Given this information, the adjacency relations between different components of the surface can be naturally derived from the topological data [1,2,4]. In classical surface mesh generation, each surface region is meshed individually. This means that points must be generated on the bounding curves and these points will be part of the mesh for the surface region. In many cases, the bounding curves are of no physical significance and small thin patches often result in the creation

*

Corresponding author. E-mail address: [email protected] (K. Morgan).

0045-7825/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.10.028

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of sliver, or distorted, triangles [3,18]. For this reason, the development of a method that enables the generation of high quality, patch independent, surface meshes, which conform to the input surface representation, is also a desirable objective. In this paper, we attempt to meet these objectives by describing a surface mesh generation method, termed EQSM, which aims to avoid the shortcomings traditionally associated with direct and indirect methods and enables the efficient, automatic, generation of high quality surface grids, using remeshing techniques [19–25]. The geometry definition formats that are supported by the current implementation of the method include composite parametric surfaces, with analytic definitions, or discrete composite parametric Ferguson curves and surfaces or NURBS. With the EQSM method, for a geometry defined in terms of composite parametric mappings, a triangulation of the points defining the intersection curves of each surface region is generated in the parametric domain. These planar meshes are then mapped onto the surface and assembled into a global surface conforming mesh. This mesh can be regarded as an initial surface triangulation. Basic remeshing operations, such as edge splitting/contraction, coupled with edge swapping, are performed to produce refinement or coarsening of this initial mesh. Metric controlled edge lengths are computed with respect to surface curvature and coupled with a user-specified mesh spacing control function. Grid points are added, by edge splitting, and are accurately located on the surface, using the surface definitions. Finally, the surface mesh is optimized by utilizing nodal smoothing and edge swapping techniques. Edge splitting/contraction, based upon G1-interpolation, is conducted locally to remove any sliver triangles which may have been created by the existence of small sliver patches in the input surface definition. As these operations require only local information, the EQSM method is highly efficient in terms of its CPU demands, e.g., for a complex geometry, such as an aircraft configuration, over one million, high quality, elements can be generated in a few minutes on a modern PC. In this paper, geometry modelling and the input data format are discussed in Section 2. An overview of the EQSM surface mesh generation method is given in Section 3. Details of the construction of the initial surface triangulation are presented in Section 4, while Section 5 describes the mesh refinement and coarsening techniques that are employed. The process of optimization of the final grid is described in Section 6 and a few practical examples are included in Section 7. 2. Geometry modelling and input data format In engineering design and analysis, CAD systems are normally used to generate the geometric definition of the system under consideration. Commercial and research solid modelers often represent boundaries as composite parametric surfaces, or employ a discrete representation in terms of triangular facets. For the purposes of this work, it will be assumed that the geometry is defined in terms of composite parametric surfaces. Composite parametric surfaces are composed of sets of patches, together with intersection curves. Each patch is represented by an analytically defined mapping or in terms of spline composite curves and tensor-product surfaces, such as Ferguson, Bezier or Nurbs [1,14,26,28–30]. This forms the so-called support surface [2,4,26], which may be shared by several surface regions. The patch technique utilizes a bi-cubic spline interpolation for the mapping. A set of intersection curves which bound the domain to be generated are also defined in the form of cubic splines. Standard data import formats, such as IGES and STEP, or our in-house FLITE [2,4] fall into this category. Although other parametric curve and surface definitions may be used in practice, for the present implementation of the EQSM method, the geometry definition must satisfy the following conditions: (1) The global surface is defined in terms of a number of conforming sub-surfaces, or patches, together with a set of intersection curves. (2) A mapping from physical space to a two-dimensional parametric space exists. This mapping can be analytical or of any other kind, such as the tensor-product of splines. (3) The intersection curves are defined by parametric representations. (4) The mapping of each patch is bijective almost everywhere, with singular points appearing only on the boundary. (5) The boundary of a surface region is formed by one or more closed loops of intersection curves. Intersection curves forming a loop connect with each other only at their end points. If the boundary of a surface region is formed by more than one nested loops, then the loop enclosing the largest area in the parametric domain is taken to be the outer loop. As an example, for a complete aircraft configuration, the support surfaces and the intersection curves bounding the generated surfaces are shown in Fig. 1(a) and (b), respectively. 3. Overview of the EQSM method In the EQSM method, surface remeshing is used to address issues such as efficiency, robustness, quality and the constraint which arises from the boundary representation. This provides a general approach for generating surface grids

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Fig. 1. Modelling the surface of an aircraft showing: (a) the support surfaces; (b) the intersection curves.

(a)

(b)

(d)

(c)

(e)

Fig. 2. Illustration of the EQSM method for a sphere: (a) initial surface triangulation; (b) surface triangulation after 1 remeshing step; (c) surface triangulation after 12 remeshing steps; (d) surface triangulation after 20 remeshing steps; (e) final mesh after optimization.

for composite parametric surfaces represented by various CAD systems and lead to a high quality triangular surface mesh. The main stages in the EQSM method are: (1) The generation of an initial surface mesh by triangulating the points defining the intersection curves bounding each surface region. Fig. 2(a) shows such an initial surface triangulation for a sphere. (2) The local remeshing of the initial surface mesh to satisfy element spacing requirements. These requirements can either be user specified or determined automatically, based on the surface curvature. Remeshing is accomplished by utilizing local mesh modifications such as edge splitting/contraction and edge swapping. Fig. 2(b)–(d) shows the surface triangulation achieved for a sphere after different numbers of remeshing steps. (3) The improvement of the element quality and point distribution by optimization. The triangulation of each surface region is subjected to point smoothing and further edge swapping. This is followed by a patch independent surface optimization, based on G1 interpolation, to remove any badly shaped triangles which are present because of the boundary definition constraint. Fig. 2(e) displays the final mesh for a sphere surface after optimization. These stages are described in detail in the following three sections. 4. Initial surface triangulation For a composite parametric surface, the generation of the initial surface triangulation consists of three steps. These steps are the computation of the parametric coordinates of the support points of the intersection curves, the initial triangulation of each surface region and the processes of assembly, orientation and ridge identification for the surface triangulation. The Ferguson patch technique is utilized to obtain the continuous parametric mapping required for the support surface [4,14,26], with the intersection curves parameterized with spline interpolation in a conforming fashion [26]. Using this

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representation, curvature data, which can be used to compute isotropic or anisotropic curvature-controlled mesh size, may be readily evaluated. Commercial modelers may also be used to produce equivalent data. The starting point of the procedure is the formation of an initial mesh in the parametric domain. Before any two-dimensional triangular mesh generation technique is applied, points defining the intersection curves of a surface region need to be mapped to the parametric domain. 4.1. Computation of the parametric coordinates of the support points The determination of the parametric coordinates of a vertex can be formulated as a point projection problem or, more precisely, a minimization problem. In general, the generated vertices of an intersection curve do not lie exactly on the two neighboring support surfaces. To determine the position of these intersection curve support points on the parametric plane, a simple steepest descent method is used [4]. The efficiency and success of this method depend upon the initial estimation and the search direction. The estimation process proceeds as follows: (1) A check is made to ensure that the point does not coincide with any singular point. (2) Starting from the four corners of the boundary of the support surface in turn, a search along the boundary is performed to find the minimum distance to the target point [27]. (3) If the minimum distance is not within an acceptable tolerance, the point on the boundary with the minimum distance is taken as the initial guess with a new search direction. (4) If the above steps fail to locate the target point, a slow projection procedure is performed to obtain the closest point to the target point.

4.2. Initial triangulation of each surface region The initial triangulation of each surface region may be achieved by employing, in the parametric domain, any twodimensional triangulation method, such as a Delaunay or an advancing front method. This is a planar, boundary constrained, triangulation of the edges constructed by joining the support points of the intersection curves of the surface region. Initially, an unconstrained triangulation of all the support points is obtained and then edge swapping is applied for boundary recovery. At this stage, element quality is not a major concern. Fig. 3 illustrates the procedure for generating an initial mesh for a typical surface region. After the planar triangulation, the surface triangulation of each region can be obtained by lifting the two-dimensional vertices into three dimensions, by using the mapping determined by the patch or by the parametric definition. 4.3. Assembly, orientation and ridge identification for the surface triangulation The triangulations of all the surface regions are assembled and common vertices are merged. A consistent surface orientation is constructed using surface integrals [1]. The surface which encloses the maximum volume is taken to be the outer boundary and all other closed loops are taken to be internal. The orientation is determined from the requirement that the normals to each triangle are pointing towards the domain. For composite parametric surfaces, each intersection edge is initially considered to be a ridge. Any point connecting more than two ridges is taken to be a corner point. The end points of each intersection curve are also considered to be corner points. No ridges are allowed to be swapped and no corner points are allowed to be smoothed in the subsequent remeshing.

(a)

(b)

(c)

(d)

Fig. 3. Illustration of two-dimensional constrained triangulation for a surface region: (a) the boundary of the surface region; (b) a triangulated rectangle used to enclose all the boundary points; (c) mesh produced when the points are inserted in a simple manner; (d) final mesh following edge swapping for boundary recovery.

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Fig. 4. Illustration of the initial surface triangulation of an aircraft configuration: (a) only support points of the intersection curves included; (b) with support surface points also included.

An example of an assembled initial surface triangulation for a complete aircraft configuration, based upon the triangulation of the support points of the intersection curves only, is shown in Fig. 4(a). 4.4. Enhanced initial surface triangulation The initial triangulation of each surface region consists only of the points located on the bounding intersection curves, with no use of the support points. The support surfaces in the FLITE format [2] provide a natural structured grid and the points lying within the surface region can be efficiently included in the initial surface triangulation at a very low cost. Fig. 4(b) shows the initial triangulation that is produced for the complete aircraft configuration when the support surface points are also included. It is apparent that this produces an improved initial surface mesh. If the support surfaces can be re-interpolated in accordance with the given mesh spacing, this surface mesh will be closer to the desired mesh, and this will result in the consequent speed up of the EQSM method. 5. Remeshing based on local operations The initial surface triangulation will not be suitable for use in a numerical simulation, as most triangles will not meet the mesh size or quality requirement. To produce an appropriate mesh, mesh enhancement is performed [14,19,20,23–25,31]. The enhancement process employs mesh refinement and mesh coarsening, in addition to local mesh operations such as edge swapping. Node relocation is also applied to improve element quality [1,19–21]. The edge length required at a point can be controlled in a number of ways. A user specified size can be defined by the use of a background mesh, coupled with point, line and triangular sources [2,32]. A curvature controlled metric can be used to determine the size required to achieve a prescribed accuracy for the surface representation [5,19,33,34]. The main radii of the curvature are used to construct a geometric metric and this metric prescribes mesh sizes as well as element stretching directions at mesh vertices [19]. For isotropic meshing, the largest radius is used, while for the anisotropic case, the two radii are used. For a composite parametric surface representation, the curvatures are computed directly from the mapping [1]. Using the geometric metric and the prescribed mesh size, a normalized edge length is computed [19,33]. This length is pffiffiffi used as a criterion for the refinement or coarsening of the given edge. Usually, the value C ¼ 2 is assigned to the splitting s pffiffiffi parameter, and the value C c ¼ 1= 2 is given to the coarsening parameter. If the edge length is larger than Cs then a splitting operation is performed, while contraction is applied if the edge length is less than Cc. To achieve a smooth variation of the generated mesh, the desired size is graded using the H-correction procedure [35]. This correction reduces the magnitude of the ratio of neighboring element edge lengths. 5.1. Edge splitting For any edge which has a normalized edge length larger than the given splitting criterion, a new point is introduced into the triangulation and the edge will be split into two smaller edges. The two triangles adjacent to the edge will be split into four triangles.

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D*

D

A

C B (a)

A*

M*

D

C*

A

P

B* (b)

C

B (c)

Fig. 5. Illustration of the process of edge splitting, using midpoint mapping and iteration.

When the geometry is represented by composite parametric surfaces, the mapping and the parametric coordinates of the two end points of an edge are used to find an optimal location for the new point P. In Fig. 5(a), the surface edge BD is to be split, with its image in the parametric domain denoted by B*D*. Suppose M* is the midpoint of edge B*D*, as shown in Fig. 5(b). The mapping of M*, denoted by P0, is chosen to be the initial guess for P. If the normalized edge lengths P0B and P0D are larger than Cc, P0 is taken as the optimal location for insertion. If this is not the case, then an iterative point location process is followed until this edge length criterion is satisfied. The configuration, after the point insertion, is illustrated in Fig. 5(c). Neighboring edge swapping is performed to improve the local configuration, in terms of both approximation of the geometry and the element quality. For edges which lie on the intersection curves forming the surface region, the parametric representation of the curve is used to map the proposed point to the physical space. After obtaining the physical coordinate of the point on the intersection curve, its parametric coordinates on the two support surfaces are then obtained using iteration [2]. 5.2. Edge contraction When the metric controlled length of an edge is less than Cc, the edge is collapsed. In this process, either the two end points are merged to create one vertex or a new vertex is created. A new vertex is located at the image of the mid point of the parametric edge. In practice, both options are checked and the configuration which produces the largest minimum angle is adopted. This is illustrated in Fig. 6. The contraction process also requires the deletion of the two triangles adjacent to the edge. Various implementation schemes have been proposed to perform edge contraction [19,22–25]. The process of checking the validity of the new local configuration is not simple, but here we employ the robust procedure that is described by the steps: (1) Suppose AB is the edge to be collapsed and that vertex A is to be moved to B. The local configuration is checked initially to determine if there is a 3-division configuration neighboring the edge, as illustrated in Fig. 7. In this case, contraction is not allowed. (2) The areas of the triangles in the new configuration are calculated in the parametric domain. Any negative area indicates that overlapping has occurred and, in this case, contraction is not allowed. (3) A further check is performed in the physical space, to ensure that the new configuration will not contain triangles with minimum angle tending to zero. In addition, the dihedral angles between all the neighboring triangles in the new configuration are determined and any configuration resulting in an angle of larger than 45° is rejected.

A B

M

B

(a)

(b)

(c)

Fig. 6. Illustration of the use of alternative edge contraction schemes.

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D

D A

5627

E

E C

C

B

B

(a)

(b)

Fig. 7. Illustration of the effect of collapsing a 3-division configuration: (a) initial mesh; (b) the overlapped mesh that would be produced.

After contraction, neighboring edge swapping is applied to improve the shape of the new elements and the geometry approximation. As in the case of edge splitting, contraction of a ridge edge follows a different procedure. In the checking, the intersection curve parametric coordinates are used to evaluate the location of the merged node. 5.3. Edge swapping The edge swapping process is illustrated in Fig. 8. The edge BD in Fig. 8(a) is identified as a non-ridge edge that connects the surface triangles ABD and BCD. The objective of edge swapping is to replace the connection BD by the diagonal edge AC. The configuration after the swapping is shown in Fig. 8(b). If the new configuration has better quality than the original one, the swapping operation is performed. Different measures may be adopted to analyse the quality of a configuration. Mesh quality measures can be classified as being either measures of element shape [1,36] or measures of geometric approximation. Geometric approximation can be measured with great accuracy, as there is a definition for each surface which can be used for comparison. In Fig. 9(b), B*D* is the inverse, in the parametric domain, of the edge BD shown in Fig. 9(a). If the midpoint of B*D* is denoted by M*, then the distance, d, between the midpoint P of BD and the mapping M of M* is computed as the distance between the edge and the given surface, as shown in Fig. 9(c). This midpoint distance is a useful measure of the quality of the geometry approximation [1,2].

D

D

N1

N1

N2

C

A

C

N2

A

B

B (a)

(b)

Fig. 8. Illustration of the edge swapping process showing: (a) the initial configuration; (b) an alternative configuration.

D

D *

D C

A

A*

M

C

B*

B (a)

M

* *

A

d

P

C

B (b)

(c)

Fig. 9. Illustration of the assessment of the quality of the geometry approximation in terms of the distance, d, of the midpoint from the surface.

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In the swapping procedure, the validity of both the surface triangulation and its inverse, the parametric triangulation, must be maintained. This is accomplished by using procedures similar to those employed when edge contraction is performed. Normally, in the initial refinement and coarsening, more attention is paid to the geometry approximation, while element quality becomes the main concern in the final optimization. 5.4. Point smoothing Point smoothing, or relocation, modifies the position of a vertex without changing the topology of the triangles meeting at the vertex. The objective is to find an optimal position on the surface for the vertex, in the sense that any alternative configuration should have better quality, according to criteria such as minimal angle, shape quality, size conformity or others [1]. Here, only the element shape quality is considered and various smoothing methods can be used [1,5,37]. Smoothing may be performed in either the parametric space, the physical space or the tangential plane. These options have all been evaluated and the use of a spring system equilibrium model [1], with under relaxation, in the parametric space is found to perform best, in terms of computational time requirements. 5.5. Refinement/coarsening procedure The refinement/coarsening procedure modifies, iteratively, an existing surface triangulation through edge splitting or contraction. The objective is to ensure elements that are in better conformity with the size specification. Edge lengths computed with respect to size or metric are compared with the given splitting parameter Cs and contraction parameter Cc. If the initial mesh is almost isotropic, sequential processing of the edges of the current mesh is feasible and efficient. However, the initial surface triangulation obtained from connecting the support points of the intersection curves often contains very large aspect ratio triangles. In this case, the result is, frequently, to produce distorted local configurations, with clusterings of many stretched triangles. Performing further operations, such as edge swapping and edge collapsing, while maintaining the surface integrity, will not always be possible. To overcome this problem, a grouping technique is applied, in which the edges are sorted into groups according to their lengths. Within each group, there is no strict order. The group which contains the longest edges is processed first and each group is updated dynamically. This guarantees that, in global terms, the longest edges are refined first. The grouping technique results in a more stable and robust procedure for refinement/coarsening of the initial mesh. As both splitting and coarsening are local operations, the computational time required increases linearly with the number of nodes added and edges merged. With the grouping technique, it is found that the total number of contraction operations is far less than the total number of splitting operations. Hence, the complete procedure is very efficient. Practical examples demonstrate that almost one million elements can be generated in less than 1 min on a 2 GHz PC. The element shape quality of the refined surface triangulation is good, as each edge possesses a normalized edge length of around unity. 6. Mesh optimization Due to the possible appearance of sliver patches in the geometry definition and the distortion sometimes caused by the irregular definition of support surfaces in CAD systems, a triangular surface mesh that has been generated by the EQSM method may contain some badly shaped triangles. To enhance the quality of the surface mesh, the procedures of individual surface region optimization and global surface optimization are applied successively. 6.1. Individual surface region optimization For each surface region, a combination of edge swapping, point smoothing and node connectivity optimization or mesh relaxation [21,23,33,38,39] is performed iteratively. Intersection edges, or ridges, are not swapped and only interior points and non-end intersection curve points can be smoothed. Each local optimization operation is based on the previously described mapping. In practice, three or four loops of edge swapping, coupled with point smoothing, are performed first. Mesh relaxation and point smoothing may be coupled for further improvement. 6.2. Global surface optimization In individual surface region optimization, no edge swapping occurs across surface regions. To remove any distorted triangles, a ridge relaxation method is applied. In this method, the dihedral angle a in degrees of the two adjacent triangles is

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checked for all ridge edges and, if 160 < a < 200, the edge is reconsidered as a non-ridge. Classical G1 interpolation [19], based on edge splitting/contraction together with edge swapping, is applied to the reclassified edges and to their area of influence. Bad elements, with small angles or very large aspect ratio, that are adjacent to these edges can be deleted naturally. The combined application of these two post processing techniques is found to be very effective in improving the regularity and smoothness of the surface grid, with almost all points connected to six triangles. Element shape quality statistics for practical configurations are presented in the following section. 7. Application examples EQSM has been applied to three aircraft configurations that are defined in terms of composite parametric surfaces in the FLITE format, using Ferguson patches [2]. Details of the surface meshes produced for generic Gulf 2, F6 and F16 configurations are shown in Figs. 10–12, respectively. The element size distribution in each example is governed by prescribed sources, coupled with curvature control. In addition, the size is smoothed by gradations. To test the efficiency of EQSM, the CPU time required for each example is compared with the time required by an advancing front based method [2] in Tables 1 and 2. For the relatively small scale meshes considered in Table 1, it is apparent that the method is almost as efficient as the advancing front method. For the F6 model, the advancing front method requires 75 s to generate 125,130 elements, while EQSM needs 88 s to produce a mesh of 140,812 elements. The results for the Gulf 2 and F16 models show similar trends. However, a different behaviour is produced when relatively larger surface meshes are generated, as demonstrated by the figures presented in Table 2. For example, for the F6 model, when the number of elements generated is increased to 703,302, the CPU time needed by the advancing front method is 620 s. This is more than 8 times the figure shown in Table 1 for the smaller mesh for the same configuration. However, the time required to generate 667,432 elements for this model by EQSM is only twice the corresponding figure presented in Table 1. Similar behaviour is apparent for both the Gulf 2 and the F16 models. This difference in meshing efficiency behaviour can be explained by recognising that the number of operations required by the advancing front method in generating n elements is Oðn log nÞ. For small values of n, this is very similar to the OðnÞ operations required by EQSM, once the initial surface triangulation has been accomplished. For larger values of n, the discrepancy in the operation count between the two methods becomes significant, especially for geometries involving surface regions which are meshed with a large number of elements, such as the Gulf 2 and the F6. This is confirmed when the CPU time required by the two methods to generate meshes on the surface of a hemisphere, defined as a single surface region, is considered. This comparison is made in Fig. 13, which displays the variation in the CPU time as the number of elements generated is increased. For the F16, where there are 524 input patches, which are all comparatively small, the difference in the efficiency of the two methods is not so great. For general large scale surface grid generation, involving an arbitrary number of patches, we can expect EQSM to be significantly more efficient that the advancing front approach, independent of the size of the geometry. The quality of the surface mesh elements generated by the two methods for the Gulf 2 and F16 configurations has also been analysed. The meshes generated for the F6 configuration have been excluded from this comparison, as its surface definition contains fewer of the sliver-like patches which challenge the process of high quality meshing. The quality of a surface triangle, e, is measured in terms of the element quality, Qe, defined as pffiffiffi Qe ¼ 4 3Ae

X 3

 i 2 Le ;

i¼1

where Ae is the area of triangle e and Lie , i = 1, 2, 3, denote the element edge lengths. It is well known that the advancing front method can produce a high quality distribution of points and, consequently, high quality surface meshes. These meshes are often superior to the meshes generated by alternative approaches, especially in terms of average element quality. For the two examples considered here, it is found that the average quality of the meshes generated by EQSM is very similar to that for the meshes produced by the advancing front method. For most numerical simulations, it is the element quality or the minimum element angle that is significant when assessing the mesh quality. For this reason, we have computed the distribution of minimum element angle, in the range from 0° to 25°, and the element quality. These distributions are shown in Fig. 14 for the Gulf 2 configuration and in Fig. 15 for the F16. Due to the geometric constraint of input sliver patches, the surface mesh generated by the advancing front method has a number of small angled elements. This number is significantly reduced in the mesh created by EQSM. The elimination of the small angle elements is achieved by the global surface optimization, through ridge relaxation. The resulting improvement can be clearly seen in Figs. 14(b) and 15(b).

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Fig. 10. Details showing a high quality surface triangulation for the Gulf 2 model.

Fig. 11. Details showing a high quality surface triangulation for the F6 model.

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Fig. 12. Details showing a high quality surface triangulation for the F16 model.

Table 1 Small scale surface meshing: comparison of the CPU time requirements for an advancing front method and for the EQSM method Configuration

F6 Gulf 2 F16

Advancing front

EQSM

Elements

CPU (s)

Elements

CPU (s)

125,130 63,376 309,672

75 25 120

140,812 75,502 367,842

88 20 130

Table 2 Larger scale surface meshing: comparison of the CPU time requirements for an advancing front method and for the EQSM method Configuration

F6 Gulf 2 F16

Advancing front

EQSM

Elements

CPU (s)

Elements

CPU (s)

703,302 521,230 1,024,324

620 812 684

667,432 500,760 1,201,312

184 143 314

8. Conclusions The EQSM method for triangular surface grid generation is proposed for composite parametric surfaces. The method is based on the local mesh modifications of edge splitting/contraction and edge swapping. The method consists of three main

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10000 Advancing Front EQSM

9000 8000

CPU time (seconds)

7000 6000 5000 4000 3000 2000 1000 0

0

1

2

3 4 Number of elements

5

6

7 5

x 10

Fig. 13. Comparison of the variation of CPU time required, with the number of elements generated on the surface of a hemisphere, by an advancing front method and EQSM.

0.12

0.06 Advancing Front EQSM

0.1

0.05

0.08

0.04

Percentage

Percentage

Advancing Front EQSM

0.06 0.04 0.02

0.03 0.02 0.01

0 0

5

(a)

10

15

20

0 0

25

0.1

(b)

Minimum angle

0.2

0.3

0.4

0.5

Element quality

Fig. 14. Surface meshing of the Gulf 2 configuration by an advancing front method and the EQSM method showing: (a) the minimum angle distribution; (b) the element quality distribution.

0.1

0.14 Advancing Front EQSM

0.12

Advancing Front EQSM

0.09 0.08 0.07

Percentage

Percentage

0.1 0.08 0.06

0.06 0.05 0.04 0.03

0.04

0.02 0.02 0

(a)

0.01 0

5

10

15

Minimum angle

20

0

25

(b)

0

0.1

0.2

0.3

0.4

0.5

Element quality

Fig. 15. Surface meshing of the F16 configuration by an advancing front method and the EQSM method showing: (a) the minimum angle distribution; (b) the element quality distribution.

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stages: the generation of an initial surface triangulation, the refinement/coarsening of the mesh applying local mesh modification, and mesh optimization. Ridge relaxation and G1 interpolation based local mesh modifications are applied for global surface grid optimization. EQSM is highly efficient as all operations are performed locally. The statistics presented for the application examples demonstrated that, for complicated geometries, EQSM is an efficient robust procedure for the generation of high quality surface meshes. Future directions of research include the extension to the anisotropic case, appropriate support surface re-interpolation and the application to adaptive meshing including the treatment of moving boundaries. References [1] J. Thompson, B.K. Soni, N.P. Weatherill, Handbook of Grid Generation, CRC Press LLC, Boca Raton, FL, 1999. [2] J. Peiro´, J. Peraire, K. 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