Reasonably ecient Delaunay based mesh. generator in 3 dimensions. H. Borouchaki, F. Hecht, E. Saltel and P.L. George.

Transcription

1 Reasonably ecient Delaunay based mesh generator in 3 dimensions H. Borouchaki, F. Hecht, E. Saltel and P.L. George August 23, 1995 INRIA, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex, France. ABSTRACT. This paper describes an automatic mesh generator suitable in general for nite element simulations. The mesh generator is of the Delaunay type and the paper focuses on recent improvements relative to this a priori well known method. 1 Introduction Reliable and ecient mesh generation algorithms are an essential prerequisite for P.D.E. (such as C.F.D.) simulations using the nite element method. In the context of unstructured meshes, several methods have been addressed in the twenty last years including advancing-front, octree and Delaunay based methods. The mesh generation algorithm we would like to propose in this paper is of the Delaunay type. A great attention has been paid to Delaunay algorithms both by the computational geometry people and by the P.D.E. groups. Delaunay, himself, has proposed, in 1934, a work which is regarded as the basis of today's methods. Nevertheless, one has to wait until the early 80's for constructive algorithms essentially proposed by Bowyer, Watson and Hermeline whose works are recognized as the basic foundations for eective algorithms. Recent improvements have been proposed to obtain reliable and ecient implementation of such algorithms. The paper we would like to propose clearly takes place in such a context. Our goal is to discuss an algorithm for the triangulation problem, subject to a geometrical, a mesh quality and a CPU requirement constraints. The data necessary for a Delaunay based algorithm is a discrete form of the boundary of the domain. This form is a triangulation of the surface dening the domain seen as a conformal triangular mesh of the latter. The rst constraint relies in maintaining exactly this surface mesh in the 3D resulting mesh. To be suitable for nite element computations, a mesh must enjoy a good quality. The

2 latter is dened as the ratio h where h is the element diameter and is the element in-radius. The second constraint is to obtain a mesh whose elements oer a good quality. Due to the size of the meshes used in the computations, in terms of number of elements, the third constraint isthe capability of creating large size meshes in very short times. The paper will focus on these three aspects and aims to show that the proposed method has the capability to produce one million elements in a few minutes on a reasonably fast work station. To achieve this, new algorithms are proposed to deal both with the Delaunay aspect of the method (section 2) and with the other parts of the global algorithm including the regeneration of the initial surface mesh (section 3), the way in which the eld points are created (section 4) and the optimization procedures used at the end of the process (section 5). Application examples will be presented (section 6) and a short conclusion ends the paper. 2 Insertion point process One of the main aspect of the mesh generation method is a generator of specied points such as Delaunay. Let fp k g be a set of points in < 3. The Delaunay method consists in creating a triangulation composed of tetrahedra or convex polyedra whose vertices are the members of fp k g. An initial triangulation can be obtained as the dual of the Voronoi cells, V i, dened as: V i = fp such that k P? P i kk P? P j k; 8j 6= ig V i is a closed convex polyhedron; these cells cover the space and do not overlap, they are known as the Dirichlet tesselation of the entire space including the initial points. It is then obvious to obtain the Delaunay triangulation associated with the Voronoi cells by constructing the elements formed by joining the points of two cells sharing a side. Thus, we obtain the triangulation of the convex hull of the specied points consisting of tetrahedra and convex polyhedra, see [5]. The latter can be easily split into simplices, leading to a non unique Delaunay triangulation of the convex hull. 2.1 Connection from points to points The above method for constructing a Delaunay triangulation of a set of points proves to be ineective in practice, being very sensitive to numerical aspects such as round-o errors. This is why alternative methods have been proposed to solve the problem. The incremental method depicted below was introduced, almost simultaneously, by Hermeline ([10]) and Watson ([12]) in the early 80's. Let fp k g = fp 1 ; :::; P n g be a set of n distinct points, and T old a Delaunay triangulation with element vertices the rst i points of fp k g, then triangulation T new can be derived from T old in such a way that P i+1 is an element vertex.

3 For simplicity and without loss of generality we assume that there exists an enclosing triangulation T 0, created by hand, such that all given points fall within T 0. A constructive method for creating T new can be as follows: T new = (T old? C) [ B i+1 (1) where C is the set of elements in T old, the so called cavity, whose circumballs contain P i+1, F 1 :::F p are the external sides of this set and B i+1 = S j=1;pff j ; P i+1 g is the ball associated with point P i+1. P 4 F 4 P 5 F3 F 5 P 3 P 6 F 6 P F 2 P 8 P 2 P 7 F 7 F 1 P 1 Figure 1: Insertion of point P,alias P i+1 (after Hermeline). Thus the element vertices of the resulting mesh T new are the rst (i + 1) points of set fp k g. This mesh is obtained by replacing the cavity C by the ball B i+1, i.e. by joining P i+1 to the edges F j (see gure 1 in two dimensions). The points of fp k g are inserted one-at-a-time and the nal triangulation T new covers T 0 and is Delaunay, see for example [10]. The sole condition for which process (1) produces a valid mesh if that C results in a suitable B i+1. It means that the key of the process is the correct denition of C. 2.2 Robust insertion point method Unfortunately, the construction of C is based on numerical computations involving spheres and thus is very sensitive to round-o errors, so there is no guarantee about

4 its correct denition. Two situations frequently encountered when constructing C using the classical Delaunay criterium must be corrected. The rst one consists in nding an element whose relevant face is too close to the point under consideration. The other case consists in enclosing a previously inserted point in C. As these two cases result in an incorrect mesh, an alternative way to construct the triangulation, not necessarily strictly Delaunay (this does not matter as the boundary integrity must be preserved), has been developed. We rst introduce an enclosing triangulation T 0. Then the only thing to do is to nd in T old a suitable connected set C; i.e. such that: - its external sides are visible from P i+1 ; - it does not contain points other than P i+1. The main interest of this alternative method is that, as will be seen below, it involves only exact computations as will be seen. Thus, the following algorithm is proposed: 1. Initialize C with the element(s) in T old enclosing P i+1 (the base); 2. Recursively construct the cavity by visiting the neighbors of its elements and checking if point P i+1 satises the sphere criterium. If an element is stacked, return to 2. This operation, due to round-o errors, results in a set C possibly not valid; 3. Find F 1 :::F p the external sides of C. Dene the p simplices, K j, formed by joining the sides F j with P i+1 and denote by Det(K j ) the volume of K j ; Dene n F to be the number of endpoints of F j and n S to be that of vertices in C; 4. Set S = C. Loop for j = 1; p: { If Det(K j ) < 0, S = S? K where K = K j and go to 3 { If Det(K j ) = 0, S = S [ K where K is the simplex of T old, with side F j, not in S and return to If n F and n S are equal, set C = S and apply algorithm (1); If not, set S = S? K and return to 3 (K is any "possible" simplex (i.e. not in the base) in T old having a vertex, distinct from P i+1, in S and not in the F j s); Computation involved in the process are of the integer type (for n F and n S ) and oating type for the volumes used to compute the relevant Det(K j ). Thus, it obviously produces a valid triangulation since the computation of volumes is exact; to obtain this, the coordinates of the points are converted to integers. 2.3 Numerical aspects To be suitable and to lead to a reasonable eciency, the steps involved by the algorithm as well as the data structure used must be dened carefully. The data structure we propose consist of the following (ne is the number of elements, np is that of points): - the element vertices, simp(4,ne), - the vicinity relationships,

5 neigh(4,ne), indicating the (4) elements sharing a face with a given element, - the circumcentre coordinates, cent(3,ne), - the circumsphere radii, radii(ne), - the vertex coordinates, coor(3,np), - a table indicating, for each point, a simplex having it as vertex, start(np), and, - a background grid where the points are encoded. With such a structure, the eective implementation of this algorithm can be seen as follows: - construction of the base. It consists in nding the element(s) within which falls a given point. Thank to T 0, we are in a convex domain and a classical searching method can be employed. To obtain some eciency, we use the background grid to nd a point close to the point under consideration, then table start gives an initial guess forthe searching process. - construction of the cavity. A guest cavity is constructed by comparing the distance between the point and the centre of the examined element with its circumradius, then, using the previous algorithm, it is corrected. - eective insertion of the point under consideration including the enumeration of the element vertices, the update of the vicinity relationships and the computation of the centre and radius of the new elements: the vicinity relationships are constructed by hashing the edges of the ball (they may be obtained directly as they inherit from the old ones but this property cannot be easily exploited) while new centres and radii clearly inherit from the old values and are obtained at a low cost. Remark 1: in the case of a nite element application, a constrained variation must be employed to preserve the specied items that could be removed when inserting a point. Thus, steps 2 and 4 of the process are modied to ensure that a specied face, already in the current mesh, cannot be traversed. On the other hand, it is well known that a Delaunay type method may produce, in three dimensions, very undesirable elements in terms of aspect ratio, among them are the so called slivers. The process can be designed so that such elements are avoided. 2.4 Performances The following table reports the performance, in terms of elements per minute, of the method. np denotes the number of vertices, ne is that of elements, t is the time (using an HP 9000/ Mhz workstation) while v is the number of elements created within one minute. The method depicted is a constrained version (see the above remarks) of the algorithm. - np ne t (in sec.) v Mesh Mesh Mesh Mesh Table 1 : Constrained Delaunay point insertion method.

6 3 Boundary integrity In the context of nite element simulations, domains are commonly known via their boundaries. The latter are formed by a set of edges in < 2 and a set of triangles in < 3. A natural idea is to constitute the set of all the endpoints of these edges (faces) as set fp k g and to apply the above method. This results in the creation of a triangulation whose element vertices are the boundary points. Such a triangulation does not a priori include the items of the initial boundaries (see gure 2 where a simple example is depicted in two dimensions). The rest of this section discusses the way in which these missing items can be regenerated. A 1 A 2 B 2 B 1 Figure 2: Edges are missing while their endpoints exist. 3.1 Boundary problem in two dimensions Several methods are popular to deal with the boundary integrity problem. Some are based on the creation of the midpoint of all missing boundary edges, see [13] or on the creation of one internal point for every boundary edge. What we propose uses only the diagonal swapping. First, all specied points are inserted and the missing boundary edges are found. Afterwhich, we dene the pipes associated with each of them. A pipe associated with an edge, say P i P j, is the following elements: one having vertex P i and such that P i P j intersects the edge opposite to P i ; one with vertex P j and such that the edge opposite to P j intersects P i P j ; and those possessing two edges intersected by P i P j. We dene the diagonal swapping as the local modication of two elements sharing an edge and forming a convex quadrilateral into two new elements by using as common edge the other diagonal of the quadrilateral. Then, the algorithm consists of processing each pipe in the mesh: while the pipe contains more than two members, swap randomly an undesirable edge, else swap them, End.

7 While the sole operation used is the diagonal swapping, it has been proved that this algorithm converges (due to its random aspect). 3.2 Boundary problem in three dimensions The three methods discussed for the two-dimensional case extend to this situation and prove to produce the desired solution. Only the third method is now depicted. First, as in < 2, all specied points are inserted and the lists of the missing boundary items are established. We dene the two following sets of elements: the pipes associated with a missing edge and the local shells associated with an edge. Similarly as in < 2, the pipe associated with an edge P i P j is the following elements: one having vertex P i and such that P i P j intersects the face opposite to P i ; one with vertex P j and such that the face opposite to P j intersects P i P j ; and those with two faces intersected by P i P j. While a local shell associated with P i P j is composed of all the elements sharing an edge intersected by P i P j. Considering one missing edge, the elements having at least one edge or one face intersected is either a pipe, or a shell in the case where at least one element exists with one edge,, intersected by the missing edge. These are the sole possible situations in which a line, whose endpoints are element vertices, passes into the tetraedra. The proposed algorithm consists in visiting all pipes and shells in the mesh and applying the appropriate transformation (see section 5) as far as the pipe contains more than two elements or the shell can be reduced. It should be noticed that in some cases, it is necessary to create one internal point (one of the main diculty of the process), the so called Steiner point, to be abble to to reduce a conguration. This algorithm converges, see [8]. Remark 2: the cpu time required for the boundary regeneration is connected to dierent parametres. Clearly a surface composed of well shaped triangles is very easy to regenerate. Once the boundary items have been regenerated, it is easy to dene the elements which are inside the domain (assumed to be non convex). In the case where the domain includes dierent connected components, it is possible to enumerate all of them. Note that the deletion of outside elements is generally performed at the end of the generation process in order to remain in a convex domain at the time the eld points will be considered. 4 Field points In the case where the data consist only of the domain boundaries, it is required to create and insert internal points. On contrary, internal points are known and must only be connected using a constrained version of algorithm (1).

8 One of the advantages of the Delaunay method is that a mesh of the domain is known since the boundary integrity has been completed. Thus, this mesh can be used as a background to help in the eld point creation. Two major classes of methods are popular: one consists in creating and inserting, element by element, a certain number of internal points as long as some criterion is not reached; the other involves a dierent method (advancing-front, algebraic, octree,...) to create the points and then uses the Delaunay method to insert them. In this section, we assume that only data of geometrical nature is known. Thus, the creation of internal points can only involve geometrical ideas (the case where physical information is available is discussed in [9]). We have retained the second type of method and we propose the following which is based on a simplied algebraic method for the creation of points and uses algorithm (1) to insert them. A local (isotropic) stepsize h is associated with each data point. Then, consider the elements in the mesh, and for every non boundary edge, e i, of the element: compare its length with h, the local stepsize of its endpoints; determine n i the number of points to create on edge e i such that they follow a given distribution varying smoothly from these two h; then stack the so-created points and compute their h if they are not too close to another point (using the background grid). This method has proved to be ecient both in < 2 and < 3. It is that we prefer in terms of eciency. It produces in average good locations for the eld points so that at the time the mesh is optimized, we have to pay a relatively low cost. Remark 3: as the eld points are created in a (large) stack, the point insertion process can be randomized so that its cost is \optimal". Here are some indications regarding the way in which the eld points are created. The example of an arithmetic distribution is considered. As mentioned above, the edges in the current mesh are examined by comparing their lengths, say d, with the stepsizes of their endpoints. The aim of the method is to determine n the number of points we need to create in the visited edge so that the edge is, on the one hand, saturated and, in the other hand, the distribution of these points is smoothly balanced in an arithmetic manner. Let us denote by h 0 the stepsize of one endpoint, say P 0, and by h n+1 the stepsize of P n+1 the other endpoint. Thus we dene the series i by 0 = h 0 + r, n = h n+1? r and i = D(P i ; P i+1 ) where D is the distance between P i and P i+1, r is the ratio of the distribution. The problem turns into solving the following system: nx i=0 i = d and i+1 = i + r: this leads to (n must be converted into integer) n = 2d h 0 + h n+1? 1 and r = kh n+1? h 0 k n + 2

9 This process is repeated for all the edges in the mesh. 5 Smoothing procedures Good mesh quality is a major key to obtain precise solution (and/or to facilitate the solution step of the computation). The mesh quality is dened in terms of element qualities. There exist several ways to measure the quality of the elements in a mesh, among which is: Q = h M where h M = max i=1;6 h i, h i being the length of edge i, is the in-radius and is a scaling factor ensuring that an equilateral element's value is 1. This measure varies from 1, a well shaped elements, to 1, a very ill shaped element. The mesh quality is then appreciated with regard to three aspects: Q mesh = max i=1;ne Q, ne being the number of elements in the mesh, the distribution of the elements in terms of their quality, Q mesh compared with a target value, T arget. This value is the quality of the best element that can be constructed from the worst face in the data. The aim is to obtain a mesh with a good distribution and such that Q mesh is in the range of T arget. To achieve this, several local processes can be envisaged which are now shortly mentionned (see also [3]). 5.1 Relocating the vertices This local tool consists in processing the balls associated with the free vertices in the mesh by relocating them in such a way as to enhanced the mesh quality. Let P be such a vertex, relocating P consists in moving P, step by step, (via a given coecient!) towards an "optimal" point P opt which is determined by simulating the ideal elements lying on the external faces of B P : P = P + ~ d with ~ d =! ~ P P opt and P opt = nx j=1 j P idj : where n is the number of elements in B P, P idj is the "ideal" location of P with regard to face number j of B P and j is the weight associated with point P idj. 5.2 Local transformations This local tool consists in processing the shells (see section 3) associated with the free edges in the mesh that can be removed in order to enhance the quality.

10 M 4 M 4 M 4 M 4 M 4 M 2 M 3 M 2 M 3 M 2 M 3 M 2 M 3 M 2 M 3 Figure 3: The 5 triangulations for 5 points. Let be such an edge. Its shell is written as (M i M i+1 ) i=1;n. If it is convex, it is possible to re-mesh it as (M j M k M l ) (M j M k M l ). Thus, such a process results in the re-meshing of the M i s "polygon", after which the so formed triangles are linked with and (gures 4 and 3). Let n be the number of elements in the initial shell, then N n the number of possible re-meshings is (see[3]) N n = P n N i=3 i?1n n+2?i with N 2 = 2: M 4 M 4 M 2 M 3 M 2 M 3 Figure 4: Shell C consisting of 5 elements. Several tricks allow us to reduce the computational eort by sorting properly the dierent possible combinations. 6 Mesh generator scheme and examples As a result, we now propose the following scheme for the automatic mesh generator. Firstly, we create the enclosing cuboid associated with the given points and mesh this box using ve tetrahedra; afterwhich we insert the points of the initial set to obtain a mesh including these points as element vertices; we regenerate the boundaries and dene the outside elements, create the internal points (if desired)

11 and insert them, remove the outside elements and nally, to complete the process, apply smoothing tools. Z Z O X Y O X Y Figure 5: SDRC). Mesh 1 (courtesy of Figure 6: Mesh 2 (courtesy of Dassault-Aviation). We have selected only 3 examples of meshes created using the present method. These examples claim to be signicant in terms of dierent aspects. The rst case corresponds to \small volumes" used when simulating solid mechanical problems, resulting in a relatively small mesh. The two following examples are CFD meshes with large size. The following illustrates the capability of the method in terms both of cpu requirements and mesh quality. - np ne t (in sec., HP 9000/735) v Del Mesh Mesh Mesh Table 2: GHS3D: cpu requirements. np(ne) is the number of vertices (elements) in the mesh, t the cpu time including i/o and v the number of elements created within one minute, while Del indicates the Delaunay part, in percentage, of the generator. In terms of mesh quality, for application example 3, the T arget is 4.72 while Q mesh is The part, in percentage, of elements with a quality falling between 1 and 3, is 99 and only 11 elements are worse than T arget. It can be observed in table 2 that the larger is the mesh, the smaller in proportion is the time necessary. Our experience on various examples indicates that most of the elements enjoys a good aspect ratio and that T arget compares well with Q mesh. These two features lead us to think that the method is, in some sense, optimal. We have presented some improvements leading to a better eciency of a Delaunay type mesh generation method. Among them have been discussed a robust

12 an ecient insertion point process suitable in the case where constraints are to be followed, a way to regenerate the boundary items and a simple and ecient method for locating the eld points leading to a minimization when smoothing the resulting mesh. Future works include, in short, the extension of the method in the case where a desired density or anisotropic specications are specied in advance leading to a method governed not only by the boundary discretization. References [1] T.J. Baker, Generation of tetrahedral meshes around complete aircraft, Numerical grid generation in computational uid mechanics'88, Miami, [2] H. Borouchaki, S.H. Lo, Fast Delaunay triangulation in three dimensions, Comp. Meth. in Appl. Mech. and Eng., to appear. [3] E. Briere de l'isle, P.L. George, Optimization of tetrahedral meshes, IMA vol. in Math. and Appl., vol 75, pp , [4] J.C. Cavendish, Automatic triangulation of arbitrary planar domains for the nite element method, Int. Jour. Num. Meth. Eng. 8, pp , [5] H.S.M. Coxeter, L. Few, C.A. Rogers, Covering space with equal spheres, Mathematika 6, pp , [6] D.A. Field, Implementing Watson's algorithm in three dimensions, Proc. of Second Annual ACM Symp. on Comp. Geom., pp , [7] P.L. George, Automatic mesh generation. Application to Finite Element Methods, Wiley, [8] P.L. George, F. Hecht, E. Saltel, Automatic mesh generator with specied boundary, Comp. Meth. in Appl. Mech. and Eng., 92, pp , [9] P.L. George, F. Hecht, M.G. Vallet, Creation of internal points in Voronoi's type method. Control and adaptation, Adv. in Eng. Soft. and. Work., vol 13, 5/6, pp , [10] F. Hermeline, Une methode automatique de maillage en dimension n, Thesis lect., Universite Paris 6, Paris, [11] C.L. Lawson, Generation of a triangular grid with application to contour plotting, California institute of technology, JPL, 299, [12] D.F. Watson, Computing the n-dimensional Delaunay Tesselation with applications to Voronoi polytopes, Computer Journal 24, n o 2, p , [13] N.P. Weatherill, The integrity of geometrical boundaries in the two-dimensional Delaunay triangulation, Com. in Appl. Num. Meth., vol 6, pp , 1990.