NEXT-GENERATION SWEEP TOOL: A METHOD FOR GENERATING ALL-HEX MESHES ON TWO-AND-ONE-HALF DIMENSIONAL GEOMTRIES

Transcription

1 NEXT-GENERATION SWEEP TOOL: A METHOD FOR GENERATING ALL-HEX MESHES ON TWO-AND-ONE-HALF DIMENSIONAL GEOMTRIES PATRICK M. KNUPP PARALLEL COMPUTING SCIENCES DEPARTMENT SANDIA NATIONAL LABORATORIES M/S 0441, P.O. BOX 5800 ALBUQUERQUE, NM PKNUPP@SANDIA.GOV Abstract. Placement of interior node points is a crucial step in the generation of quality meshes in sweeping algorithms. Two new algorithms were devised for node point placement and implemented in Sweep Tool, the first based on the use of linear transformations between bounding node loops and the second based on smoothing. Examples are given that demonstrate the effectiveness of these algorithms. 1. Introduction It has been demonstrated that geometries that are two-and-one-half dimensional (e.g., generalized cylinders) can be meshed with all-hexahedral finite elements [1], [4]. Because all-hexahedral mesh generation on general three-dimensional geometries remains an elusive goal, algorithms to mesh two-and-one-half dimensional geometries, generally referred to as sweeping or projecting methods, continue to be important. Although mesh sweeping is simple in concept, the first-generation algorithms lacked robustness and often produced poor mesh quality. There are several approaches to mesh generation via sweeping but common to all is the idea of identifying surfaces on a volume to serve as sources or caps and a complemetary set to serve as linking sides. Source surfaces may be arbitrarily meshed with quadrilaterals via paving or multi-block structured meshing and then swept along the linking sides towards a target surface which may or may not be pre-meshed. This is feasible provided the linking side surfaces are meshed with a type of multi-block, structured mesh known as submap [5]. Identification of these source and linking surfaces has been automated in the CUBIT code [6]. Interior mesh connectivity is entirely determined once these surfaces have been identified and meshed. Placement of the interior nodes in space is then a crucial step in generating a quality mesh. For geometries that are sweepable via source mesh translation or rotation, the spatial location of the interior nodes is unambiguous. For more general geometries, however, there is no uniquely correct location for the nodes. The guiding principle in the general case is that nodes are placed so that hexahredral elements of a swept mesh are of good quality. As a minimal requirement, the Jacobian at the eight corners of all hexahedrons should be positive [2]. Another reasonable requirement is that qualities of the source surface mesh such as biasing or the relative areas of quadrilaterals should be transferred, in so far as possible, to the interior layer meshes and to the target. Even with these requirements, flexibility remains in the placement of the interior nodes in sweeping general geometries. This paper reports on a robust method for uniquely specifying the interior node positions in a way that directly addresses the mesh quality issue. SANDIA IS A MULTIPROGRAM LABORATORY OPERATED BY SANDIA CORPORATION, A LOCKHEED MARTIN COMPANY, FOR THE UNITED STATES DEPARTMENT OF ENERGY UNDER CONTRACT DE-ACO4-94AL

2 The outline of this paper is as follows: section 2 describes how point placement was done in the first-generation Project Tool [4] in order to illustrate the challenge posed by general two-and-one-half dimensional geometries. Section 3 treats the point placement algorithm based on linear transformations that is used in Sweep Tool while Section 4 describes results of layer smoothing as a point placement method, both in Sweep and Project Tools. Of course, interior point placement is not the only issue that arises in designing a good sweeping algorithm, so Section 5 briefly covers some additional topics of importance. In section 6 we offer our conclusions. 2. Interior Point Placement in the Project Scheme Interior points are located in the Project scheme by an advancing front mechanism that builds a layer of hexahedrons on the front from the previous layer, begining with the source surface meshes. An individual hexahedron on the front is built using the four nodes x 0, x 1, x 2, and x 3 on the previous layer plus three other nodes x 4, x 5, and x 6 belonging to the hexahedron under construction. The latter are known from the boundary data on the linking surface and from pre-existing nodes on the layer containing the advancing front. In the original algorithm, the final node x 7 of the hexahedron was computed by averaging the three vectors x 4 x 0, x 5 x 1, and x 6 x 2 that approximately point in the direction in which the front advances: (1) x 7 = x {(x 4 x 0 ) + (x 5 x 1 ) + (x 6 x 2 )}. This approach gives the desired nodal position under a translation: if x 4+k = x k +c for some constant vector c R 3 and k = 0,1,2, it immediately follows that x 7 = x 3 + c. However, not all meshes on sweepable geometries can be translated so equation (1) is inadequate for many important cases. Foremost among these are rotatable geometries; rotations are not preserved under (1) and very poor meshes result. An improved planar-intersection algorithm was suggested in [4] in which the point x 7 was determined by the intersection of the three planes containing the points (x 4,x 5,x 6 ), (x 0,x 3,x 4 ), and (x 2,x 3,x 6 ), respectively. While this alternate algorithm improved the quality of the projected meshes, it still did not work effectively on rotatable geometries. Two approaches to solve this difficulty were suggested in [4]. In the first approach, a separate program module (known as scheme Rotate) was developed for geometries whose source meshes can be rotated. The planar-intersection algorithm was discarded in favor of one that works only in the rotational case. The result was a highly efficient module that consistently achieved good quality meshes on a limited set of geometries. A minor disadvantage of adding Rotate to the suite of tools was that there were then two modules to maintain when only one is actually needed. More importantly, however, is that other commonly encountered geometries such as a frustum needed development of additional special modules because the heuristically-based planarintersection algorithm failed to give quality meshes on them as well. To overcome this piecemeal approach, the authors of Project tried mesh smoothing. In some respects this was premature because, as is demonstrated in the next section, the planar-intersection algorithm can be replaced with a unifying technique involving linear transformations that handles all geometries which require translation, rotation, and scaling of layer meshes.

3 3. Interior Point Placement via Linear Transformations Given a layer mesh of points p j R 3 with bounding loop L consisting of the points {x k }, k = 1,2,...,K with K 3, and given a second loop L consisting of K points { x k } in R 3, we seek a vector b R 3 and a 3 3 non-singular linear transformation T between loops such that (2) x k = T x k + b for all k. Then, for any point p j belonging to the layer bounded by loop L, we determine the point p j in the layer bounded by loop L by p j = T p j + b. The vector b can be readily determined from the loop data. Define loop center points c = 1 (3) x k, K (4) c = 1 K k x k. It is not necessary that these center points lie inside the loops. If we sum (2) over all k, we find: (5) thus (2) becomes (6) k b = c T c ; x k c = T (x k c). For convennience, let u k = x k c and ũ k = x k c so that ũ k = T u k. Define K matrices U k = [u k,u k+1,u k+2 ] and Ũk = [ũ k,ũ k+1,ũ k+2 ]. 1 The former matrices will be non-singular provided the three column vectors are linearly independent, i.e., each portion of the loop must be non-planar. If U k is non-singular, then we can uniquely define K linear transformations T k = Ũk U 1 k which satisfy ũ k = T k u k. In general, however, there may not be a single linear transformation T between the loops. If T exists, then T k = T for all k. Since, in general, such a transformation between arbitrary loops does not exist, we perform a least-squares fit to the bounding loop data by minimizing the non-negative function F(T ) = 1 K (7) ũ k T u k 2. 2 k=1 Clearly, if T in (2) exists, then it mimimizes F. To find T, set F/ T ij = 0 to get three uncoupled linear systems that can be written as (8) with (9) M T M = F = k (u k u k ), (10) F = k (u k ũ k ), 1 Since the loops are closed we can achieve periodicity by letting x k+k = x k and x k+k = x k.

4 where is the vector outer product. T is determined provided M is non-singular but, unfortunately, M is singular for the important case of planar loops. Definition: a loop L is planar if there exists n R 3 such that n u k = 0 for all k. Suppose that n exists for loop L. Then Mn = (u k n)u k = 0. Hence the null-space of M is non-trivial and so M is singular. To circumvent the problem of planar loops, redefine the set of vectors u k, ũ k by (11) (12) u k = x k (2c c), ũ k = x k c. It is easy to show that u k = ũ k = K ( c c). Note that u k = (x k c) + ( c c), giving a clear geometric interpretation. Note that for the sweep problem there will not be a vector n that is normal to all the u k unless the geometry is pathologic. We wish to apply the least-squares fit above to find a relation of the form (13) ũ k = T u k. Summing this relation over k we find that T ( c c) = c c. Putting (11-12) into (13), we find that (14) x k c = T (x k c). We have thus arrived at relationship (6), which in turn implies (2), as desired. Note that if T is the identity matrix, then one gets loop translation. Note also that if F is not zero at the minimum, then T does not necessarily send {x k } to { x k }. A new linear transformation is calculated in Sweep Tool for each layer of the advancing front based on consecutive boundary loops from the linking surfaces. 4. Layer Smoothing The algorithm outlined in the previous section was implemented in Sweep Tool and was found effective on a wide variety of problems, including rotatable geometries (see Figure 1). Despite the effectiveness of linear transformations, another algorithm is needed to sweep general two-and-one-half dimensional geometries. To handle general geometries, we resort to layer smoothing techniques in which each layer of quadrilaterals on the advancing front is smoothed independently of connections to nodes on the previous layer. Although fully three-dimensional smoothing is appropriate for the hexahedral meshes generated in sweep, it would be very inefficient to incorporate such a procedure into the advancing front method. An alternative is layer smoothing which provides a relatively efficient, if somewhat less robust, approach. A variety of layer smoothing methods, including Laplacian and isoparametric, were tried in Project with mixed success. Although Project smoothing was generally an improvement over the Project point placement algorithm described in section 2, two difficulties adversely impact mesh quality. The first difficulty is illustrated by the example of sweeping a half-torus with the source surface meshed with a radially-biased circle primitive (Figure 2). Because the Project algorithms of section 2 produce meshes with negative Jacobian elements on rotatable geometries, layer smoothing is necessary. Laplace layer smoothing eliminates the negative Jacobians but the biasing on the source mesh is lost. More importantly, distorted hexahedral elements are created on the first layer of the advancing front

5 Y X Z Fig. 1. Sweep Tool on Rotatable Geometry via Linear Transformation Approach because there is not a smooth transition between the source and target meshes. In this example, the need for the original biasing of the source mesh to be preserved by the layer smoothing is obvious. The second difficulty occurs on volumes having non-convex or non-simply connected source surfaces. Layer smoothing is absolutely necessary on the example in Figure 3 because no linear transformation between the source and target loops exists. Project with Laplacian layer smoothing produces a bad target mesh in this example (note the compression of the quadrilaterals next to the inner boundary). The second difficulty arises in many other important sweepable geometries. Both difficulties are overcome by devising a robust smoother for Sweep Tool that takes mesh biasing into account and performs mesh copying/morphing 2. For example, the mesh in Figure 1 can be obtained with Sweep Tool, with or without layer smoothing. Sweep Tool smoothing also solves the diffulty illustrated in Figure 2. Fig- 2 Space limitations prevent discussion of the smoothing algorithm here. See [3] for a detailed presentation

6 X Z Y Fig. 2. Project Tool: Loss of Mesh Biasing due to Smoothing (Target on Right) ure 4 shows the result of sweeping with Sweep Tool layer smoothing on the example in Figure 3. Experience with a series of realistic problems has shown it is possible to sweep a wide variety of general two-and-one-half dimensional geometries with this smoother. If the curvature of a source surface is significantly different than the target surface, poor mesh quality may result even with layer smoothing. Curvature is somewhat preserved by layer smoothing if the bounding loop reflects the surface curvature but, in general, this is not the case. Research on methods to achieve good mesh quality for curved source and target surfaces is presently being pursued. The approach takes into account both source and target meshes. Sections 3 and 4 give two alternative approaches to placement of the interior nodes, one involving linear transformations and the other layer smoothing. Although linear transformations are less general than layer smoothing, they remain useful because, when they exist, it is computationally much faster to generate them than to perform layer smoothing. Fortunately, (7) gives a direct measure of whether or not such a transformation exists: if F = 0, then the linear transformation exists and smoothing need not be used because good quality is guaranteed, as shown in the following result. Proposition. Assume that a layer mesh has positive Jacobian at all mesh nodes. Let T be a linear transformation with positive determinant. Then T applied to this layer in the manner of (14) will result in another layer mesh which also has positive Jacobians at all of its nodes. Proof. For each quadrilateral of the mesh let U be the matrix [x 1 x 0,x 2 x 0,n] for some unit normal. We have T U = Ũ with the determinant of U positive by assumption. Then the determinant of Ũ is positive because it is the product of the

7 X Z Y Fig. 3. Project Tool: Poor Quality Target Mesh (right) Resulting from Laplacian Layer Smoothing determinant of two matrices which each have positive determinants. There are thus two criteria to determine whether or not smoothing is needed. If (i) F < ǫ where ǫ is some tolerance criterion and (ii) the linear transformation has positive determinant, then smoothing is not needed. Occasionally, layer smoothing will fail to produce adequate mesh quality even with the new smoother because, although the geometry and topology are two-andone-half dimensional, the mesh is fully three-dimensional. When layer smoothing fails, the situation can sometimes be salvaged by invoking the fully three-dimensional smoother as a post-processing procedure. 5. Additional Topics We briefly discuss in this section some Sweep topics not directly related to that of interior node placement. Poor quality meshes on the linking surfaces generally result in poor quality hexadrons generated by sweeping. It is therefore crucial to generate good meshes on the source and linking surfaces. Because surface meshing is outside the control of Sweep Tool, Sweep Tool checks the Jacobians of all the quadrilaterals on the linking surfaces before sweeping. If any Jacobians are negative, then sweep aborts because negative hexahedron Jacobians will surely result. Even when there are no bad quadrilaterals on the linking surfaces, one can still sweep out bad hexahedra due to problems with the linking surface meshes. Highly skewed meshes on the linking surface is a common cause of this problem. Sweep Tool contains a topology-based mesh-matching algorithm because in some

8 X ZY Fig. 4. Sweep Tool: Good Quality Target Mesh (right) Resulting from new Layer Smoothing Algorithm instances it is advantageous to have the target surface meshed prior to sweeping. If the target mesh topology is incompatible with the source mesh topology, the sweep is aborted, otherwise each quadrilateral on the last layer of the advancing front is matched with the appropriate quadrilateral of the target mesh. Multiple source surfaces are allowed in sweep, but multiple targets are not presently allowed. Mesh generation rates of up to 3000 hexahedrons per second have been generated with Sweep Tool on a 300 MHz workstation, compared to 1800 hexahdrons per second for Project Tool. It is possible to achieve this through careful memory-speed tradeoffs to eliminate quadratic-time calculations. 6. Summary and Conclusions. Sweep Tool provides a relatively fast and semi-automatic means of generating volume meshes on general two-and-one-half dimensional geometries. Interior point placement is crucial to the success of the algorithm. Two approaches for the placement of interior node points have been suggested that are robust and highly effective. The linear transformation approach is fast and gives high quality meshes on translatable, rotatable, and scalable geometries. The layer smoothing approach is slower but can give high quallity on more general geometries. Two criteria are given which form the basis for an automatic method for determining if smoothing is needed. Some geometries are border-line sweepable, i.e., even though the geometry is two-and-one-half dimensional, meshes on the linking surfaces conspire to make layer smoothing inadequate. Full 3D smoothing applied as a post-processing step may give

9 the desired mesh quality. Further development of Sweep Tool technology is underway to improve mesh quality in the case of highly curved source and target surfaces. Research on improved algorithms for sweeping to multiple targets is also in progress. Of course, not all geometries are two-and-one-half dimensional so research must continue on meshing the general problem. Acknowledgements The author would like to thank the members of the CUBIT team and the codes users at Sandia National Laboratories for many helpful suggestions. REFERENCES [1] T. Blacker, The Cooper Tool, Proceedings of the 5th International Meshing RoundTable, pp13-29, Pittsburg, Pennsylvania, October 10-11, [2] P. Knupp, On the invertibility of the isoparametric map, Comp. Meth. Appl. Mech. Engr., 78, p , [3] P. Knupp, Applications of Mesh Smoothing: Copy, Morph, and Sweep on Unstructured Quadrilateral Meshes, submitted for publication. [4] L. Mingwu, S. Benzley, G. Sjaardema, T. Tautges, A Multiple Source and Target Sweeping Method for Generating All Hexahedral Finite Element Meshes, Proceedings of the 5th International Meshing RoundTable, p , Pittsburg, Pennsylvania, October 10-11, [5] D. White, L. Mingwu, S. Benzley, G. Sjaardema, Automated Hexahedral Mesh Generation, Proceedings of the 4th International Meshing RoundTable, pp , October 16-17, Albuquerque, New Mexico, [6] David White, private communication.