Basic LOgical Bulk Shapes (BLOBs) for Finite Element Hexahedral Mesh Generation

Transcription

1 Basic LOgical Bulk Shapes (BLOBs) for Finite Element Hexahedral Mesh Generation Shang-Sheng Liu and Rajit Gadh Department of Mechanical Engineering University of Wisconsin - Madison Madison, Wisconsin research assistant, shang-sh@smartcad.me.wisc.edu, URL: assistant professor, gadh@me.engr.wisc.edu, URL: Abstract. This article presents an approach for automatically generating hexahedral meshes from solid models. The mesh generating method presented in this paper involves four major steps. First, objects called Basic LOgical Bulk shapes (BLOBs) are determined from the solid model of a given part. Second, these BLOBs are used to decompose the solid model into its various sub-volumes. Third, a multiple-block structure (MBS), which is a group of hexahedral objects, is constructed to approximate the solid model. Finally, transfinite mapping is employed to project the faces of the MBS onto the surfaces of a model to generate the finite element meshes. 1. Introduction Mesh generation has become an important research topic because finite element methods are used intensively in some applications, like heat transfer, fluid mechanics, structural analyses, etc. Finite element analysis (FEA) requires a finite element mesh as input. This input can be created using a variety of mesh generation methods. Some research has been done for hexahedral mesh generation, including: a grid-based approach[1], plastering[2], whisker weaving[3][4], etc. Some other research suggests that any given solid model can be decomposed into a number of simpler objects. Creating meshes for these simpler objects is easier than creating one for the entire model. ICEM AutoHexa[5] defines four primitives (blocks, cylinders, discs, and triangular prisms) and requires an input file which contains a description of the model with these primitives. However, it can be very time-consuming to create such a input file when the model is complex. In order to avoid creating the primitives manually like AutoHexa does, volume submapping[6] uses pseudo or virtual geometry to decompose complex volumes into mappable sub-volumes. Armstrong[7][8][9] et. al propose another method to partition the region of a solid model into subregions by using the medial-axis transform (MAT). In the present research, we propose an automatic hexahedral mesh generation approach by the use of the notion of Basic LOgical Bulk shapes (BLOBs). BLOB is the term we use to designate the simple objects of solid model decomposition. Figure 1 shows two typical kinds of BLOBs -- a protrusion and a depression. A BLOB is typically the result of the absence or presence of material on an existing surface. For example, in Figure 1 the protrusion is a BLOB that represents the presence of material; the depression is a BLOB that represents the absence of material. Both the protrusion and the depression are known as shape or form features in standard Computer-aided Design (CAD) terminology. The BLOB, therefore, represents an abstraction of a form feature as applied to finite element mesh generation. Figure 2 illustrates the decomposition of a solid model into its BLOBs using BLOB

2 determination and BLOB decomposition. BLOB determination marks the surfaces of the model into different BLOBs. BLOB decomposition separates the feature shapes from the original model as independent volumes. protrusion depression Figure 1: Protrusion and depression BLOBs BLOB Determination BLOB Decomposition solid model BLOBs Figure 2: Solid model decomposition into simpler shapes via BLOB determination and BLOB decomposition In the present research, we use the notion of BLOB to decompose an object into subregions. Once the BLOBs of an object are determined, a number of hexahedral blocks are created to approximate the geometry of the object. The uniform hexahedral mesh is generated by applying transfinite mapping to the geometry of the blocks and the object. 2. Present research In the research presented in this paper, we use decomposition, a process which allows us to determine and then separate the BLOBs of a solid model. There are four basic steps to our mesh generation approach, illustrated in Figure 3. Hexahedral elements for solid models are produced by: 1) BLOB determination, 2) BLOB decomposition, 3) MBS (multiple-block structure) construction, and 4) transfinite mapping. Step 1 (Figure 3 (i) and (ii)) determines the BLOBs of the solid model; step 2 (Figure 3 (iii)) decomposes it into its BLOBs; in step 3 (Figure 3 (iv)), a group of hexahedral blocks, which approximate the geometry of the BLOB, are constructed for each BLOB; in step 4 (Figure 3 (v)), the faces of the BLOB MBSs are mapped to the surfaces of the solid model to generate the finite element meshes.

3 solid model (i) BLOB determination (ii) BLOB decomposition multiple-block structure construction transfinite mapping FEA (iii) (iv) (v) 2.1 BLOB determination Figure 3: The diagram for mesh generation approach The present approach to automatic mesh generation utilizes the notion of the BLOB, which allows determination of the essential characteristics of a shape in terms of protrusions and depressions. In general, determining protrusions and depressions is a difficult problem, especially when geometry and topology are combined. (Geometry refers to the shape of a surface; topology to the relationships between different surfaces, edges, and vertices of a boundary representation (BREP) solid model.) Prismatic BLOBs such as protrusions and depressions (Figure 1) can be classified using entities called C- loops (Convex or Concave loops) for different shapes. C-loops, a set of convex or concave edges, are used to define four basic types of features: protrusions, blind depressions, through depressions, and bridges. These C-loop definitions of features types are used as the basis of BLOB determination. Prior to the discussion of BLOB determination, it is necessary to define the C-loop and associated terms. Angle of an Edge: The angle between the two surfaces of an edge as measured from inside the solid. In Figure 4, θ 1 is the angle of Edge 1, measured between Face 1 and Face 2 from inside the solid. Similarly, θ 2 is the angle of Edge 2, measured between Face 2 and Face 3 from inside the solid. Convex Edge: An edge whose angle is less than 180 degrees. In Figure 4, Edge 1 is a convex edge. Concave Edge: An edge whose angle is greater than 180 degrees. In Figure 4, Edge 2 is a concave edge.

4 Edge 2 Edge 1 Face 2 Face 3 Face 1 θ 1 (Angle of Edge 1) < 180 o θ 2 (Angle of Edge 2) > 180 o Edge 1 is a convex edge Edge 2 is a concave edge θ 1 < 180 o θ 2 > 180 o Model A Cross section of Model A Figure 4: Angles of convex and concave edges C-loop: A C-loop is a closed, connected set of edges in which the edges are either all convex or all concave. Thus, a C-loop is either a convex C-loop or a concave C-loop. Figure 5 shows some examples of convex and concave C-loops. or or Convex C-loops Concave C-loops Figure 5: Convex and concave C-loops C-loops have been used to define four basic classes of BLOBs[10][11], all shown in Figure Protrusion: encloses material and consists of one concave C-loop. 2. Blind Depression: does not enclose material and consists of one convex C-loop. 3. Through Depression: does not enclose material and consists of two convex C-loops. 4. Bridge: encloses material and consists of two concave C-loops. C-loops provide a method to represent a class of BLOBs in a uniform manner even though they may have different shapes (e.g. the two protrusion BLOBs in Figure 6). To determine BLOBs, the boundaries of BLOBs must be decided. Table 1 shows the boundaries of different BLOB types. The boundary surfaces of every BLOB are also shown in Figure 6.

5 Bridge Through Depression Protrusion Blind Depression or or Convex C-loops Concave C-loops outer surface inner surface surface between two concave C-loops surface between two convex C-loops Figure 6: Four C-loop-based BLOB classes BLOB type Boundaries Protrusion the concave C-loop and the outer surfaces Blind Depression the convex C-loop and the inner surfaces Through Depression the two convex C-loops and the surfaces in between Bridge the two concave C-loops and the surfaces in between 2.2 BLOB decomposition Table 1: The boundaries of BLOBs The BLOBs determined in the previous section can be classified as positive volume BLOBs (protrusions and bridges) and negative volume BLOBs (blind depressions and through depressions). When positive volume BLOBs are determined, their volumes need to be separated from the solid model. The surfaces used to separate the positive volume BLOBs are called cutting surfaces. The cutting surfaces can be determined by covering BLOBboundary concave C-loops with surfaces. Figure 7 illustrates the BLOB decomposition for a protrusion BLOB. In Figure 7 (i), the solid model s BLOBs are shown (the protrusion is shaded). Figure 7 (ii) shows the concave C-loop for the protrusion. In Figure 7 (iii), the cutting surface for the concave C-loop is determined by surfacing the concave C-loop. Figure 7 (iv) shows the cutting surfaces being inserted in the solid model, with the result that its BLOBs become separated.

6 BLOB solid model C-loop cutting surfaces (i) (ii) (iii) (iv) concave C-loop 2.3 MBS construction Figure 7: An example of BLOB decomposition Once an object is decomposed into BLOBs (like the one shown in Figure 9) each BLOB needs to be mapped to a collection of hexahedral objects. This collection of hexahedral objects, synthesized from a BLOB, is called a Multiple Block Structure (MBS). The objective of MBS construction is to determine a group of hexahedral objects which approximates the geometry of the solid model s part in such a way that the surfaces of the hexahedral object are as close as possible to a square (in terms of the angles of the polygonal surfaces). Since BLOBs are simpler than the entire solid model, an MBS is created for every BLOB. The final MBS for the entire part can be produced by connecting the MBSs of all the BLOBs. (The connection of BLOBs MBSs is discussed later in this section.) Based on previous mesh generation research[12], we define five simple objects called basic elements and their corresponding MBSs (Figure 8). More basic elements can be defined for other new shapes if necessary, e.g. when the current five basic elements cannot cover the geometry of an object. The basic elements MBSs in the geometric domain is also shown in Figure 8. An example of determining the geometric domain of a tetrahedral element is shown in Figure 9. First, some faces of the MBS (indicated using bi-directional arrows) are glued together as shown in Figure 9 (i) with the result that the topology of the MBS is equivalent to the topology needed to form the geometrically mapped structure for the tetrahedral element. Second, the faces labeled f 1 of the MBS in Figure 9 (ii) are projected to the face labeled f 2 of the tetrahedral element (Figure 9 (ii-a)). The MBS in the geometric domain is determined by applying a similar projection from the other faces of the original MBS to the corresponding faces of the tetrahedral element (Figure 9 (ii-b).

7 basic element: MBS: geometric domain: (i) (ii) (iii) (iv) (v) Figure 8: Basic elements and their corresponding MBSs and meshes f 1 f 2 f 1 f1 tetrahedral element MBS (a) (b) (c) (i) (a) the projection from faces f 1 to face f 2 (ii) (b) MBS in geometric domain Figure 9 (i): Glueing the faces of MBS. (ii): The projection from the faces of MBS to the corresponding faces of a tetrahedral element If a BLOB corresponds to one of the basic elements defined earlier, its corresponding MBS is obtained by the mapping shown in Figure 8. If a BLOB is not one of the basic elements, an additional step is required, in which the BLOB volumes are broken down into basic elements. The MBSs of such BLOBs can be determined by combining the MBSs of the basic elements out of which they are made. The algorithm for decomposing BLOBs into basic elements is limited to swept BLOBs. A swept BLOB is one which can be represented by sweeping a surface (called a swept surface) along a path. The sweeping surface can be modified slightly during the sweeping as long as the topology of the sweeping surface remains unchanged. Two examples of swept BLOBs are shown in Figure 10. Both BLOBs can be represented by sweeping Surface 1 along a

8 path from Point 1 to Point 2. The path can be linear (Figure 10 (i)) or nonlinear (Figure 10(ii)). For a swept BLOB, the sweeping surface can be decomposed into quadrilateral elements[13] and the MBS can be created by extruding the quadrilateral elements (Figure 10 (iii)). Surface 1 Point 1 Surface 1 Point 1 quadrilateral elements Path 1 Point 2 Path 2 Point 2 MBS (i) (ii) (iii) Figure 10: The creation of MBS of BLOBs which can be represented by sweeping a surface along a path Figure 11 illustrates an example of the MBS construction for a solid model. Here, two BLOBs are found by BLOB determination (Figure 11 (I)). Subsequently, BLOB decomposition is performed (Figure 11 (ii)). Then, MBSs are constructed for these two BLOBs (Figure 11 (iii)). Finally, the final MBS of the solid model is produced by connecting the MBSs of the BLOBs. A BLOB s MBS construction sometimes depends on the other BLOBs MBSs. For example, in Figure 11 the BLOB 2 s MBS depends on the BLOB 1 s MBS. In order to understand the relation between BLOBs, we must first discuss the order of MBS construction. BLOB 1 BLOB 2 MBS MBS connection (i) solid model (ii) BLOBs (iii) MBSs for BLOBs (iv) final MBS Figure 11: Example of MBS construction for a solid model Although BLOB volumes need to be separated from the object, the MBSs of two adjacent BLOBs are not independent; they have to match the MBS of adjacent BLOBs along the boundary between these two MBSs. Therefore, the order of creating the MBSs of BLOBs becomes very important. Figure 12 and Figure 13 show the importance of the order in which the BLOB MBSs for a part (with three BLOBs: B1, B2, and B3) are created. The order for determining the MBSs of BLOBs shown in Figure 12 is B1, B2, and B3. The creation or modification of the MBSs is indicated in every step with the mark. The MBSs of B1, B2, and B3 are created in Step 1, Step 2, and Step 4, respectively. However, the table in Figure 12 shows that Step 3 constitutes a modification of the MBS of B1 due to the creation of the MBS of B2 (that was done in Step 2). In step 4 the MBS of B3 is created, which results in the modification of the MBS of B2, shown in Step 5. Due to the

9 modification of the MBS of B2 in Step 5, the MBS of B1 must be modified once again (shown in Step 6). The MBS of B1 is modified twice due to the creation and modification of B2 s MBS. However, the number of steps can be reduced by using a different sequence for determining the MBSs. Figure 13 shows the MBS construction of the same part with a different sequence: B3, B2, B1. In this case, the MBS does not need modification after creation. As a result, only three steps are required. It is, therefore, important to order the BLOBs before creating the MBS. B3 B2 B1 B3 B2 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 creation creation modification B1 creation modification modification Figure 12: The creation of the MBS of a part with the order: B1, B2, and B3 B3 B2 Step 1 Step 2 Step 3 creation B1 creation creation Figure 13: The creation of the MBS of a part with the order: B3, B2, and B1 For the purpose of determining the construction sequence of the BLOBs MBSs, BLOBs are classified as child-blobs and parent-blobs. If BLOB A is on the surface of BLOB B, BLOB A is the child-blob of BLOB B and BLOB B is the parent-blob of BLOB A. A BLOB without a parent-blob is a root-blob; a BLOB without a child-blob is a leaf-blob.

10 Figure 14 provides an example of the relationship between BLOBs. BLOB 1 is the parent-blob of BLOB 2 and BLOB 3. BLOB 3 is the parent-blob of BLOB 4. BLOB 1 is root-blob. BLOB 2 and BLOB 4 are leaf- BLOBs. An acyclic digraph is the most appropriate structure to represent the BLOB relationships. Figure 15 shows the acyclic digraph for the model in Figure 14. For the acyclic digraph, the arrow-head points from child-blobs to parent-blobs. BLOB 1 BLOB 2 BLOB 3 BLOB 4 BLOB 1 is the parent-blob of BLOB 2 and BLOB 3. BLOB 3 is the parent-blob of BLOB 4. BLOB 1 is a root-blob. BLOB 2 and BLOB 4 are leaf-blobs. Figure 14: The relationship between BLOBs BLOB 1 BLOB 2 BLOB 3 root-blob BLOB 4 leaf-blob Figure 15: The acyclic digraph for the model in Figure 14 Since a child-blob is always on the surface of its parent-blob(s), the MBSs of a child-blob will always affect the MBS(s) of its parent-blob(s). Sometimes, the MBSs of a parent-blob will affect the MBS(s) of its child-blob(s). Figure 16 illustrates an example of such a case. Figure 16 (i) shows an object with one parent- BLOB and one child-blob. In Figure 16 (ii), the original child-blob s MBS contains only one block and the parent-blob s MBS contains twelve blocks. Since the child-blob s MBS is adjacent to two blocks of its parent- BLOB s MBS, the child-blob s MBS is changed to two blocks. In general, the MBSs of leaf-blobs are created first, followed by the MBSs of their parent-blob(s); the MBS of the root-blobs are constructed last. The MBSs of child-blobs should be modified as they are affected by the MBSs of their parent-blobs.

11 child-blob parent-blob (i) MBS new MBS child-blob affects MBS parent-blob (ii) 2.4 Transfinite mapping Figure 16: A parent-blob s MBS affects its child-blob s MBS Once the MBS has been constructed, a method called transfinite mapping can be applied to generate hexahedral elements for every block. The transfinite mapping method maps a unit square (cube) into a quadrilateral (hexahedral) region in 2D (3D). Figure 17 shows an example of the mapping method in 3D. The transfinite mapping method allows the edges of the region to be any curve[14][15][16]. The same idea can be applied to 2D or 3D mapping. unit cube object mesh Figure 17: Mapping method: The mesh is generated by mapping a unit cube to the object The transfinite mapping method has been very popular in commercial mesh generators because of the simplicity with which it can be implemented and its capacity to produce high quality meshes[9]. Figure 18 gives an example of mesh generation for a cylinder. The MBS of the cylinder can be represented by five blocks (Figure 18 (ii-a)). Subsequently the side open-faces in Figure 18 (ii-a) are glued together to form the part shown in Figure 18 (ii-b). Finally, the four faces (the faces labeled f in Figure 18 (ii-b)) of the blocks are projected onto the surface of the cylinder (Figure 18 (ii-c)), and the final mesh is created by transfinite mapping (Figure 18 (ii-d)).

12 (i) f f f f (a) (b) (c) (d) (ii) Figure 18: The mesh generation for a cylinder 3. Example of hexahedral mesh generation This section provides an example of a hexahedral mesh for a part with five BLOBs (Figure 19(i)). The MBS is also shown in Figure 19 (ii). (i) Hexahedral mesh (ii) MBS Figure 19: A part s hexahedral mesh and MBS 4. Discussion In this paper, we propose an automatic hexahedral mesh generation approach for solid models. The hexahedral meshes can then be used to perform finite element analyses. Complex models are decomposed into a number of simple sub-objects called BLOBs prior to mesh generation. Creating meshes for these BLOBs is easier than creating one for the entire model. The concept of multiple-block structure is introduced in our approach. A block is a hexahedral object. When a number of hexahedral blocks can be created to approximate the geometry of the object, an uniform hexahedral mesh can be generated by applying transfinite mapping to these blocks. One potential problem with this approach lies in its inability to handle BLOBs that cannot be defined by C- loops. In such cases, the information about concave edges become very important. While BLOB decomposition for

13 BLOBs representable with C-loops has been extensively developed, an approach that uses information regarding concave edges to determine BLOBs needs to be researched further. In addition, while the current proposal addresses BLOBs that are prismatic or approximately prismatic, another area for future research is developing a method for meshing more complex BLOB shapes. This may involve decomposing the complex BLOB shapes into sub-blobs that are simple in shape or determining completely new meshing techniques for each class of BLOB shape. 5. Summary The present research proposes a systematic approach to generating a hexahedral element from solid models of part designs. We use the notion of the BLOB (an abstraction of the shape of the part), which is determined using the concept of C-loops to decompose a solid model into simple shapes. The multiple-block structure is then created using these simple shapes. Finally, a hexahedral mesh is generated by applying transfinite mapping to the blocks of the multiple-block structure. Acknowledgements The authors gratefully acknowledge the financial support received from ALCOA Technical Center and the software support received from XYZ Scientific Applications, Inc. References [1] Schneiders, R. A grid-based algorithm for the generation of hexahedral element meshes submitted for publication in Engineering with Computer (1995) [2] Blacker, T. D. and Meyers, R. J. Seams and Wedges in Plastering: A 3-D Hexahedral Mesh Generation Algorithm Engineering with Computers Vol. 9 pp (1993) [3] Tautges, T. J., Blacker, T., and Mitchell, S. A. The Whisker Weaving Algorithm: A Connectivity-Based Method for Constructing All-Hexahedral Finite Element Meshes submitted to Int. J. Num. Meth. Eng. (1995) [4] Tautges, T. J. and Mitchell, S. Whisker Weaving: Invalid Connectivity Resolution and Primal Construction Algorithm 4th International Meshing Roundtable, pp (Albuquerque, NM, Oct , 1995) [5] Hohmeyer, M. E. and Christopher, W. Fully-Automatic Object-Based Generation of Hexahedral Meshes 4th International Meshing Roundtable, pp (Albuquerque, NM, Oct , 1995) [6] White, D. R., Lai, M., Benzley, S. E., and Sjaardema, G. D. Automated Hexahedral Mesh Generation by Virtual Decomposition 4th International Meshing Roundtable, pp (Albuquerque, NM, Oct , 1995) [7] Armstrong, C. G., Tam, T. K. H., Robinson, D. J., McKeag, R. M., and Price, M. A. Automatic Generation of Well Structured Meshes using Medial Axis and Surface Subdivision ASME Design Automation Conf. Vol. 2 (Miami, USA, Sept. 1991) [8] Li, T. S., McKeag, R. M., and Armstrong, C. G. Hexahedral Meshing using Midpoint Subdivision and Integer Programming Computer Meth. App. Mech. Eng., to appear (1994) [9] Li, T. S. Armstrong, C. G., and McKeag, R. M. Quad Mesh Generation for k-sided Faces and Hex Mesh Generation for Trivalent Polyhedra Technical report of The Queen s University of Belfast, Northern Ireland. (1995) [10] Gadh, R. and Prinz, F. B. Reduction of Geometric Forms Using the Differential Depth Filter Computer-Aided Design Butterworth-Heinemann Publishers Vol. 24 No. 11 pp (November 1992) [11] Gadh, R. and Prinz, F. B. A Computationally Efficient Approach to Feature Abstraction in Design- Manufacturing Integration Journal of Engineering for Industry Vol. 17 No. 1 pp (January 1995) [12] A Tutorial by XYZ Scientific Applications, Inc. [13] Liu, S.-S. and Gadh, R. Automatic Quadrilateral Meshing in 2D regions Technical Report #I-CARVE ,I-CARVE Lab, University of Wisconsin-Madison (1996)

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