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Preparation of Geometry Models for Mesh Generation and CFD 6 www.cadence.com The bounding (limiting) of surfaces in a B-Rep topology is accomplished through trimming. This operation imprints curve(s) into a surface's parametric space and limits the surface to the portion of the parametric space bounded by that curve and others (aka, the trimming loop). The resulting trimmed surface has a non-rectangular parametric space which affords it great flexibility in modeling complex shapes. Surface-surface intersection curves (Figure 7) are essential to using B-Rep topology when modeling complex objects. For comprehensive insight into how geometry models are trimmed, see Marussig & Hughes [27]. Volumetric Representation Volumetric representation describes a solid object explicitly from a series of solid, space-filling primitives. The primitives can be overlapping or non-overlapping. Constructive Solid Geometry Constructive Solid Geometry (CSG), for example, uses overlapping primitives, whereas explicit volume meshes, and voxel or spatial occupancy representations use non-overlapping primitives (i.e., the mesh cells). CSG is the method of taking solid primitives and combining them hierarchically using the standard Boolean operations: union, intersection, and difference. The individual solid primitives are typically trivial shapes such as spheres and blocks, but can be arbitrarily complex. The key here is that the volume of space occupied by the final solid is implicitly defined, and can be queried using ray casting, point sampling, or other methods. CSG models can be converted into B-Rep form by converting each primitive into its boundary representation, and repeatedly performing explicit Boolean operations until the final boundary representation is realized. Spatial Occupancy Modeling a shape with a spatial occupancy technique involves "digitizing" the region of interest into pixels (2D) or voxels (3D). The relative properties of adjacent pixels define boundaries within the region. X-ray, MRI, and CT scans are examples of spatial occupancy models of the human body. Many techniques and tools are available to convert the pixels or voxels to geometry. Figure 4: Example of a geometry model (B) of a cerebral aneurysm derived from a 3D rotational angiograph (A). Image from Castro et al. [25]. Image used with the permission of the author. Note that spatial occupancy can also refer to how a CFD flow solver discretizes the flowfield. As an alternative to body-con- forming meshes, a spatial occupancy technique can flood the region of interest with a 3D mesh (typically a Cartesian aligned mesh with adaptive refinement) and identify the geometric boundaries by an in-out test for each cell in the mesh. There are a variety of techniques to handle the mesh cells that intersect the geometry. Regardless, a geometry model of some represen- tation – analytic or discrete - must exist for this spatial occupancy technique to work. Implicit Modeling Implicit geometry modeling is a technique that avoids the computational complexity of B-Rep models in which the compo- nents are explicitly defined. As an explicitly defined geometry model becomes more realistic, the computational load on the modeling software becomes increasingly heavy and potentially more fragile due to the calculations required to trim and assemble a patchwork of surfaces into a solid. As the name implies, implicit modeling defines a shape by an implicit function that evaluates to zero on the shape's surface, a negative value on its interior, and a positive value elsewhere (Figure 5). Similar to CSG, multiple shapes can be combined in an implicit model. The result is that the object is defined simply by the minimum value of each shape's defining implicit function. In addition to the ease in evaluating whether a point is on the model's surface, computing intersections and making shape changes are easier than similar operations on B-Reps. A brief introduction to implicit modeling is available in [28].