Issue link: https://resources.system-analysis.cadence.com/i/1472078
Preparation of Geometry Models for Mesh Generation and CFD 5 www.cadence.com Discrete Geometry Geometry models in explicit surface meshes, independent of degree or element type, whether created initially for rendering, 3D printing, simulation, etc., are referred to as discrete boundary representations. Meshes Discrete representations model shapes with a definitive collection of points (i.e., discrete geometric locations). If the points are unconnected, the model is referred to as a point cloud. If the points are connected to form a wireframe of (usually) trian- gular facets, it is referred to often as a faceted model. The use of discrete geometry representations is not a new development. Mesh generation for CFD in the 1970s and 1980s relied predominantly on wireframe geometry models (faceted, 3D, quadrilateral surfaces). Yet it was the rise of 3D printing in the 1990s and the introduction of the STL file format [24] for 3D Systems' stereolithography printers popularized triangular facets as geometry models for CFD meshing. Extant meshes from prior simulations are sometimes the only geometry repre- sentation available after the original geometry model has been lost or forgotten. [It is worth noting an apparent paradox. The primary source of discrete boundary representation models is CAD software, from which a design in an analytic boundary representation is tessellated and exported to a discrete format. There is a loss of precision associated with replacing a continuous, analytical representation with a faceted one; the model's approximation of reality has gotten worse. This tradeoff can be explained simply and practically. A discrete representation will suffice if the downstream application (mesh generation or CFD flow solver) cannot work with analytic boundary representations. Furthermore, because a discrete model is basically a mesh, something with which mesh generators and CFD flow solvers are very familiar, there is a natural affinity to use that type of representation regardless of its relative drawbacks.] 3D Scans Discrete models can be produced by most CAD software (by tessellating an analytic model). Of course, there are other situa- tions where the source of discrete models are 3D scans and extant meshes. 3D scans of an object allow the as-built versus the as-designed object to be represented for simulation. 3D scans also capture an object in its loaded configuration such as the upward flex of an aircraft's wings in flight. [The difference between as-designed and as-flown geometry can be significant as in the case of an X-38 flight test vehicle that performed an unexpected 360-degree roll immediately upon being dropped from its B-52 mothership during a test flight. This particular flight vehicle, V-131-R, was found to have a 2.5-inch asymmetry in the fin root region that wasn' t present in the design. The resulting aerodynamic forces induced a roll that the flight controls weren' t programmed to counter [26]. Complementary to CAD software that converts an analytic representation to a discrete one, techniques and tools exist to do the opposite. The benefits of discrete to analytic conversion are a) to convert a 3D scan or mesh into a form that can be brought into the CAD software for design retention and update and b) convert the discrete form to something more amenable for use by the meshing or CFD software. Not surprisingly, the conversion of a discrete model to an analytic one is highly dependent on the accuracy of the discretization, thereby affecting its suitability for a particular purpose. Features in the original source model that were smaller than the characteristic edge length of the discrete model were not adequately captured and therefore, cannot be recovered by any reasonable extrapolation of the discrete model. [It is important to understand that analytic geometry has an effectively unlimited resolution, but discrete geometry is limited to the resolution of the point density used to describe the shape. In other words, you can evaluate a NURBS surface anywhere and get coordinates that lie on the surface, but when you evaluate a discrete surface, you get a shape defined by linear inter- polation between the known discrete points. https://www.pointwise.com/theconnector/2012-September/Analytic-vs- Discrete-Geometry.html.] Boundary Topology The modeling of complex shapes (e.g., the outer mold line (OML) of an aircraft, the underbody of a car) generally requires more than just geometry. Topology is a mapping of logical connections that unifies a collection of subsets of geometric entities into a whole, often called a solid model or just a solid. Boundary representation (B-Rep) topology defines bounded (aka limited) portions of the geometry by introducing regions (oriented, bounded portion of a volume), faces (oriented, bounded portion of a surface), edges (oriented, bounded portion of a curve), and vertices. The hierarchy of these 3-, 2- 1-, and 0-D primitives, respectively, form a complete partition of three-di- mensional space.