Fidelity Pointwise Remedy Drawbacks Working with Analytic and Discrete Geometry Types

Most engineers are of the view that all mesh generators use an underlying geometry that is discrete in nature, but in fact, Fidelity Pointwise can import and mesh both analytic and discrete geometry. Analytic geometry defines curves and surfaces with mathematical functions. These functions allow users to retrieve specific points in space. Non-uniform rational basis spline (NURBS) curves and surfaces form the foundation for the most common analytic geometry representations. In contrast, discrete geometry (also known as faceted geometry) describes a shape as a mesh, with discrete points usually connected to form triangles.

An Analogy to Geometry Representation

Using digital image representation, images can be described analytically with vectors (PostScript) or discretely with raster graphics (JPEG, PNG). The resolution of a vector-based image does not change with magnification, whereas a raster graphics image has a fixed resolution. At normal magnification, the vector version and raster version of the bottle on the left of Figure 1 look the same. Magnifying a corner of the label to 7x clearly illustrates the differences between the image formats.

Figure 1. The differences between vector and raster image formats are a good analogy for the differences between analytic and discrete geometry.

Pros and Cons of the Two Geometry Types

Analytic Geometry

Pros:

• The precision of analytic geometry is only constrained by the precision of the CAD system that generated it. This makes analytic description the ideal candidate for design-to-manufacturing processes, as every step in the process may require different precision.

• The mathematical description allows enormous control of the shape. The NURBS definition enables a great deal of flexibility by only defining the location of the boundary points and using control points with slope definition to define the internal shape of curves and surfaces. The NURBS curve illustrates this concept where the curve shape is not explicitly defined by the four internal points but by slope definition.

Cons:

• Analytic geometries can be computationally intensive to process and modify.
• There are many different formats, typically specific to the CAD package that created them, and translating among formats can be error-prone and imprecise.
• Topology introduces additional drawbacks related to the tolerance with which curves and surfaces are defined to be adjacent.

 Discrete Geometry

Pros:

• The main benefit of discretely defining geometry is speed. This representation is used for video game and movie rendering, where the surfaces are represented as triangles to which colors, lighting, and textures are applied, enabling the photorealistic look to which we are accustomed.

Cons:

• Having a fixed resolution can be limiting for design and manufacturing. For example, if an insufficient number of points are used to define a fillet when milling a piece of metal, the fillet will not look round but may have a crinkly look. As fillets are commonly used to reduce stress concentrations in structurally important components, introducing sharp corners would negate the benefit of the fillet.
• Can be difficult to manipulate because the surface relationships and topology are not recovered during reverse engineering and are lost when converting from analytic representations.

Figure 2. Feature recovery is difficult on the Stanford bunny, a common 3D discrete test geometry= represented as a single faceted surface.

How Does Fidelity Pointwise Handle the Drawbacks of Working with Both Discrete and Analytic Geometries?

Analytic geometry is imported using many common CAD formats, such as IGES, STEP, Parasolid, ACIS, CATIA V4/V5, Pro/Engineer, NX, and SolidWorks. While, discrete geometry is imported using the STL and VRML formats, as well as a few additional mesh formats. Fidelity Pointwise meshing software has tools that can help remedy the drawbacks of working with both geometry types.

Pointwise can read a variety of open and proprietary analytic CAD formats. Beyond the native CAD import, Pointwise utilizes a geometry kernel that is customized for the needs of CAE geometry analysis. The customized geometry kernel is designed to be both fast and efficient for analytic geometry manipulations. In addition, Pointwise's solid meshing suite of tools that includes modeling and quilting helps users work with analytic geometry more easily. The modeling feature can not only repair gaps and overlaps in imported models but also can create models where there were none to begin with. 

Pointwise provides a feature extraction tool for recovering topology from discrete geometry. This tool depends on the relative turning angle between adjacent facets. The second and perhaps, more notable drawback is that because the resolution of the geometry is fixed, under-resolved regions can pose problems with no recourse but to return to whatever method was used to create the original geometry in an attempt to increase its resolution. If the analyst had no ability to improve the resolution of the discrete geometry, a potential solution might be to create high-order interpolation between the discrete points instead of a linear one.

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To learn more about the differences between discrete and analytic geometry types, read the article Analytic vs. Discrete Geometry by clicking the button below.