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Mesh Size Control

Three levels of mesh size specification are considered - the global mesh size specification, local mesh size specification, and adaptive mesh size specification. The global mesh size specification uses global weight functions to control the mesh size description over a domain. The local mesh size specification concept prescribes the desired mesh size variation on individual model entities. In the adaptive mesh size specification strategy, various mesh size sources (typically a background mesh) built according to the preceding problem analysis are used. Only the local mesh size control is directly related to the underlying model.

The local mesh size specification consists of two parts - the required mesh size specification and the curvature-based mesh size control. The former concept is used to explicitly prescribe the mesh size at individual model entities. The mesh size specification is stored at each vertex and at each control point of any curve or surface. These values are used to extract the mesh size specification at any location on a curve or surface. Moreover, each model entity (except vertices) stores an upper bound limit on the mesh size which is not allowed to be exceeded. Similar expressions to the ones describing the geometry of rational Bezier curves and surfaces are used to interpolate the local mesh size specification at control points over the curve or surface. The mesh size extracted from a curve mesh size specification has the form

(2.25)


where is the required mesh size at point on the curve, are the mesh size specifications at Bezier control points, and the other symbols have the same meaning as in Eq. (2.1). A similar formula can be written for the extraction of the required mesh size on a surface

(2.26)


where is the required mesh size at point on the surface and are the mesh size specifications at Bezier control points. The remaining variables have the the same meaning as in Eq. (2.3).

The curvature-based mesh size control is employed to enable an accurate representation of a curve or surface by its discretization even if no particular mesh size is required. The criterion is based on the ratio between the appropriate mesh size and the radius of curvature at a given location on the curve or surface. The radius of the first (flexural) curvature on a parameterized curve is given by

(2.27)


with

(2.28)


(2.29)


where is the positional vector of a point on the curve and stands for the length of curve arc. Particularly in the case of rational Bezier curves, the derivatives of with respect to may be written as

(2.30)


where

(2.31)
   

(2.32)
   

The principal radii of curvature and on a parameterized surface are given by the solution of the quadratic equation

(2.33)


Only the smaller radius of curvature is relevant for the curvature-based mesh size control. The symbols , , and , , represent coefficients of the first and second fundamental form of a surface and are given by

(2.34)


(2.35)


where

(2.36)


The unit normal vector of the surface can be evaluated as

(2.37)


In the actual implementation, it is more convenient to express , , and as

(2.38)


where

(2.39)


The derivatives of , in the case of rational Bezier surfaces, are given by

(2.40)


(2.41)


(2.42)


where

(2.43)
   

(2.44)
   

(2.45)
   
   

(2.46)
   
   

(2.47)
   
   
   
   

(2.48)
   
   
   
   

The derivatives of Bernstein polynomials in Eqs (2.32), (2.33), and (2.44) - (2.49) can be expressed recursively as

(2.49)


where

(2.50)




Next: Mesh Description Up: Model Description Previous: Geometrical Operations

Daniel Rypl
2005-12-07