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Linear finite element techniques


Let us decompose the surface into planar triangles and assume that the radiosity variation is linear on these triangles. Thus, each vertex i of the triangle mesh will correspond to a ``tent shaped'' basis function tex2html_wrap_inline2630 that is 1 at this vertex and linearly decreases to 0 on the triangles incident to this vertex.

Placing the center of the coordinate system into vertex i, the position vector of points on an incident triangle can be expressed by a linear combination of the edge vectors tex2html_wrap_inline2677 :


with tex2html_wrap_inline2679 .

Figure: Linear basis function in three dimensions

Thus, the surface integral of some function F on a triangle can be written as follows:



If tex2html_wrap_inline2685 is a polynomial function, then its surface integration can be determined in closed form by this formula.

The basis function which is linearly decreasing on the triangles can be conveniently expressed by tex2html_wrap_inline2687 coordinates:


where k, k' and k'' are the three vertices of the triangle.

Let us consider the general equation (equation 1.79) defining the weights of basis functions; that is the radiosities at triangle vertices for linear finite elements. Although its integrals can be evaluated directly, it is worth examining whether further simplification is possible. Equation 1.79 can also be written as follows:


The term enclosed in brackets is a piecewise linear expression according to our assumption if tex2html_wrap_inline2697 is also linear. The integration of the product of this expression and any linear basis function is zero. That is possible if the term in brackets is constantly zero, thus an equivalent system of linear equations can be derived by requiring the closed term to be zero in each vertex k (this implies that the function will be zero everywhere because of linearity):


As in the case of piecewise constant approximation, the diffuse coefficient tex2html_wrap_inline2514 is assumed to be equal to tex2html_wrap_inline2659 at vertex k, and using the definitions of the normalized radiosities we can conclude that:


Substituting this into equation 1.90 and taking into account that tex2html_wrap_inline2630 is zero outside tex2html_wrap_inline1826 , we get:


Let us introduce the vertex-patch form factor tex2html_wrap_inline2711 :


If the diffuse coefficient can be assumed to be (approximately) constant on the triangles adjacent to vertex i, then:


The linear equation of the vertex radiosities is then:


This is almost the same as the linear equation describing the piecewise constant approximation (equation 1.85), except that:

The vertex-patch form factor can be evaluated by the techniques developed for patch-to-patch form factors taking account also the linear variation due to tex2html_wrap_inline2630 . This integration can be avoided, however, if linear approximation of tex2html_wrap_inline2719 is acceptable. One way of achieving this is to select the subdivision criterion of surfaces into triangles accordingly.

A linear approximation can be based on point-to-point form factors between vertex k and the vertices of triangle tex2html_wrap_inline2723 . Let the tex2html_wrap_inline2725 values of the possible combinations of point tex2html_wrap_inline2727 and the vertices be tex2html_wrap_inline2729 respectively. A linear interpolation of the point-to-point form factor between tex2html_wrap_inline2727 and tex2html_wrap_inline2733 is:


Using this assumption the surface integral defining tex2html_wrap_inline2711 can be expressed in closed form.

next up previous index
Next: Global element approach - Up: Higher order radiosity approximation Previous: Piecewise constant radiosity approximation

Szirmay-Kalos Laszlo
Mon Oct 21 14:07:41 METDST 1996