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 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 :

with .

  
Figure: Linear basis function in three dimensions

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

If 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 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 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 is assumed to be equal to 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 is zero outside , we get:

Let us introduce the vertex-patch form factor :

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:

• Unknown parameters represent now vertex radiosities rather than patch radiosities. According to Euler's law, the number of vertices of a triangular faced polyhedron is half of the number of its faces plus two. Thus the size of the linear equation is almost the same as for the number of quadrilaterals used in the original method.
• There is no need for double integration and thus the linear approximation requires a simpler numerical integration to calculate the form factors than constant approximation.

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 . This integration can be avoided, however, if linear approximation of 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 . Let the values of the possible combinations of point and the vertices be respectively. A linear interpolation of the point-to-point form factor between and is:

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