### Natural/Volumn coordinates in element matrix

#### by Yi Zhang

When evaluating element matrix, one needs perform integration over local element domain, during which coordinate transform usually happens. For example, for shape function where subscript is for local node index in a heat equation with constant conductivity, one does the integral during applying variational principle:

In a tetrahedron the natural/volumn coordinate can be described as:

where is the “standard” tetrahedron coordinate between which and the element a transform is constructed:

By this partial derivatives can be performed like:

where is the Jacobian matrix of the transformation , etc. In turn we have

and similar expressions for . This is equal to:

In summary, the integral in calculating element matrix can be performed always in the standard element :

For example, in linear element, the shape functions are exactly the same as natural coordinates: . Then we have:

while in quadratic element, as the shape function shown in last post, the gradient of against natural coordinates would be: