Interpolation on a Triangular/Quad Facet
The r,s coordinates of a triangular facet can be determined from the use
of interpolation (shape) functions. Let
uivi=(1−r−s)u1+ru2+su3=(1−r−s)v1+rv2+sv3 These can be rewritten as
(ui−u1)(vi−v1)=(u2−u1)r+(u3−u1)s=(v2−v1)r+(v3−v1)s and then
rs=(v3−v1)(u2−u1)−(u3−u1)(v2−v1)(v3−v1)(ui−u1)−(u3−u1)(vi−v1)=(u3−u1)(ui−u1)−(u2−u1)r Then
fi=(1−r−s)f1+rf2+sf3 On a quadrilateral facet using the interpolation functions gives
uivi=41(1−r)(1−s)u1+41(1+r)(1−s)u2+41(1+r)(1+s)u3+41(1−r)(1+s)u4=41(1−r)(1−s)v1+41(1+r)(1−s)v2+41(1+r)(1+s)v3+41(1−r)(1+s)v4 these can be rewritten
uivi=ua+ubr+ucs+udrs=va+vbr+vcs+vdrs Using the first of these gives
s=(uc+udr)((ui−ua)−ubr) which can be substituted into the second to give
vi=va+vbr+(vc+vdr)(uc+udr)((ui−ua)−ubr) This can then be solved for r & s and then the interpolation function
used as for the triangular facet.