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Interpolation on a Triangular/Quad Facet

The r,sr,s coordinates of a triangular facet can be determined from the use of interpolation (shape) functions. Let

ui=(1rs)u1+ru2+su3vi=(1rs)v1+rv2+sv3\begin{aligned}u_{i} &= (1 - r - s)u_{1} + ru_{2} + su_{3} &\\v_{i} &= (1 - r - s)v_{1} + rv_{2} + sv_{3}\end{aligned}

These can be rewritten as

(uiu1)=(u2u1)r+(u3u1)s(viv1)=(v2v1)r+(v3v1)s\begin{aligned}\left( u_{i} - u_{1} \right) &= \left( u_{2} - u_{1} \right)r + \left( u_{3} - u_{1} \right)s &\\\left( v_{i} - v_{1} \right) &= \left( v_{2} - v_{1} \right)r + \left( v_{3} - v_{1} \right)s\end{aligned}

and then

r=(v3v1)(uiu1)(u3u1)(viv1)(v3v1)(u2u1)(u3u1)(v2v1)s=(uiu1)(u2u1)r(u3u1)\begin{aligned}r &= \frac{\left( v_{3} - v_{1} \right)\left( u_{i} - u_{1} \right) - \left( u_{3} - u_{1} \right)\left( v_{i} - v_{1} \right)}{\left( v_{3} - v_{1} \right)\left( u_{2} - u_{1} \right) - \left( u_{3} - u_{1} \right)\left( v_{2} - v_{1} \right)} &\\s &= \frac{\left( u_{i} - u_{1} \right) - \left( u_{2} - u_{1} \right)r}{\left( u_{3} - u_{1} \right)}\end{aligned}

Then

fi=(1rs)f1+rf2+sf3f_{i} = (1 - r - s)f_{1} + rf_{2} + sf_{3}

On a quadrilateral facet using the interpolation functions gives

ui=14(1r)(1s)u1+14(1+r)(1s)u2+14(1+r)(1+s)u3+14(1r)(1+s)u4vi=14(1r)(1s)v1+14(1+r)(1s)v2+14(1+r)(1+s)v3+14(1r)(1+s)v4\begin{aligned}u_{i} &= \frac{1}{4}(1 - r)(1 - s)u_{1} + \frac{1}{4}(1 + r)(1 - s)u_{2} + \frac{1}{4}(1 + r)(1 + s)u_{3} + \frac{1}{4}(1 - r)(1 + s)u_{4} &\\v_{i} &= \frac{1}{4}(1 - r)(1 - s)v_{1} + \frac{1}{4}(1 + r)(1 - s)v_{2} + \frac{1}{4}(1 + r)(1 + s)v_{3} + \frac{1}{4}(1 - r)(1 + s)v_{4}\end{aligned}

these can be rewritten

ui=ua+ubr+ucs+udrsvi=va+vbr+vcs+vdrs\begin{aligned}u_{i} &= u_{a} + u_{b}r + u_{c}s + u_{d}rs &\\v_{i} &= v_{a} + v_{b}r + v_{c}s + v_{d}rs\end{aligned}

Using the first of these gives

s=((uiua)ubr)(uc+udr)s = \frac{\left( \left( u_{i} - u_{a} \right) - u_{b}r \right)}{\left( u_{c} + u_{d}r \right)}

which can be substituted into the second to give

vi=va+vbr+(vc+vdr)((uiua)ubr)(uc+udr)v_{i} = v_{a} + v_{b}r + \left( v_{c} + v_{d}r \right)\frac{\left( \left( u_{i} - u_{a} \right) - u_{b}r \right)}{\left( u_{c} + u_{d}r \right)}

This can then be solved for rr & ss and then the interpolation function used as for the triangular facet.