The l.h.s integral I can be approximated by the Q - point Gauss quadrature ^{[1,}^{2,}^{3,}^{4]}
1 | 1 | Q | ||||
I = | ∫ | dL_{1} | ∫ | dL_{2} |det J| f(L_{1}, L_{2}, L_{3}) ∼ | ∑ | f_{q}(L_{1}, L_{2}, L_{3})W^{q} |
0 | 0 | q=1 |
where W_{q} denotes the weights for q - points of the numerical integration, and can be found in the Table 5.3 in^{[1]}. As it was already said, a set of N_{k}(L_{1}, L_{2}, L_{3}) shape functions where k = 1, 2, 3 can be used to evaluate each f function in the interpolation series which, for instance, in the highest order 10 – nodal cubic triangular element has the following form
3 | 9 | ||||
f(L_{1}, L_{2}, L_{3}) = | ∑ | N_{k}(L_{1}, L_{2}, L_{3})f^{k} + | ∑ | N_{k}(L_{1}, L_{2}, L_{3})Δf^{k} + | N_{10}ΔΔf^{10} |
k=1 | k=4 |
where f_{k} are nodal values of f function, f_{k} denote departures from linear interpolation for mid – side nodes, and f_{10} is departure from both previous orders of approximation for the central node_{C}^{[1]}. For linear triangular elements only the first term is important which gives an approximation
f(L_{1}, L_{2}, L_{3}) = | ∑ | L_{k}f^{k} |
k=1 |
Note, that the r.h.s sum does not include the jacobian j det J_{j} that should be incorporated by the weights W_{q} but it is not (in their values given in Table 5.3 from^{[1]}). Thus let's add the triangle area to the above – recalled formula
|det J|/(2Δ) | ∑ | f_{q}(L_{1}, L_{2}, L_{3})W^{q} |
q=1 |
and in that way we end up with the final expression for the Q – point Gauss quadrature.