I'm working with the Boundary Element Method and want to integrate an expression over a triangular region. I would like to use Gauss Integration to do this, but I'm having trouble since the triangular element is zero order(1 degree lower than a constant stress triangle). Unlike the higher order elements, the zero order triangle has only one node at its centroid. Normally, Gauss Integration is carried out over a triangular element by defining a natural coordinate system and corresponding shape functions/Jacobin. In the case of the zero order element, however, their is one shape function = 1. Therefore the shape functions can not be expressed in terms of the natural coordinates, and by extension, in terms of the Gauss Points.

Does anyone know how to integrate an expression over a zero order triangular element using Gauss Integration? If not does anyone know a different numerical integration scheme to integrate over a triangular region by just plugging in points and weights. I don't want to write out both integrals and then apply a numerical method to each one. I would also like to avoid defining the limits of integration.

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    $\begingroup$ If there is only one shape function, then the solution is constant, and the integral over a triangle is just the value of the solution times the area of the triangle. Or did I miss a detail? $\endgroup$
    – Jesse Chan
    Apr 28 '15 at 1:08
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    $\begingroup$ And @JesseChan's answer is equivalent to taking a single quadrature point at an arbitrary location inside the triangle (e.g., at the centroid) and letting the weight be the area of the triangle. $\endgroup$ Apr 28 '15 at 4:02
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    $\begingroup$ In other words, it's a one-point Gauss formula. $\endgroup$ Apr 28 '15 at 4:02
  • $\begingroup$ This makes sense. I didn't realize I could pull everything outside the integral. As you say though with no shape function their are no variables, and I don't need numerical integration. I see how I can do the Gauss Integration now as well. Thanks to both of you. I made something really simple seem really complicated. $\endgroup$ Apr 28 '15 at 13:33

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