In need to calculate the integral
$I = \iiint_\Omega \exp(i \vec{k} \cdot \vec{x} ) dV$
where $\Omega$ is a finite domain, bounded by piecewise-continuous continuous surfaces $P_i$, namely $\partial \Omega = \cup_i P_i$. In the simplest example it is just a polyhedron, but I require it to be a little more general, allowing the faces of the polyhedron to be slightly curved.
I am interested to use as little resources as possible to compute many such integrals. Thus it is best to reduce the dimension of integration as much as possible. First step is trivial using divergence theorem
$I = i\iiint_\Omega \vec{c} \cdot \nabla \exp(i \vec{k} \cdot \vec{x} ) dV = i\iint_{\partial \Omega} \vec{c} \cdot \vec{n}(\vec{x}) \exp(i \vec{k} \cdot \vec{x} ) dS = i\sum_{P_i} \iint_{\vec{x} \in P_i} \vec{c} \cdot \vec{n}(\vec{x}) \exp(i \vec{k} \cdot \vec{x} ) dS $
where $\vec{c}$ is such that $\vec{c} \cdot \vec{k} = -1$.
Is it possible to do one more step, and convert the last integral to an integral over the perimeter of each surface $P_i$