# Numerical integration of a region bounded by an ellipse and a circle

Consider an ellipse (say with major axis $a$ and minor axis $b$) centered at origin with a concentric circle of radius $R$. Area of the region between the circle and the ellipse is $$A = \pi R^2 - \pi a b.$$ I need to integrate a function numerically inside this region with the usual Gauss-Legendre quadrature within the classical Finite Element framework. Let the region between ellipse and the circle be divided in 4 finite elements and consider the finite element in the first quadrant. I use the following for computing the $x$, $y$ coordinates of a Gauss point in the finite element,

$$\theta_{gauss} = \theta_0 + \eta (\theta_2 - \theta_0)$$ where, $\theta_1$ and $\theta_2$ are the angles made by the horizontal and vertical edges of the finite element considered with the $X$-axis, $\theta_0=\frac{\theta_2-\theta_1}{2}$ and $\eta$ is the first Gauss abscissa. Radius of a point that lies on the ellipse and makes an angle $\theta_{gauss}$ with the $X$-axis is simply, $$r_e = \left(a^2 \cos^2 (\theta_{gauss}) + b^2 \sin^2 (\theta_{gauss}) \right)^{1/2}.$$

In order to place the Gauss integration points such that they are placed from the ellipse moving towards the circle, I use, $$r_0 = \frac{ R + r_e }{2} \hspace{0.1in} \mbox{and},\\ r_{gauss} = r_0 + \xi \left(\frac{R- r_e}{2}\right),$$

where, $\xi$ is the second Gauss abscissa. Finally, the $x$ and $y$ coordinates of the Gauss integration points are now, $$x_{gauss} = r_{gauss} \cos (\theta_{gauss}), y_{gauss} = r_{gauss} \sin (\theta_{gauss}).$$

The problem with this though is that since I have an elliptical radius $r_e$ and a circular radius $R$ used to compute the effective radius of the Gauss point, the distribution of the integration points is not uniform. Secondly, I clearly miss a considerable area (can be seen as white space at $45^{\circ}$) and this results in erroneous results.

For this figure, a 30 x 30 Gauss rule is used in the first finite element and the red dots are the Gauss integration points (the points that are seen clustering towards the end of the ellipse do not cross over into the ellipse). Am I doing this correctly? How do I get my integration points correctly distributed in the annular region between ellipse and the circle. If this is not possible due to elliptic boundary on one side and circular on the other, is it possible to at least position the Gauss points such that the the entire annular region is accounted for (i.e. the numerically integrated area is correct).

• What exactly is your Gauss-Legendre rule? I am not familiar with Gauss rules that work on regions like the one you describe. Most Gauss rules I know are only derived for standard shapes like circles and regular polygons. – Christopher A. Wong Feb 22 '14 at 0:14
• @Wong: The Gauss integration abscissas can be chosen to be equally spaced in $\xi$ and $\eta$ directions over a canonical square $[-1,1] \times [-1,1]$. Since the element shown is not a square, I need to compute the Jacobian based on the formulas used for $x,y \rightarrow \xi,\eta$ transformation. – nawidgc Feb 22 '14 at 0:32
• Yes, probably the numerical quadrature can be used for a circle but I wonder how one models a region using a circular finite element - I think only a quadratic interpolation will be meaningful in such a case. – nawidgc Feb 22 '14 at 0:39
• Just a clarifying question: if you're going to take a square and then map it to this element and use a change of variables, how are you numerically integrating the Jacobian? – Christopher A. Wong Feb 23 '14 at 4:16