Adaptative subdivision to find a polygonal approximation of curved shapes

The input of curved shapes and curves is desired
Curved shapes still are not implemented, and there's no prevision for that
Reduce the quantity of required information (as mesh) from the user

Method

Inspired by the adaptative quadrature, a good idea to discretize a curved domain is by adaptative subdivision:

Define a metric (of value $I$) and a tolerance (of value $\varepsilon$)
Start with a division of the domain
Iterate to reduce the error over $I$

For now, let $a$ and $b$ be two positive integer values. We define the metric by

$$I{a, b} = \int{\Omega} x^a \ y^b \ dx \ dy$$

As described in the theory, this integral is converted into a boundary integral.

$$I{a, b} = \int{\Gamma} \square \ ds$$

Sice the boundary curves are described by nurbs, a knotvector is known and knots are used for initial division

$$I{a, b} = \sum{k=0}^{n-1} I_k; \ \ \ \ \ \ \ \ \ \ \ \ \ Ik = \int{tk}^{t{k+1}} \square \ dt$$

The criterea is to use adaptive quadrature to compute $\overline{I}_k$ such

$$\left[\overline{I}{a,b,k} - I{a,b,k}\right] < \dfrac{\varepsilon}{n}$$

Polygonal domains

As described in the theory, the value of $I_{a, b}$ for a polygonal domain can be computed exactly as

$$(a+b+2)(a+b+1)\binom{a+b}{a} I{a,b} = \sum{k=0}^{n-1} J_k$$

$$Jk = \left(x{k}y{k+1}-y{k}x{k+1}\right) \sum{i=0, \ j=0}^{a, \ b} \binom{i+j}{j} \binom{a+b-i-j}{b-j} x{k}^{a-i}x{k+1}^{i}y{k}^{b-j}y{k+1}^{j}$$

But as $xk$, $y{k}$, $x{k+1}$, $y{k+1}$ are obtained from the evaluations of the curve at points $tk$ and $t{k+1}$, then a new approximation can be obtained by evaluating at middle point

Algorithm

The main idea is:

Get the knots from the knotvector of the curve
For a pair of knots (tk0, tk1):
- Evaluate $A = I(tk0, tk1)$
- Evaluate $B = I(tk0, 0.5tk0 + 0.5tk1)$
- Evaluate $C = I(0.5tk0 + 0.5tk1, tk1)$
- If $|B+C-A| > eps$, recursively computes $B$ and $C$
- Else, continue to the next pair of knots

compmec / section