CGAL users discussion list

[杨成林 <>]

It is better to asume the density is linear in a triangle, as Finite
Element Method does. This improves the approximation a lot though the
calculation of the area of a triangle is more complex.

2013/5/29, Tapadi 
<>:
>
> Here, you just found a failure case of the checking method we used so far
> here. Let's remember:
>
> 1. We assume the density function to be constant over each triangle of the
> polygon's discretization (that is an approximation)
> 2. To check the validity of this approximation, we compare the values at
> each triangle's vertices. If they are equal, we deduce that the function
> has
> good chances to be actually constant in the triangle.
>
> But this is not always true, your case is a good example. The function may
> have equal values at each vertice, but still have huge variations between
> those vertices. How to overcome this case? There are two straight
> solutions:
>
> a. Decide of an maximum area each triangle should have. If any triangle has
> an area greater than this max value, then subdivide it no matter what are
> the values at vertices. This max area should be chosen in function of the
> maximum frequency you can find in your function. We can discuss this point
> later if you want, for now just choose a max area value and decrease it
> until you get a good integral result.
>
> b. In addition to the comparison of function's values at each vertice, you
> can check for the function's partial derivatives at each vertice. If they
> are not near zero, subdivide the triangle. You can generalize this approach
> by checking for zero values of function's second, third derivatives, etc.
> Actually, this approach is considering the Taylor series of the function
> and
> checking for zero values till a given rank.
>
> Here you are!
> Best regards,
> Hugo Loi
> PhD student at Inria - Maverick team
>
>
>
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-- 
杨成林
Yang Chenglin