Problem with ImplicitRegion in three dimensions

Questions
1

The Problem is the following, I want to use NDSolveValue to solve for specific Dirichlet Conditions, therefore I need to construct a region with ImplicitRegion which has those boundaries I need for my Dirichlet conditions. While it works very good in two dimensions, I fail in three dimensions, because the interior boundaries are not good. For example, let’s construct a hollow cylinder with some width:

\[ScriptCapitalR] = 
 ImplicitRegion[
  10 >= x^2 + y^2 >= 1 && 0 <= z <= 10, {x, y, 
   z}]; RegionPlot3D[\[ScriptCapitalR]]

Looking on the graph, one can see that the interior has not the form of a cylinder, thus defining a Dirichlet condition with x^2 + y^2 == 1 will lead to an error (effectively being ignored) while x^2+y^2 ==10 will be processed. Interestingly, the same problem arises if one wants to construct a hollow cylinder. Has anyone a solution for my problem?

2

It looks mostly like a cylinder to me, except that the section is a polygon instead of a circle. I don’t know how to increase tha number of sides in the polygon.

3

That’s the problem, the outside looks like a cylinder, but the interior hollow part not. The problem is that when I want to impose a dirichlet condition that for example the voltage at the interior cylindric surface is 1 Volt it gives me an error because it cannot find those points on that surface (since the boundary is now some polygonic shape).

4

You have to discretize the Dirichlet condition, I am afraid. I wonder if we can build our own discretization, bypassing the built-in automatic algorithms.

5

8281re01.png
8281re01.png719×386 21.7 KB