Rolf H. Höhring
Abstract
We investigate a problem arising in the computer-aided design of cars, planes, ships, trains, and other motor vehicles and machines: refine a mesh of curved polygons, which approximates the surface of a workpiece, into quadrilaterals so that the resulting mesh is suitable for a numerical analysis. This mesh refinement problem turns out to be strongly NP-hard
In commercial CAD systems, this problem is usually solved using a gree dy approach. However, these algorithms leave the user a lot of patchwork to do afterwards. We introduce a new global approach, which is based on network flow techniques. Abstracting from all geometric and numerical aspects, we obtain an undirected graph with upper and lower capacities on the edges and some additional node constraints. We reduce this problem to a sequence of bidirected flwo problems (or, equivalently, to b-matching problems). For the first time, network flow techniques are applied to a mesh refinement problem.
This approach avoids the local traps of greedy approaches and yields solutions that require significantly less additional patchwork.
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Index Terms
- Mesh refinement via bidirected flows: modeling, complexity, and computational results
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Reviews
Paul Cull
It may surprise some readers that applied mathematics, including numerical analysis, is as much an art as a science. This paper gives a glimpse of some of the real difficulties associated with applying mathematics to computer-aided design. The problem is to choose the right (or a good enough) grid to allow efficient analysis by the finite element method. From this vague problem, the authors abstract an exact combinatorial problem and then show that their problem is NP-complete. They then develop a heuristic technique based on the idea of network flows to find (they hope) a reasonable grid and<__?__Pub Caret> present examples in which their method proves superior to some other methods. There are a number of lessons and a question from this paper. If the authors' method works in their problem domain, could one simplify (restrict) the problem so that there is a polynomial-time algorithm__?__ The lesson here is that the real problem is not well-specified, and the major effort is in abstracting the right problem. The authors mention that they ignored the fact that their graphs could have “blossoms.” The lesson of this is not to complicate your heuristic; if you cannot really solve a problem, getting a pretty good answer quickly is probably better than spending a lot of time on finding an answer that may not be much better. The authors build an escape into their method so that it always produces a grid. The lesson of this is that heuristics should have soft failures, so that a usable, if not good, solution can be produced. The major question left open is how one should evaluate heuristics. Is comparing one method with another on a handful of examples the best we can do__?__
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