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A linearization technique for solving general 3-D shape optimization problems in spherical coordinates. (English) Zbl 1410.49049
Summary: Regarding the some useful advantages of spherical coordinates for some special problems, in this paper, based on Radon measure properties, we present a new and basic solution method for general shape optimization problems defined in spherical coordinates. Indeed, our goal is to determine a bounded shape located over the $$(x,y)$$-plane, such that its projection in the $$(x,y)$$-plane and its volume is given and also it minimizes some given surface integral. To solve these kinds of problems, we somehow extend the embedding process in Radon measures space. First, the problem is converted into an infinite-dimensional linear programming one. Then, using approximation scheme and a special way for discretization in spherical region, this problem is reduced to a finite-dimensional linear programming one. Finally, the solution of this new problem is used to construct a nearly optimal smooth surface by applying an outlier detection algorithm and curve fitting. More than reducing the complexity, this approach in comparison with the other methods has some other advantages: linear treatment for even nonlinear problems, and the minimization is global and does not depend on initial shape and mesh design. Numerical examples are also given to demonstrate the effectiveness of the new method, especially for classical and obstacle problems.
##### MSC:
 49Q10 Optimization of shapes other than minimal surfaces 93B51 Design techniques (robust design, computer-aided design, etc.) 90C05 Linear programming
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