Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed
Eveline Rosseel 1, Herbert De Gersem 2, Stefan Vandewalle 1*1 Computer Science Department, K. U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium.
2 Faculty of Sciences, K. U. Leuven, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium.
Received 2 July 2009; Accepted (in revised version) 21 December 2009
Available online 31 May 2010
The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. Implementation issues of the stochastic Galerkin method are addressed and a numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.AMS subject classifications: 35K60, 65N35, 65C50, 78M25
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Key words: Nonlinear PDE with random coefficients, polynomial chaos, stochastic collocation method, stochastic Galerkin method, electromagnetics.
Email: Eveline.Rosseel@cs.kuleuven.be (E. Rosseel), Herbert.DeGersem@kuleuven-kortrijk.be (H. De Gersem), Stefan.Vandewalle@cs.kuleuven.be (S. Vandewalle)