Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed (Communications in Computational Physics, 2010)
The stochastic Galerkin and stochastic collocation method are two state-of-the-artmethods for solving partial differential equations (PDE) containing random coefficients.While the latter method, which is based on sampling, can straightforwardlybe applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkinmethod and approximations are required. In this paper, both methods are used forconstructing high-order solutions of a nonlinear stochastic PDE representing the magneticvector potential in a ferromagnetic rotating cylinder. This model can be usedfor designing solid-rotor induction machines in various machining tools. A precisedesign requires to take ferromagnetic saturation effects into account and uncertaintyon the nonlinear magnetic material properties. Implementation issues of the stochasticGalerkin method are addressed and a numerical comparison of the computational costand accuracy of both methods is performed. The stochastic Galerkin method requiresin general less stochastic unknowns than the stochastic collocation approach to reacha certain level of accuracy, however at a higher computational cost.