Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Abstract

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. 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. AbstractThe 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. 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.

Analysis of FE

“…Afterwards, this sample set is used to calculate the PC coefficients; this approach commonly is referred to as non-intrusive stochastic collocation (Babuška et al, 2010). The authors of (Rosseel et al, 2010) offer a comparison of both approaches, concluding that the implementation of intrusive methods is tedious and error-prone while they do not yield any significant advantage in comparison to the non-intrusive, sampling-based approaches.…”

Stochastic models for the evaluation of magnetisation faults

“…Tools for the propagation of uncertainties across electromagnetic models recently came into the focus of research and development and are available, as for instance the spectral stochastic finite element (FE) method (Ghanem and Spanos, 2003), applying stochastic Galerkin (Rosseel et al, 2010) or non-intrusive projection/regression methods (Sudret, 2007). Nevertheless, for some input variables, e.g.…”

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

“…These methods enable a black-box reuse of deterministic simulation codes and do not require modifications for solving nonlinear stochastic problems. Although very good results can be obtained with the stochastic collocation method, its convergence rate is typically somewhat slower than the stochastic Galerkin convergence, in terms of the number of deterministic PDEs to be solved [7,24]. In terms of computational cost, the success of the stochastic Galerkin method depends on the solution method for the high-dimensional deterministic Galerkin systems.…”