Chebyshev spectral method for superconductivity problems

Abstract

We solve several applied superconductivity problems using the series in Chebyshev polynomials. Although this method may be less general than the usually employed finite element methods, accuracy of the obtained solutions is often much higher due to the very fast convergence of the Chebyshev expansions. First, as an introduction, we consider the thin strip magnetization and transport current problems for which the analytical solutions are known. Then, assuming the Bean critical-state model, we apply this method to two-dimensional problems for stacks and pancake coils made of coated conductors. Finally, we extend this approach to evolutionary stack problems with an arbitrary current-voltage relation. Simulation of coils with a large number of turns is also discussed.

Fast solution of the superconducting dynamo benchmark problem

“…However, as shown in our work, for this specific benchmark and similar problems the Chebyshev polynomial-based method is more than an order of magnitude faster than all finite element methods in [11]. Partly, this is related to the efficient treatment of the singular Cauchy integral term in equation ( 4), based on the relation (12).…”

“…As in [12], we seek an approximation to the sheet current density in the form of a weighted expansion in Chebyshev polynomials of the first kind, ( , ) 1…”

“…Here we show that an accurate numerical solution of this benchmark problem can be obtained using the method of lines for integration in time and Chebyshev polynomials for the approximation in space. For HTS strips, strip stacks, and pancake coils a similar approach has been proposed in [12]. As in [12], our choice of Chebyshev polynomials for spacial approximation is based on two factors: remarkably fast convergence of the interpolation by Chebyshev polynomial expansions [13] and the ease of using such expansions for solving integral equations with a singular Cauchy-type kernel (see, e.g., [14,15] ).…”

“…For HTS strips, strip stacks, and pancake coils a similar approach has been proposed in [12]. As in [12], our choice of Chebyshev polynomials for spacial approximation is based on two factors: remarkably fast convergence of the interpolation by Chebyshev polynomial expansions [13] and the ease of using such expansions for solving integral equations with a singular Cauchy-type kernel (see, e.g., [14,15] ). The derived method is much faster than all methods in [11].…”

Thin Shell Model of a Coated Conductor With a Ferromagnetic Substrate

“…This system (with any nonlinear currentvoltage relation for the superconductor) can be solved numerically using an efficient and highly accurate Chebyshev spectral method. Such methods make possible simple analytical treatment of the integral kernel singularities, converge quickly, and have recently been applied to other superconductivity problems in our works [19][20][21]. The 1d model proposed in this work contains a dimensionless parameter, / a  …”