Quasi-3-D spectral wavelet method for a thermal quench simulation

Jonas Bundschuh,Herbert De Gersem,Laura A M D’Angelo

doi:10.1186/s13362-021-00113-8

Jonas Bundschuh, Herbert De Gersem + Show 1 more

Open Access

https://doi.org/10.1186/s13362-021-00113-8

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Abstract

The finite element method is widely used in simulations of various fields. However, when considering domains whose extent differs strongly in different spatial directions a finite element simulation becomes computationally very expensive due to the large number of degrees of freedom. An example of such a domain are the cables inside of the magnets of particle accelerators. For translationally invariant domains, this work proposes a quasi-3-D method. Thereby, a 2-D finite element method with a nodal basis in the cross-section is combined with a spectral method with a wavelet basis in the longitudinal direction. Furthermore, a spectral method with a wavelet basis and an adaptive and time-dependent resolution is presented. All methods are verified. As an example the hot-spot propagation due to a quench in Rutherford cables is simulated successfully.

Highlights

A standard finite element method (FE method) allows to discretize and solve differential equations on arbitrary geometries [1]
2 Introduction to wavelets 2.1 General statements The multiresolution analysis (MRA) [11] consists of closed spaces Vj that are nested subspaces of each other, i.e. it holds that Vj ⊂ Vj–1
8 Conclusion A spectral method using Daubechies scaling functions has been formulated and verified. This has been combined with a 2-D FE method with a nodal basis into a quasi-3-D method

Summary

Introduction

A standard finite element method (FE method) allows to discretize and solve differential equations on arbitrary geometries [1]. Within the spectral methods of this work, scaling functions and wavelets of the DB-N kind are used, because they are compactly supported and constructed in a way to have a maximum number of vanishing moments [12]. One has to distinguish between decreasing and increasing the resolution (see Fig. 4) For the former case, after every time step those basis functions whose coefficients are smaller than τ are removed from. Whenever a coefficient in ud(t) changes from one time step to another by a certain factor α, the resolution in the interval of the support of the corresponding basis function ψj,k is increased. Does not exceed the minimum scale jmin , i.e. the added wavelets must not be removed for a time interval of length β because the coefficients may need a certain time to exceed the tolerance τ. The norm · max is used to compare θwith an analytical solution as a function, whereas · ∞ compares the vector u with a vector uana that contains the coefficients of the analytical solution

Spectral scaling function method

Conclusion