The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

Lipnikov, Konstantin; Manzini, Gianmarco; Brezzi, Franco; Buffa, Annalisa

doi:10.1016/j.jcp.2010.09.007

Cited by 58 publications

(35 citation statements)

References 46 publications

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“…On polyhedrons the present approach could also be seen as being close to previous works on Mimetic Finite Differences (the ancestor of Virtual Elements) like [16] or [22]. Here however the approach is more simple and direct, allowing a thorough analysis of convergence properties.…”

Section: 2)supporting

confidence: 55%

Lowest order Virtual Element approximation of magnetostatic problems

Veiga

Brezzi

Dassi

et al. 2018

Computer Methods in Applied Mechanics and Engineering

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We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B = µH). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind Nédélec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.

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Section: 2)supporting

confidence: 55%

Lowest order Virtual Element approximation of magnetostatic problems

Veiga

Brezzi

Dassi

et al. 2018

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…It is unclear to which extent this approximation error, which is far from negligible at larger distances from the quadrupole axis, entails a limitation for the accuracy that tracking algorithms can achieve. Mimetic methods like those presented in [10] guarantee that a discrete equivalent of ∇ × ∇ × A = 0 is exactly satisfied, so that they could represent a potentially interesting alternative to the techniques described in this section. As a final remark we notice that, in the use of equation 20, particular attention has to be devoted to the computation of the input harmonics.…”

Section: Number Of Evaluations Ratio Hfc/afmentioning

confidence: 99%

High order time integrators for the simulation of charged particle motion in magnetic quadrupoles

Simona

Bonaventura

Pugnat

et al. 2019

Computer Physics Communications

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Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the particle beam crossing a quadrupole, a large number of particle revolutions in the accelerator must be simulated, thus leading to the necessity to preserve numerically invariants of motion over a long time interval and to a substantial computational cost, mostly related to the repeated evaluation of the magnetic vector potential. In this paper, in order to reduce this cost, we first consider a specific gauge transformation that allows to reduce significantly the number of vector potential evaluations. We then analyze the sensitivity of the numerical solution to the interpolation procedure required to compute magnetic vector potential data from gridded precomputed values at the locations required by high order time integration methods. Finally, we compare several high order integration techniques, in order to assess their accuracy and efficiency for these long term simulations. Explicit high order Lie methods are considered, along with implicit high order symplectic integrators and conventional explicit Runge Kutta methods. Among symplectic methods, high order Lie integrators yield optimal results in terms of cost/accuracy ratios, but non symplectic Runge Kutta methods perform remarkably well even in very long term simulations. Furthermore, the accuracy of the field reconstruction and interpolation techniques are shown to be limiting factors for the accuracy of the particle tracking procedures.

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“…Among them the mimetic finite difference (MFD) method that has been successfully applied to a wide range of scientific and engineering applications (see, for instance, [17,28,33,29] and the references therein). However, the construction of high-order MFD schemes is still a challenging task even for two and three dimensional second order elliptic problems.…”

Section: Introductionmentioning

confidence: 99%

The nonconforming virtual element method

2016

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Abstract. We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods.

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The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

Cited by 58 publications

References 46 publications

Lowest order Virtual Element approximation of magnetostatic problems

Lowest order Virtual Element approximation of magnetostatic problems

High order time integrators for the simulation of charged particle motion in magnetic quadrupoles

The nonconforming virtual element method

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