Meshless method for solving fixed boundary problem of plasma equilibrium

Imazawa, Ryota; Kawano, Y.; Itami, K.

doi:10.1016/j.jcp.2015.03.035

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…The Localized Collocation Meshless Method is a numerical technique for the solution of partial differential equations (PDE's), derived from an interpolation strategy utilizing radialbasis functions (RBF) [18][19][20] . This method has several inherent benefits that make it preferable to traditional modeling techniques [19][20][21][22][23] . This is especially true in inverse problems, where the iterative nature of the solution methods can leverage the pre-computing strategy of derivative interpolators in the localized method [24][25][26] .…”

Section: Meshless Methodsmentioning

confidence: 99%

“…Meshless techniques utilizing RBF, particularly multiquadric RBF, benefit from spectral convergence in globally collocated methods [19,33] . Although the claim to spectral convergence is lost in localized collocation, local methods are less susceptible to ill-conditioning without the expense of domain decomposition and still retains a degree of accuracy unmatched by traditional meshed methods [23,24,30] .…”

Section: On Efficiencymentioning

confidence: 99%

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An Analytical Validation of the Localized Collocation Meshless Method Formulation for Transdermal Drug Delivery

Khoury¹,

Golubev²,

Divo³

2021

14th WCCM-ECCOMAS Congress

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The Localized Collocation Meshless Method (LCMM) is applied to the solution of a set of partial differential equations describing the diffusive transport of a dermally injected compound. This efficient and accurate numerical technique then provides the framework for the estimation of pharmacokinetic (PK) parameters by the comparison of two models, both containing information derived from experimental results, using the iterative Golden Section search algorithm. This comparison is quantified using a norm between the two solution spaces. While this method does provide an estimate of the effective-diffusion coefficient, it ultimately demonstrates that a simple, free-diffusion model is inadequate for quantifying the spatial and temporal distribution of a dermally-injected compound if elimination effects are not accounted for properly.

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Section: Meshless Methodsmentioning

confidence: 99%

Section: On Efficiencymentioning

confidence: 99%

An Analytical Validation of the Localized Collocation Meshless Method Formulation for Transdermal Drug Delivery

Khoury¹,

Golubev²,

Divo³

2021

14th WCCM-ECCOMAS Congress

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“…Additionally, these basis polynomials satisfy p k (ξ, η) ∈ H(∇×, Ω), see [22,56]. It can be shown, [22,56], that if h p (ξ, η) is expanded in terms of edge polynomials, as in (39), then the expansion of ∇ × h p in terms of the volume polynomials, (32), is…”

Section: The Finite Dimensional Basis Functionsmentioning

confidence: 99%

“…ASTRA [60], CORSICA [17], CRONOS [3], JETTO [14], RAPTOR [20], TRANSP [13]).The different schemes to numerically solve the MHD equilibrium problem can either compute the flux function, ψ, on a prescribed mesh in the (r, z) coordinate system (Eulerian or direct solvers, e.g. CEDRES++, CHEASE, CREATE-NL+, ECOM, [29,39,41]) or employ a flux-based mesh and compute the physical coordinates (r, z) from the plasma geometry and ψ (Lagrangian or indirect solvers, e.g. EEC, ESC, [42,49]).…”

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confidence: 99%

A mimetic spectral element solver for the Grad–Shafranov equation

Palha

Koren

Felici

2016

Journal of Computational Physics

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In this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad-Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators (∇, ∇×, ∇·) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulation is an arbitrary order method even on highly curved meshes. Additionally, the integral of the toroidal current J φ is exactly equal to the boundary integral of the poloidal field over the plasma boundary. This property can play an important role in the coupling between equilibrium and transport solvers. The proposed solver is tested on a varied set of plasma cross-sections (smooth and with an X-point) and also for a wide range of pressure and toroidal magnetic flux profiles. Equilibria accurate up to machine precision are obtained. Optimal algebraic convergence rates of order p + 1 and geometric convergence rates are shown for Soloviev solutions (including high Shafranov shifts), field-reversed configuration (FRC) solutions and spheromak analytical solutions. The robustness of the method is demonstrated for non-linear test cases, in particular on an equilibrium solution with a pressure pedestal.The Grad-Shafranov equation, (1), is a non-linear elliptic partial differential equation. Its non-linear character stems from the non-linear dependence of P and f on the unknown flux function ψ and from the fact that the plasma domain Ω p is also an unknown, that in general can only be determined once the flux function is known. These two characteristics make the solution of this equation a challenging task.The literature on the numerical solution of MHD equilibria is extensive. A detailed review of different formulations and codes prior to 1991 is presented in [67] and a shorter review up to 1984 is given in [5]. More recently, several other approaches have been proposed and are used by different research groups, e.g. CEDRES++ [32], CHEASE [50, 51], CREATE-NL+ [2], ECOM [46, 59], EEC [47], ESC [71], HELENA [37], NIMEQ [35], SPIDER [40], etc.The solution of MHD equilibria can be grouped into two distinct classes: (i) fixed-boundary (e.g. CHEASE, ECOM, EEC, ESC, HELENA, NIMEQ, SPIDER) and (ii) free-boundary (e.g. CEDRES++, CREATE-NL+). In the fixed-boundary case the plasma domain is prescribed together with ψ = constant at its boundary, ∂Ω p . Equation (1) is then used to find ψ inside the plasma. The free-boundary approach requires the solution of the MHD equilibrium in an infinite domain with homogeneous boundary conditions, ψ = 0, at infinity and taking into consideration the current flowing in a set of external coils and the Grad-Shafranov equation in the plasma region, (1). In this situation, both the plasma domain and the flux function, ψ, are unknowns tha...

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“…Some other recent alternatives that have attracted attention include the hybrid approach EEC-ESC that couples Hermite elements near the plasma edge with Fourier decomposition methods in the plasma core [20], the use of meshless methods [21,22], the use of approximate particular solutions [23], and the method of fundamental solutions [24].…”

Section: Introductionmentioning

confidence: 99%