CREATE-FXB, a Fixed Boundary Code Based on Finite Element Methods for the Solution of the Grad–Shafranov Equation and Optimization of Equilibrium Currents

Abstract

CREATE-FXB is a Finite Element Method solver specifically developed to address the fixed boundary problem, namely the determination of the Magnetohydrodynamic equilibrium of an axisymmetric plasma confined within a toroidal nuclear fusion device. Although several solvers are already available and widely used, CREATE-FXB represents a valuable alternative with further improvements, as it introduces a set of distinctive capabilities that extend its applicability and accuracy in modelling this class of problems. Its key features include the design of plasma scenarios in tokamaks, the derivation of linearized plasma response with high accuracy, the treatment of a wide class of current density profiles including non-smooth distributions, and an improved capability of interfacing with other existing codes.

1. Introduction

Numerical models solving the Grad–Shafranov equation in magnetically confined toroidal plasmas are crucial both for the design phase of plasma scenarios [1] and for the development of active magnetic control systems [2]. For these applications, a variety of procedures and associated codes for the calculation of Magnetohydrodynamic (MHD) equilibria are widely used.

Under the axisymmetric assumption, MHD codes deal with forward and inverse problems. Forward codes analyze the fixed boundary problem, i.e., the resolution of the Grad–Shafranov equation inside a prescribed plasma region, and the free boundary problem, where the results provide the overall equilibrium configuration including the plasma shape. Inverse MHD codes are aimed at the optimization of the external currents which maintain a desired plasma shape in equilibrium [3], or the identification of the plasma configuration starting from magnetic measurements [4]. Several solvers adaptable to one or more categories are available in the literature, and their range of applications is very vast. The typical numerical schemes involved are the finite difference method (FD) or the Finite Element Method (FEM).

Some remarkable examples of FD-based codes are Lackner’s code [3], FIESTA [5], essentially used for analysis and control purposes, and the SPIDER module [6], included in scenario simulator FENIX [7]. Their domain of resolution is typically a rectangular grid.

On the other hand, Finite Element Method (FEM) schemes are usually intensely adopted in combination with other codes. The first example is the self-consistent SCED code [8], which couples the FEM analysis of the free boundary MHD problem via FEM to one-dimensional transport equations. The MAXFEA code [9] evaluates the plasma motion and supplies it to codes for electromechanical analyses on the external conductors. The CREATE-NL+ [10] free boundary solver makes transient analyses and also provides the outputs to the CREATE-L code [11] to derive plasma response linearized models.

Unlike the above-mentioned procedures, the present work focuses exclusively on fixed boundary problems. The main application is the evaluation of linearized plasma response models. Indeed, thanks to the perturbative fixed boundary approach proposed in [12], the determination of the plasma response to external poloidal fields can be carried out without explicitly defining the external coil system. Consequently, linear models can be derived without any information about geometry and number of poloidal field coils, improving flexibility. A dramatic reduction in computational effort is the main consequence, since only the plasma region is meshed.

This paper presents a new tool called CREATE-FXB, a FEM solver for fixed boundary problems. With respect to the procedure presented in [12], the code incorporates innovative numerical features that make it particularly suited for the derivation of robust and flexible linearized models. In particular, the implemented novel nodal numerical formulation makes the subsequent linearization procedure robust in the treatment of generic current density profiles, such as those ones provided by transport codes [13,14], no matter about the presence of points of non-regularity. Secondly, as well as the previous tools, the code can work using plasma parameters instead of current density profile as input, i.e., total plasma current, poloidal beta, and internal inductance, but the current density parametrization allows the procedure to match equilibria in a wide range of initial plasma parameters, still preserving a simple parametric expression with few degrees of freedom (DoFs) like in [15]. Finally, using the procedure depicted in [16], both magnetic flux and field are provided with a second-order accuracy.

These aspects make the code highly suitable for the design of plasma scenarios, including the estimation of the equilibrium currents once the geometry of the magnetic system is fixed.

The paper is organized as follows. Section 2 describes the main assumptions. Section 3 presents the Grad–Shafranov equilibrium equation and its numerical resolution with the formulation used by CREATE-FXB. Section 4 illustrates some applications of the procedure. Finally, Section 5 draws the conclusions and discusses potential future developments compatible with the model.

2. Assumptions

2.1. Basic Equations

At a generic point of the plasma region ${\Omega }_{pl}$, static (zero plasma velocity v = 0) and stationary (∂/∂t = 0) plasma equilibria are well described by the following basic equations below, obtained by means of simplifications of the more complex MHD model [17,18]:

$$\nabla \times \mathit{B}={\mu }_0\mathit{J}$$

(1a)

$$\nabla ·\mathit{B}=0$$

(1b)

$$\mathit{J}\times \mathit{B}=\nabla p$$

(1c)

where $\mathit{J}$ is the current density, $\mathit{B}$ is the magnetic field, ${\mu }_0$ is the vacuum magnetic permeability assumed in the plasma, and p is the kinetic pressure.

Under the 2D axisymmetric assumption in a cylindrical reference coordinate system $\left(r,\phi ,z\right)$, the magnetic poloidal flux per radian $\psi \left(r,z\right)$ and the poloidal current function $f\left(r,z\right)$ are defined as

$$\psi \left(r,z\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^r{B}_{z}rdr$$

(2)

$$f\left(r,z\right)={\displaystyle \frac{{\mu }_0}{2\pi }}{\int }_0^r{J}_{z}rdr$$

(3)

Using Ampere’s law (1a) and the divergence-free condition for the magnetic field (1b), both $\mathit{J}$ and $\mathit{B}$ can be expressed in terms of the two above scalar functions:

$$\mathit{B}={\displaystyle \frac{\nabla \psi \times {\mathit{i}}_{\phi }}{r}}+{\displaystyle \frac{f}{r}}{\mathit{i}}_{\phi }$$

(4)

$$\mathit{J}={\displaystyle \frac{\nabla \left(\frac{f}{\mu }\right)\times {\mathit{i}}_{\phi }}{r}}+L\psi {\mathit{i}}_{\phi }$$

(5)

where $L$ is a 2nd order elliptic operator defined by:

$$L\psi =-{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{1}{{\mu }_{0}r}}{\displaystyle \frac{\partial \psi }{\partial r}}\right)-{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{1}{{\mu }_{0}r}}{\displaystyle \frac{\partial \psi }{\partial z}}\right)$$

(6)

Referring to the plasma region ${\Omega }_{pl}$, by substituting (4) and (5) in (1c) and noting that (5) implies:

$$L\psi =J_{\phi }$$

(7)

the Grad–Shafranov (G-S) equation is derived:

$$L\psi =r{\displaystyle \frac{dp}{d\psi }}+{\displaystyle \frac{1}{{\mu }_{0}r}}{\displaystyle \frac{d\left(\frac{f^2}{2}\right)}{d\psi }}$$

(8)

The functions $dp/d\psi$ and $d\left(f^2/2\right)/d\psi$ are named free functions.

Two classes of differential problems can be distinguished. In free boundary problems [3,6,10], (8) is coupled with (7), in which the current density is known outside the plasma, namely prescribed in the external conductors, and zero in the vacuum. The problem has to be solved in the whole poloidal plane rz, but the plasma–vacuum interface (the last closed magnetic surface that does not intersect solid walls) is a result of the calculation and is not known in advance.

Instead, in fixed boundary problems [12], the plasma region is assigned. Thus, the differential problem is closed by coupling the PDE to some Dirichlet boundary conditions at the plasma boundary $\partial {\mathsf{\Omega }}_{pl}$:

$${\left.\psi \left(r,z\right)\right|}_{\partial {\mathsf{\Omega }}_{pl}}={\psi }_b, {\left.p\left(r,z\right)\right|}_{\partial {\mathsf{\Omega }}_{pl}}=p_b, {\left.f\left(r,z\right)\right|}_{\partial {\mathsf{\Omega }}_{pl}}=f_b$$

(9)

The problem resolution requires the expression of $J_{\phi }$. Thus, the free functions must be assigned or derived in some way.

2.2. Current Density Modelling

The current density modelling passes through the derivation of the free functions $dp/d\psi$ and $d\left(f^2/2\right)/d\psi$. Usually, they are given as function of the non-dimensional variable $\overline{\psi }\in \left[\mathrm{0,1}\right]$ defined as follows:

$$\overline{\psi }={\displaystyle \frac{\psi -{\psi }_a}{{\psi }_b-{\psi }_a}}$$

(10)

where ${\psi }_a$ and ${\psi }_b$ are the values of the flux per radian at the magnetic axis and at the plasma boundary, respectively.

These functions can be obtained by coupling a simplified one-dimensional version of the Grad–Shafranov equilibrium to the transport equations [13,14], adopting iterative schemes to solve the non-linear system and derive the plasma pressure $p\left(\psi \right)$ and poloidal current function $f\left(\psi \right)$. Afterwards, these profiles are given to specific 2D axisymmetric codes coupled to circuit equations like CREATE-NL+ [10] to obtain the time evolution of the poloidal flux map.

An alternative procedure takes $dp/d\psi$ and $d\left(f^2/2\right)/d\psi$ from the suitable family of functions $g\left(\overline{\psi },\mathit{\alpha }\right)$. The $\mathit{\alpha }$ variable is a set of shape parameters necessary to model the features of the current density profile as shown in [15]. They represent the DoFs of the parameterization $g$, linked to the main plasma quantities, namely plasma current $I_{pl}$, poloidal beta ${\beta }_{pol}$, and internal inductance $l_i$, which can be defined as follows:

$$I_{pl}={\int }_{{\mathsf{\Omega }}_{pl}}{J}_{\phi }\mathrm{d}\mathsf{\Omega }, {\beta }_{pol}=4{\int }_{V_{pl}}{\displaystyle \frac{p}{{\mu }_0{R}_0{I_{pl}}^2}}\mathrm{d}\mathrm{V}, l_i=4{\int }_{V_{pl}}{\displaystyle \frac{B_{pol}^2}{{{2\mu }_0}^2{R}_0{I_{pl}}^2}}\mathrm{d}\mathrm{V}$$

(11)

where ${\mathit{B}}_{pol}$ is the magnetic field component in the poloidal plane $rz$.

However, different definitions of ${\beta }_{pol}$ and $l_i$ can also be employed [18]. The choice of this parameterization is motivated by the fact that the flux surfaces outside the plasma boundary are nearly identical for two equilibria with different current density profiles but the same values of $I_{pl}$, ${\beta }_{pol}$, and $l_i$ [15]. Therefore, the magnetic coupling of the plasma with the external conductors can be modelled by means of a suitable choice of the $\mathit{\alpha }$ parameters.

These $\mathit{\alpha }$ parameters are calculated using iterative methods, i.e., by solving (7)–(9) until the desired values of plasma quantities (11) are obtained. The computational effort depends on the number of DoFs. Therefore, the choice of $g$ should be made so as to ensure coverage of the widest possible range of plasma quantities with a limited amount of computational effort.

Several methods have been proposed to model the current density profile in terms of $\overline{\psi }$. For instance, power functions can be employed for both $dp/d\psi$ and $d\left(f^2/2\right)/d\psi$, as in the CREATE-L code [11], which usually implements the following formulation:

$$J_{\phi }=\lambda \left[\beta {\displaystyle \frac{r}{R_0}}+\left(1-\beta \right){\displaystyle \frac{R_0}{r}}\right]{\left(1-\overline{\psi }\right)}^{{\alpha }_m}$$

(12)

which corresponds to bell-shaped current density profiles due to the term ${\left(1-\overline{\psi }\right)}^{{\alpha }_m}.$ In this formula, the $\lambda$ is a scale factor to match the desired plasma current, while the $\beta$ and ${\alpha }_m$ quantities are mainly used to control the $\left(r,z\right)$ coordinates of the maximum value of current density, and the internal inductance, respectively.

This bell-shaped expression is relatively simple, but it cannot be used for plasma equilibria characterized by low values of $l_i$, where the current density typically exhibits a hollow profile and tends to concentrate near the boundary. A new family of functions must be determined to be non-monotonic with $\overline{\psi }$ for specific values of α correspondent to a low internal inductance. Moreover, it should be as simple as possible, to guarantee convergence with sufficient rate.

The CREATE-FXB code can also use the following parameterization in terms of the four parameters ${[\alpha }_0$,${\beta }_0, {\gamma }_0, {\delta }_0]$ derived by modifying the McCarthy class of profiles [19]:

$$J_{\phi }=\left[r{\beta }_0\left({\displaystyle \frac{1}{{\mu }_0}}+{\displaystyle \frac{{\gamma }_0}{{\mu }_0}}\overline{\psi }\right)+{\displaystyle \frac{\left(1-{\beta }_0\right)}{{\mu }_{0}r}}\left({\alpha }_0\overline{\psi }+1\right)\right]{\psi }_f, {\psi }_f=\left\{\begin{array}{ll}1& \overline{\psi }\le {\delta }_0 \\ 1-{\displaystyle \frac{{\left(\overline{\psi }-{\delta }_0\right)}^2}{{\left(1-{\delta }_0\right)}^2}}& \overline{\psi }>{\delta }_0\end{array}\right.$$

(13)

which enables the treatment of a wide class of plasmas, as shown in Section 4. It is worth noticing that the filter function ${\psi }_f$ plays a key role in shaping the plasma current density.

In some cases, the profiles of both $dp/d\psi$ and $d\left(f^2/2\right)/d\psi$ provided by transport codes may be highly rough and irregular. Examples of treatment of these kinds of profiles are also shown in Section 4, where an innovative technique based on information from the mesh nodes has been adopted.

All these features represent a challenge more for the linearization procedure than for the numerical resolution of the fixed boundary problem: indeed, the coupling between standard FEM with linear triangular elements with iterative fixed-point scheme (Picard iterations) is quite robust to face the non-linearity in the computation of the plasma equilibrium quantities. On the other hand, the derivation of the linearized model becomes challenging, because the linearization involves the numerical calculation of profiles derivatives, which cannot be rigorously evaluated in the non-regularity points. For these reasons, adequate meshing processes and suitable numerical techniques to treat the irregularities of the profile are crucial for derivation of the linear model. In this context, the so-called nodal formulation, a numerical technique formalized in the subsequent paragraphs, represents a suitable way to overcome the limitations of conventional methods. Additionally, a procedure of adaptive refinement of mesh can also be used.

3. Finite Element Method Formulation in CREATE-FXB

3.1. Picard Iterations

Equation (8) is a non-linear elliptical partial differential equation, perfectly suited for the resolution using the standard FEM. The first step is to convert it into the weak formulation, by multiplying all the members by a weight function $w\left(r,z\right)\in H_0^1\left({\mathfrak{R}}^2\right)$ and applying integration by parts:

$${\int }_{{\mathsf{\Omega }}_{pl}}{\displaystyle \frac{\nabla w·\nabla \psi }{{\mu }_{0}r}}d\mathsf{\Omega }-{\int }_{\partial {\mathsf{\Omega }}_{pl}}{\displaystyle \frac{w}{{\mu }_{0}r}}·{\displaystyle \frac{\partial \psi }{\partial \mathit{n}}}d\mathrm{S}={\int }_{{\mathsf{\Omega }}_{pl}}w·J_{\phi }\left(r,\psi \right)d\mathsf{\Omega }$$

(14)

The non-linearity appears in the right-hand side, since the toroidal current density $J_{\phi }$ depends on the unknown $\psi$.

The plasma domain is discretized into a set of triangular elements with N nodes, corresponding to the vertices of the triangles. The unknown solution $\psi \left(r,z\right)$ is approximated via a linear combination of shape functions ${\varphi }_l\left(r,z\right), l=1,\dots ,N$:

$$\psi \left(r,z\right)={\sum }_{l=1}^N{\psi }_l{\varphi }_l\left(r,z\right)$$

(15)

where ${\varphi }_l\left(r,z\right)=1$ at the $l$-th node, is zero at the other nodes, and is piecewise linear within the elements containing the l-th node.

The unknown variable can thus be represented as a numerical vector of length N rather than as a function, so that $N$ independent equations are required to find the solution. These are obtained from weak formulation (14) by applying Galerkin’s approach, which selects $w_l\equiv {\varphi }_l, \forall l=1,\dots ,N$:

$${\sum }_{m=1}^N\left({\int }_{{\mathsf{\Omega }}_{pl}}\nabla {\varphi }_l·{\displaystyle \frac{1}{{\mu }_{0}r}}\nabla {\varphi }_m d\mathsf{\Omega }\right)d\mathsf{\Omega }={\int }_{{\mathsf{\Omega }}_{pl}}{\varphi }_l·J_{\phi }\left(r,\psi \right) d\mathsf{\Omega }+{\int }_{\partial {\mathsf{\Omega }}_{pl}}{\displaystyle \frac{{\varphi }_l}{{\mu }_{0}r}}·{\displaystyle \frac{\partial \psi }{\partial \mathit{n}}} d\mathrm{S}$$

(16)

which, in a compact form, become the matrix relation:

$$K\mathit{\psi }=\mathit{F}\left(\psi \right)+\mathit{r}$$

(17)

where

$$K_{lm}={\int }_{{\mathsf{\Omega }}_{pl}}{\displaystyle \frac{\nabla {\varphi }_l·\nabla {\varphi }_m}{{\mu }_{0}r}} d\mathsf{\Omega }, F_l={\int }_{{\mathsf{\Omega }}_{pl}}{\varphi }_l·J_{\phi }\left(r,\psi \right) d\mathsf{\Omega }, r_l={\int }_{\partial {\mathsf{\Omega }}_{pl}}{\displaystyle \frac{{\varphi }_l}{{\mu }_{0}r}}·{\displaystyle \frac{\partial \psi }{\partial \mathit{n}}} d\mathrm{S}$$

(18)

At this point, Dirichlet’s boundary conditions must be imposed on (17), according to (9). In this case the last term $r_l$, which would be needed in case of Neumann boundary conditions, becomes redundant. To this end, the set of nodes is partitioned into two subsets: $N_i$, corresponding to internal nodes, and $N_b$, related to the boundary nodes, marked by indices ii and ib, respectively. The same operation is performed for the nodal values of $\psi$. In system (17) K is then rewritten in block format:

$$\left[\begin{array}{cc}{K}_{ii,ii}& K_{ii,ib}\\ K_{ib,ii}& K_{ib,ib}\end{array}\right]\left[\begin{array}{c}{\mathit{\psi }}_{\mathit{i}\mathit{i}}\\ {\mathit{\psi }}_{\mathit{i}\mathit{b}}\end{array}\right]=\left[\begin{array}{c}{\mathit{F}}_{\mathit{i}\mathit{i}}\\ {\mathit{F}}_{\mathit{i}\mathit{b}}\end{array}\right]+\left[\begin{array}{c}{\mathit{r}}_{\mathit{i}\mathit{i}}\\ {\mathit{r}}_{\mathit{i}\mathit{b}}\end{array}\right]$$

(19)

where clearly ${\mathit{r}}_{\mathit{i}\mathit{i}}=\mathbf{0}$ due to properties of ${\varphi }_l$ functions, ${\mathit{r}}_{\mathit{i}\mathit{b}}$ can be ignored since only the first subset of $ii$ equations is required for the next step, ${\mathit{\psi }}_{\mathit{i}\mathit{b}}$ is known (Dirichlet’s boundary condition), and $\mathit{F}$ depends on $\mathit{\psi }$.

Therefore, a Picard scheme is adopted to solve the non-linearity, with the unknown vector ${\mathit{\psi }}_{\mathit{i}\mathit{i}}$ uniquely determined by solving, at each step $k$:

$${\mathit{\psi }}_{\mathit{i}\mathit{i}}^{\left(k\right)}=K_{ii,ii}^{-1}·\left({\mathit{F}}_{\mathit{i}\mathit{i}}\left({\mathit{\psi }}^{(k-1)}\right)-K_{ii,ib}·{\mathit{\psi }}_{\mathit{i}\mathit{b}}\right)$$

(20)

until a convergence criterion is satisfied.

Additional information on the well-posedness of system (17) can be found in [20].

3.2. Nodal Formulation for $J_{\phi }$

A critical aspect in solving the Grad–Shafranov equation is the accurate evaluation of the current density $J_{\phi }$, especially in the presence of steep gradients or discontinuities in the equilibrium profiles. Standard approaches use linear triangular elements and Gauss integration, e.g., in the CREATE-NL+ code [10], where the term ${\mathit{F}}_{\mathit{i}\mathit{i}}\left({\mathit{\psi }}^{(k-1)}\right)$ in (18) is computed iteratively from the values of $J_{\phi }(r_g^{\left(e\right)},z_g^{\left(e\right)})$ and the shape functions ${\varphi }_l(r_g^{\left(e\right)},z_g^{\left(e\right)})$ in each triangle barycentre $e=1,\dots ,N_e$ as well as the $\overline{{\psi }_g}$ map are determined in these points, while the free function values are read from the diagrams ${\displaystyle \frac{dp}{d\psi }}\left(\overline{\psi }\right)$ and $f{\displaystyle \frac{df}{d\psi }}\left(\overline{\psi }\right)$. However, the main issue arises when the profiles present discontinuities. In such cases, if the $\overline{\psi }$ level at which $J_{\phi }$ is discontinuous falls into a bend of elements, the integrals are done with a wrong value of $J_{\phi }\left({\overline{\psi }}_g\right)$. To address this limitation, a nodal formulation is introduced, in which the current density is reformulated in terms of nodal quantities, ensuring a smoother and more robust treatment of discontinuities.

The main issue arises when the profiles present discontinuities. In such cases, if the $\overline{\psi }$ level at which $J_{\phi }$ is discontinuous falls into a bend of elements, the integrals are calculated with a wrong value of $J_{\phi }\left({\overline{\psi }}_g\right)$.

In order to properly handle the high steep current density regions within the plasma domain, as well as the possible discontinuities of $J_{\phi }$ with respect to $\overline{\psi }$, the current density is reformulated as follows:

$$L\psi =r{\displaystyle \frac{\nabla p·\nabla \psi }{{\left|\nabla \psi \right|}^2}}+{\displaystyle \frac{1}{{\mu }_{0}r}}{\displaystyle \frac{\nabla \left(\frac{f^2}{2}\right)·\nabla \psi }{{\left|\nabla \psi \right|}^2}}, \mathrm{with} p=p\left(\overline{\psi },\mathit{\alpha }\right), f=f\left(\overline{\psi },\mathit{\alpha }\right)$$

(21)

where $\nabla p$ and $\nabla \left({\displaystyle \frac{f^2}{2}}\right)$, are computed from $p$ and $f$ defined at the mesh nodes according to their respective profile functions. In this way, spatial discontinuities resulting from discontinuities in ${\displaystyle \frac{dp}{d\psi }}$ and ${\displaystyle \frac{d\left(\frac{f^2}{2}\right)}{d\psi }}$ are smoothed out. As a result, the computation of $J_{\phi }$ (the flux source) becomes more accurate, and consequently, so does the evaluation of the flux itself. The singularity in the right-hand side of (21) wherever $\left|\nabla \psi \right|=0$ can easily be handled. Indeed, the flux gradient is typically zero for well-defined values of $\overline{\psi }$, namely at the magnetic axis ($\overline{\psi }=0$) and at the null points at the plasma boundary ($\overline{\psi }=1$).

The nodal formulation is particularly suited in two cases that are beyond the scope of this paper, namely,

• The free boundary problem in cases where the current density is non-zero at the plasma edge;
• The derivation of the linearized plasma response to external currents and profile variations.

3.3. Accurate Calculation of the Magnetic Flux and Field

The above MHD equilibrium calculation with linear triangles provides a piecewise constant value of the flux gradient and yields magnetic field discontinuities across elements.

To tackle these issues, especially when the derivative of the magnetic field is needed, two different actions are undertaken. The first action is a mesh refinement wherever the discontinuities in the flux gradient and/or the gradients of the current density exceed prescribed thresholds.

The other method is a post-processing algorithm that requires very limited computational effort while ensuring the regularity of field derivatives [21], which is crucial for certain applications. This method is based on Helmholtz’s theorem and employs the Finite Element Method (FEM) with linear shape functions to calculate $\partial \psi /\partial r$ and $\partial \psi /\partial z$ in the second-order calculation. With the technique proposed in [16], the poloidal magnetic field can be determined by post-processing the standard FEM flux map as the solution to the minimization problem:

$$min{\displaystyle \frac{1}{2}}\left\{{\int }_{\Omega }\left[{{\alpha }_D\left(\nabla ·{\mathit{B}}_{pol}\right)}^2+{{\alpha }_C\left(\nabla \times {\mathit{B}}_{pol}-{\mu }_0{J}_{\phi }{\mathit{i}}_{\mathit{\phi }}\right)}^2\right]r^{2}d\Omega +{\int }_{\partial \mathsf{\Omega }}{\alpha }_B{\left(\mathit{n}·{\mathit{B}}_{pol}-{\displaystyle \frac{\partial \psi /\partial t}{r}}\right)}^{2}rd\mathsf{\Gamma }\right\}$$

(22)

where from (4), ${\mathit{B}}_{pol}={\displaystyle \frac{\nabla \psi \times {\mathit{i}}_{\mathit{\phi }}}{r}}$, while ${\alpha }_D, {\alpha }_C,\mathrm{a}\mathrm{n}\mathrm{d} {\alpha }_B$ are suitable non-dimensional constants.

The first and second terms in (22) are related to (1a,b), whereas the third term takes into account the boundary condition. The quantities $J_{\phi }$ and $\partial \psi /\partial t$, which is the tangential component of $\nabla \psi$, are obtained from the first-order FEM solution of the G-S equation. This formulation ensures continuity and achieves a convergence rate of order O(h²) also for ${\mathit{B}}_{pol}$, where h is the mesh size. These properties are highly useful, offering a high level of accuracy and enabling the derivation of reliable linearized models with limited computational effort, especially when compared to other methods, such as [22].

3.4. Optimization of Poloidal Field Coil Currents

The optimization of the number, position, and dimension of the Poloidal Field (PF) coils in a tokamak is a challenging task, due to linear and non-linear constraints related to maintenance requirements, and technological limits on electric power, current density, magnetic fields, and vertical forces [13,14,17,23,24]. Here, the problem of the optimization of the currents of the PF coils in air core tokamaks while keeping their positions fixed has been addressed [9,10,11,12]. It is based on the linearization of the GS equation around the desired plasma equilibrium. An iterative optimization problem, subject to both linear and quadratic constraints, is then solved.

The total flux per radian ${\psi }_b$ at the plasma boundary is the sum of two terms:

$${\psi }_b={\psi }_{bpl}+{\psi }_{bPF}$$

(23)

Here ${\psi }_{bPF}$, the flux pre radian due to the PF coils, can readily be computed using Green functions based on elliptic integrals [25]. The calculation of the plasma contribution ${\psi }_{bpl}$ using elliptic integrals may be cumbersome, since the boundary is in contact with the source currents. A solution having the same degree of accuracy as FEM can be obtained by surrounding the fixed mesh boundary (i.e., the plasma contour) with a halo region ${\mathsf{\Omega }}_h$ made of a few layers of elements, and solving:

$$L{\psi }_{pl}=J_{\phi }\mathrm{in} {\mathsf{\Omega }}_{pl}\cup {\mathsf{\Omega }}_h, \mathrm{with} {\psi }_{pl}={\psi }_{p{l}_G}\mathrm{on} \partial {(\mathsf{\Omega }}_{pl}\cup {\mathsf{\Omega }}_h)$$

(24)

where $J_{\phi }$ is given by the solution of Section 2.1 in ${\mathsf{\Omega }}_{pl}$, and vanishes in ${\mathsf{\Omega }}_h$.

The boundary condition ${\psi }_{p{l}_G}$ on $\partial {(\mathsf{\Omega }}_{pl}\cup {\mathsf{\Omega }}_h)$ is computed using Green functions from $J_{\phi }$. This method gets rid of the singularities, since the minimum distance between the points of the boundary and the sources is the thickness of the layer.

The PF coil currents are then optimized using one of the methods described in [13,14,17,23,24]. Since the optimized flux per radian is hardly equal to the difference between ${\psi }_b$ and ${\psi }_{bpl}$, it is necessary to check whether the free boundary equilibrium is sufficiently close to the desired fixed boundary configuration. This task is fulfilled by a specific version of the CREATE-NL+ code, solving the free boundary equilibrium code in a semicircle ${\mathsf{\Omega }}_{CNL+}$ of the r-z plane in the first and fourth quadrants, adding further elements to the mesh used for solving the fixed boundary problem. In this version of the free boundary code, the PF equilibrium currents are treated as a set of filaments at the same locations used for Gauss integration of elliptic integrals.

3.5. Linearization

The formulation described in this paper can effectively be used to derive a linearized plasma response model.

By using the perturbative fixed boundary approach of [12], combined with the nodal formulation of $J_{\phi }$ in (21), the equilibrium linearized variations can be computed without restrictions on the class of $dp/d\psi$ and $fdf/d\psi$ profiles. The smoothing feature of the nodal formulation, indeed, makes the evaluation of the plasma response (based on the evaluation of Jacobian matrices) robust to each non-regularity of the profiles, such as high-gradient regions, cusps, and discontinuities. Moreover, only the plasma region must be meshed to derive the response model. As a consequence, the linearization procedure is very accurate and flexible to treat any magnetic flux perturbation due to geometry variations in the external conductors, without requiring a mesh in the region outside the plasma. The new linearization procedure can efficiently be employed for plasma identification, magnetic control, optimization of PF coil current waveforms, and all the phases of the design of tokamak magnetic plasma scenarios.

4. Results and Discussion

This section shows the flexibility of parameterization (13) and demonstrates the reliability of the CREATE-FXB code with a number of benchmark cases, obtaining a very good agreement with available analytical solutions and other numerical results in the literature.

4.1. Envelopes for Current Density Parameterization

This section compares the limits of validity of three parameterization choices for the current density in the ${\beta }_{pol}$-$l_i$ plane for an elliptic plasma (Figure 1a).

The CREATE-FXB parameterization proposed here in (13) is able to obtain plasmas with a wide range of ${\beta }_{pol}$ and $l_i$ values. Figure 1 shows that the upper bounds (more than 20 for $l_i$ and more than 4 for ${\beta }_{pol}$) are clearly larger than those obtained with the other parameterizations. As stated in Section 2, the CREATE-FXB parameterization (13) is also able to obtain low values of $l_i$, i.e., down to about 0.2 like McCarthy’s family, whereas the corresponding lowest value for CREATE-L is about 0.5.

4.2. Circular Plasma with a Large Aspect Ratio and Discontinuous Current Density Profile

The robustness of the nodal formulation is shown by comparing the standard and new numerical solutions with the analytical one, in a circular plasma case with very large aspect ratio:

$$AR={\displaystyle \frac{R_0}{a}}$$

with $R_0$ and $a$ as the major and minor radius, respectively. Under this assumption, the two-dimensional differential problem (8) and (9) reduces to a 1D one [17]. The test is carried out by comparing the straightforward analytical solution of the 1D problem with the FEM solution returned by the 2D axisymmetric solver. A discontinuous current density profile is defined with the following input data:

$$I_{pl}=1 MA, {\displaystyle \frac{d\left(\frac{f^2}{2}\right)}{d\psi }}=0, {\displaystyle \frac{dp}{d\psi }}=\left\{\begin{array}{l}\begin{array}{c}{\displaystyle \frac{J_0}{R_0}} \overline{\psi }\le 0.4191 \end{array}\\ \begin{array}{c} \\ 0 \overline{\psi }>0.4191\end{array}\end{array} \right.$$

(25)

Figure 2 shows the azimuthal field in the plasma region evaluated both using the standard and the nodal formulation of the current density $J_{\phi }$ in the numerical solver. Moreover, for each formulation, the second-order procedure has been applied.

As shown in Figure 2d, the continuous red line (second-order nodal result) practically coincides with the continuous green one (analytical result) for R > 0.5.

Two aspects are highlighted in

Table 1. Circular plasma with a large aspect ratio and discontinuous current density profile: numerical errors.

	Case 1	Case 2	Case 3
$\mathrm{Maximum} \mathrm{step} \mathrm{size} h_{max} \left[\mathrm{c}\mathrm{m}\right]$	$10$	$5$	$1$
$\mathrm{Number} \mathrm{of} \mathrm{nodes} N_{nodes}$	936	3727	98,076
$\mathrm{Number} \mathrm{of} \mathrm{elements} N_{elements}$	1783	7288	195,118
${\epsilon }_{\psi }$ (first-order, nodal formulation) 1	0.51%	0.20%	0.15%
${\epsilon }_{\psi }$ (second-order, nodal formulation)	1.93%	0.35%	0.08%
${\epsilon }_B$ (first-order, nodal formulation) 2	4.24%	2.32%	0.48%
${\epsilon }_B$ (second-order, nodal formulation)	1.30%	0.71%	0.24%
${\epsilon }_B$ (first-order, standard formulation)	4.16%	2.34%	0.46%
${\epsilon }_B$ (second-order, standard formulation)	1.65%	0.71%	0.30%

¹ ${\epsilon }_{\psi }$ = $\max \left|\left|{\psi }_{an}-{\psi }_{num}\right|\right|/\left|\right|{\psi }_{an}\left|\right|$ at $\overline{\psi }=0.4191$. ² ${\epsilon }_B$ =$max\left|\right|B_{\theta ,an}\left(R,0\right)-B_{\theta ,num}\left(R,0\right)\left|\right|/max \left|\right|B_{\theta ,an}\left(R,0\right)\left|\right|$.

, which reports the values of the numerical error. Firstly, the nodal formulation provides better results when the discretization is coarse. Secondly, higher accuracy and smoothness of the magnetic field is obtained by post-processing the FEM solution with the method introduced in [16] and summarized in Section 3.3, no matter the involved $J_{\phi }$ formulation.

4.3. Non-Circular Low-Aspect-Ratio Plasma Equilibrium

The procedure has been applied also to a number of test cases for which analytical solutions are available. The reference case shown here is a low-aspect-ratio plasma equilibrium described in [26], with

$${\displaystyle \frac{dp}{d\psi }}={\displaystyle \frac{8{\psi }_0}{{\mu }_0{R}_0^4}}\left(1+{\alpha }^2\right), f{\displaystyle \frac{df}{d\psi }}=0$$

where ${\psi }_0$ is the poloidal flux per radian at the magnetic axis, $R_0$ is the radial coordinate of the magnetic axis, and $\alpha$ is a non-dimensional constant (there is a typo in [26] in the exponent of $R_0$).

The analytic solution of the Grad–Shafranov equation is given by:

$$\psi \left(R,Z\right)={\displaystyle \frac{{\psi }_0{R}^2}{R_0^4}}\left(2R_0^2-R^2-4{\alpha }^2{Z}^2\right)$$

By fixing the previous parameters as follows:

$${\psi }_0=5{\displaystyle \frac{Vs}{rad}}, R_0=2m, \alpha =1$$

and choosing as boundary flux per radian:

$${\psi }_b=1{\displaystyle \frac{Vs}{rad}}$$

the $\left(R,Z\right)$ coordinates of the separatrix $\psi \left(R,Z\right) =$ ${\psi }_b$ are shown in Figure 3a.

In this case the nodal and the standard formulations yield exactly the same results, since the current density is linear in each triangular element. Figure 3b shows the trend of the numerical error on the flux with the infinity norm as a function of the maximum element size $h_{max}$, which scales with $h_{max}^2$, as shown in [16].

4.4. Optimization of PF Equilibrium Currents in DTT

The fixed boundary solver has been used to determine the set of PF current needed to keep a single plasma in equilibrium in DTT, standing for Divertor Tokamak Test facility, one of the next generation tokamak devices scheduled in the EU European roadmap for nuclear fusion [27].

Figure 4 shows the target plasma boundary at 27 s in the DTT plasma scenario with $I_{pl} = 5.5$ MA, ${\psi }_b=0.70 \mathrm{V}\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$, ${\beta }_{pol}=0.10$, and $l_i=0.80$ [28]. The CREATE-FXB code has firstly calculated the fixed boundary equilibrium in ${\mathsf{\Omega }}_{pl}$, and then the flux per radian due to the plasma ${\psi }_{pl}$, obtained in the region ${\mathsf{\Omega }}_{pl}\cup {\mathsf{\Omega }}_h$, using the procedure described in Section 3.4.

Afterwards, the CREATE-FXB code minimizes ${‖\mathsf{\Delta }{\mathsf{\psi }}_{\mathrm{P}\mathrm{F}}‖}_{\infty }$, which is the difference between ${\psi }_{PF}$ (flux per radian due to the PF system at the plasma boundary) and the desired equilibrium value, namely ${\psi }_b$–${\psi }_{bpl}$. The PF system includes 12 circuits and 24 coils. The details are given in [28]. Figure 5a shows the ${‖\mathsf{\Delta }{\mathsf{\psi }}_{\mathrm{P}\mathrm{F}}‖}_{\infty }$ as a function of the maximum current in the circuits ${‖{\mathrm{I}}_{\mathrm{P}\mathrm{F}}‖}_{\infty }$. Figure 5b shows the optimal currents in comparison with those reported in [28]. The slight improvement of CREATE-FXB is due to a different choice of the figure of merit, since the procedure used in [28] optimizes the entire scenario, adding further constrains.

The validity and the effectiveness of the CREATE-FXB resulting PF currents is demonstrated by solving a free boundary problem using these currents as input data. The test consists of determining, via FEM, the separatrix line confining the plasma region as output from the free boundary solver and verifying the matching with the plasma boundary of the fixed boundary procedure.

The result is shown in Figure 5c, where the free boundary equilibrium obtained by the CREATE-NL+ code using the PF circuit currents obtained by CREATE-FXB provides a plasma boundary almost coincident with the desired shape, since the maximum distance is less than 9 mm.

4.5. Software Performance

The procedure here presented, has been implemented as a new software called CREATE-FXB. The programming language is MATLAB^®. Its copyright is being transferred to Consorzio CREATE. Requests to obtain the code should be addressed to create@unina.it with raffaele.albanese@unina.it in cc. The problem described in Section 4.3 has been analyzed with meshes of different sizes. The numerical results have been obtained with an Intel (R) Core(TM) i7–10750H processor at 2.6 GHz and a 32 GB RAM using the MATLAB^® version R2023b. With reference to cases 1, 2, and 3 of Table 1, the first-order solution requires 37 ms, 90 ms, and 2.261 s, respectively. The corresponding computational times of the second-order solutions are instead 437 ms, 526 ms, and 5.468 s.

5. Conclusions

In this work, CREATE-FXB is presented as a new fixed boundary solver for the Grad–Shafranov equation, based on the Finite Element Method with an innovative technique. The code provides an efficient and accurate tool for plasma equilibrium analysis in tokamaks, offering robust handling of non-smooth current density profiles, second-order accuracy in the evaluation of magnetic flux and field, and an integrated framework for PF coil current optimization.

Its accuracy has been demonstrated via comparison with various available analytic solutions. Validation against the free-boundary code CREATE-NL+ has confirmed the reliability of the approach for scenario design and equilibrium current estimation. Future extensions will focus on the systematic use of the derived linearized models for plasma identification and control, and on the integration of plasma response-based scenario optimization strategies.

Overall, CREATE-FXB constitutes a flexible and accurate complement to the CREATE code family, enhancing its capability to address fixed boundary problems in tokamak applications.