An introduction to MEQ
In this example we will use the 'anamak' tokamak, a tokamak with a simplified geometry description for educational and code testing purposes.
Contents
Codes In MEQ
- FBTE (fbt) - Fit desired equilibrium description
- LIUQE (liu) - Equilibrium reconstruction from measurements
- FGS (fgs) - Forward Grad-Shafranov Static
- FGE (fge) - Forward Grad-Shafranov Evolutive
Each code is treated in more detail in later tutorials.
The MEQ structures
MEQ uses several standardized structures that contain information. It is good to be familiar with these structures since they appear in various places in the code.
If we call any code of the meq suite with only one output argument, the code is not run but only the so-called L structure is calculated. This structure contains parameters, geometry, and pre-calculated data used to run the code. Each code produces a different L structure although some fields will be shared.
Let's first get an L structure by calling fbt. The first parameter is the tokamak, where we choose 'ana' for the anamak tokamak. The second argument represents the shot number, which will be discussed later.
L = fbt('ana',1);
We see that this structure contains two substructures, P and G as well as other quantities. Let's first look at those.
The P structure
L.P;
This contains parameters that will be used to run the code. These parameters are documented in: * meqp.m for general parameters * meqp<tok>.m for tokamak-specific parameters (e.g. meqpana.m) * fbtp.m for code-specific parameters general to all tokamaks * fbtp<tok>.m for code-specific and tokamak-specific parameters, for example fbtpana.m
help meqp help meqpana help fbtp help fbtpana
MEQP MEQ default algorithm configuration parameters
P = MEQP defines default algorithm configuration parameters common to
MEQ.
.tokamak Full name of tokamak
.tok Short name of tokamak
.shot #shotnum
.dasm Minimal distance (grid points) between X and axis points [asxy]
.lasy Search axes and X-points also outside the limiter contour
.xdoma Angle of X-point polygon edges with perpendicular (>0 values to get a smaller polygon) [pdom]
.isaddl If ~isaddl, ignore X-points when searching LCFS
.ilim If 0, ignore limiter when searching LCFS. If 1, use limiter points, if 2, use quadratic interpolation, if 3: cubic splines (requires icsint)
.idoublet If idoublet, allow doublet equilibria
.ihole If ihole (default), allow axes with current densities opposite to Ip when multiple axes are found
.r0 default major radius used for outputs/normalizations
.b0 default toroidal field used for outputs/normalizations
.pq If not empty provides user-defined radial grid (excl LCFS and axis)
.pqu maximum rho-psi for radial grid [active only if pq is empty]
.pql minimum rho-psi for radial grid [active only if pq is empty]
.npq number of radial grid points (excl LCFS and axis) [active only if pq is empty]
.noq size of theta-grid mesh for contours
.raS r/a values of flux surfaces for which to return r,z, coordinates
.ri [m] Inner boundary of computational grid [active only if selx is empty for TCV]
.ro [m] Outer boundary of computational grid [active only if selx is empty for TCV]
.zl [m] Lower boundary of computational grid [active only if selx is empty for TCV]
.zu [m] Upper boundary of computational grid [active only if selx is empty for TCV]
.nr Set radial grid points, true grid will be of size nr+1 [active only if selx is empty for TCV]
.nz Set vertical grid points, true grid will be of size 2*nz+1 [active only if selx is empty for TCV]
.selu Generalized vessel description selector string e.g. 's','e','v';
.nu Number of vessel variables (eigenmodes, segments)
.cde CDE used (also includes static CDEs for fgs), empty for no
cde, otherwise string or function handle for cde type
.ilackner flag for boundary flux computation in meqfbp: 0: use Mby*Iy 1: use Lackners trick, 2: Use a precomputed Mby representing Lackners trick
.nFW Number of fluxes to track for wall gaps. >1 gives secondary separatrices
.rn [m] Radial coordinate for points for flux, field interpolations
.zn [m] Z coordinate for points for flux, field interpolations
.bfct Basis function handle
.bfp Basis function parameters
.infct Interpolation function handle
.inp Interpolation function parameters
.interpmethod Interpolation method for data (default: nearest)
.iterq Iterations of contour algorithm, enables contour-dependent calculations
.tolq Tolerance for convergence of contour algorithm, normalized to the radial grid size
.iqR Value of 1/q for which to return r/a location (only if iterq>0)
.naR Number of occurrences of each 1/q surfaces to track (size of raR will be [nR,naR])
.ifield Calculate Br,Bz fields as additional outputs
.ivacuum Calculate vacuum fluxes and fields Br0x, Bz0x, F0x (excluding plasma contribution)
.izgrid Additional 'z' grid for post-processing flux, field calculations
.izxoverlap Force partial overlapping of z and x grids, allows re-use of Fx in Fz
.relax Relaxation parameter for SQP and Newton methods
.Ipmin Minimum plasma current for which GS solution will be attempted
.icsint If icsint, use cubic spline interpolation for domain id. and contouring (default: false)
.tolcs Tolerance on position of axes, saddle points, limiter points [active only if icsint is true]
.itercs Maximum number of Newton-Raphson iterations for asxycs/flcs [active only if icsint is true]
.nthreadscs Maximum number of threads to be used in bspsum MEX-file for cubic spline evaluation [active only if icsint is true]
.mdsserver MDS server hostname for parameters/measurements loading
.gsxe Switch for treatment of external currents inside computational grid
.gsxetol Tolerance for mapping of external currents inside computational grid to y grid (when gsxe=2)
.gfile If non-empty, attempt to load geometry data from this file (see meqgload/meqgsave)
.LYall If true, LY contains all time slices, converged or not
.ohcoils Names of coils used for driving Ohmic current (from G.dima)
.compcoils Names of coils used for compensation of stray fields from Ohmic transformer coils
.ohequal Force currents of coils for OH induction to be equal
.debug debug flag
.debugplot level of debug plotting
.debugfail trigger extra debugging upon encontering failed slices/equilibria
.argout Additional output arguments (syntax: '[file:]fld[=exp]')
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
Default parameters for Anamak - an analytical test tokamak
Anamak vessel, coils, probes, flux loops are distributed along
concentric elliptical shapes.
.ac PF coil minor radius
.af Flux loop minor radius
.al Limiter minor radius
.am Magnetic probe minor radius
.av Vessel minor radius
.cappav Vessel elongation
.nc Number of PF coils
.nf Number of flux loops
.nl Number of limiter points
.nm Number of magnetic probes
.nv Number of vessel elements
.nW Number of wall gaps
.wc Coil width
.fastcoil Internal fast coil (true/false)
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
FBTP FBTE default configuration parameters
P = FBTP(P,'PAR',VAL,...) fills the structure P with default
configuration parameters, optionally replacing or adding parameters with
specified values.
FBT-specific parameters:
.fbtagcon Internal profile constraints (see meqagcon for details)
.limm Margin factor for coil current limits, values lower than 1 are more conservative. Can be a scalar or a column vector with as many elements as limits.
.mergex If true, merge redundant X-point constraints in LX
.circuit If true, solve time-dependent problem with circuit equations (GSpulse algorithm)
.itert If not 0, re-use previous Iy from previous time as starting guess
.iterfrz Number of iterations before which plasma domain is frozen to ease convergence
.interpx Interpolation method for LX time base. Options: 'none', 'nearest','linear','previous','next'
.MHyinterp If true, use qintmex interpolation of Mxy from meqFxjac for computing control point sensitivity MHy
.qpsolver Solver for contrained QP problem. 'quadprog' (default if available), or 'ipm-optimized' (fallback) or 'ipm' (see HELP IPM).
.useSQP If true, use sequential quadratic programming
.wsSQP Number of picard iterations to perform to warm-start the SQP iterations
.initineq If true, inequality constraints will be included to compute the initial guess
.algoNL Plasma convergence method
.xscal If true, scale internal plasma state as in FGS
.tolstep Tolerance for step size
.timeder Type of stencil for time derivatives, 'centered' or 'backward'
.jacobian_handle Use function handle form of jacobian
.jacobian_sparse Use sparse-matrix form of jacobian (algoNL=Newton-GS)
SQP Solver parameters
.userowmask Reduce Newton problem size when possible
.algoGMRES Algorithm variant for Block-GMRES method (direct_inversion,sim,giv,qr)
.mkryl Maximum krylov subspace dimension for Block-GMRES
.epsilon_res Residual tolerance for Block-GMRES
.usepreconditioner If true, use Picard approx. preconditioner for Block-GMRES
Quantities which will end up in LX
.tol Tolerance for residual
.capaj High capaj means stronger peaking of initial plasma current density guess
.niter Maximum number of iterations
if bffbt basis function is used, additional parameters exist which are documented in BFPFBT
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
FBTPANA ANA FBT configuration parameters
P = FBTPANA(SHOT,'PAR',VAL,...) returns a structure P with configuration
parameters for ANA tokamak, optionally replacing or adding
parameters with specified values. See also FBTP.
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
These parameters can be customized by entering parameter, value pairs in the call to the code (after the first three arguments being tokamak, shot and time), for example
fprintf('iterq=%d\n',L.P.iterq) % old L.P.iterq value L = fbt('ana',1,[],'iterq',1); % set another value fprintf('iterq=%d\n',L.P.iterq) % new L.P.iterq value
iterq=0 iterq=1
The G structure
The G structure contains the geometry description of the tokamak, including many useful quantities such as mutual inductances, grids, coil locations and more.
L.G;
We can plot the geometry using meqgplot.m
clf; meqgplot(L.G)
Or we can plot quantites we want manually by hand:
figure; clf; hold on; plot(L.G.rw,L.G.zw,'sr','MarkerFaceColor','r'); % coil windings plot(L.G.rv,L.G.zv,'ob'); % vessel plot(L.G.rl,L.G.zl,'.k'); % limiter plot(L.G.rm,L.G.zm,'sm','MarkerFaceColor','m'); % magnetic probes plot(L.G.rf,L.G.zf,'oc','MarkerFaceColor','c'); % flux loops axis equal;
and add the computational grid. Note that we use the derived quantities L.nzx,|L.nrx| stored in the L structure which are the grid sizes
mesh(L.G.rx,L.G.zx,zeros(L.nzx,L.nrx),... 'edgecolor','k','facecolor','none'); view(2); shg axis equal
Fluxes and fields
We can use the L structure already to do very useful things. For example, we can compute the flux generated by each circuit on the computational grid, as follows
Ia = zeros(L.G.na,1); % initialize current vector ia = 3; % index of circuit to plot Ia(ia) = 1000; % put 1kA of current Fx = reshape(L.G.Mxa*Ia,L.nzx,L.nrx); % compute flux, and reshape it into right size for plotting clf; hold on; plot(L.G.rl,L.G.zl,'.'); contour(L.rx,L.zx,Fx*1e3,21); axis equal; colorbar; title(sprintf('Flux [mWb] produced by a %2.2f kA current in coil %s',Ia(ia)/1e3,L.G.dima{ia}),... 'Interpreter','none');
Note how L.G.dima contains a label for the circuit.
Similarly we can easily plot the radial, vertical and poloidal field, but first we need to add the relevant matrices to G. (They are not included by default since they are not always needed).
L.G = meqg(L.G,L.P,'Brxa','Bzxa'); % get required matrices Br = reshape(L.G.Brxa*Ia,L.nzx,L.nrx); Bz = reshape(L.G.Bzxa*Ia,L.nzx,L.nrx); Bp = sqrt(Br.^2 + Bz.^2); clf; hold on; vars = {'Fx','Br','Bz','Bp'}, units = {'mWb','mT','mT','mT'}; for ii=1:4 subplot(2,2,ii); hold on; var = eval(vars{ii}); % get variable plot(L.G.rl,L.G.zl,'.'); contour(L.rx,L.zx,var*1e3,21); axis equal; colorbar; title(sprintf('%s [%s]',... vars{ii},units{ii}),... 'Interpreter','none'); end
vars =
1x4 cell array
{'Fx'} {'Br'} {'Bz'} {'Bp'}
Coordinate system
Since we just plotted fluxes and fields, we should talk about coordinate systems. MEQ uses coordinate system as defined by COCOS=17 O. Sauter and S. Y. Medvedev, Comput. Phys. Commun., 184(2), 293–302, (2013)
Specifically it * Uses a right-handed 
coordinate system * Defines flux [in Wb] as 
. * Uses a right-handed toroidal coordinate system 
meq data
Calling a meq code with two arguments gives the input data structure LX required for the code to run. We separate the data from the structure L, where LX contains (time/shot-dependent) data while L is intended to contain fixed tokamak/parameter information. Let's use another code for this: the equilibrium reconstruction code liuqe
[L,LX] = liu('ana',1,0);
LX
LX =
struct with fields:
tokamak: 'anamak'
shot: 1
t: 0
Bm: [16x1 double]
Ff: [16x1 double]
Ft: 2.6832e-04
Ia: [8x1 double]
Ip: 2.0000e+05
Is: [0x1 double]
Iu: [200x1 double]
Uf: [16x1 double]
Xt: 2.6832e-04
aW: []
aq: []
rBt: 1
The meaning of these quantities is documented in
help liux
LIUX LIUQE diagnostic measurements LX = LIUX<TOK>(SHOT,T,L) returns a structure with diagnostic measurements at time T for SHOT .t [s] (t) Time base .Ff [Wb] (:,t) Flux loop flux .Bm [T} (:,t) Magnetic probes .Ia [A} (:,t) Active coil currents .Uf [V] (:,t) Flux loop voltage .Ft [Wb] (1,t) Diamagnetic flux .Xt [Wb] (1,t) Diamagnetic flux measurement (excludes Iy correction) .rBt [m T] (1,t) Vaccum toroidal field *r .Is [A] (:,t) Vessel segment current .Ip [A] (1,:) Plasma current from discrete Rogowski [+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
In the case of liuqe, the data structure contains measurements that can be used to reconstruct the equilibrium, for example Bm are the fields measured at the probes and Ff are fluxes measured at flux loops. From LX.t you can see that only one time point is provided for this (ficticious) shot.
Let's plot some quantities
figure; hold on; subplot(131); barh(1:L.G.nm,LX.Bm); set(gca,'YTick',1:L.G.nm,'YTickLabels',L.G.dimm,'TickLabelInterpreter','none'); subplot(132); barh(1:L.G.nf,LX.Ff); set(gca,'YTick',1:L.G.nf,'YTickLabels',L.G.dimf,'TickLabelInterpreter','none'); subplot(133); barh(1:L.G.na,LX.Ia); set(gca,'YTick',1:L.G.na,'YTickLabels',L.G.dima,'TickLabelInterpreter','none');
With our previous example, we can plot the vacuum flux generated by the measured coil currents, showing a dominantly vertical field.
Fx = reshape(L.G.Mxa*LX.Ia,L.nzx,L.nrx); figure; hold on; plot([L.G.rl;L.G.rl(1)],[L.G.zl;L.G.zl(1)],'-k'); % plot closed limiter contour contour(L.rx,L.zx,Fx,21); axis equal;
My first meq equilibrium
Let's now calculate our first meq equilibrium using the FBT code. As you know, this code determines the coil currents required to maintain a desired equilibrium. We now call the function with 3 output arguments to force the calculation.
[L,~,LY] = fbt('ana',1);
We now get an output structure LY that contains the equilibrium. In this case only 1 time slice. The parameters contained in LY are documented in meqt.m for generic quantities, and |<code>t.m| for code-specific quantities, for example fbtt.m
help meqt help fbtt
MEQT helper function, lists meaning of output variables stored in LY
Code-specific output quantities are documented in code-specific files,
see FBTT, LIUT, FGET, FGST
Dimensions: Q: normalized sqrt poloidal flux including axis (L.pQ)
q: normalized sqrt poloidal flux excluding axis (L.pq)
rx,zx: computational grid L.rx,L.zx
ry,zy: inner computational grid excluding boundary: L.ry,L.zy
o: poloidal angle theta grid (L.oq)
g: basis functions
D: plasma domain (>1 for e.g. doublets)
W: wall gaps, nFW: number of flux values being tracked
R: rational 1/q surfaces, nR: number of 1/q surfaces being tracked
S: flux surfaces with requested outboard r/a values, nS: number of tracked values
t: time
NB: In the code the q and Q suffixes can be used to designate 2D (rho,theta) quantities.
For clarity they are broken down here into q/Q for rho and o for theta.
.shot Shot number
.t Time base (t) [s]
Main equilibrium and integral quantities:
.ag Basis function coefficients (g,t) [-]
.Bm Simulated magnetic probe measurements (*,t) [T]
.Um Simulated magnetic field time derivative measurements (*,t) [T/s]
.bp Beta poloidal (t) [-]
.bpli2 Betap + li/2 (t) [-]
.bt Beta toroidal (t) [-]
.Ff Simulated flux loop poloidal flux (*,t) [Wb]
.Uf Simulated flux loop voltage measurements (=d/dt(Ff)) (*,t) [V]
.FR Ratio of normalized X-point/boundary flux (D,t) [-]
.Ft0 Vacuum contribution to the toroidal flux (t) [Wb]
.Ft Plasma contribution to the toroidal flux (t) [Wb]
.Fx Plasma poloidal flux map on (rx,zx,t) [Wb]
.Ia Fitted poloidal field coil currents (*,t) [A]
.Ip Total plasma current (t) [A]
.Is Vessel segment currents (*,t) [A]
.Iu Vessel currents in generalized description (*,t) [A]
.Iv Vessel filament currents (*,t) [A]
.Iy Plasma current distribution (ry,zy,t) [A]
.Opy Plasma domain index (ry,zy,t) [-]
.li Normalised internal inductance (t) [-]
.mu Normalised diamagnetism (t) [-]
.nA Number of magnetic axes found (t) [-]
.PA Pressure on axis (D,t) [Pa]
.qA q on axis using small diamagnetic approximation (D,t) [-]
.FA Poloidal flux on magnetic axis (D,t) [Wb]
.rA r position of magnetic axis (D,t) [m]
.zA z position of magnetic axis (D,t) [m]
.dr2FA 2nd derivative wrt r of poloidal flux on axis (D,t) [Wb/m^2]
.drzFA 2nd derivative wrt r,z of poloidal flux on axis (D,t) [Wb/m^2]
.dz2FA 2nd derivative wrt z of poloidal flux on axis (D,t) [Wb/m^2]
.nB Number of plasma domain boundaries found (t) [-]
.lB Boolean flag indicating if domain boundary has been found (D,t) [-]
.FB Poloidal flux at pt. defining plasma domain boundary (D,t) [Wb]
.rB r position of point defining plasma domain boundary (D,t) [m]
.zB z position of point defining plasma domain boundary (D,t) [m]
.F0 Poloidal flux at domain center point (D,t) [Wb]
.F1 Poloidal flux at domain boundary point (D,t) [Wb]
.rBt Vacuum toroidal field * r (t) [T.m]
.rIp current distribution weighted by r grid (t) [A.m]
.zIp current distribution weighted by z grid (t) [A.m]
.rY r position of centroid of current distribution (=rIp/Ip) (t) [m]
.zY z position of centroid of current distribution (=zIp/Ip) (t) [m]
.lX Boolean flag indicating if domain is diverted (D,t) [-]
.nX Number of X points found (t) [-]
.FX Flux at X points (*,t) [Wb]
.rX r position of X points (*,t) [m]
.zX z position of X points (*,t) [m]
.dr2FX 2nd derivative wrt r of poloidal flux at X-points (*,t) [Wb/m^2]
.drzFX 2nd derivative wrt r,z of poloidal flux at X-points (*,t) [Wb/m^2]
.dz2FX 2nd derivative wrt z of poloidal flux at X-points (*,t) [Wb/m^2]
.PQ Pressure profile (Q,D,t) [Pa]
.TQ T = r*Btor profile (Q,t) [T.m]
.iTQ 1/TQ (if P.smalldia: with small diagmagnetic approximation) (Q,t) [1/(T.m)]
.Wk Plasma kinetic energy (t) [J]
.WN Normalisation energy (t) [J]
.Wp Plasma poloidal magnetic energy (t) [J]
.Wt0 Vacuum toroidal magnetic energy (t) [J]
.Wt Plasma toroidal magnetic energy (t) [J]
.Vp Plasma volume (t) [m^3]
% Integral quantities per plasma domain
.bpD Beta poloidal per domain (D,t) [-]
.btD Beta toroidal per domain (D,t) [-]
.FtD Plasma toroidal flux per domain (D,t) [Wb]
.Ft0D Vacuum toroidal flux per domain (D,t) [Wb]
.IpD Plasma current per domain (D,t) [A]
.liD Normalised internal inductance per domain (D,t) [-]
.TpDg Integral of basis function values per domain (g,D,t) [-]
.ITpDg Integral of primitives of basis function values per domain (g,D,t) [-]
.rYD r position of centroid of current distribution (=rIpD./IpD) (D,t) [m]
.zYD z position of centroid of current distribution (=zIpD./IpD) (D,t) [m]
.rIpD current distribution weighted on r grid (D,t) [A.m]
.zIpD current distribution weighted on z grid (D,t) [A.m]
.VpD Plasma volume per domain (D,t) [m^3]
.WkD Plasma kinetic energy per domain (D,t) [J]
.WpD Plasma poloidal magnetic energy per domain (D,t) [J]
.WtD Plasma toroidal magnetic energy per domain (D,t) [J]
.Wt0D Vacuum toroidal magnetic energy per domain (D,t) [J]
.WND Normalisation energy per domain (D,t) [J]
% Integral quantities on rho grid per plasma domain
.FtQ Plasma toroidal flux contained in flux surface (Q,D,t) [Wb]
.Ft0Q Vacuum toroidal flux contained in flux surface (Q,D,t) [Wb]
.FtPQ Total toroidal flux contained in flux surface (FtQ+Ft0Q) (Q,D,t) [Wb]
.IpQ Plasma current contained in flux surface (Q,D,t) [A]
.OpQ Number of grid points contained in flux surface (Q,t) [-]
.PpQg p' base functions multiplied by their coefficients (Q,D,g,t) [Pa/Wb]
.PpQ p' profile (Q,D,t) [Pa/Wb]
.TTpQg TT' base functions multiplied by their coefficients (Q,D,g,t) [T^2.m^2/Wb]
.TTpQ TT' profile (Q,D,t) [T^2.m^2/Wb]
.VpQ Plasma volume contained in flux surface (Q,D,t) [m^3]
.WkQ Plasma kinetic energy contained in flux surface (Q,D,t) [J]
.WpQ Plasma poloidal magnetic energy contained in flux surface (Q,D,t) [J]
% Contour-related quantities on rho grid per plasma domain D, only available for iterq>0
.aq distance between point on flux surface and magnetic axis (o,q,D,t) [m]
.zq z position of flux surface points (o,q,D,t) [m]
.rq r position of flux surface points (o,q,D,t) [m]
.raQ Outboard r/A value on pQ grid (Q,D,t) [-]
.iqQ 1/q profile where q is the safety factor (Q,D,t) [-]
.jtorQ Toroidal current density defined as R0*<jphi/R> (Q,D,t) [A/m2]
.raqmin Outboard r/a value of location of minimum safety factor (t) [-]
.qmin Minimum safety factor (t) [-]
.q95 safety factor at 95% normalized poloidal flux surface (D,t) [-]
.Q0Q <1/R> (Q,D,t) [1/m]
.Q1Q -dpsi/dV (Q,D,t) [T/m]
.Q2Q <1/R^2> (Q,D,t) [1/m^2]
.Q3Q <|grad psi|^2/R^2> (Q,D,t) [T^2]
.Q4Q <|grad psi|^2> (Q,D,t) [T^2.m^2]
.Q5Q <|grad psi|/(2pi)> (Q,D,t) [T.m]
.ItQ Plasma current contained in flux surface (Q,D,t) [A]
.LpQ Length of the poloidal contour: \oint dl (Q,D,t) [m]
.rbQ Radial location of contour barycenter: (\oint R dl) / dl (Q,D,t) [m]
.VQ Plasma volume contained in flux surface (Q,D,t) [m^3]
.AQ Cross-section area of flux-surface (Q,D,t) [m^2]
.SlQ Lateral area of flux-surface (Q,D,t) [m^2]
.raR Outboard r/a value of location of requested 1/q values (R,t) [-]
.rS r position of flux surfaces with requested outboard r/a (S,t) [-]
.zS z position of flux surfaces with requested outboard r/a (S,t)
% Shape related quantities, only available for iterq>0
.aminor Minor radius: (Rmax-Rmin)/2 (Q,D,t) [m]
.epsilon Aspect ratio: aminor/rgeom (Q,D,t) [-]
.delta Triangularity: (deltal+deltau)/2 (Q,D,t) [-]
.deltal Lower triangularity: (r(zmin )-rgeom)/aminor (Q,D,t) [-]
.deltau Upper triangularity: (r(zmmax)-rgeom)/aminor (Q,D,t) [-]
.kappa Elongation: (Zmax-Zmin)/(Rmax-Rmin) (Q,D,t) [-]
.lp Length of the LCFS poloidal contour: \oint dl (D,t) [m]
.rbary Radial location of LCFS barycenter: (\oint R dl) / dl (D,t) [m]
.rgeom r position of geom. center of flux surface: (Rmax+Rmin)/2 (Q,D,t) [m]
.zgeom z position of geom. center of flux surface: (Zmax+Zmin)/2 (Q,D,t) [m]
% Wall gap information available only for iterq>0
.aW Gap distance (defined by L.G.rW,.zW and angle .oW) (W,nFW,t) [m]
.FW Flux values that are being tracked by gap algorithm (1,nFW,t) [Wb]
% Magnetic fields (ifield=true) and vacuum fluxes (ivacuum=true)
% on x and z grids (z if izgrid=true)
.Brx Radial magnetic field (rx,zx,t) [T]
.Bzx Vertical magnetic field (rx,zx,t) [T]
.Brz Radial magnetic field on z grid (rz,zz,t) [T]
.Bzz Vertical magnetic field on z grid (rz,zz,t) [T]
.Br0x Radial magnetic field (rx,zx,t) [T]
.Bz0x Vertical magnetic field (rx,zx,t) [T]
.Br0z Radial magnetic field on zgrid (rz,zz,t) [T]
.Bz0z Vertical magnetic field on zgrid (rz,zz,t) [T]
.Fx0 Vacuum flux (rx,zx,t) [Wb]
.Fz0 Vacuum flux on z grid (rz,zz,t) [Wb]
% Quantities on interpolation points (when using 'infct')
.Fn Flux at interpolation points L.P.rn, L.P.zn (*,t) [Wb]
.Un Loop voltage at interpolation points (*,t) [V]
.Brn Br at interpolation points (*,t) [T]
.Bzn Bz at interpolation points (*,t) [T]
.Brrn dBr/dr at interpolation points (*,t) [T/m]
.Brzn dBr/dz at interpolation points (*,t) [T/m]
.Bzzn dBz/dz at interpolation points (*,t) [T/m]
.Bzrn dBz/dr at interpolation points (*,t) [T/m]
% Quantities related to evolutive equations
.Va Voltage for active circuits (*,t) [V]
.IniD Non-inductive plasma current (D,t) [A]
% Solver output:
.err sum of error flags (slx only):
err=(|Ip|<L.P.Ipmin)+2*(nA==0)+4*(lB==0)+8*(PA<0) (t)[-]
.isconverged Boolean flag indicating if code has converged (t) [-]
.niter Number of iterations done (t) [-]
.res Solver residual (t) [-]
.rese Residual of circuit equations (t) [-]
.resy Residual norm of Iy (t) [-]
.resp Residual norm of plasma current equation (t) [-]
.resC Residual norm of ag constraints (t) [-]
.resFx Residual norm of Fx (t) [-]
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
FBTT FBT inverse equilibrium solver
LY = FBTT(L,LX[,'PAR',VAL,...]) calculates equilibrium using the parameters
in L as obtained by FBTC, optionally replacing or adding parameters with
specified values. It returns a structure LY with the calculated
equilibria.
For help on output structure: see meqt.m
FBT-specific outputs:
.Sp Sum of surfaces of all grid points containing plasma [m^2]
.Ep Plasma box elongation
.chi Root-mean-square of cost function residual
.chie Root-mean-square of constraint residual
.werr Weighted cost function error
.uerr Unweighted cost function error
.ID Equation id for each cost function term
.Fb Boundary flux (fitting parameter)
The current version extends both the original FBT algorithm [Hofmann CPC
1988] as well as the GSPD algorithm [J. Wai et al, arXiv 2023].
The optimization algorithm is described in MEQOPT.
See also FBTLSQINIT FBTINIT MEQOPT
[+MEQ MatlabEQuilibrium Toolbox+] Swiss Plasma Center EPFL Lausanne 2022. All rights reserved.
we can plot the key quantities for illustration:
figure; subplot(121); hold on; contourf(L.rx,L.zx,LY.Fx,21); contour(L.rx,L.zx,LY.Fx,LY.FB*[1 1],'linecolor','w','linewidth',2); % LCFS plot(LY.rA,LY.zA,'or'); % magnetic axis plot([L.G.rl;L.G.rl(1)],[L.G.zl;L.G.zl(1)],'-k','linewidth',2); % plot closed limiter contour axis equal; title('flux distribution') subplot(122); hold on; axis xy imagesc(L.ry,L.zy,LY.Iy); % current distribution (note different grid) contour(L.rx,L.zx,LY.Fx,LY.FB*[1 1],'linecolor','w','linewidth',2); % LCFS plot(LY.rIp/LY.Ip,LY.zIp/LY.Ip,'*r'); % current centroid plot([L.G.rl;L.G.rl(1)],[L.G.zl;L.G.zl(1)],'-k','linewidth',2); % plot closed limiter contour title('current distribution'); axis equal;
As you can see this is a rather dull circular equilibrium. Let's plot something more exciting, and use the function meqplott that does all the plotting for us.
[L,~,LY] = fbt('ana',2);
figure; meqplott(L,LY);
If you want to see more details of the flux in the region around the coils, as well as the coil currents, use meqplotfancy function. This needs a specific parameter izgrid to be set first to compute the flux on a secondary grid.
[L,~,LY] = fbt('ana',2,[],'izgrid',true); figure; meqplotfancy(L,LY);
