up

Magnetodynamics/Lib_MagDyn_av_2D_Cir.pro

+// Lib_MagDyn_av_2D_Cir.pro
+//
+// Template library for 2D magnetostatic and magnetodynamic problems in terms
+// of the magnetic vector potential a (potentially coupled with the electric
+// scalar potential v), with optional circuit coupling.
+
+// Default definitions of constants, groups and functions that can/should be
+// redefined from outside the template:
+
+DefineConstant[
+  Flag_FrequencyDomain = 1, // frequency-domain or time-domain simulation
+  Flag_CircuitCoupling = 0, // consider coupling with external electric circuit
+  Flag_NewtonRaphson = 1, // Newton-Raphson or Picard method for nonlinear iterations
+  CoefPower = 0.5, // coefficient for power calculations
+  Freq = 50, // frequency (for harmonic simulations)
+  TimeInit = 0, // intial time (for time-domain simulations)
+  TimeFinal = 1/50, // final time (for time-domain simulations)
+  DeltaTime = 1/500, // time step (for time-domain simulations)
+  FE_Order = 1, // finite element order
+  Val_Rint = 0, // interior radius of annulus shell transformation region (Vol_Inf_Mag)
+  Val_Rext = 0 // exterior radius of annulus shell  transformation region (Vol_Inf_Mag)
+  NL_tol_abs = 1e-6, // absolute tolerance on residual for noninear iterations
+  NL_tol_rel = 1e-6, // relative tolerance on residual for noninear iterations
+  NL_iter_max = 20 // maximum number of noninear iterations
+];
+
+Group {
+  DefineGroup[
+    // The full magnetic domain:
+    Vol_Mag,
+
+    // Subsets of Vol_Mag:
+    Vol_C_Mag, // massive conductors
+    Vol_S0_Mag, // stranded conductors with imposed current densities js0
+    Vol_S_Mag, // stranded conductors with imposed current, voltage or circuit coupling
+    Vol_NL_Mag, // nonlinear magnetic materials
+    Vol_V_Mag, // moving massive conducting parts (with invariant mesh)
+    Vol_M_Mag, // permanent magnets
+    Vol_Inf_Mag, // annulus where a infinite shell transformation is applied
+
+    // Boundaries:
+    Sur_Neu_Mag, // boundary with Neumann BC (flux tube with n x h = nxh[])
+    Sur_Perfect_Mag, // boundary of perfect conductors (non-meshed)
+    Sur_Imped_Mag // boundary of conductors approximated by an impedance (non-meshed)
+  ];
+  If(Flag_CircuitCoupling)
+    DefineGroup[
+      SourceV_Cir, // voltage sources
+      SourceI_Cir, // current sources
+      Resistance_Cir, // resistors (linear)
+      Inductance_Cir, // inductors
+      Capacitance_Cir, // capacitors
+      Diode_Cir // diodes (treated as nonlinear resistors)
+    ];
+  EndIf
+}
+
+Function {
+  DefineFunction[
+    nu, // reluctivity (in Vol_Mag)
+    sigma, // conductivity (in Vol_C_Mag and Vol_S_Mag)
+    br, // remanent magnetic flux density (in Vol_M_Mag)
+    js0, // source current density (in Vol_S0_Mag)
+    dhdb, // Jacobian for Newton-Raphson method (in Vol_NL_Mag)
+    nxh, // n x magnetic field (on Sur_Neu_Mag)
+    Velocity, // velocity of moving part (in Vol_V_Mag)
+    Ns, // number of turns (in Vol_S_Mag)
+    Sc, // cross-section of windings (in Vol_S_Mag)
+    CoefGeos, // geometrical coefficient for 2D or 2D axi model (in Vol_Mag)
+    Ysur // surface admittance (on Sur_Imped_Mag)
+  ];
+  If(Flag_CircuitCoupling)
+    DefineFunction[
+      Resistance, // resistance values
+      Inductance, // inductance values
+      Capacitance // capacitance values
+    ];
+  EndIf
+}
+
+// End of definitions.
+
+Group{
+  // all linear materials
+  Vol_L_Mag = Region[ {Vol_Mag, -Vol_NL_Mag} ];
+  // all volumes + surfaces on which integrals will be computed
+  Dom_Mag = Region[ {Vol_Mag, Sur_Neu_Mag, Sur_Perfect_Mag, Sur_Imped_Mag} ];
+  If(Flag_CircuitCoupling)
+    // all circuit impedances
+    DomainZ_Cir = Region[ {Resistance_Cir, Inductance_Cir, Capacitance_Cir} ];
+    // all circuit sources
+    DomainSource_Cir = Region[ {SourceV_Cir, SourceI_Cir} ];
+    // all circuit elements
+    Domain_Cir = Region[ {DomainZ_Cir, DomainSource_Cir} ];
+  EndIf
+}
+
+Jacobian {
+  { Name Vol;
+    Case {
+      { Region Vol_Inf_Mag ;
+        Jacobian VolSphShell {Val_Rint, Val_Rext} ; }
+      { Region All; Jacobian Vol; }
+    }
+  }
+  { Name Sur;
+    Case {
+      { Region All; Jacobian Sur; }
+    }
+  }
+}
+
+Integration {
+  { Name Gauss_v;
+    Case {
+      { Type Gauss;
+        Case {
+          { GeoElement Point; NumberOfPoints  1; }
+          { GeoElement Line; NumberOfPoints  5; }
+          { GeoElement Triangle; NumberOfPoints  7; }
+          { GeoElement Quadrangle; NumberOfPoints  4; }
+          { GeoElement Tetrahedron; NumberOfPoints 15; }
+          { GeoElement Hexahedron; NumberOfPoints 14; }
+          { GeoElement Prism; NumberOfPoints 21; }
+        }
+      }
+    }
+  }
+}
+
+// Same FunctionSpace for both static and dynamic formulations
+FunctionSpace {
+  { Name Hcurl_a_2D; Type Form1P; // 1-form (circulations) on edges
+                                  // perpendicular to the plane of study
+    BasisFunction {
+      // \vec{a}(x) = \sum_{n \in N(Domain)} a_n \vec{s}_n(x)
+      //   without nodes on perfect conductors (where a is constant)
+      { Name s_n; NameOfCoef a_n; Function BF_PerpendicularEdge;
+        Support Dom_Mag; Entity NodesOf[All, Not Sur_Perfect_Mag]; }
+
+      // global basis function on boundary of perfect conductors
+      { Name s_skin; NameOfCoef a_skin; Function BF_GroupOfPerpendicularEdges;
+        Support Dom_Mag; Entity GroupsOfNodesOf[Sur_Perfect_Mag]; }
+
+      // additional basis functions for 2nd order interpolation
+      If(FE_Order == 2)
+        { Name s_e; NameOfCoef a_e; Function BF_PerpendicularEdge_2E;
+          Support Vol_Mag; Entity EdgesOf[All]; }
+      EndIf
+    }
+    GlobalQuantity {
+      { Name A; Type AliasOf; NameOfCoef a_skin; }
+      { Name I; Type AssociatedWith; NameOfCoef a_skin; }
+    }
+    Constraint {
+      { NameOfCoef a_n;
+        EntityType NodesOf; NameOfConstraint MagneticVectorPotential_2D; }
+
+      { NameOfCoef I;
+        EntityType GroupsOfNodesOf; NameOfConstraint Current_2D; }
+
+      If(FE_Order == 2)
+        { NameOfCoef a_e;
+          EntityType EdgesOf; NameOfConstraint MagneticVectorPotential_2D_0; }
+      EndIf
+    }
+  }
+}
+
+FunctionSpace {
+  // Gradient of Electric scalar potential (2D)
+  { Name Hregion_u_2D; Type Form1P; // same as for \vec{a}
+    BasisFunction {
+      { Name sr; NameOfCoef ur; Function BF_RegionZ;
+        // constant vector (over the region) with nonzero z-component only
+        Support Region[{Vol_C_Mag, Sur_Imped_Mag}];
+        Entity Region[{Vol_C_Mag, Sur_Imped_Mag}]; }
+    }
+    GlobalQuantity {
+      { Name U; Type AliasOf; NameOfCoef ur; }
+      { Name I; Type AssociatedWith; NameOfCoef ur; }
+    }
+    Constraint {
+      { NameOfCoef U;
+        EntityType Region; NameOfConstraint Voltage_2D; }
+      { NameOfCoef I;
+        EntityType Region; NameOfConstraint Current_2D; }
+    }
+  }
+
+  // Current in stranded coil (2D)
+  { Name Hregion_i_2D; Type Vector;
+    BasisFunction {
+      { Name sr; NameOfCoef ir; Function BF_RegionZ;
+        Support Vol_S_Mag; Entity Vol_S_Mag; }
+    }
+    GlobalQuantity {
+      { Name Is; Type AliasOf; NameOfCoef ir; }
+      { Name Us; Type AssociatedWith; NameOfCoef ir; }
+    }
+    Constraint {
+      { NameOfCoef Us;
+        EntityType Region; NameOfConstraint Voltage_2D; }
+      { NameOfCoef Is;
+        EntityType Region; NameOfConstraint Current_2D; }
+    }
+  }
+}
+
+If(Flag_CircuitCoupling)
+  // UZ and IZ for impedances
+  FunctionSpace {
+    { Name Hregion_Z; Type Scalar;
+      BasisFunction {
+        { Name sr; NameOfCoef ir; Function BF_Region;
+          Support Domain_Cir; Entity Domain_Cir; }
+      }
+      GlobalQuantity {
+        { Name Iz; Type AliasOf; NameOfCoef ir; }
+        { Name Uz; Type AssociatedWith; NameOfCoef ir; }
+      }
+      Constraint {
+        { NameOfCoef Uz;
+          EntityType Region; NameOfConstraint Voltage_Cir; }
+        { NameOfCoef Iz;
+          EntityType Region; NameOfConstraint Current_Cir; }
+      }
+    }
+  }
+EndIf
+
+
+// Static Formulation
+Formulation {
+  { Name MagSta_a_2D; Type FemEquation;
+    Quantity {
+      { Name a; Type Local; NameOfSpace Hcurl_a_2D; }
+      { Name ir; Type Local; NameOfSpace Hregion_i_2D; }
+    }
+    Equation {
+      Integral { [ nu[] * Dof{d a} , {d a} ];
+        In Vol_L_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      If(Flag_NewtonRaphson)
+        Integral { [ nu[{d a}] * {d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+        Integral { [ dhdb[{d a}] * Dof{d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+        Integral { [ - dhdb[{d a}] * {d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+      Else
+        Integral { [ nu[{d a}] * Dof{d a}, {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+      EndIf
+
+      Integral { [ - nu[] * br[] , {d a} ];
+        In Vol_M_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ - js0[] , {a} ];
+        In Vol_S0_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ - (js0[]*Vector[0,0,1]) * Dof{ir} , {a} ];
+        In Vol_S_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ nxh[] , {a} ];
+        In Sur_Neu_Mag; Jacobian Sur; Integration Gauss_v; }
+    }
+  }
+}
+
+// Dynamic Formulation (eddy currents)
+Formulation {
+  { Name MagDyn_a_2D; Type FemEquation;
+    Quantity {
+      { Name a; Type Local; NameOfSpace Hcurl_a_2D; }
+      { Name A_floating; Type Global; NameOfSpace Hcurl_a_2D [A]; }
+      { Name I_perfect; Type Global; NameOfSpace Hcurl_a_2D [I]; }
+
+      { Name ur; Type Local; NameOfSpace Hregion_u_2D; }
+      { Name I; Type Global; NameOfSpace Hregion_u_2D [I]; }
+      { Name U; Type Global; NameOfSpace Hregion_u_2D [U]; }
+
+      { Name ir; Type Local; NameOfSpace Hregion_i_2D; }
+      { Name Us; Type Global; NameOfSpace Hregion_i_2D [Us]; }
+      { Name Is; Type Global; NameOfSpace Hregion_i_2D [Is]; }
+
+      If(Flag_CircuitCoupling)
+        { Name Uz; Type Global; NameOfSpace Hregion_Z [Uz]; }
+        { Name Iz; Type Global; NameOfSpace Hregion_Z [Iz]; }
+      EndIf
+    }
+    Equation {
+      Integral { [ nu[] * Dof{d a} , {d a} ];
+        In Vol_L_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      If(Flag_NewtonRaphson)
+        Integral { [ nu[{d a}] * {d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+        Integral { [ dhdb[{d a}] * Dof{d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+        Integral { [ - dhdb[{d a}] * {d a} , {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+      Else
+        Integral { [ nu[{d a}] * Dof{d a}, {d a} ];
+          In Vol_NL_Mag; Jacobian Vol; Integration Gauss_v; }
+      EndIf
+
+      Integral { [ - nu[] * br[] , {d a} ];
+        In Vol_M_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      // Electric field e = -Dt[{a}]-{ur},
+      // with {ur} = Grad v constant in each region of Vol_C_Mag
+      Integral { DtDof [ sigma[] * Dof{a} , {a} ];
+        In Vol_C_Mag; Jacobian Vol; Integration Gauss_v; }
+      Integral { [ sigma[] * Dof{ur} / CoefGeos[] , {a} ];
+        In Vol_C_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ - sigma[] * (Velocity[] /\ Dof{d a}) , {a} ];
+        In Vol_V_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ - js0[] , {a} ];
+        In Vol_S0_Mag; Jacobian Vol; Integration Gauss_v; }
+
+      Integral { [ nxh[] , {a} ];
+        In Sur_Neu_Mag; Jacobian Sur; Integration Gauss_v; }
+
+      Integral { DtDof [  Ysur[] * Dof{a} , {a} ];
+        In Sur_Imped_Mag; Jacobian Sur; Integration Gauss_v; }
+      Integral { [ Ysur[] * Dof{ur} / CoefGeos[] , {a} ];
+        In Sur_Imped_Mag; Jacobian Sur; Integration Gauss_v; }
+
+      // When {ur} act as a test function, one obtains the circuits relations,
+      // relating the voltage and the current of each region in Vol_C_Mag
+      Integral { DtDof [ sigma[] * Dof{a} , {ur} ];
+        In Vol_C_Mag; Jacobian Vol; Integration Gauss_v; }
+      Integral { [ sigma[] * Dof{ur} / CoefGeos[] , {ur} ];
+        In Vol_C_Mag; Jacobian Vol; Integration Gauss_v; }
+      GlobalTerm { [ Dof{I} *(CoefGeos[]/Fabs[CoefGeos[]]) , {U} ]; In Vol_C_Mag; }
+
+      Integral { DtDof [ Ysur[] * Dof{a} , {ur} ];
+        In Sur_Imped_Mag; Jacobian Sur; Integration Gauss_v; }
+      Integral { [ Ysur[] * Dof{ur} / CoefGeos[] , {ur} ];
+        In Sur_Imped_Mag; Jacobian Sur; Integration Gauss_v; }
+      GlobalTerm { [ Dof{I} *(CoefGeos[]/Fabs[CoefGeos[]]) , {U} ]; In Sur_Imped_Mag; }
+
+      // js[0] should be of the form: Ns[]/Sc[] * Vector[0,0,1]
+      Integral { [ - (js0[]*Vector[0,0,1]) * Dof{ir} , {a} ];
+        In Vol_S_Mag; Jacobian Vol; Integration Gauss_v; }
+      Integral { DtDof [ Ns[]/Sc[] * Dof{a} , {ir} ];
+        In Vol_S_Mag; Jacobian Vol; Integration Gauss_v; }
+      Integral { [ Ns[]/Sc[] / sigma[] * (js0[]*Vector[0,0,1]) * Dof{ir} , {ir} ];
+        In Vol_S_Mag; Jacobian Vol; Integration Gauss_v; }
+      GlobalTerm { [ Dof{Us} / CoefGeos[] , {Is} ]; In Vol_S_Mag; }
+      // Attention: CoefGeo[.] = 2*Pi for Axi
+
+      GlobalTerm { [ - Dof{I_perfect} , {A_floating} ]; In Sur_Perfect_Mag; }
+
+      If(Flag_CircuitCoupling)
+	GlobalTerm { NeverDt[ Dof{Uz} , {Iz} ]; In Resistance_Cir; }
+        GlobalTerm { NeverDt[ Resistance[] * Dof{Iz} , {Iz} ]; In Resistance_Cir; }
+
+	GlobalTerm { [ Dof{Uz} , {Iz} ]; In Inductance_Cir; }
+	GlobalTerm { DtDof [ Inductance[] * Dof{Iz} , {Iz} ]; In Inductance_Cir; }
+
+	GlobalTerm { NeverDt[ Dof{Iz} , {Iz} ]; In Capacitance_Cir; }
+	GlobalTerm { DtDof [ Capacitance[] * Dof{Uz} , {Iz} ]; In Capacitance_Cir; }
+
+	GlobalTerm { NeverDt[ Dof{Uz} , {Iz} ]; In Diode_Cir; }
+	GlobalTerm { NeverDt[ Resistance[{Uz}] * Dof{Iz} , {Iz} ]; In Diode_Cir; }
+
+	GlobalTerm { [ 0. * Dof{Iz} , {Iz} ]; In DomainSource_Cir; }
+
+	GlobalEquation {
+	  Type Network; NameOfConstraint ElectricalCircuit;
+	  { Node {I};  Loop {U};  Equation {I};  In Vol_C_Mag; }
+	  { Node {Is}; Loop {Us}; Equation {Us}; In Vol_S_Mag; }
+	  { Node {Iz}; Loop {Uz}; Equation {Uz}; In Domain_Cir; }
+	}
+      EndIf
+
+    }
+  }
+}
+
+Resolution {
+  { Name MagDyn_a_2D;
+    System {
+      { Name Sys; NameOfFormulation MagDyn_a_2D;
+        If(Flag_FrequencyDomain)
+          Type ComplexValue; Frequency Freq;
+        EndIf
+      }
+    }
+    Operation {
+      If(Flag_FrequencyDomain)
+        Generate[Sys]; Solve[Sys]; SaveSolution[Sys];
+      Else
+        InitSolution[Sys]; // provide initial condition
+        TimeLoopTheta[TimeInit, TimeFinal, DeltaTime, 1.]{
+          // Euler implicit (1) -- Crank-Nicolson (0.5)
+          Generate[Sys]; Solve[Sys];
+          If(NbrRegions[Vol_NL_Mag])
+            Generate[Sys]; GetResidual[Sys, $res0];
+            Evaluate[ $res = $res0, $iter = 0 ];
+            Print[{$iter, $res, $res / $res0},
+              Format "Residual %03g: abs %14.12e rel %14.12e"];
+            While[$res > NL_tol_abs && $res / $res0 > NL_tol_rel &&
+                  $res / $res0 <= 1 && $iter < NL_iter_max]{
+              Solve[Sys]; Generate[Sys]; GetResidual[Sys, $res];
+              Evaluate[ $iter = $iter + 1 ];
+              Print[{$iter, $res, $res / $res0},
+                Format "Residual %03g: abs %14.12e rel %14.12e"];
+            }
+          EndIf
+          SaveSolution[Sys];
+        }
+      EndIf
+    }
+  }
+  { Name MagSta_a_2D;
+    System {
+      { Name Sys; NameOfFormulation MagSta_a_2D; }
+    }
+    Operation {
+      InitSolution[Sys];
+      Generate[Sys]; Solve[Sys];
+      If(NbrRegions[Vol_NL_Mag])
+        Generate[Sys]; GetResidual[Sys, $res0];
+        Evaluate[ $res = $res0, $iter = 0 ];
+        Print[{$iter, $res, $res / $res0},
+          Format "Residual %03g: abs %14.12e rel %14.12e"];
+        While[$res > NL_tol_abs && $res / $res0 > NL_tol_rel &&
+              $res / $res0 <= 1 && $iter < NL_iter_max]{
+          Solve[Sys]; Generate[Sys]; GetResidual[Sys, $res];
+          Evaluate[ $iter = $iter + 1 ];
+          Print[{$iter, $res, $res / $res0},
+            Format "Residual %03g: abs %14.12e rel %14.12e"];
+        }
+      EndIf
+      SaveSolution[Sys];
+    }
+  }
+}
+
+// Same PostProcessing for both static and dynamic formulations (both refer to
+// the same FunctionSpace from which the solution is obtained)
+PostProcessing {
+  { Name MagDyn_a_2D; NameOfFormulation MagDyn_a_2D;
+    PostQuantity {
+      // In 2D, a is a vector with only a z-component: (0,0,az)
+      { Name a; Value {
+          Term { [ {a} ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      // The equilines of az are field lines (giving the magnetic field direction)
+      { Name az; Value {
+          Term { [ CompZ[{a}] ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name b; Value {
+          Term { [ {d a} ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name norm_of_b; Value {
+          Term { [ Norm[{d a}] ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name h; Value {
+          Term { [ nu[] * {d a} ]; In Vol_Mag; Jacobian Vol; }
+          Term { [ -nu[] * br[] ]; In Vol_M_Mag; Jacobian Vol; }
+        }
+      }
+      { Name js; Value {
+          Term { [ js0[] ]; In Vol_S0_Mag; Jacobian Vol; }
+          Term { [  (js0[]*Vector[0,0,1])*{ir} ]; In Vol_S_Mag; Jacobian Vol; }
+	  // to force a vector result out of sources
+          Term { [ Vector[0,0,0] ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name j; Value {
+          Term { [ -sigma[] * (Dt[{a}]+{ur}/CoefGeos[]) ]; In Vol_C_Mag; Jacobian Vol; }
+          Term { [ js0[] ]; In Vol_S0_Mag; Jacobian Vol; }
+          Term { [ (js0[]*Vector[0,0,1])*{ir} ]; In Vol_S_Mag; Jacobian Vol; }
+          Term { [ Vector[0,0,0] ]; In Vol_Mag; Jacobian Vol; }
+          // Current density in A/m
+          Term { [ -Ysur[] * (Dt[{a}]+{ur}/CoefGeos[]) ]; In Sur_Imped_Mag; Jacobian Sur; }
+        }
+      }
+      { Name JouleLosses; Value {
+          Integral { [ CoefPower * sigma[]*SquNorm[Dt[{a}]+{ur}/CoefGeos[]] ];
+            In Vol_C_Mag; Jacobian Vol; Integration Gauss_v; }
+          Integral { [ CoefPower * 1./sigma[]*SquNorm[js0[]] ];
+            In Vol_S0_Mag; Jacobian Vol; Integration Gauss_v; }
+	  Integral { [ CoefPower * 1./sigma[]*SquNorm[(js0[]*Vector[0,0,1])*{ir}] ];
+            In Vol_S_Mag; Jacobian Vol; Integration Gauss_v; }
+          Integral { [ CoefPower * Ysur[]*SquNorm[Dt[{a}]+{ur}/CoefGeos[]] ];
+            In Sur_Imped_Mag; Jacobian Sur; Integration Gauss_v; }
+	}
+      }
+      { Name U; Value {
+          Term { [ {U} ]; In Vol_C_Mag; }
+          Term { [ {Us} ]; In Vol_S_Mag; }
+          If(Flag_CircuitCoupling)
+            Term { [ {Uz} ]; In Domain_Cir; }
+          EndIf
+        }
+      }
+      { Name I; Value {
+          Term { [ {I} ]; In Vol_C_Mag; }
+          Term { [ {Is} ]; In Vol_S_Mag; }
+          If(Flag_CircuitCoupling)
+            Term { [ {Iz} ]; In Domain_Cir; }
+          EndIf
+        }
+      }
+    }
+  }
+
+  { Name MagSta_a_2D; NameOfFormulation MagSta_a_2D;
+    PostQuantity {
+      { Name a; Value {
+          Term { [ {a} ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name az; Value {
+          Term { [ CompZ[{a}] ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name b; Value {
+          Term { [ {d a} ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name h; Value {
+          Term { [ nu[] * {d a} ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+      { Name j; Value {
+          Term { [ js0[] ]; In Vol_S0_Mag; Jacobian Vol; }
+          Term { [ (js0[]*Vector[0,0,1])*{ir} ]; In Vol_S_Mag; Jacobian Vol; }
+          Term { [ Vector[0,0,0] ]; In Vol_Mag; Jacobian Vol; }
+        }
+      }
+    }
+  }
+}