Boundary conditions

Like the variable initialisation, boundary conditions can be set for each variable in individual sections, with default values in a section [All]. Boundary conditions are specified for each variable, being applied to variable itself during initialisation, and the time-derivatives at each timestep. They are a combination of a basic boundary condition, and optional modifiers.

When finding the boundary condition for a variable var on a boundary region, the options are checked in order from most to least specific:

  • Section var, bndry_ + region name. Depending on the mesh file, regions of the grid are given labels. Currently these are core, sol, pf and target which are intended for tokamak edge simulations. Hence the variables checked are bndry_core, bndry_pf etc.
  • Section var, bndry_ + boundary side. These names are xin, xout, yup and ydown.
  • Section var, variable bndry_all
  • The same settings again except in section All.

The default setting for everything is therefore bndry_all in the All section.

Boundary conditions are given names, with optional arguments in brackets. Currently implemented boundary conditions are:

  • dirichlet - Set to zero
  • dirichlet(<number>) - Set to some number e.g. dirichlet(1) sets the boundary to \(1.0\)
  • neumann - Zero gradient
  • robin - A combination of zero-gradient and zero-value \(a f + b{{\frac{\partial f}{\partial x}}} = g\) where the syntax is robin(a, b, g).
  • constgradient - Constant gradient across boundary
  • zerolaplace - Laplacian = 0, decaying solution (X boundaries only)
  • zerolaplace2 - Laplacian = 0, using coefficients from the Laplacian inversion and Delp2 operator.
  • constlaplace - Laplacian = const, decaying solution (X boundaries only)

The zero- or constant-Laplacian boundary conditions works as follows:

\[\begin{split}\nabla_\perp^2 f &= 0 \\ &\simeq g^{xx}\frac{\partial^2 f}{\partial x^2} + g^{zz}\frac{\partial^2 f}{\partial z^2}\end{split}\]

which when Fourier transformed in \(z\) becomes:

\[g^{xx}\frac{\partial^2 \hat{f}}{\partial x^2} - g^{zz}k_z^2 \hat{f} = 0\]

which has the solution

\[\hat{f} = Ae^{xk_z\sqrt{g^{zz}/g^{xx}}} + Be^{-xk_z\sqrt{g^{zz}/g^{xx}}}\]

Assuming that the solution should decay away from the domain, on the inner \(x\) boundary \(B = 0\), and on the outer boundary \(A = 0\). Boundary modifiers change the behaviour of boundary conditions, and more than one modifier can be used. Currently the following are available:

  • relax - Relaxing boundaries. Evolve the variable towards the given boundary condition at a given rate
  • shifted - Apply boundary conditions in orthogonal X-Z coordinates, rather than field-aligned
  • width - Modifies the width of the region over which the boundary condition is applied

These are described in the following subsections.

Relaxing boundaries

All boundaries can be modified to be “relaxing” which are a combination of zero-gradient time-derivative, and whatever boundary condition they are applied to. The idea is that this prevents sharp discontinuities at boundaries during transients, whilst maintaining the desired boundary condition on longer time-scales. In some cases this can improve the numerical stability and timestep.

For example, relax(dirichlet) will make a field \(f\) at point \(i\) in the boundary follow a point \(i-1\) in the domain:

\[.{{\frac{\partial f}{\partial t}}}|_i = .{{\frac{\partial f}{\partial t}}}|_{i-1} - f_i / \tau\]

where \(\tau\) is a time-scale for the boundary (currently set to 0.1, but will be a global option). When the time-derivatives are slow close to the boundary, the boundary relaxes to the desired condition (Dirichlet in this case), but when the time-derivatives are large then the boundary approaches Neumann to reduce discontinuities.

By default, the relaxation rate is set to \(10\) (i.e. a time-scale of \(\tau=0.1\)). To change this, give the rate as the second argument e.g. relax(dirichlet, 2) would relax to a Dirichlet boundary condition at a rate of \(2\).

Shifted boundaries

By default boundary conditions are applied in field-aligned coordinates, where \(y\) is along field-lines but \(x\) has a discontinuity at the twist-shift location. If radial derivatives are being done in shifted coordinates where \(x\) and \(z\) are orthogonal, then boundary conditions should also be applied in shifted coordinates. To do this, the shifted boundary modifier applies a \(z\) shift, applies the boundary condition, then shifts back. For example:

bndry_core = shifted( neumann )

would ensure that radial derivatives were zero in shifted coordinates on the core boundary.

Changing the width of boundaries

To change the width of a boundary region, the width modifier changes the width of a boundary region before applying the boundary condition, then changes the width back afterwards. To use, specify the boundary condition and the width, for example

bndry_core = width( neumann , 4 )

would apply a Neumann boundary condition on the innermost 4 cells in the core, rather than the usual 2. When combining with other boundary modifiers, this should be applied first e.g.

bndry_sol = width( relax( dirichlet ), 3)

would relax the last 3 cells towards zero, whereas

bndry_sol = relax( width( dirichlet, 3) )

would only apply to the usual 2, since relax didn’t use the updated width.

Limitations:

  1. Because it modifies then restores a globally-used BoundaryRegion, this code is not thread safe.
  2. Boundary conditions can’t be applied across processors, and no checks are done that the width asked for fits within a single processor.

Examples

This example is taken from the UEDGE benchmark test (in examples/uedge-benchmark):

[All]
bndry_all = neumann # Default for all variables, boundaries

[Ni]
bndry_target = neumann
bndry_core = relax(dirichlet(1.))   # 1e13 cm^-3 on core boundary
bndry_all  = relax(dirichlet(0.1))  # 1e12 cm^-3 on other boundaries

[Vi]
bndry_ydown = relax(dirichlet(-1.41648))   # -3.095e4/Vi_x
bndry_yup   = relax(dirichlet( 1.41648))

The variable Ni (density) is set to a Neumann boundary condition on the targets (yup and ydown), relaxes towards \(1\) on the core boundary, and relaxes to \(0.1\) on all other boundaries. Note that the bndry_target = neumann needs to be in the Ni section: If we just had

[All]
bndry_all = neumann # Default for all variables, boundaries

[Ni]
bndry_core = relax(dirichlet(1.))   # 1e13 cm^-3 on core boundary
bndry_all  = relax(dirichlet(0.1))  # 1e12 cm^-3 on other boundaries

then the “target” boundary condition for Ni would first search in the [Ni] section for bndry_target, then for bndry_all in the [Ni] section. This is set to relax(dirichlet(0.1)), not the Neumann condition desired.

Boundary regions

The boundary condition code needs ways to loop over the boundary regions, without needing to know the details of the mesh.

At the moment two mechanisms are provided: A RangeIterator over upper and lower Y boundaries, and a vector of BoundaryRegion objects.

// Boundary region iteration
virtual const RangeIterator iterateBndryLowerY() const = 0;
virtual const RangeIterator iterateBndryUpperY() const = 0;

bool hasBndryLowerY();
bool hasBndryUpperY();

bool BoundaryOnCell; // NB: DOESN'T REALLY BELONG HERE

The RangeIterator class is an iterator which allows looping over a set of indices. For example, in src/solver/solver.cxx to loop over the upper Y boundary of a 2D variable var:

for(RangeIterator xi = mesh->iterateBndryUpperY(); !xi.isDone(); xi++) {
  ...
}

The BoundaryRegion class is defined in include/boundary_region.hxx

Boundary regions

Different regions of the boundary such as “core”, “sol” etc. are labelled by the Mesh class (i.e. BoutMesh), which implements a member function defined in mesh.hxx:

// Boundary regions
virtual vector<BoundaryRegion*> getBoundaries() = 0;

This returns a vector of pointers to BoundaryRegion objects, each of which describes a boundary region with a label, a BndryLoc location (i.e. inner x, outer x, lower y, upper y or all), and iterator functions for looping over the points. This class is defined in boundary_region.hxx:

/// Describes a region of the boundary, and a means of iterating over it
class BoundaryRegion {
  public:
  BoundaryRegion();
  BoundaryRegion(const string &name, int xd, int yd);
  virtual ~BoundaryRegion();

  string label; // Label for this boundary region

  BndryLoc location; // Which side of the domain is it on?

  int x,y; // Indices of the point in the boundary
  int bx, by; // Direction of the boundary [x+dx][y+dy] is going outwards

  virtual void first() = 0;
  virtual void next() = 0; // Loop over every element from inside out (in X or
Y first)
  virtual void nextX() = 0; // Just loop over X
  virtual void nextY() = 0; // Just loop over Y
  virtual bool isDone() = 0; // Returns true if outside domain. Can use this
with nested nextX, nextY
};

Example: To loop over all points in BoundaryRegion *bndry , use

for(bndry->first(); !bndry->isDone(); bndry->next()) {
  ...
}

Inside the loop, bndry->x and bndry->y are the indices of the point, whilst bndry->bx and bndry->by are unit vectors out of the domain. The loop is over all the points from the domain outwards i.e. the point [bndry->x - bndry->bx][bndry->y - bndry->by] will always be defined.

Sometimes it’s useful to be able to loop over just one direction along the boundary. To do this, it is possible to use nextX() or nextY() rather than next(). It is also possible to loop over both dimensions using:

for(bndry->first(); !bndry->isDone(); bndry->nextX())
  for(; !bndry->isDone(); bndry->nextY()) {
    ...
  }

Boundary operations

On each boundary, conditions must be specified for each variable. The different conditions are imposed by BoundaryOp objects. These set the values in the boundary region such that they obey e.g. Dirichlet or Neumann conditions. The BoundaryOp class is defined in boundary_op.hxx:

/// An operation on a boundary
class BoundaryOp {
 public:
  BoundaryOp() {bndry = NULL;}
  BoundaryOp(BoundaryRegion *region)

  // Note: All methods must implement clone, except for modifiers (see below)
  virtual BoundaryOp* clone(BoundaryRegion *region, const list<string> &args);

  /// Apply a boundary condition on field f
  virtual void apply(Field2D &f) = 0;
  virtual void apply(Field3D &f) = 0;

  virtual void apply(Vector2D &f);

  virtual void apply(Vector3D &f);

  /// Apply a boundary condition on ddt(f)
  virtual void apply_ddt(Field2D &f);
  virtual void apply_ddt(Field3D &f);
  virtual void apply_ddt(Vector2D &f);
  virtual void apply_ddt(Vector3D &f);

  BoundaryRegion *bndry;
};

(where the implementations have been removed for clarity). Which has a pointer to a BoundaryRegion object specifying which region this boundary is operating on.

Boundary conditions need to be imposed on the initial conditions (after PhysicsModel::init()), and on the time-derivatives (after PhysicsModel::rhs()). The apply() functions are therefore called during initialisation and given the evolving variables, whilst the apply_ddt functions are passed the time-derivatives.

To implement a boundary operation, as a minimum the apply(Field2D), apply(Field2D) and clone() need to be implemented: By default the apply(Vector) will call the apply(Field) functions on each component individually, and the apply_ddt() functions just call the apply() functions.

Example: Neumann boundary conditions are defined in boundary_standard.hxx:

/// Neumann (zero-gradient) boundary condition
class BoundaryNeumann : public BoundaryOp {
 public:
  BoundaryNeumann() {}
 BoundaryNeumann(BoundaryRegion *region):BoundaryOp(region) { }
  BoundaryOp* clone(BoundaryRegion *region, const list<string> &args);
  void apply(Field2D &f);
  void apply(Field3D &f);
};

and implemented in boundary_standard.cxx

void BoundaryNeumann::apply(Field2D &f) {
  // Loop over all elements and set equal to the next point in
  for(bndry->first(); !bndry->isDone(); bndry->next())
    f[bndry->x][bndry->y] = f[bndry->x - bndry->bx][bndry->y - bndry->by];
}

void BoundaryNeumann::apply(Field3D &f) {
  for(bndry->first(); !bndry->isDone(); bndry->next())
    for(int z=0;z<mesh->LocalNz;z++)
      f[bndry->x][bndry->y][z] = f[bndry->x - bndry->bx][bndry->y -
bndry->by][z];
}

This is all that’s needed in this case since there’s no difference between applying Neumann conditions to a variable and to its time-derivative, and Neumann conditions for vectors are just Neumann conditions on each vector component.

To create a boundary condition, we need to give it a boundary region to operate over:

BoundaryRegion *bndry = ...
BoundaryOp op = new BoundaryOp(bndry);

The clone function is used to create boundary operations given a single object as a template in BoundaryFactory. This can take additional arguments as a vector of strings - see explanation in Boundary factory.

Boundary modifiers

To create more complicated boundary conditions from simple ones (such as Neumann conditions above), boundary operations can be modified by wrapping them up in a BoundaryModifier object, defined in boundary_op.hxx:

class BoundaryModifier : public BoundaryOp {
 public:
  virtual BoundaryOp* clone(BoundaryOp *op, const list<string> &args) = 0;
 protected:
  BoundaryOp *op;
};

Since BoundaryModifier inherits from BoundaryOp, modified boundary operations are just a different boundary operation and can be treated the same (Decorator pattern). Boundary modifiers could also be nested inside each other to create even more complicated boundary operations. Note that the clone function is different to the BoundaryOp one: instead of a BoundaryRegion to operate on, modifiers are passed a BoundaryOp to modify.

Currently the only modifier is BoundaryRelax, defined in boundary_standard.hxx:

/// Convert a boundary condition to a relaxing one
class BoundaryRelax : public BoundaryModifier {
 public:
  BoundaryRelax(BoutReal rate) {r = fabs(rate);}
  BoundaryOp* clone(BoundaryOp *op, const list<string> &args);

  void apply(Field2D &f);
  void apply(Field3D &f);

  void apply_ddt(Field2D &f);
  void apply_ddt(Field3D &f);
 private:
  BoundaryRelax() {} // Must be initialised with a rate
  BoutReal r;
};

Boundary factory

The boundary factory creates new boundary operations from input strings, for example turning “relax(dirichlet)” into a relaxing Dirichlet boundary operation on a given region. It is defined in boundary_factory.hxx as a Singleton, so to get a pointer to the boundary factory use

BoundaryFactory *bfact = BoundaryFactory::getInstance();

and to delete this singleton, free memory and clean-up at the end use:

BoundaryFactory::cleanup();

Because users should be able to add new boundary conditions during PhysicsModel::init(), boundary conditions are not hard-wired into BoundaryFactory. Instead, boundary conditions must be registered with the factory, passing an instance which can later be cloned. This is done in bout++.cxx for the standard boundary conditions:

BoundaryFactory* bndry = BoundaryFactory::getInstance();
bndry->add(new BoundaryDirichlet(), "dirichlet");
...
bndry->addMod(new BoundaryRelax(10.), "relax");

where the add function adds BoundaryOp objects, whereas addMod adds BoundaryModifier objects. Note: The objects passed to BoundaryFactory will be deleted when cleanup() is called.

When a boundary operation is added, it is given a name such as “dirichlet”, and similarly for the modifiers (“relax” above). These labels and object pointers are stored internally in BoundaryFactory in maps defined in boundary_factory.hxx:

// Database of available boundary conditions and modifiers
map<string, BoundaryOp*> opmap;
map<string, BoundaryModifier*> modmap;

These are then used by BoundaryFactory::create():

/// Create a boundary operation object
BoundaryOp* create(const string &name, BoundaryRegion *region);
BoundaryOp* create(const char* name, BoundaryRegion *region);

to turn a string such as “relax(dirichlet)” and a BoundaryRegion pointer into a BoundaryOp object. These functions are implemented in boundary_factory.cxx, starting around line 42. The parsing is done recursively by matching the input string to one of:

  • modifier(<expression>, arg1, ...)
  • modifier(<expression>)
  • operation(arg1, ...)
  • operation

the <expression> variable is then resolved into a BoundaryOp object by calling create(<expression>, region).

When an operator or modifier is found, it is created from the pointer stored in the opmap or modmap maps using the clone method, passing a list<string> reference containing any arguments. It’s up to the operation implementation to ensure that the correct number of arguments are passed, and to parse them into floats or other types.

Example: The Dirichlet boundary condition can take an optional argument to change the value the boundary’s set to. In boundary_standard.cxx:

BoundaryOp* BoundaryDirichlet::clone(BoundaryRegion *region, const list<string>
&args) {
  if(!args.empty()) {
    // First argument should be a value
    stringstream ss;
    ss << args.front();

    BoutReal val;
    ss >> val;
    return new BoundaryDirichlet(region, val);
  }
  return new BoundaryDirichlet(region);
}

If no arguments are passed i.e. the string was “dirichlet” or “dirichlet()” then the args list is empty, and the default value (0.0) is used. If one or more arguments is used then the first argument is parsed into a BoutReal type and used to create a new BoundaryDirichlet object. If more arguments are passed then these are just ignored; probably a warning should be printed.

To set boundary conditions on a field, FieldData methods are defined in field_data.hxx:

// Boundary conditions
  void setBoundary(const string &name); ///< Set the boundary conditions
  void setBoundary(const string &region, BoundaryOp *op); ///< Manually set
  virtual void applyBoundary() {}
  virtual void applyTDerivBoundary() {};
 protected:
  vector<BoundaryOp*> bndry_op; // Boundary conditions

The FieldData::setBoundary() method is implemented in field_data.cxx. It first gets a vector of pointers to BoundaryRegions from the mesh, then loops over these calling BoundaryFactory::createFromOptions() for each one and adding the resulting boundary operator to the FieldData::bndry_op vector.