Boundary Conditions Reference

Boundary conditions in Exasim have two layers:

1. pdeapp.txt or frontend fields map geometry to integer boundary IDs.
2. Model functions dispatch on the integer boundary ID and compute boundary values or fluxes.

Boundary Geometry

pdeapp.txt uses:

boundaryconditions = [1, 2, 3, 4];
boundaryexpressions = [
  "x < 1e-8",
  "x > 1-1e-8",
  "y < 1e-8",
  "y > 1-1e-8"
];

Each boundary expression corresponds to one boundary condition code. The meaning of the code is model-defined.

Boundary Function Interfaces

Ubou

Defines boundary state/value data.

function Ubou(x, uq, v, w, uhat, n, tau, eta, mu, t)
  output_size(ub) = ncu;
  ...
end

Fbou

Defines LDG boundary flux.

function Fbou(x, uq, v, w, uhat, n, tau, eta, mu, t)
  output_size(fb) = ncu;
  ...
end

FbouHdg

Defines HDG boundary residual/flux.

function FbouHdg(x, uq, v, w, uhat, n, tau, eta, mu, t)
  output_size(fb) = ncu;
  ...
end

Direct C++ functions receive an additional int ib boundary ID and dispatch on it.

Boundary Types

Exasim does not enforce universal meanings for integer boundary codes. Common application conventions include:

Type	Typical implementation
Dirichlet	Ubou returns imposed state; FbouHdg penalizes trace/state mismatch.
Neumann	Boundary flux is set directly in Fbou or FbouHdg.
Robin	Boundary flux combines state and flux terms.
Wall	Model-specific no-penetration/no-slip/thermal condition.
Inflow	Prescribed physical state or characteristic inflow.
Outflow	Pressure, extrapolation, or characteristic outflow.
Farfield	Freestream or characteristic boundary model.

The exact formulas belong to the PDE model.

Periodic Boundaries

Periodic boundaries are configured with:

periodicboundaries1 = [1];
periodicexprs1 = ["x"];
periodicboundaries2 = [2];
periodicexprs2 = ["x - 1.0"];

The preprocessing layer matches periodic faces; model boundary functions should not implement periodicity manually.

Coupled Interfaces

Coupled interfaces use:

• interfaceconditions
• interfacefluxmap
• extFhat
• extUhat
• extStab

These enable advanced interface flux hooks through provider callbacks such as Fint, Fext, Fhat, Uhat, and Stab.

Common Errors

Symptom	Likely issue
Boundary faces unassigned	Expression does not match mesh coordinates within tolerance.
Wrong physical boundary behavior	Boundary code ordering does not match model dispatch.
HDG Newton failure near boundary	FbouHdg Jacobians inconsistent with FbouHdg.
MPI periodic mismatch	Periodic boundary expressions or node matching inconsistent.