8. Boundary and interface conditions

Chapter status

Illustrative · compile-only. The boundary components target the current 0.1.0a7 Fluid and Solid packages.

A domain is not isolated by default

Interior equations describe behavior inside a region. Boundary conditions say how that region meets its environment; interface conditions say how two modeled regions meet each other. These relations are part of the mathematical model, not settings to leave for a solver to invent.

Learning outcomes

After this chapter, you should be able to:

• distinguish prescribed trace data from prescribed normal flux or traction;
• read the standard fluid and solid boundary components;
• explain continuity and action/reaction at an interface; and
• diagnose missing or contradictory boundary assumptions.

Trace and flux form a boundary pair

An essential condition prescribes a field trace, such as velocity or displacement. A natural condition prescribes its conjugate flux, such as traction. The names describe mathematical roles; whether a condition is physically appropriate depends on the modeled boundary.

The standard fluid package includes no-slip and traction-free components:

eqiora

public component NoSlip2d(
  support body: volume(ambient_dimension = 2),
  support face: boundary(parent = body),
) {
  public port mechanical:
    conserving mechanics.VelocityTractionBoundary over face;

  relation prescribed_velocity on face {
    trace(mechanical) = 0;
  }
}

public component TractionFree2d() {
  // same support and port structure
  relation prescribed_traction on face {
    flux(mechanical) = 0;
  }
}

NormalPressureOutlet2d supplies a nonzero normal traction from an exterior pressure. NormalVelocityInlet2d reads a scalar speed Field and prescribes the corresponding inward parent-normal velocity. The solid package uses the same pattern for FixedDisplacement2d and its own TractionFree2d.

Read the fluid boundary components · read the solid boundary components.

Interfaces exchange equal and opposite action

At an interface between compatible regions, the trace variable is continuous and the outward fluxes balance. With outward normals defined separately for the two regions, a compact statement is

$$\operatorname{tr}u_1 = \operatorname{tr}u_2, \qquad f_1 + f_2 = 0.$$

The first relation is compatibility. The second is action/reaction. Conserving ports let components expose this pair without deciding in advance which side computes the trace and which computes the flux.

The package interface components connect the interior field to a boundary port:

eqiora

relation boundary_interface[boundary in exterior] on boundary {
  trace(velocity)
    - trace(mechanical[boundary = boundary]) = 0;
  normal(
    2 * dynamic_viscosity * symmetric_part(grad(velocity))
    - isotropic_lift(pressure)
  ) - flux(mechanical[boundary = boundary]) = 0;
}

Choose conditions from the phenomenon

A wall may justify no slip; a free solid surface may justify zero traction; an outlet may justify a prescribed normal pressure. These are different physical assumptions even when they use the same boundary support machinery. A complete set must constrain the intended problem without prescribing incompatible data.

Deliberate failure: fix and free the same quantity

Suppose the same solid face is declared both fixed and subject to a prescribed nonzero displacement. Each relation can be dimensionally valid on its own, but together they contradict one another. Conversely, omitting every condition that removes rigid translation can leave an elasticity problem undetermined.

Count and interpret the boundary freedoms before choosing a realization. A successful syntax check cannot decide whether the physical boundary story is complete.

Exercises

1. Classify no slip, fixed displacement, traction free, and normal pressure as trace or flux conditions.
2. Explain why outward tractions on two sides of an interface sum to zero rather than being textually identical.
3. Assign conditions to the inlet, outlet, walls, and cylinder in the checked-in steady-flow example. State the physical assumption behind each choice.
4. For an elastic body in two dimensions, explain why traction-free boundaries alone do not remove rigid translation.
5. Propose one fluid–solid interface pair: which trace quantity should be compatible, and which actions should balance?

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