8. Boundary and interface conditions

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.

Can the same wall have two different solutions?

The source-free steady slab obeys $T''=0$ , so its general solution is $T=a+bx$ . Two independent constants remain. Prescribing $T(0)=T_h$ and $T(L)=T_c$ determines both, giving the familiar linear profile. If instead the left surface is held at $T_h$ and the right is insulated, $T'(L)=0$ forces $b=0$ : the whole slab reaches $T_h$ . The interior material and balance are unchanged; the environment changed the answer.

A boundary condition expresses how the surroundings act on the system. It is part of the physical question, even if software presents it in a different panel from the interior equations.

Values, fluxes, and exchange

For Fourier conduction, outward heat flux is $j_n=-\kappa\nabla T\cdot\boldsymbol n$ . Three common idealizations are:

Boundary statement	Physical picture	Mathematical data
$T=T_b$	a controlled-temperature reservoir	prescribed temperature
$j_n=q_b$	a controlled outward heat flux	prescribed normal flux
$j_n=h(T-T_\infty)$	exchange with a surrounding fluid	a relation between flux and temperature

The last is a Robin condition: the surface temperature is solved rather than prescribed. The transfer coefficient $h$ has units $\mathrm{W/(m^2K)}$ . An insulated surface uses $q_b=0$ ; it does not require zero tangential gradient. The terms “essential” and “natural” below refer to a chosen variational formulation, so their physical meaning should always be stated too.

For an illustrative slab with $T(0)=T_h$ and convection at $x=L$ , the steady heat flux is

j=\frac{T_h-T_\infty}{L/\kappa+1/h}.

Derive this by adding the slab drop $jL/\kappa$ and surface-to-room drop $j/h$ . As $h$ grows, the surface approaches the room temperature. As $h$ tends to zero, heat exchange vanishes and the slab approaches $T_h$ . These limits connect the boundary law back to the thermal-resistance network.

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:

public connector VelocityTractionBoundary {
  trace velocity: m / s;
  flux traction: kg / (m * s ^ 2);
  shape spatial_vector;
  frame spatial;
  pairing euclidean_boundary_duality;
  orientation parent_outward;
}

The connector’s vector size follows the dimension of the region that owns the boundary. Its member names mark the trace and flux roles; here shape and frame describe the connector itself rather than the type of a Field.

public component NoSlip2d(
  support body: volume(ambient_dimension = 2),
  support face: boundary(parent = body),
  port mechanical: mechanics.VelocityTractionBoundary over face
) {
  relation prescribed_velocity on face {
    mechanical.velocity = 0;
  }
}

public component TractionFree2d(
  support body: volume(ambient_dimension = 2),
  support face: boundary(parent = body),
  port mechanical: mechanics.VelocityTractionBoundary over face
) {
  relation prescribed_traction on face {
    mechanical.traction = 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:

public component VelocityTractionInterface2d(
  support body: volume(ambient_dimension = 2),
  support exterior: complete_exterior(parent = body),
  variable velocity: vector<m / s, 2> on body in smooth,
  variable pressure: kg / (m * s ^ 2) on body in h1,
  parameter dynamic_viscosity: kg / (m * s),
  port mechanical[boundary in exterior]: mechanics.VelocityTractionBoundary over boundary
) {
  relation boundary_interface[boundary in exterior] on boundary {
    trace(velocity)
      - mechanical[boundary = boundary].velocity = 0;
    normal(
      2 * dynamic_viscosity * symmetric_part(grad(velocity))
      - isotropic_lift(pressure)
    ) - mechanical[boundary = boundary].traction = 0;
  }
}

Join two materials without creating energy

At a perfectly bonded thermal interface with no surface storage, temperature is continuous and outward fluxes balance:

T_1=T_2,\qquad j_{n,1}+j_{n,2}=0.

The normals are opposite. The same rightward physical heat flux therefore has opposite outward signs on the two regions. If conductivities differ, normal temperature gradients generally differ too: continuity of temperature does not mean continuity of its derivative.

The resistance calculation from the algebraic chapter now has a continuum interpretation. Each layer obeys Fourier’s law and energy balance; the interface adds compatibility and exchange. For finite contact resistance per area $r_c$ , a temperature jump can occur, with rightward flux $j=(T_1-T_2)/r_c$ . That is a changed interface model, while conservation of total heat remains intact.

In mechanics, the corresponding pair is displacement or velocity and traction. Which trace should be continuous depends on the physics: a perfectly bonded solid interface prevents separation and slip, while a frictional contact needs a different law. Recognizing this distinction is what lets you judge whether a bundled boundary component fits your problem.

A boundary condition can leave a freedom

If both ends of a source-free slab prescribe heat flux, first check that the total outward power is zero. Otherwise no steady solution exists. Even when fluxes balance, adding a constant to $T$ leaves its gradient unchanged, so the absolute temperature is undetermined without another reference or constraint. For two insulated ends, every constant temperature is a steady solution.

A similar freedom appears in an unloaded elastic body: translating or rotating it rigidly need not change its strain. Boundary conditions must remove the unwanted rigid motions if a unique displacement is required. A solver cannot supply the missing physical reference.

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 though they attach to boundaries in the same way. 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

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

Exercise guidance

For exercise 1, no slip and fixed displacement prescribe traces; traction-free and prescribed normal traction constrain flux variables. For exercise 2, draw the two outward normals before writing the sum. For exercise 4, a constant displacement changes position without changing displacement gradient, so a stress law cannot determine that translation.

Closing investigation. Use two slab layers of thicknesses $L_1,L_2$ and conductivities $\kappa_1,\kappa_2$ , with fixed temperatures at the exterior ends. Derive the steady heat flux and interface temperature, then recover the network formula using $R_i=L_i/(\kappa_i A)$ . Add a finite contact resistance and predict which temperature becomes discontinuous. Before running a spatial model, list one balance, one constitutive assumption, and one interface assumption you would check against its result.

References

Gilbert Strang and Edwin “Jed” Herman, Calculus, volume 3 (OpenStax, 2016), section 6.8, The divergence theorem, for outward orientation in flux integrals.
Carol Livermore, Henrik Schmidt, James H. Williams, and Simona Socrate, Mechanics & Materials I, MIT OpenCourseWare 2.001 (2006), lecture notes, for equilibrium, compatibility, and the stress/strain boundary-value setting.

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