5. Fields and spatial domains

A field needs somewhere to live

A scalar such as pressure assigns one value to each point of a region. A velocity field assigns a spatial vector. Writing only a quantity and its unit is therefore incomplete: a spatial model must also say which support carries the field and what shape its values have.

Eqiora separates the semantic support from concrete geometry. A volume says that a field lives in a region; its boundaries identify the places where the interior model meets its surroundings. Meshes and discretization choices come later.

Learning outcomes

After this chapter, you should be able to:

distinguish a field’s role, support, dimension, and value shape;
read grad, div, trace, and normal by their mathematical roles;
identify which quantities live in a volume and which are compared on its boundary; and
diagnose an operator applied to the wrong kind of field.

Name the region and its boundaries

The steady cylinder example declares one two-dimensional fluid volume and four named parts of its boundary:

component FlowRegion(
  support fluid: volume(ambient_dimension = 2),
  support inlet: boundary(parent = fluid),
  support outlet: boundary(parent = fluid),
  support walls: boundary(parent = fluid),
  support cylinder: boundary(parent = fluid),
) {
  variable velocity: vector<m / s, 2> on fluid;
  variable pressure: kg / (m * s ^ 2) on fluid;
  // Interior and boundary equations complete the component.
}

velocity and pressure share the support fluid, but their value shapes differ. The four boundary supports do not create four new fluids; they name where distinct boundary relations apply.

Read spatial operators by where they act

For a scalar field $p$ and vector field $\boldsymbol{u}$ :

Expression	Meaning	Where the result is used
$\nabla p$	spatial rate of change of a scalar	in the volume
$\nabla\!\cdot\boldsymbol{u}$	net outward tendency per unit volume	in the volume
$\operatorname{tr}\boldsymbol{u}$	interior value restricted to a boundary	on the boundary
$\boldsymbol{T}\boldsymbol{n}$	normal action of a tensor	on the boundary

The operators also transform dimensions. If velocity has dimension $L/T$ , its divergence has dimension $1/T$ . If pressure has dimension force per area, its gradient has dimension force per volume. Shape and dimension are separate checks, and both must make sense.

The model uses these roles directly:

relation incompressibility on fluid {
  div(velocity) = 0;
}

relation wall_velocity on walls {
  trace(velocity) = 0;
}

The first relation is an interior statement. The second compares the boundary trace of the interior velocity with a prescribed zero velocity.

Read the complete steady-flow model.

A spatial equation with a known solution

For a scalar field on the unit square, a useful checking problem is

-\operatorname{div}(\operatorname{grad} u) =2\pi^2\sin(\pi x)\sin(\pi y), \qquad u=0\text{ on all four sides}.

The source is chosen so the solution is known: $u(x,y)=\sin(\pi x)\sin(\pi y)$ . This is a scalar Poisson problem, not the cylinder-flow model above. Knowing the answer in advance lets a numerical comparison measure error instead of relying on a plausible-looking picture.

The Poisson verification package contains its Eqiora declarations, and the registered Poisson case describes the numerical comparison. For a complete public spatial workflow, follow the cylinder Gallery case, whose equations and execution both refer to the same flow problem.

Deliberate failure: divergence of a scalar

Replace the incompressibility relation with div(pressure) = 0. Pressure is a scalar, while divergence here expects a spatial vector or tensor. The field is still on the right support and its unit is still valid, but the operator has no matching shape. Repair the mathematical statement; changing the mesh or solver cannot make the expression meaningful.

Exercises

State the support, dimension, and value shape of velocity and pressure.
Explain why trace(velocity) belongs in a relation on walls, not in the volume relation for incompressibility.
If temperature has dimension $\Theta$ , what dimension does its gradient have?
Add a named symmetry boundary to the support declarations. What physical relation would still need to be supplied before the name has meaning?

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