Constraints and interactions

A Constraint expresses a relation between fields through weak terms. It owns the multiplier field and contributes the relation to the residual rows of the multiplier and its participants. An Interaction is the broader category for relation-specific terms that may also own geometry, search, transfer operators, or native-provider state.

The multiplier is an algebraic field on an independently selected finite- element space. It does not borrow a participant’s DoFHandler. All weak terms in one constraint use the same unregistered multiplier field:

auto lambda = multiplier_space.field("lambda");
auto first = weak_term(value(u), test(lambda));
auto second = weak_term(value(v), test(lambda));
auto constraint = make_constraint(first - second, rhs);
auto fields = adapter.add(constraint, "continuity");

The multiplier row receives the signed weak-term sum minus rhs. Participant rows receive the transpose reactions. The same construction handles scalar and vector fields; the FE extractor determines the value type.

The weak-term backend chooses direct cell traversal, paired DoFHandler traversal, or nonmatching point search from the spaces and geometry. This choice does not change the constraint API. Distributed point ownership and search use deal.II’s distributed particle facilities where the coupling requires them.

The MetricFlowX vessel-wall example expresses its cell-volume pressure feedback as a WeakTerm and its kinematic relation as a Constraint. The application adds both Problems and the constraint to the same IDAAdapter.