DAE Lowering and Structural Analysis

DAE Lowering (rumoca-phase-dae)

DAE lowering eliminates the Modelica-specific surface and produces the MLS Appendix B canonical system (see The Four IRs). The transformations with the most moving parts:

• pre() elimination. Every pre(x) becomes an explicit __pre__.* parameter slot; the runtime writes those slots at event entry. This must hold in all partitions (f_x, f_z, f_m, f_c) — pre exists only in AST and Flat.
• when/reinit lowering. When-clauses become event conditions plus guarded discrete update equations; reinit(x, e) becomes a guarded state update over current/pre slots.
• Relation extraction. Every relational expression that can generate an event gets a relation/condition variable; conditions.relations is the single owner of that surface. Synthetic numeric roots (from abs, sign) live separately in events.synthetic_root_conditions.
• sample/clocks. Scheduled events are explicit DAE metadata, not expressions the runtime has to discover.
• assert/terminate. Lowered to guarded events.event_actions with source spans, keeping the compute graphs pure.

A positive validation gate (appendix_b_validation) rejects any surviving source temporal operator — the contract is enforced, not assumed.

Structural Analysis (rumoca-phase-structural)

Between the DAE and a runnable system sits structural preparation:

1. Matching. Assign each equation to the unknown it will compute (maximum bipartite matching). Failure here is the structurally singular system diagnostic users see, which names unmatched equations and unknowns.
2. Sorting (BLT). Order the matched system into block lower-triangular form: a sequence of scalar assignments and strongly connected components (simultaneous blocks).
3. Tearing. Within coupled blocks, choose tearing variables so a small nonlinear core is iterated while the rest is evaluated explicitly.
4. Index reduction. Higher-index DAEs are reduced with the Mattsson–Söderlind dummy-derivative method: constraint equations are differentiated and some derivative-defined states are demoted to dummy variables. State selection decides which candidates remain integrator states.
5. Scalarization (when a backend requires it) expands symbolic arrays using shape metadata — never by parsing display strings.

The analysis products (matching, BLT blocks, tearing choices, state selection reports) are returned as separate artifacts — they are inputs to Solve lowering and to --inspect structure, and deliberately not stored in the DAE.

Watching It Work

rumoca compile Model.mo --inspect structure   # matching, BLT, SCCs, tearing
rumoca sim Model.mo --inspect eval            # values + derivatives at a point
rumoca sim Model.mo --inspect jacobian --at "x=1@0"

These are the same tools the user guide teaches for debugging models — as a compiler developer you will mostly use them to verify that a lowering or matching change did what you intended on a specific model.