4. Interface reduction techniques

In the previous chapter, the Craig-Bampton (CB) method reduced each substructure's interior DOFs while retaining all physical interface DOFs. For a large CAE assembly, that interface can still carry thousands of DOFs — a bolted flange with 200 nodes has 1200 interface DOFs before any reduction. Interface reduction attacks this remaining bottleneck: it compresses the boundary set itself while preserving the coupling accuracy that makes CB useful in the first place.

4.1 Characteristic Constraint Modes

Characteristic Constraint (CC) modes are the eigenmodes of the substructure's constraint modes, computed with the interface DOFs free and the interior DOFs fixed. The constraint mode matrix ΨC maps unit interface displacements to interior response. Its associated interface stiffness and mass are:

K_ii = Ψ_C^T · K_ii · Ψ_C
M_ii = Ψ_C^T · M_ii · Ψ_C

Solving the small eigenproblem K_ii · Φ_CC = M_ii · Φ_CC · Λ yields the CC modes Φ_CC. The reduced interface basis is Φ_CC,r (first r modes), and the physical interface DOFs are recovered as u_i ≈ Φ_CC,r · q_CC. Typical practice retains enough CC modes to capture 90–99% of the interface modal mass, often 10–30 modes per interface regardless of mesh density.

The key advantage: CC modes are interface-local. Two substructures sharing a flange each compute their own CC basis, and the assembled reduced model uses Φ_CC,r as the coupling DOF set. This decouples interface size from mesh refinement — refining the flange mesh no longer inflates the coupled system.

4.2 Dual Craig-Bampton

Classical CB enforces interface compatibility by keeping physical interface DOFs and assembling them. Dual Craig-Bampton (DCB) takes the opposite route: it treats interface forces as the unknowns and enforces compatibility weakly through a dual assembly. Each substructure is reduced with free-interface normal modes plus residual flexibility attachment modes, and the interface is represented by a set of dual interface modes derived from the interface flexibility.

The practical consequence is that DCB avoids the artificial stiffening that CB can exhibit when interface DOFs are over-constrained, and it converges faster for structures with many redundant interface connections (e.g., spot-welded or bolted joints). The trade-off is that DCB produces a non-positive-definite assembled system and requires a dual solver or a transformation back to primal form. In CAE practice, DCB is preferred when interface compliance dominates — thin flanges, adhesive joints, or compliant bushings — while CB remains simpler for rigid interfaces.

4.3 Interface Enrichment

Interface enrichment adds local basis vectors to the interface reduction to capture high-gradient interface behavior that global CC modes miss. Typical enrichment vectors include:

• Static enrichment: residual attachment modes from unit loads at critical interface points (bolt locations, weld nuggets).
• Dynamic enrichment: local eigenmodes of the interface region computed with the interior fixed.
• Data-driven enrichment: vectors extracted from a full-order solution or from measured interface deformation.

Enrichment is most effective when the interface has a small number of high-stress or high-curvature regions. Adding 3–8 enrichment vectors per critical region often recovers 95%+ of the full-order interface response at a fraction of the cost.

4.4 Practical Implementation and Verification

Consider a representative CAE example: a two-part bolted flange assembly with 100,000 total DOFs. The interface has 200 nodes (1,200 DOFs). Using CB with 60 fixed-interface modes per substructure and 20 CC modes per interface, the reduced model has 145 DOFs. Adding 5 enrichment vectors per bolt (4 bolts) increases the interface basis to 25 DOFs, giving 155 total DOFs.

Verification follows a standard modal assurance criterion (MAC) check. Compute the first 20 elastic modes of the full-order model and the reduced model, then evaluate MAC between corresponding mode pairs. A well-tuned CC + enrichment basis should achieve MAC > 0.95 for the first 15 modes and frequency errors below 1% up to 500 Hz. If MAC drops below 0.90 for a specific mode, inspect the interface deformation shape: if it shows high curvature between bolts, add enrichment vectors at those locations.

4.5 Choosing the Right Technique

The central engineering judgment is to match the interface basis to the interface physics. CC modes handle the global interface deformation; enrichment handles local gradients; DCB handles compliance-dominated interfaces. Combining them selectively — rather than applying all three everywhere — keeps the reduced model small and the results trustworthy.