2. Full-order assembly models

Before any reduced-order model can be trusted, it must be traceable back to a full-order assembly model that faithfully represents the physical structure. In large-scale CAE, that full-order model is rarely a single monolithic mesh. It is almost always an assembly of components connected at shared boundaries, and the way those boundaries are discretized and coupled determines both the accuracy of the baseline solution and the quality of every subsequent reduction. This section examines the three ideas that make large assemblies tractable: substructuring, superelements, and interface degrees of freedom (DOFs).

Why the Full-Order Model Is Not Monolithic

A full-order model (FOM) is the highest-fidelity discretization you intend to solve directly, without modal truncation or static condensation. For a vehicle body, an aircraft wing box, or a wind turbine drivetrain, the FOM may contain tens of millions of DOFs. Solving it as one matrix is possible in principle but wasteful in practice, because the same components reappear across design variants and load cases. Substructuring exploits this repetition by partitioning the assembly into physical subdomains, each meshed and assembled independently, then coupling them through their shared boundaries.

The partition is not arbitrary. A good substructure boundary follows a physical joint, a bolted flange, a weld line, or a mating surface, because those locations already carry the load transfer you must preserve. Cutting through a homogeneous region introduces an artificial interface that has no physical meaning and often degrades accuracy.

Substructuring and the Assembly Equation

Once the structure is partitioned into n_s substructures, each substructure s contributes its own stiffness and mass matrices, K^(s) and M^(s), partitioned into interior DOFs (i) and interface DOFs (b):

K^(s) = [ K_ii  K_ib ]
        [ K_bi  K_bb ]

M^(s) = [ M_ii  M_ib ]
        [ M_bi  M_bb ]

The assembled system is formed by summing contributions at shared interface DOFs, exactly as in standard finite element assembly. The difference is that interior DOFs of each substructure are private and can be eliminated before the global solve. This is the essence of substructuring: keep the interface, condense the interior.

Superelements: Condensed Component Representations

A superelement is the reduced operator that results from eliminating a substructure's interior DOFs while retaining its interface behavior. Two condensation routes dominate practice.

Static condensation (Guyan reduction). Setting the interior force vector to zero gives the static relation u_i = −K_ii⁻¹K_ibu_b. Substituting back yields the condensed stiffness

K_bb* = K_bb - K_bi * inv(K_ii) * K_ib

and a correspondingly condensed mass. The result is exact for static loading and approximate for dynamics, because inertia associated with interior motion is only partially captured.

Craig–Bampton (CB) reduction. The interior displacement is expanded in a set of fixed-interface normal modes Φ_i plus constraint modes Ψ_i = −K_ii⁻¹K_ib that describe the static response to unit interface motion:

u_i = Psi_i * u_b + Phi_i * q_m

Here q_m are modal coordinates. The CB superelement retains the full interface DOF set plus a truncated set of interior modes, which makes it the standard choice for dynamic substructuring and for CMS-based reduction in later chapters.

Interface Degrees of Freedom

Interface DOFs are the translational and rotational components retained at the boundary nodes where substructures connect. Their count and distribution control both cost and fidelity. Three practical considerations matter.

• Mesh compatibility. Conforming interfaces share identical nodes, so interface DOFs map one-to-one. Non-conforming interfaces require interpolation or tied-contact constraints, which introduce additional constraint equations and can soften or stiffen the connection if the interpolation is poor.
• Rotational DOFs. Solid elements carry no rotational DOFs, so a bolted joint modeled with solids transfers moment only through the distributed translational interface. Shell-to-solid transitions must handle this mismatch explicitly, often with kinematic coupling or RBE2/RBE3 elements.
• Interface mode count. For CB superelements, the number of fixed-interface modes retained per component sets the frequency range of validity. A common rule is to retain modes up to roughly twice the highest frequency of interest, then verify convergence by adding modes and checking that interface responses change by less than a chosen tolerance.

Worked Example: Two-Component Bracket Assembly

Consider a simplified bracket: a steel plate (component A) bolted to an aluminum housing (component B). The plate is meshed with shell elements, the housing with solids. The bolted flange is the interface.

Step 1 — Partition. Define the flange as the interface. Assign interface DOFs to the flange nodes on both sides. Because the plate uses shells and the housing uses solids, the interface is non-conforming in rotational DOFs; introduce a kinematic coupling at the flange to transfer plate rotations to the solid face.

Step 2 — Condense each component. For the housing, run CB reduction retaining fixed-interface modes up to 1500 Hz (assumed target: first global mode near 400 Hz). For the plate, static condensation is sufficient if the plate is stiff relative to the housing.

Step 3 — Assemble and solve. Sum the condensed operators at the shared interface DOFs and solve the reduced eigenproblem. Compare the first three interface-referenced natural frequencies against the FOM.

These values are illustrative and would need to be reproduced on the actual model. The pattern is typical: low modes converge quickly, higher modes drift as truncation error grows. If Mode 3 deviation exceeded an acceptance threshold such as 2%, the remedy is to retain more fixed-interface modes on the housing, not to refine the mesh.

How the Full-Order Model Feeds Reduction

The full-order assembly model is the ground truth against which every reduced model is validated. Substructuring organizes it into components; superelements compress each component while preserving interface behavior; interface DOFs define the coupling that must survive reduction. Get these three right, and the reduced-order model inherits the physics of the assembly rather than approximating it blindly.

The diagram summarizes the flow: two components with distinct mesh types are condensed by different methods, coupled at a shared interface, assembled into a reduced system, and validated against the full-order reference. The interface is the critical junction — it is where physics is transferred and where most modeling errors surface.