4. Load path topology and topology optimization

In a lightweight body-in-white (BIW), the load path is not a single beam but a network: rocker-to-floor-to-tunnel-to-dash-to-A-pillar-to-roof-rail, with the shock towers and subframe mounts acting as junctions. Topology optimization answers a deceptively simple question — where should material exist at all? — before gauge, section shape, or material grade are decided. This chapter focuses on density-based topology optimization (SIMP and its variants) and how to run it credibly for two very different load cases: static stiffness and crash.

Why topology comes first in the load-path tradeoff

Body structures are usually developed in a sequence: topology (material layout) → topography (bead and rib patterns on panels) → gauge and section sizing → material grade and joining. Each later stage is constrained by the earlier one. If the topology omits a shear path between the rocker and the rear shock tower, no amount of gauge increase will recover the torsional stiffness efficiently — you will simply add mass to a structurally wrong layout. This is why topology studies are cheap insurance: they run on coarse meshes, tolerate idealized loads, and reveal the dominant load paths before CAD geometry hardens.

The tradeoff is that topology optimization is only as good as its boundary conditions. A stiffness model with a rigidly constrained suspension pickup and a point-loaded shock tower will produce a beautiful, wrong answer. The practical discipline is to keep the load introduction and reaction regions physically representative (spreader plates, bush stiffness, subframe compliance) even when the design domain itself is coarse.

Density-based formulation: what the solver actually does

In density-based methods, every element in the design domain receives a pseudo-density ρe ∈ [0, 1]. The stiffness is interpolated with the SIMP power law:

E(ρ) = E_min + ρ^p · (E_0 − E_min),   p = 3 (typical),  E_min ≈ 1e-6 · E_0

The optimizer minimizes compliance C = fᵀu subject to a mass fraction constraint Σρeve ≤ f·V0, driving densities toward 0 or 1. The exponent p = 3 penalizes intermediate densities so the result is nearly binary. Two numerical controls matter in practice:

  • Filter radius rmin: a mesh-independent density filter (typically 1.5–3 × element size) that suppresses checkerboarding and enforces a minimum member size. Set rmin to the smallest feature you can actually manufacture — for stamped steel, roughly 15–25 mm; for cast nodes, 8–12 mm.
  • Continuation: start with p = 1 and increase to 3 over the first 20–30 iterations to avoid premature local minima.

A typical run for a full BIW torsion domain on a 10 mm coarse mesh converges in 60–120 iterations. Watch the compliance history: a monotonic decrease to a plateau is healthy; oscillation usually means the filter is too small or the mass constraint is too tight relative to the load paths available.

Stiffness load cases: multiple-load-case compliance

Body stiffness is rarely a single objective. The standard set is global torsion, global bending, and one or two local stiffness cases (e.g., shock-tower lateral stiffness, tailgate opening stiffness). The multi-load-case formulation is a weighted sum:

min  C = Σ_k w_k · f_kᵀ u_k
s.t. Σ_e ρ_e v_e ≤ f · V_0,   0 ≤ ρ_e ≤ 1

Weights are not arbitrary. Normalize each case by its baseline compliance so that wk reflects engineering priority rather than unit scale. A defensible starting set for a mid-size SUV body is wtorsion = 0.4, wbending = 0.3, wlocal = 0.3, with a mass fraction f = 0.30 of the design domain. The output is a density field; the engineering interpretation is the load-path skeleton — the ridges of ρ ≈ 1 that connect load introduction points to reactions.

Read the density result as a path map, not a part. Extract the ridge lines, then ask: does each ridge correspond to a real closed section (rocker, rail, pillar) or a shear panel? A ridge that runs diagonally across the floor is telling you a shear path is needed — that becomes a floor cross-member or a bead pattern, not a new beam.

Crash load cases: why plain SIMP is not enough

Crash is a nonlinear, transient, large-deformation problem. Compliance is meaningless there; the objective is usually absorbed energy or peak force, subject to intrusion limits. Density-based crash topology optimization (often called HCA — hybrid cellular automata — or equivalent density-based crash methods) works differently: it targets a uniform internal energy density field across the design domain, iteratively updating ρe so that each element contributes equally to energy absorption. The mass constraint is enforced by scaling the target energy density.

Practical crash topology runs are expensive, so they are usually done on a sub-assembly — front longitudinal, bumper beam and crash box, or rear rail — with a simplified barrier and a prescribed impact velocity (e.g., 50 km/h full frontal, or 64 km/h ODB). Key settings:

  • Element size 8–12 mm in the crush zone; the topology result cannot resolve features smaller than the mesh.
  • Explicit time step controlled by the smallest element; mass scaling kept below 2% to avoid distorting the energy balance.
  • Objective: maximize SEA (specific energy absorption) or meet a force–displacement corridor; constraint: intrusion ≤ target at the firewall or pedal box.
  • Run 3–5 design iterations, not 50 — crash topology is used to identify where crush initiators and load-spreading members belong, not to converge a final shape.

Combining stiffness and crash in one tradeoff

The two objectives pull in opposite directions. Stiffness wants short, direct, triangulated paths; crash wants long, progressive, collapsible paths with stable folding. Running them separately and overlaying the density fields is the pragmatic approach: regions where both fields are dense are non-negotiable structure; regions dense only for stiffness become closed sections or shear panels; regions dense only for crash become thin-walled crush members with triggers.

Density-based topology optimization workflow for stiffness and crash load cases Diagram showing a design domain split into stiffness and crash branches, each with its own objective and solver settings, converging into a combined density field that is interpreted as load-path skeleton, shear panels, and crush members. Design domain BIW coarse mesh 10 mm elements ρ ∈ [0,1] Stiffness branch (SIMP) Objective: min Σ wₖ fₖᵀuₖ Cases: torsion 0.4 / bending 0.3 / local 0.3 p = 3, r_min = 20 mm Mass fraction f = 0.30 Crash branch (HCA) Objective: uniform energy density Case: 50 km/h frontal, 64 km/h ODB Constraint: intrusion ≤ target 8–12 mm crush-zone mesh 3–5 design iterations Combined density field overlay ρ_stiff and ρ_crash ridge extraction Both dense non-negotiable closed sections + crush members Stiffness only shear panels beads, ribs Crash only thin-walled crush members Neither candidate for mass removal Stiffness path Crash path Shared structure

Interpreting the result: from density field to body structure

The density field is not a deliverable; the load-path map is. A practical interpretation sequence:

  1. Extract ridges. Threshold at ρ ≥ 0.5, skeletonize, and trace continuous paths between load introduction and reaction. Each path is a candidate member.
  2. Classify each path. Axial paths become closed sections (rails, pillars); diagonal paths across panels become shear panels or bead fields; short stubs near junctions become gussets or cast nodes.
  3. Check manufacturability. Any member thinner than the filter radius or the stamping minimum flange is not real. Re-run with a larger rmin rather than "cleaning up" the result by hand.
  4. Validate on a fine model. Rebuild the interpreted layout as a coarse CAD concept, re-mesh at 5 mm, and re-run the stiffness cases plus a crash pulse. Compare compliance and intrusion against the topology prediction. A 10–20% loss is normal; a 50% loss means the interpretation broke a path.
Do not feed a topology density field directly into a gauge optimization. The intermediate densities (ρ ≈ 0.3–0.7) have no physical thickness, and the optimizer will exploit them. Always convert to discrete members and panels first, then size.

Common failure symptoms and fixes

Symptom Likely cause Fix
Checkerboard pattern in density field Filter radius too small or disabled Set rmin ≥ 1.5 × element size; verify filter is mesh-independent
Compliance oscillates, never plateaus Mass constraint too tight for available paths; p too high at start Relax f by 0.05; use continuation from p = 1 to 3
All mass migrates to one corner Unrealistic point load or rigid constraint Add spreader plates and bush stiffness at load/reaction regions
Crash topology shows no crush initiators Energy density target uniformized too aggressively Reduce iterations; add a force corridor constraint; check mass scaling
Fine-model validation loses 50% stiffness Ridge extraction broke a path at a junction Re-trace paths through junctions; add gusset at the break

The output of this stage is a load-path map with classified members, a documented set of weights and constraints, and a validated coarse concept. That map becomes the fixed input to topography and gauge studies — and the reason those later studies converge quickly instead of chasing mass around a structurally wrong layout.

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