topology-optimization

name	topology-optimization
description	Topology optimization — SIMP method, compliance minimization, volume constraints, manufacturing constraints (overhang, symmetry, draw direction), density filter, lattice optimization, tools.
metadata	{"priority":7,"promptSignals":{"phrases":["topology optimization","topology opt","SIMP","compliance minimization","OptiStruct","nTopology","density filter","lattice optimization","structural optimization","material distribution"],"minScore":3}}

Topology Optimization — Complete Skill

Problem Formulation

Standard Compliance Minimization

Minimize: C = {U}ᵀ[K]{U} = Σ u_eᵀ k_e u_e (global compliance = strain energy) Subject to: V(ρ) = Σ ρ_e v_e ≤ V* (volume constraint) 0 < ρ_min ≤ ρ_e ≤ 1 (density bounds) [K(ρ)]{U} = {F} (equilibrium)

C = compliance (reciprocal of stiffness — lower C = stiffer structure) ρ_e = element density (design variable, 0 = void, 1 = solid) V* = prescribed volume fraction × total volume (e.g., V* = 0.3 means 30% material)

SIMP (Solid Isotropic Material with Penalization)

Element stiffness: E_e = ρ_e^p × E₀ + (1-ρ_e^p) × E_min p = penalization factor (typically p=3) E_min = small value to avoid singularity (typically 10⁻⁹ E₀)

Sensitivity (compliance gradient): ∂C/∂ρ_e = -p × ρ_e^(p-1) × u_eᵀ k_0 u_e Negative sensitivity → increasing density reduces compliance → keep high-sensitivity elements

Optimality Criteria (OC) Update

ρ_e^(new) = ρ_e × (B_e^η) clamped between [max(ρ_min, ρ_e - m), min(1, ρ_e + m)] B_e = -∂C/∂ρ_e / (λ × ∂V/∂ρ_e) (ratio of strain energy density to Lagrange multiplier) η = damping exponent (typically 0.5) m = move limit (typically 0.2) λ = Lagrange multiplier found by bisection to satisfy volume constraint

Density Filter

Sensitivity/Density Filtering

Density filter (Bourdin): ρ̃_e = Σ H_ei ρ_i / Σ H_ei H_ei = max(0, r_min - dist(e,i)) (cone weighting) r_min = filter radius (typically r_min = 3 × element size)

Purpose: eliminates checkerboard patterns; imposes minimum length scale No filter → checkerboard artifacts (one-node-connected alternating solid/void)

Projection (threshold): ρ̄_e = tanh(βη) + tanh(β(ρ̃_e - η)) / (tanh(βη) + tanh(β(1-η))) η = threshold (0.5 typical), β = sharpness (increase from 1 to 32 during iterations) Converts grey elements to near-0/1 → cleaner final geometry

Minimum Length Scale Controls

Minimum solid member: filter radius r_min controls solid minimum (solid ≥ r_min) Minimum void: projection with dilated filter (eroded + dilated envelope method)

Manufacturing Constraints

Minimum Member Size

Solved by density filter: r_min = desired minimum feature size / 2 Elements smaller than r_min are smeared out → no thin features

Symmetry Constraints

Map density: if symmetry about x-axis: ρ(x,y) = ρ(x,-y) Apply after density update; symmetry reduces design variables

Draw Direction (Castability)

Constraint: no isolated voids in casting draw direction Implementation: density monotonicity ρ(x, y+Δy) ≤ ρ(x,y) (no re-entrant overhangs in draw direction) Or: overhang filter — void below solid is penalized

Overhang Constraint (Additive Manufacturing)

45° self-supporting: void below solid at angle < 45° is penalized Heaviside projection filter in build direction: ρ_e,support = min(ρ_e, ρ_below) — cannot be more solid than support below it Slope constraint: ρ_e ≤ max(ρ_e1, ρ_e2, ρ_e3) of neighbors in build direction

Extrusion Constraint

Constant cross-section: ρ(x,y,z) = ρ(x,y) — all z layers equal Apply by averaging densities along extrusion axis: ρ_e = (1/N_z) Σ ρ_(e,k)

Multiple Load Cases

Weighted Compliance

Minimize: C_total = Σ_j w_j C_j (weighted sum of compliances for each load case j) Each load case gets separate FEA solve; sensitivities summed (weighted)

Minimax formulation: Minimize: C_max = max_j C_j (minimize worst-case compliance) Use smooth approximation: C_max ≈ (1/p) × log(Σ exp(p×C_j)) (p-norm, p=15-20)

Stress Constraints

Stress Constrained Topology Optimization

Qp-aggregation: g(ρ) = (Σ (σ_e/σ_y)^P × v_e)^(1/P) ≤ 1 (P-norm) P-norm parameter P = 8-20 (higher P → closer to true maximum)

Challenge: large number of local stress constraints; singularity phenomenon Singularity: void elements have zero stress → stress constraint always satisfied → can eliminate needed material

ε-relaxation: σ̃_e = σ_e / (ρ_e + ε) — scale stress by density to avoid singularity

Lattice Optimization

Graded Lattice

Map density to lattice relative density: ρ̄(x) → strut diameter or wall thickness Gyroid, Schwartz-D, BCC, FCC: different stiffness/density curves Gibson-Ashby: E_lattice/E_solid = C × ρ̄^n (n=2 for bending-dominated, n=1 for stretch-dominated)

Workflow:

1. Topology optimization → coarse density field
2. Map density to lattice element size → variable lattice
3. Extract implicit surface (level-set) or explicit strut model
4. Refine with FEA on actual lattice geometry

Infill Optimization

Shell + lattice core: solid shell (for surface finish, fatigue resistance) + graded lattice inside Define: shell thickness (constant), infill density (from TO result)

Software Tools

Commercial

Altair OptiStruct: mature, SIMP + shape + size; manufacturing constraints; widely used ANSYS Mechanical (Topology Opt. Module): integrated with FEA; good for coupled thermal-structural Simulia TOSCA (Abaqus): sequential response surface method; stress constraints SolidWorks Topology Study: entry-level; easy setup, limited constraints

Open Source / Research

88-line MATLAB code (Sigmund 2001): classic educational; FDM elements; no manufacturing constraints Top3d (Liu 2014): 3D extension; 169 lines; educational OpenLSTO: level-set method; shape derivatives; manufacturing constraints nTopology: commercial with lattice + TO integration; CAD-ready output

Workflow Best Practices

1. Pre-processing: define loads, BCs, keep-out zones (holes for fasteners), symmetry
2. Run TO: start at 30-40% volume fraction with p=1; ramp p to 3; then add projection
3. Interpret: smooth with CAD (spline-fit isosurface); maintain minimum wall thickness
4. Validate: re-run full FEA on interpreted/smoothed geometry; check stress ≤ allowable
5. Iterate: if stress exceeded, add local material; if too conservative, reduce volume fraction

Typical Results

Mass reduction vs. reference solid or intuitive design: Simple bracket: 30-60% mass reduction at same stiffness Airframe component: 20-50% Medical implant: 40-70% (plus improved osseointegration) Maximum stiffness-to-weight: Michell truss patterns (arch, tension web) for point loads

Output

Provide: volume fraction recommendation [%], minimum member size guidance [mm], dominant load case identification, manufacturing constraint settings (overhang/draw/symmetry), expected mass reduction [%] vs. solid, validation FEA re-check results [MPa].