Multidisciplinary Optimization

Multidisciplinary Design Optimization (vehicle-design/mdo/multidisciplinary-optimization)

Use when the task is multidisciplinary design optimization (MDO) for an aircraft or spacecraft: the design variables, the objective function, and the constraints, the disciplines (aerodynamics, structures, propulsion, trajectory) and their coupling variables, the monolithic or distributed architecture, and the analysis loop that iterates the coupling to convergence before the design point is accepted.

Domain quick reference

• MDO formulation: minimize f(x, y) subject to g(x, y) <= 0 and h(x, y) = 0 over the design variables x, with the coupling variables y fixed by the discipline residuals R_i(x, y) = 0; f is the objective, g the inequality constraints, h the equality constraints.
• Aero-structural coupling: the aerodynamic discipline sees CL = CL_alpha * (alpha_geom - delta), where the structural deflection index delta reduces the effective angle of attack, and the structures discipline sees delta = k_def * q * CL, with q the dynamic pressure in Pa. Both hold at the fixed point CL* = CL_alpha * alpha_geom / (1 + CL_alpha * k_def * q).
• Worked anchor: CL_alpha = 5.0 1/rad, alpha_geom = 4.0 deg, q = 2000 Pa, k_def = 2.0e-5 1/Pa gives the contraction factor r = 0.2, CL* = 0.290888, delta* = 0.0116355 rad, converged in about 15 fixed-point iterations at tolerance 1e-10.
• Fixed-point iteration CL_{n+1} = CL_alpha * (alpha_geom - k_def * q * CL_n) converges when the contraction factor r = CL_alpha * k_def * q < 1; each iteration multiplies the error by r, so the iteration count grows as log(tol / error_0) / log(r).
• Optimizer anchor: minimize f(x) = (x - 2.0)^2 subject to x >= 4.0; the unconstrained optimum x = 2.0 is infeasible, and the exterior penalty 1.0e6 * (4.0 - x)^2 for x < 4.0 moves the grid-search optimum to x = 4.0 with f = 4.0.
• Architecture: monolithic runs one optimizer over one coupled model; distributed coordinates per-discipline optimizers through the coupling variables. Family context: MDO is the optimization discipline over the conceptual, sizing, mass-properties, cost-estimation, and structures-integration leaves; each leaf contributes a discipline model or an objective term to the loop.
• FAR-25 (14 CFR Part 25) and CS-25 set the certification context (loads, structural margins) that the MDO constraints encode; the formulation is common engineering optimization practice.

Workflow

1. Define the design variables x and their bounds.
2. Define the objective f(x, y) and the constraints g and h.
3. Identify the disciplines and the coupling variables y, and write each discipline residual R_i(x, y) = 0.
4. Choose the architecture: monolithic or distributed.
5. Run the fixed-point coupling loop with aero_structural_fixed_point to the convergence tolerance, and check the contraction factor.
6. Optimize the design variable with grid_search_optimize, penalizing infeasible points.
7. Check the sensitivity with finite_difference_gradient, then accept or refine the design point.

Pitfalls

• Confusing MDO with ws-tw-trade: ws-tw-trade sets one binding-constraint design point at a given wing loading on the matching chart; MDO closes the coupled loop and optimizes over the design space, and the matching chart result enters as a constraint or a start point.
• Confusing MDO with wing-planform-sizing: wing-planform-sizing produces the planform geometry from the sizing point; MDO treats the planform dimensions as design variables and re-evaluates the coupled disciplines at every candidate.
• Confusing MDO with the mass-budget: mass-budget allocates the weight statement; MDO consumes the mass estimate as a discipline output and couples it to the structural and aerodynamic responses.
• Confusing MDO with trade-study-analysis: a trade study varies one factor at a time against fixed alternatives; MDO optimizes simultaneously over the design variables with the coupling closed.
• Confusing MDO with engine-sizing: engine-sizing sets the thrust requirement from the matching chart; MDO folds propulsion in as a discipline whose outputs (thrust, fuel flow) feed the objective and the constraints.
• Iterating a fixed point without checking the contraction factor: r = CL_alpha * k_def * q must stay below 1 or the loop diverges; check r before running.
• Optimizing without a constraint penalty: the unconstrained optimum can be infeasible; penalize and verify that the returned point is feasible.
• Mixing units: alpha in degrees with CL_alpha in 1/rad, or q in non-SI units; keep alpha_geom in degrees, CL_alpha in 1/rad, q in Pa, k_def in 1/Pa, delta in rad.

Behavior contract (gate 3)

The aero-structural fixed-point coupling, the grid-search optimizer with the constraint penalty, and the sensitivity check are exercised by the gate 3 contract test: scripts/test_mdo_logic.py against scripts/mdo_logic.py (stdlib unittest, offline). Run: python3 skills/vehicle-design/mdo/multidisciplinary-optimization/scripts/test_mdo_logic.py

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain) and CS-25 is a free EASA download; the MDO formulation and the coupling equations are common engineering optimization methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.