Size Optimization
Size Optimization: Theoretical Foundations
Size Optimization
Professor, is size optimization the simplest type of optimization?
Yes. It optimizes using plate thickness, cross-sectional dimensions, and material properties as design variables. It does not change the shape or topology.
Summary
The Prototype of Size Optimization Dates Back to Before Michell in the 1800s
One of the oldest examples of size optimization using cross-sectional dimensions as design variables is the analytical solution for minimum weight trusses published by Rankine in 1858 in "Manual of Applied Mechanics." Rankine's method, which algebraically determines the optimal cross-sectional area for each member given load conditions and material strength, is considered the prototype of modern linear programming-based size optimization. In the 1960s, Dorn, Gomory, and Greenberg reformulated it as a linear programming problem, forming the foundation for size optimization in the computer era.
Computational Methods for Size Optimization
FEM for Size Optimization
Nastran SOL 200:
```
DESVAR, 1, T_FLANGE, 10., 5., 30. $ Design Variable: Flange thickness 10mm (5~30mm)
DVPREL1, 1, PSHELL, 1, T $ Relates to PSHELL plate thickness
DRESP1, 1, STRESS, STRESS, , , , MAX
DCONSTR, 1, 1, , 250. $ Stress constraint ≤ 250 MPa
```
Summary
KKT Conditions are the Optimality Criteria for Nonlinear Size Optimization
The optimality conditions for nonlinear size optimization are formulated using the Karush-Kuhn-Tucker (KKT) conditions. The KKT conditions, independently proven by Karush in his 1939 master's thesis (unpublished for many years) and by Kuhn & Tucker in a 1951 Berkeley conference presentation, are the first-order necessary conditions for constrained optimization problems with inequality constraints. NASTRAN SOL 200 uses the KKT conditions as convergence criteria, judging a solution as optimal when the KKT residuals for all constraints fall below a threshold (default 0.005).
Size Optimization in Practice
Size Optimization in Practice
Aircraft panel thickness optimization, automotive frame cross-section optimization.
Practical Checklist
Bridge Girder Cross-Section Design is the Most Classical Size Optimization
Size optimization of plate thickness and flange width for I-shaped steel girders in road bridge superstructures is one of the oldest practical applications, implemented by civil engineering design firms since the 1980s. In the era of Allowable Stress Design (ASD), the minimum weight section satisfying section modulus constraints could be found analytically. However, modern Limit State Design (LSD) includes nonlinear constraints like buckling and fatigue, making Nonlinear Programming (NLP) essential. The 2005 revision of the JSSC (Japan Society of Steel Construction) design guidelines includes size optimization application examples as reference material.
Size Optimization: Software & Solver Comparison
Tools for Size Optimization
NASTRAN SOL200 is a Size Optimization Feature Active for Over 40 Years
MSC Nastran's size optimization feature "SOL 200 (Design Sensitivity and Optimization)" began development in the late 1970s with NASA funding, with its first release in 1982. Continuously improved for over 40 years, the 2023 version enables coupling with ML surrogates and custom constraint definition via Python scripts. It reigns as the de facto standard for size optimization in aerospace, used in applications like Boeing's passenger aircraft fuselage frame cross-section optimization and Lockheed Martin's F-35 main wing spar thickness optimization.