寸法最適化
Theory and Physics
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.
$$ \min_{t_1, t_2, ...} \quad \text{Mass} \quad \text{s.t.} \quad \sigma_{max} \leq \sigma_{allow} $$
Summary
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.
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration." Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored." It cannot be omitted in impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously, the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades, right? That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior such as plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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).
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix each iteration. Exhibits quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies the full load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on overall cost-effectiveness.
Practical Guide
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.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the answer must be correct because the computer produced it."
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is often the most critical step in the entire analysis.
Software 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.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types needed for size optimization? For example, in fluids, the presence of LES support, and in structures, the ability to handle contact and large deformation make a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will it expand?": Consider future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools.
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