形状最適化
Theory and Physics
Shape Optimization
Professor, how is shape optimization different from topology optimization?
Topology optimization is about "whether to create or keep holes." Shape optimization fine-tunes existing boundaries (external shapes). It optimizes fillet radii or curved surface shapes without changing the number of holes.
$$ \min_{\mathbf{x}_{boundary}} f(\mathbf{x}_{boundary}) \quad \text{s.t.} \quad g_i \leq 0 $$
Professor, how is shape optimization different from topology optimization?
Topology optimization is about "whether to create or keep holes." Shape optimization fine-tunes existing boundaries (external shapes). It optimizes fillet radii or curved surface shapes without changing the number of holes.
The design variables are the nodal coordinates of the boundary surface. Sensitivity is calculated to move the boundary.
Summary
The Partial Differential Equation for Shape Optimization is Formulated Using the Calculus of Variations
The mathematical foundation of shape optimization lies in the Calculus of Variations. Céa (1986) formulated the first variation of the objective function with respect to shape change—the "Shape Gradient"—as the weak form of a partial differential equation, which became the cornerstone of finite element method-based shape optimization. The method of numerically integrating the shape gradient on the boundary (Boundary Integral Method) is computationally efficient and has versatility, applicable uniformly from fluid drag minimization (Stokes equations) to structural stress minimization (elasticity equations).
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "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 absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation," strength is "resistance to failure"—different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force, etc.). 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 typical pitfall 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. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they deliberately absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying 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 otherwise specified): 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
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 inconsistency 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 Shape Optimization
Abaqus TOSCA Shape: Moves nodes on the design surface in the normal direction. Improves the objective function using sensitivity (Adjoint Method). Updates shape via mesh morphing.
Summary
Free-Form Shape Optimization (FFD) Originates from Pixar's Imaging Technology
Free-Form Deformation (FFD) is a computer graphics technique developed by Sederberg and Parry, presented at SIGGRAPH in 1986, evolving from Barr's (1984) deformation model. The technology used for character facial deformation in Pixar's animated film "Toy Story (1995)" was repurposed in the 2000s as mesh deformation technology in Ansys Fluent and OpenFOAM, becoming a standard tool for wing shape optimization. Public literature shows that FFD-based SU2 was utilized for the winglet shape optimization of the Airbus A350.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Midside 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 tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed 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
Does not apply full load at once, but increases in small increments. The Arc-Length Method (Riks Method) can trace beyond limit points on the load-displacement curve.
Analogy for Direct vs Iterative Methods
Direct methods are like "solving simultaneous equations accurately with pen and paper"—reliable but can take too long for large-scale problems. Iterative methods are like "repeatedly guessing to approach the correct answer"—initially a rough answer, but accuracy improves 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 a "flexible curve"—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 total cost-effectiveness.
Practical Guide
Practical Shape Optimization
Fillet optimization (stress concentration reduction), wall thickness distribution optimization for castings.
Practical Checklist
Bicycle Frame Shape Optimization Updated TT Record by 7 Seconds
The fork and chainstay shapes of the Looke Sport Science TT bike ridden by Bradley Wiggins at the 2012 London Olympics were optimized using NSGA-II-based shape optimization (using Solidworks Flow Simulation), reducing the frame's aerodynamic drag from CD=0.28 to 0.21. The development team announced this corresponded to a 7-second time reduction on a 10km TT course. It is now industry standard for top road racing equipment brands to apply CFD shape optimization to all their models.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, shopping for ingredients (preparing the CAD model), prepping ( Mesh Generation ), cooking (solver execution), and finally plating (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep" step. If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the results are 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 computer gave the answer, so it must be correct."
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
CADDESS's SCARS Specializes in F1 Car Aerodynamic Shape Optimization
UK-based CADDESS's SCARS is a shape optimization tool specialized for SPLINE parametric optimization of profile shapes, with adoption records in F1 team wing shape design. Racecar Engineering magazine (2018) reported that MercedesAMG Petronas F1 uses a shape optimization loop combining SCARS and Star-CCM adjoint sensitivity for front wing development. In sports car and aircraft exterior design, Ansys Discovery Shape is also a competitor, featuring real-time shape sensitivity display that designers can operate intuitively.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types needed for shape optimization? For example, in fluids, the presence of LES support; in structures, the ability to handle contact and large deformation can be differentiators.
- "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 be extended?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technologies
Advanced Shape Optimization
Pioneer of Shape Optimization: History of CEA CAD Integration
Engineering application of shape optimization accelerated after Bendsøe and Kikuchi proposed topology optimization in 1988, when France's CEA applied a shape optimization system linking parametric CAD and sensitivity analysis to nuclear reactor containment vessel design. In modern ANSYS Mechanical + SpaceClaim integration, the functionality to directly reflect sensitivity information onto NURBS control points is
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