一方向流体構造連成
Theory and Physics
Concept of One-Way FSI
Professor, what's the difference between One-Way FSI and Two-Way FSI?
In One-Way FSI, the pressure and shear forces obtained from the fluid analysis are simply transferred as load conditions for the structural analysis, assuming that the structural deformation does not affect the fluid. In other words, the information flow is one-way: "Fluid → Structure".
This assumption is valid when the structural deformation does not significantly alter the fluid field. Specifically,
1. The structural displacement is very small compared to the characteristic length ($\delta/L \ll 1$)
2. The natural frequency of the structure is sufficiently separated from the fluid excitation frequency
3. The structural density is sufficiently larger than the fluid density ($\rho_s / \rho_f \gg 1$)
What kind of specific problems can it be used for?
Sufficient accuracy can be obtained with One-Way FSI in the following scenarios.
| Application | Fluid | Structure | Evaluation Items |
|---|---|---|---|
| Automotive Exterior Panels | Driving Wind | Steel Panels | Deformation & Stress due to Wind Pressure |
| Fluid Forces on Piping | Internal Flow | Piping Structure | Stress due to Pressure Load |
| Wind Load on Buildings | Wind | Steel/RC Structure | Wind Pressure Distribution, Inter-story Displacement |
| Turbo Machinery Blades | Steady Aerodynamic Forces | Blade | Static Stress from Centrifugal Force + Aerodynamics |
| Thermal Deformation of Electronics | Cooling Airflow | Substrate | Temperature Distribution → Thermal Stress |
So it's used when deformation is small and there's no need to feed it back to the flow field, right? Is the computational cost also significantly cheaper than Two-Way?
Overwhelmingly cheaper. Since you only solve CFD and FEA once each, the computation time is on the order of 1/10 to 1/100 compared to Two-Way FSI (which solves iteratively dozens of times). It's very efficient for initial design screening or rough structural strength estimation.
The Price Paid for "Omitting" One-Way FSI in Bridge Design
One-Way FSI is a valid simplification when "the influence from fluid to structure is small," but misjudging this can lead to serious consequences. In fact, there is a case where a suspension bridge was designed by completely separating fluid force analysis and structural analysis, treating the dynamic components of fluid loads with static conversion. After construction, vibrations during strong winds far exceeded design predictions. This failure was caused by the assumption that "solving separately should yield the same answer." Before choosing One-Way FSI, it is essential to quantitatively verify the validity of its assumptions.
Physical Meaning of Each Term
- Time Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? The time term describes this "period of change." The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, a basic CFD strategy is to first try solving steady-state.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far end of a room is also because the air, as a "carrier," transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while they naturally mix, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? The plunger side is high pressure, the needle tip is low pressure—this pressure difference creates the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow arises where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A common point of confusion: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it up. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result like turning on a heater in a winter room but the warm air doesn't rise.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number ≥ 0.3, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock wave capturing), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: ~1.225 kg/m³@20°C, Water: ~998 kg/m³@20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for Laminar/Turbulent Transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Data Transfer Methods
When passing CFD results to FEA, I think the meshes are different. How is the data transferred?
Data mapping from CFD mesh (usually fine) to FEA mesh (usually coarse) is required. The main methods are as follows.
| Method | Principle | Accuracy | Conservation |
|---|---|---|---|
| Nearest neighbor | Copy value from nearest point | Low | No |
| Inverse distance weighting | Weight by inverse distance | Medium | No |
| Conservative mapping | Weight by area/volume | High | Yes (Force sum conserved) |
| Profile preserving | Interpolation based on shape functions | High | No (Distribution shape preserved) |
| RBF Interpolation | Interpolation using radial basis functions | Very High | Configuration dependent |
Conservative mapping is crucial for force (pressure/shear) transfer. This guarantees that the total force on the CFD surface exactly matches the total force on the FEA surface. This option is available in Ansys System Coupling or STAR-CCM+ Co-Simulation.
What happens without conservation?
For example, when transferring a CFD result of 100 N lift to FEA, the integrated value on the FEA side becomes 95 N or 105 N. This disrupts the overall force balance of the structure, introducing errors into deformation and reaction force results. Especially when overall load is a design criterion (e.g., blade root stress), conservative mapping is essential.
Time History Data Transfer
How about when passing unsteady CFD results to structure?
There are two approaches.
1. Instantaneous Value Transfer: Input the pressure distribution from each CFD time step into FEA transient analysis. Mapping is performed for all time history steps. High accuracy but data volume becomes enormous.
2. Statistical Quantity Transfer: Transfer time-averaged pressure and pressure fluctuation RMS (or PSD). Evaluate static loads with time average and fatigue loads with RMS basis. Less data, practical for real work.
How do you get structural fatigue loads from PSD (Power Spectral Density)?
In random vibration analysis (Nastran SOL 111/112), calculate stress PSD using pressure PSD as input, then estimate fatigue life using Dirlik or Steinberg methods. This method is widely used in analyses like rocket fairing acoustic loads.
File-Based vs. Software Coupling
For One-Way FSI data exchange, is file-based transfer okay?
For One-Way FSI, file-based is sufficiently practical.
| Method | Advantages | Disadvantages |
|---|---|---|
| File-Based (CSV, CGNS) | Simple, no software dependency | Manual mapping, no conservation guarantee |
| Within Ansys Workbench | Complete within GUI, automatic mapping | Limited to Ansys products |
| System Coupling | Conservation guaranteed, automated | License required |
| Python script | Flexible, can be automated | Requires custom implementation |
For file-based, export wall pressure from Fluent in CGNS/EnSight format and load it as Imported Pressure in Ansys Mechanical or Abaqus. Mapping accuracy depends on the software's import function, so it's important to verify the total force after transfer.
"Data Mapping" Accuracy Determines the Fate of One-Way FSI
The most overlooked problem in One-Way FSI is interface data mapping. CFD meshes and structural meshes are typically created with different shapes and densities, so nodes don't match. Using nearest neighbor interpolation without care for force or pressure mapping can introduce non-trivial errors, causing integrated load values to deviate by a few percent. There are reported cases where this causes 10-20% error in structural peak stress. The perception that "data transfer is a simple task" can be the cause of unexpected accuracy degradation.
Upwind Differencing (Upwind)
1st Order Upwind: Large numerical diffusion but stable. 2nd Order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
2nd order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs. Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for Continuity, momentum, and energy equations drop by 3-4 orders of magnitude. Mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. Reduce if diverging. Increase after convergence to accelerate.
Unsteady Calculation Inner Iterations
Iterate within each time step until steady solution converges. Inner iteration count: 5~20 times is a guideline. If residuals fluctuate between time steps, review time step size.
Analogy for SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised with the corrected pressure—this catchball is repeated to approach the correct answer. Similar to two people leveling a shelf: one adjusts height, the other balances, alternating repeatedly.
Analogy for Upwind Differencing
Upwind differencing is a technique that "stands in the river flow and prioritizes upstream information." A person in the river looking downstream cannot tell where the water comes from—it reflects the physics that upstream information determines downstream. Accuracy is 1st order, but it's highly stable because it correctly captures flow direction.
Practical Guide
One-Way FSI in Ansys Workbench
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