This chapter introduces two-way (strong) fluid–structure interaction, where fluid and structural solvers exchange data in both directions within each time step until interface equilibrium is achieved. The chapter explains partitioned coupling algorithms, explicit and implicit coupling schemes, mesh motion strategies, and stabilization techniques. Special emphasis is placed on numerical stability, added-mass effects, convergence control, and industrial best practices for co-simulation.

What Makes Two-Way FSI Fundamentally Different

In two-way FSI:

Fluid loads deform or move the structure
Structural motion modifies the fluid domain
The modified flow changes the loads again

This creates a closed feedback loop that must be resolved iteratively within each time step.

Key consequence:

The solution is no longer a simple sequential execution — it is a nonlinear coupled problem.

Monolithic vs Partitioned Two-Way Coupling

3.1 Monolithic Approach (Conceptual Ideal)

In monolithic FSI:

Fluid, structure, and mesh equations are assembled into one system
All unknowns are solved simultaneously

Advantages:

Excellent numerical stability
Exact enforcement of interface conditions

Limitations:

Extremely complex
Prohibitive for industrial geometries
Rarely used outside research codes

3.2 Partitioned Approach (Industrial Standard)

In partitioned coupling:

CFD and structural solvers remain independent
Data is exchanged at the interface
Iterations enforce consistency

This is the basis of ANSYS System Coupling and nearly all industrial FSI workflows.

Interface Variables and Coupling Logic

At the fluid–structure interface:

Kinematic quantities
Displacements and velocities (structure → fluid)
Dynamic quantities
Forces and stresses (fluid → structure)

Two-way coupling alternates:

Given displacements → compute forces → update displacements → repeat

This corresponds to a Dirichlet–Neumann coupling.

Explicit vs Implicit Coupling

5.1 Explicit (Weak) Two-Way Coupling

Explicit schemes:

Perform a fixed number of solver calls per time step
Do not fully enforce equilibrium

Characteristics:

First-order accurate in time
Conditionally stable
Sensitive to added-mass effects

They are rarely sufficient for incompressible flows or light structures.

5.2 Implicit (Strong) Two-Way Coupling

Implicit schemes:

Iterate fluid–structure exchange until convergence
Recover the monolithic solution in a partitioned way

Characteristics:

Stable for strongly coupled problems
Higher computational cost
Mandatory for most practical FSI cases

System Coupling implements implicit partitioned coupling.

Partitioned Coupling Schemes (Conceptual Overview)

Several staggered schemes exist:

Serial vs parallel execution
Improved vs conventional formulations
Predictor–corrector variants

Key idea:

All schemes attempt to approximate the monolithic solution
Stability depends more on physics than on scheme choice

For practical work:

Theoretical scheme selection matters less than stabilization and time stepping.

Added-Mass Effect and Instability

7.1 Physical Origin

Added-mass instability occurs when:

Fluid inertia is comparable to or larger than structural inertia
Typical in incompressible flows with light structures

Physically:

The fluid resists structural acceleration
This resistance feeds back instantly
Explicit exchange becomes unstable

7.2 Practical Consequences

Added-mass effects cause:

Diverging oscillations
Pressure spikes
Non-convergent coupling iterations

Rule of thumb:

Incompressible flow + flexible structure ⟶ strong coupling required.

Iterative Structure of a Two-Way FSI Simulation

A transient two-way FSI simulation has three nested loops:

Time loop
Advances physical time
Coupling loop
Iterates fluid ↔ structure exchange
Field loop
Converges each solver internally

Understanding these loops is critical for diagnosing convergence problems.

Mesh Motion in Two-Way FSI

Structural motion requires mesh update in the fluid domain.

9.1 Mesh Deformation (Smoothing)

Mesh connectivity is preserved; nodes are moved smoothly.

Common methods:

Spring analogy
Elastic (pseudo-solid) analogy
Laplacian smoothing
Radial basis functions (RBF)

Used when:

Deformations are moderate
Topology does not change

9.2 Remeshing

Required when:

Mesh quality degrades
Large deformation or contact occurs

More robust but:

Computationally expensive
Can introduce numerical noise

System Coupling: Practical Implementation

10.1 What System Coupling Does

System Coupling:

Manages data exchange
Controls coupling iterations
Enforces convergence of transferred quantities

Transferred data includes:

Forces
Displacements
Temperatures
Heat flow and HTC

10.2 Conservation and Mapping

Force and heat flow transfers are conservative
Displacement mapping preserves profiles
Non-matching meshes are supported via GGI mapping

Convergence in Two-Way FSI

11.1 What “Converged” Means

A time step is converged only when:

CFD fields are converged
Structural fields are converged
Interface data transfers are converged

Neglecting any of these leads to misleading results.

11.2 Monitoring Convergence

Key indicators:

Force and displacement history at interfaces
Saw-tooth convergence pattern within time steps
Residuals alone are insufficient

Stabilization Techniques

12.1 Under-Relaxation

Reduces amplitude of exchanged updates
Useful mainly for steady problems
Limited effectiveness for strong transient instabilities

12.2 Load Ramping

Gradually applies fluid loads
Prevents violent startup transients
Very effective for flexible structures

12.3 Time Step Control

Time step selection is critical:

Too large → instability
Too small → stronger added-mass coupling

Counter-intuitive result:

Reducing the time step can worsen FSI stability .

Industrial Applications of Two-Way FSI

Typical use cases:

Aeroelastic wings and blades
Vortex-induced vibrations
Sloshing tanks
Offshore risers and jumper pipes
Fatigue and lifetime assessment

In these cases:

One-way coupling fails
Two-way coupling is mandatory for realism

Engineering Intuition

Two-way FSI is about numerical stability as much as physics
Most failures occur at startup
Initialization strategy matters more than solver settings
Start simple, then increase coupling strength