# A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems

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## Abstract

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## 1. Introduction

## 2. Fluid–Structure Interaction Problems

## 3. Approaches to Fluid–Structure Interaction Problems

#### 3.1. Monolithic Approach

#### 3.2. Segregated Approach

- Explicit algorithms: after time discretization, the coupling conditions are treated explicitly at every time-step. These algorithms, also known as weakly or loosely coupled algorithms [39], are successfully applied in aerodynamics applications (see [40,41]), but some studies (see [38,42,43]) showed that they are unstable under some physical and geometrical conditions due to the added mass effect, as we previously mentioned.
- Implicit algorithms: in these algorithms, also known as strongly coupled algorithms, the coupling conditions are treated implicitly at every time-step; see for example [44,45]. This implicit coupling represents a way to circumvent the instability problems due to the added mass effect; nevertheless, an implicit treatment of the coupling conditions leads to algorithms that are more expensive in terms of computational time.
- Semi-implicit algorithms: in these algorithms (see [46,47,48]), the continuity of the displacement is treated explicitly whereas the other coupling conditions are treated implicitly. This alternative represents a tradeoff between the computational cost of the algorithm and its stability in relation to the physical and geometrical properties of the problem. In Section 6.4, we present a reduced order method that is based on this kind of partitioned approach.

## 4. Monolithic Approach

#### 4.1. Time Discretization

#### 4.2. Space Discretization

#### Coupling Conditions through Lagrange Multipliers

#### 4.3. Lifting Function and Supremizer Enrichment

#### 4.4. Reduced Basis Generation

#### 4.5. Online Phase

## 5. Results

## 6. Partitioned Approach

#### 6.1. Time Discretization

- Extrapolation of the mesh displacement: find ${\mathit{d}}_{f}^{i+1}:{\Omega}_{f}\mapsto {\mathbb{R}}^{2}$ such that$$\left(\right)$$
- Fluid explicit step: find ${\mathit{u}}_{f}^{i+1}:{\Omega}_{f}\mapsto {\mathbb{R}}^{2}$ such that$$\left(\right)$$$$\epsilon \left({\mathit{u}}_{f}^{i+1}\right):=\nabla {\mathit{u}}_{f}^{i+1}{F}^{-1}+{F}^{-T}{\nabla}^{T}{\mathit{u}}_{f}^{i+1}.$$
- Implicit step:
- Fluid projection substep (pressure Poisson formulation): find ${p}_{f}^{i+1}:{\Omega}_{f}\mapsto {\mathbb{R}}^{2}$ such that$$\left(\right)$$$${p}_{f}^{i+1}=\overline{p}\phantom{\rule{1.em}{0ex}}\mathrm{on}\phantom{\rule{4.pt}{0ex}}{\Gamma}_{in},$$
- Structure projection substep: find ${\mathit{d}}_{s}^{i+1}:{\Omega}_{s}\mapsto {\mathbb{R}}^{2}$ such that$$\left(\right)$$

- The original Navier–Stokes problem was divided into two subproblems, namely the fluid explicit step and the fluid projection step. In the explicit step, we take care of the momentum balance of the fluid problem, whereas in the projection step, we take care of the divergence free condition: this subdivision is a peculiarity of the Chorin–Temam projection scheme; we refer the interested reader to [57,58]. The advantage of adopting such a numerical scheme for the fluid problem is given by the fact that, in this case, we can also use pairs of discrete spaces (${V}_{h}$ and ${Q}_{h}$) for the fluid velocity and pressure that do not necessarily satisfy the inf–sup condition; this represents a great advantage for the forthcoming online phase of the method, since we are able to obtain a stable approximation of the fluid pressure also without employing the supremizer enrichment technique, unlike in the monolithic approach.
- The treatment of the boundary conditions (in our specific case, the inlet boundary condition) is a delicate aspect of partitioned schemes. If the original problem of interest is provided with a boundary condition for the fluid Cauchy stress tensor ${\sigma}^{f}({\mathit{u}}_{f},{p}_{f}){\mathit{n}}_{in}={\mathit{g}}_{in}$, where ${\mathit{n}}_{in}$ is the normal outgoing the inlet boundary, then during the Chorin–Temam projection scheme, this condition can be splitted into two: a natural condition for the fluid velocity explicit step $\epsilon \left({\mathit{u}}_{f}\right){\mathit{n}}_{in}=\mathbf{0}$ and a Dirichlet condition for the pressure ${p}_{f}{\mathit{n}}_{in}={\mathit{g}}_{in}$ (see for example [58]). However, in our particular toy problem, we have a Dirichlet inlet condition for the velocity: we therefore need a Dirichlet boundary condition also for the pressure in order to obtain uniqueness of solution of the pressure Poisson problem. In this case, we have no specific indication of what value for the pressure to choose at the inlet boundary: for our test case, we decided to compute the inlet pressure value by computing the quantity ${\sigma}^{f}({\mathit{u}}_{f},{p}_{f}){\mathit{n}}_{in}$ on the inlet boundary and use the fact that, there, we have ${\mathit{u}}_{f}={\mathit{u}}_{in}$.
- In the projection step (20), we chose a pressure Poisson formulation; it is possible to use a Darcy formulation instead: find ${p}_{f}^{i+1}$ and ${\tilde{\mathit{u}}}_{f}^{i+1}$ such that$$\left(\right)$$However, in view of an efficient model order reduction, we chose to employ a Poisson formulation, since the Darcy formulation requires the introduction of an additional unknown ${\tilde{\mathit{u}}}_{f}$, which translates in a larger system, comprised of both velocity and pressure, at the implicit step.

#### 6.2. Space Discretization

- Extrapolation of the mesh displacement: find ${\mathit{d}}_{f,h}^{i+1}\in {E}_{h}^{f}$ such that $\forall {\mathit{e}}_{f,h}\in {E}_{h}^{f}$:$$\left(\right)$$
- Fluid explicit step: find ${\mathit{u}}_{f,h}^{i+1}\in {V}_{h}^{f}$ such that $\forall {\mathit{v}}_{f,h}\in {V}_{h}^{0}$:$$\left(\right)$$
- Implicit step: for any $j=0,\dots $ until convergence:
- Fluid projection substep (pressure Poisson formulation): find ${p}_{f,h}^{i+1,j+1}\in {Q}_{h}$ such that $\forall {q}_{f,h}\in {Q}_{h}^{0}$:$$\begin{array}{cc}& -\frac{{\rho}_{f}}{\Delta T}{(\mathrm{div}\left(J{F}^{-1}{\mathit{u}}_{f,h}^{i+1}\right),{q}_{f,h})}_{{\Omega}_{f}}-{\rho}_{f}{(\left({D}_{tt}{\mathit{d}}_{s,h}^{i+1,j}\right),J{F}^{-T}{\mathit{n}}_{f}{q}_{f,h})}_{{\Gamma}_{FSI}}+\hfill \\ & +{\alpha}_{ROB}{({p}_{f,h}^{i+1,j},{q}_{f,h})}_{{\Gamma}_{FSI}}={\alpha}_{ROB}{({p}_{f,h}^{i+1,j+1},{q}_{f,h})}_{{\Gamma}_{FSI}}+{(J{F}^{-T}\nabla {p}_{f,h}^{i+1,j+1},{F}^{-T}\nabla {q}_{f,h})}_{{\Omega}_{f}}.\hfill \end{array}$$
- Structure projection substep: find ${\mathit{d}}_{s,h}^{i+1,j+1}\in {E}_{h}^{s}$ such that $\forall {\mathit{e}}_{s,h}\in {E}_{h}^{s}$:$${\rho}_{s}{({D}_{tt}{\mathit{d}}_{s,h}^{i+1,j+1},{\mathit{e}}_{s,h})}_{{\Omega}_{f}}+{(P\left({\mathit{d}}_{s,h}^{i+1,j+1}\right),\nabla {\mathit{e}}_{s,h})}_{{\Omega}_{s}}=-{(J{\sigma}_{f}({\mathit{u}}_{f,h}^{i+1},{p}_{f,h}^{i+1,j+1}){F}^{-T}{\mathit{n}}_{f},{\mathit{e}}_{s,h})}_{{\Gamma}_{FSI}}+{({\mathit{b}}_{s},{\mathit{e}}_{s,h})}_{{\Omega}_{s}}$$

#### 6.3. Reduced Basis Generation

#### 6.3.1. Change of Variable for the Fluid Velocity

#### 6.3.2. Harmonic Extension of the Fluid Displacement

#### 6.4. Online Computational Phase

- Fluid projection substep: find ${p}_{f,N}^{0,i+1,j+1}\in {Q}_{N}^{0}$ such that $\forall {q}_{f,N}\in {Q}_{N}^{0}$:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& -\frac{{\rho}_{f}}{\Delta T}{(\mathrm{div}\left(J{F}^{-1}{\mathit{u}}_{f,N}^{i+1}\right),{q}_{f,N})}_{{\Omega}_{f}}-{\rho}_{f}{(\left({D}_{tt}{\mathit{d}}_{s,N}^{i+1,j}\right),J{F}^{-T}{\mathit{n}}_{f}{q}_{f,N})}_{{\Gamma}_{FSI}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +{\alpha}_{ROB}{({p}_{f,N}^{i+1,j},{q}_{f,N})}_{{\Gamma}_{FSI}}-{\alpha}_{ROB}{({\ell}_{p}^{i+1},{q}_{f,N})}_{{\Gamma}_{FSI}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -J{F}^{-T}{(\nabla {\ell}_{p}^{i+1},{F}^{-T}\nabla {q}_{f,N})}_{{\Omega}_{f}}={\alpha}_{ROB}{({p}_{f,N}^{i+1,j+1},{q}_{f,N})}_{{\Gamma}_{FSI}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +{(J{F}^{-T}\nabla {p}_{f,N}^{i+1,j+1},{F}^{-T}\nabla {q}_{f,N})}_{{\Omega}_{f}};\hfill \end{array}$$
- Structure projection substep: find ${\mathit{d}}_{s,N}^{i+1,j+1}\in {E}_{N}^{s}$ such that $\forall {\mathit{e}}_{s,N}\in {E}_{N}^{s}$:$${\rho}_{s}{({D}_{tt}{\mathit{d}}_{s,N}^{i+1,j+1},{\mathit{e}}_{s,N})}_{{\Omega}_{s}}+{(P\left({\mathit{d}}_{s,N}^{i+1,j+1}\right),\nabla {\mathit{e}}_{s,N})}_{{\Omega}_{s}}=-{(J{\sigma}_{f}({\mathit{u}}_{f,N}^{i+1},{p}_{f,N}^{i+1,j+1}){F}^{-T}{\mathit{n}}_{f},{\mathit{e}}_{s,N})}_{{\Gamma}_{FSI}}+{({\mathit{b}}_{N}^{s},{\mathit{e}}_{s,N})}_{{\Omega}_{s}},$$

## 7. Results

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Fluid–structure interaction domains. Time-dependent configuration (

**left**): fluid domain ${\Omega}_{f}\left(t\right)$ (blue) and solid deformed domain ${\Omega}_{s}\left(t\right)$ (red). Reference configuration (

**right**): the inlet boundary ${\widehat{\Gamma}}_{in}$ (magenta), the wall boundaries for the fluid ${\widehat{\Gamma}}_{walls}$ (light blue), and the Neumann (outlet) boundary ${\widehat{\Gamma}}_{N}^{f}$ (orange). The fluid arbitrary reference configuration ${\widehat{\Omega}}_{f}$ is depicted in blue, the solid reference configuration ${\widehat{\Omega}}_{s}$ is depicted in red, and ${\widehat{\Gamma}}_{D}^{s}$ is the solid Dirichlet boundary. The fluid–structure interface in the reference configuration ${\widehat{\Gamma}}_{FSI}$ is highlighted in green.

**Figure 2.**Reference configuration of the benchmark test case. The solid is depicted in red, while the fluid domain is in blue.

**Figure 3.**The outcomes of the monolithic POD: eigenvalue decay (

**a**) and retained energy (

**b**) for the first 100 modes.

**Figure 4.**The first POD modes for the fluid velocity ${u}_{f}$ and for the supremizer s (magnitude).

**Figure 8.**The deformation of the bar at time $t=0.9$ s: the FE solution (

**top left**) and the reduced order solution (

**top right**). Bottom: approximation error $|{d}_{s,h}-{d}_{s,N}|$, represented over the solid reference configuration (undeformed state). ${N}_{{d}_{s}}=21$ basis functions were used for the solid displacement. The deformation was magnified by a factor 5 for visualization purposes.

**Figure 9.**Fluid pressure at time $t=0.9$ s: the FE solution (

**top left**) and the reduced order solution (

**top right**). Bottom: approximation error $|{p}_{f,h}-{p}_{f,N}|$. ${N}_{p}=21$ basis functions were used for the fluid pressure, with the supremizer enrichment technique.

**Figure 10.**Fluid pressure approximation without the implementation of the supremizer enrichment: solution before the code diverges.

**Figure 11.**Fluid velocity at time $t=0.9$ s: the FE solution (

**top left**) and the reduced order solution (

**top right**). Bottom: approximation error $|{u}_{f,h}-{u}_{f,N}|$. ${N}_{u}=42$ basis functions were used for the fluid velocity.

**Figure 12.**Average relative error as a function of the number of basis functions used in the online system.

**Figure 14.**The outcomes of the partitioned POD: eigenvalue decay (

**a**) and energy retained (

**b**) for the first 100 modes.

**Figure 15.**The first three POD modes for the fluid velocity change in variable ${z}_{f}$ (magnitude): notice how, for the partitioned approach, the magnitude of the modes is zero not only on the inlet boundary (thanks to the lifting function) but also on the fluid–structure interface (thanks to the implementation of the change of variable).

**Figure 17.**The first three POD modes for the solid displacement ${d}_{s}$ (

**left column**) and the corresponding mesh displacement modes (

**right column**) obtained with an harmonic extension of the basis functions on the left column.

**Figure 18.**Deformation of the structure: FE solution (

**top left**), reduced order solution (

**top right**), and local approximation error (

**bottom**). The approximation was obtained with ${N}_{{d}_{s}}=13$ basis functions. The deformation was magnified by a factor 10 for visualization purposes.

**Figure 19.**Pressure Poisson recovery: FE solution (

**top left**), reduced order solution (

**top right**), and local approximation error (

**bottom**). The approximation was obtained with ${N}_{{p}_{f}}=13$ basis functions.

**Figure 20.**Fluid velocity: FE solution (

**top left**), reduced order solution (

**top right**), and local approximation error (

**bottom**). The approximation was obtained with ${N}_{{u}_{f}}=13$ basis functions.

**Figure 21.**Average relative approximation error as a function of the number N of modes used in the online phase.

**Figure 23.**Average number of iterations needed in the implicit step in order to reach convergence, as a function of the number N of reduced basis for ${\mathit{u}}_{f}$, ${p}_{f}$, ${\mathit{d}}_{s}$.

**Figure 24.**A comparison between the energy retained by the first modes for ${\mathit{z}}_{f}$ and for ${\mathit{u}}_{f}$.

Parameter | Value | Parameter | Value |
---|---|---|---|

Fluid density ${\rho}_{f}$$\left[{10}^{3}\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right]$ | 1 | Solid density ${\rho}_{s}$$\left[{10}^{3}\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}\right]$ | 1 |

Fluid kinematic viscosity ${\nu}_{f}$$\left[{10}^{-3}\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right]$ | 1 | 2nd Lamé constant (solid shear modulus) ${\mu}_{s}$$\left[{10}^{6}\frac{\mathrm{k}\mathrm{g}}{\mathrm{m}{\mathrm{s}}^{2}}\right]$ | $0.5$ |

Mean inflow velocity $\overline{U}$ $\left[\frac{\mathrm{m}}{\mathrm{s}}\right]$ | 1 | 1st Lamé constant ${\lambda}_{s}$ | $0.4$ |

Fluid external force ${\mathit{b}}_{f}$ | $(0,0)$ | Solid external force ${\mathit{b}}_{s}$ | $(0,0)$ |

Time Discretization Parameters | Value |
---|---|

Timestep $\Delta T$ | $0.01$ s |

Total number of iterations ${N}_{T}$ | ${10}^{3}$ |

$\gamma $ | $0.25$ |

$\beta $ | $0.5$ |

Space Discretization Parameters | Value |

FE velocity order | 2 |

FE pressure order | 2 |

FE displacement order (${\mathit{d}}_{f}$ and ${\mathit{d}}_{s}$) | 2 |

FE multiplier order (${\mathit{\lambda}}_{u}$ and ${\mathit{\lambda}}_{d}$) | 1 |

mesh resolution using mshr mesh generator | 128 |

Time Discretization Parameters | Value |
---|---|

$\Delta T$ | $0.0001$ s |

total number of iterations ${N}_{T}$ | ${10}^{4}$ |

Space Discretization Parameters | Value |

FE velocity order | 2 |

FE pressure order | 1 |

FE displacement order | 2 |

mesh resolution using mshr generator | 128 |

tolerance $\epsilon $ for the implicit iterations | ${10}^{-5}$ |

Monolithic Approach | Value |
---|---|

$\Delta T$ | $0.01$ s |

number of time iterations i of the RB solver | 200 |

solver for the system | Newton method |

average iterations of Newton method | 4 |

absolute tolerance for Newton method | $\left|\right|\xb7{\left|\right|}_{\infty}<6\xb7{10}^{-6}$ |

computational time to solve the online system for one time iteration | $224.9$ s |

Partitioned Approach | Value |

$\Delta T$ | $0.001$ s |

number of time iterations i of the RB solver | 2000 |

solver for explicit fluid step | Newton method |

average number of iterations of Newton method | 2 |

absolute tolerance for Newton method | $\left|\right|\xb7{\left|\right|}_{\infty}<6\xb7{10}^{-6}$ |

computational time to solve the online systems (explicit + implicit) for one time iteration | $162.68$ s |

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## Share and Cite

**MDPI and ACS Style**

Nonino, M.; Ballarin, F.; Rozza, G.
A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems. *Fluids* **2021**, *6*, 229.
https://doi.org/10.3390/fluids6060229

**AMA Style**

Nonino M, Ballarin F, Rozza G.
A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems. *Fluids*. 2021; 6(6):229.
https://doi.org/10.3390/fluids6060229

**Chicago/Turabian Style**

Nonino, Monica, Francesco Ballarin, and Gianluigi Rozza.
2021. "A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems" *Fluids* 6, no. 6: 229.
https://doi.org/10.3390/fluids6060229