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====== Incompressible fluid porous structure interaction with phase homogenization: a partitioned approach ======

In this project, we propose a new decomposition formulation for the dynamic analysis of poroelastic problems called //**y**//<sup>s</sup>//**v**//<sup>f</sup>//p//<sup>f</sup> considering geometric nonlinearity and allowing large displacements, where the superindex denotes the respective phase {//f,s//}. These guarantees full modularity of the calculation code and is particularly suitable when displacements are large. In developing this formulation, several assumptions are made to reduce the complexity of the poroelastic problem. A poroelastic solid is considered saturated with fluid, and both the liquid and solid phases are considered materially incompressible within the particle, ignoring the effects of thermal and chemical reactions. Furthermore, porosity is assumed to remain constant during the deformation process, and although large displacements are well represented, the scope is limited to small deformation problems (see e.g [1]).

The solution process for the porous solid phase is based on the positional Finite Element Method (FEM) in Lagrangian description with the Neo-Hookean constitutive model, while for the interstitial incompressible fluid flow the Pressure-Stabilized/Petrov–Galerkin method is used (PSPG) [2] using the Arbitrary Lagrangian-Eulerian (ALE) description. The generalized alpha method was used for temporal integration for both phases.

In summary, the strong coupling between the two problems occurs through the transfer of forces between the phases, in which the fluid in the following partitioned scheme transfers pressure (//p//<sup>f </sup>) and the resistance force generated due to the microscale effect, also called the Darcy force (//**f**//<sup> f </sup>), which can be seen in Figure.1.

{{:ife_ps.png?nolink&400|}}

Figure 1 - Specific fluid-solid interaction force.

In the other direction, the solid transfers a new position (//**y**//<sup> s</sup>) and updated velocity (//**v**//<sup> s</sup>=//**v**//<sup> m</sup>) to the mesh (ALE). In this scheme, a strategy was used for homogenized phases and overlapping meshes with coincident nodes, as can be understood in Figure 2.

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Figure 2 - Overlapping of computational domains of the phases.


Finally, the derivation of the partitioned solution for the coupling problem is based on the description of [3] for the direct, quasi-direct, and block-iterative solutions of fluid–structure interaction problems. One iteration //k// on the solution of a monolithic-coupled problem via the Newton–Raphson method requires finding the correction for solid {Δ//**u**//<sub> s </sub>}<sup> k</sup> and fluid {Δ//**u**//<sub> f </sub>}<sup> k</sup> variables by solving a problem of type:

|  [**H**<sub>ff</sub>]<sup>k</sup>  |  [**H**<sub>fs</sub>]<sup>k</sup>  |  {Δ//**u**//<sub> s </sub>}<sup> k</sup>  | = |  - {//**h**//<sub> f </sub>}<sup> k</sup>  |
|  [**H**<sub>sf</sub>]<sup>k</sup>  |  [**H**<sub>ss</sub>]<sup>k</sup>  |  {Δ//**u**//<sub> f </sub>}<sup> k</sup>  | = |  - {//**h**//<sub> s </sub>}<sup> k</sup>  |


where [**H**<sub>ab</sub>]<sup>k</sup> is the part of the tangent matrix given by the derivative of the discrete equations for the phase //a// concerning the variables of the phase //b//, and (//**h**//<sub> a </sub>)<sup> k</sup> is the residual of the discrete equation for the phase //b//. The fluid state variables vector //**u**//<sub>f </sub> contains the nodal values of pressure and the nodal values of velocity, while the solid state variables vector //**u**//<sub>s </sub> contains only nodal values of position.

The terms [**H**<sub>fs</sub>]<sup>k</sup> and [**H**<sub>sf</sub>]<sup>k</sup> can be difficult to obtain as different mathematical approaches and frameworks are usually employed for solid and fluid. One option is to modify the associated tangent matrix by making [**H**<sub>fs</sub>]<sup>k</sup>=[**H**<sub>sf</sub>]<sup>k</sup> = {**0**}. This approximation results in a system with two independent blocks, and still satisfies the problem solution, as the residual is still calculated consistently. However, convergence conditions of Newton-Raphson's method are not ensured.

Based on the Gauss-Seidel approach, we can improve the procedure by solving the system blocks sequentially, and updating variables after the solution of each block, so that, for each Newton-Raphson iteration //k//, at a given time step the following steps are performed:

  * Compute the residual and the tangent matrix for the fluid block ({//**h**//<sub> f </sub>}<sup> k</sup>(//**u**//<sub> f </sub><sup> k</sup>,//**u**//<sub> s </sub><sup> k</sup>) and [**H**<sub>ff</sub>]<sup>k</sup>(//**u**//<sub> f </sub><sup> k</sup>,//**u**//<sub> s </sub><sup> k</sup>), respectively), considering the values of solid and fluid variables from the last iteration (//**u**//<sub> f </sub><sup> k</sup>=//**u**//<sub> f </sub><sup> (k-1)</sup>+Δ //**u**//<sub> f </sub><sup> (k-1)</sup> and //**u**//<sub> s </sub><sup> k</sup>=//**u**//<sub> s </sub><sup> (k-1)</sup>+Δ //**u**//<sub> s </sub><sup> (k-1)</sup>). Then solve the first block: 

 [**H**<sub>ff</sub>](//**u**//<sub> f </sub><sup> k</sup>,//**u**//<sub> s </sub><sup> k</sup>) Δ//**u**//<sub> f </sub><sup> k</sup> = - {//**h**//<sub> f </sub>}(//**u**//<sub> f </sub><sup> k</sup>,//**u**//<sub> s </sub><sup> k</sup>)
 
  * Update the fluid variables by //**u**//<sub> f </sub><sup> (k+1)</sup>=//**u**//<sub> f </sub><sup> k</sup>+Δ //**u**//<sub> f </sub><sup> k</sup>  and compute the solid block residual and tangent matrix with the updated fluid variables ({//**h**//<sub> s </sub>}<sup> (k+1)</sup>(//**u**//<sub> f </sub><sup> (k+1)</sup>,//**u**//<sub> s </sub><sup> k</sup>)$ and [**H**<sub>ss</sub>]<sup>k</sup>(//**u**//<sub> f </sub><sup> (k+1)</sup>,//**u**//<sub> s </sub><sup> k</sup>), respectively. Then, solve the second block:

 [**H**<sub>ss</sub>](//**u**//<sub> f </sub><sup> (k+1)</sup>,//**u**//<sub> s </sub><sup> k</sup>) Δ//**u**//<sub> s </sub><sup> k</sup> = - {//**h**//<sub> s </sub>}(//**u**//<sub> f </sub><sup> (k+1)</sup>,//**u**//<sub> s </sub><sup> k</sup>)


  * Update solid variables by //**u**//<sub> s </sub><sup> (k+1)</sup>=//**u**//<sub> s </sub><sup> k</sup> + Δ//**u**//<sub> s </sub><sup> k</sup>, and check the user prescribed convergence criteria for solid and fluid. If the convergence criterion is satisfied, go to the next time step, otherwise, proceed to the next Newton-Raphson iteration. 

With the modifications in the tangent matrix, in strong coupled problems (where small changes in fluid produce big changes in solid and //vice-versa//), convergence can be hard, or even the solution can diverge. This problem is also found in partitioned methods for standard fluid-structure interaction problems [4]. One possible way to improve convergence is to use the Aitken relaxation as in [4,5,6]. One simpler way, proposed by [7,8,9] is to scale the solid mass matrix to prevent corrections in the solid variables with elevated magnitude during the iterative solution process, keeping the solution consistent as the residuals are not changed. 

The following example consists of a saturated porous 2D domain with little deformation of the solid matrix. The solid phase boundary conditions on the side faces are of the type //y//<sub> (1) </sub>=0 in Γ<sup> D </sup> and lower //y//<sub> (2) </sub>=0 in Γ<sup> D </sup>, and in the top part the applied boundary condition is characterized by a single pulse (Γ<sup> N </sup>), where D refers to Dirichlet and N to Neumman. As for the fluid, the condition of no-slinping boudary is applied at the lateral and base boundaries Γ<sup> D </sup>, while at the top the flow permission is considered (permeable region) in Γ<sup> N </sup>. The temporal (Figures 3,4,5 and 6) results obtained for displacement and pressure are shown below, where point A refers to a point on the top of the surface and point B an internal point of the domain close to the application of the load.

{{:ush.png?nolink&400|}}

Figure 3 - Two-dimensional saturated poroelastic domain under a single pulse: horizontal displacement field with ρ<sub> ∞ </sub> = 0.3.

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Figure 4 - Two-dimensional saturated poroelastic domain under a single pulse: vertical displacement field with ρ<sub> ∞ </sub> = 0.3.

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Figure 5 - Two-dimensional saturated poroelastic domain under a single pulse: time history of vertical vs. horizontal displacements in Point A.

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Figure 6 - Two-dimensional saturated poroelastic domain under a single pulse: time history of pressure at Point B.

This second problem, and the last one presented here (see e.g [1] for other examples), to evaluate computational approaches for large deformations in poroelasticity. It consists of a 2D square block, subjected to self-weight, leading to deformation and consolidation over time. The deformed configurations are illustrated along with the pressure field in the interstitial fluid and matrix displacement (Figures 7,8 and 9). The points evaluated for pressure //P1//, //P2// and //P3// refer to the lower extremity on the right, the central extremity of the lower base and the central extremity of the domain, respectively.

{{:displacementblock.png?nolink&400|}}

Figure 7 - 2D self-weight slumping block: deformed configuration snapshots, displacement magnitude.

{{:pressureblock.png?nolink&400|}}

Figure 8 - 2D self-weight slumping block: pressure snapshots, values in Pa.

{{:pressuretimeblock.png?nolink&400|}}

Figure 9 - 2D self-weight slumping block: time history of pressure for points P1, P2 and P3.

==== References ====

1- Paulo H. de F. Meirelles, Jeferson W.D. Fernandes, Rodolfo A.K. Sanches, and Wilson W. Wutzow. A modular finite element approach to saturated poroelasticity dynamics: Fluid–solid coupling with neo-hookean ma-
terial and incompressible flow. Finite Elements in Analysis and Design,
242:104256, 2024.

2- T.E. Tezduyar. Stabilized finite element formulations for incompressible flow computations. 28:1–44, 1992.

3- Y. Bazilevs, K. Takizawa, and T.E. Tezduyar. Computational Fluid-Structure Interaction: Methods and Applications. John Wiley & Sons, Chichester, UK, 2013.

4- Jeferson Wilian Dossa Fernandes, Humberto Breves Coda, and Rodolfo André Kuche Sanches. ALE incompressible fluid–shell coupling based on a higher-order auxiliary mesh and positional shell finite element. Computational Mechanics, 63(3):555–569, 2019.

5- Ulrich Kuttler and Wolfgang A. Wall. Fixed-point fluid–structure interaction solvers with dynamic relaxation. Computational Mechanics, 43(1):61–72, 2008.

6- Wolfgang A. Wall, Steffen Genkinger, and Ekkehard Ramm. A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Computers & Fluids, 36(1):169–183, 2007.

7- T.E. Tezduyar. Stabilized finite element methods for computation of flows with moving boundaries and interfaces. In Lecture Notes on Finite Element Simulation of Flow Problems (Basic - Advanced Course), Tokyo, Japan, 2003. Japan Society of Computational Engineering and Sciences.

8- T.E. Tezduyar. Stabilized finite element methods for flows with moving boundaries and interfaces. HERMIS: The International Journal of Computer Mathematics and its Applications, 4:63–88, 2003.

9- T.E. Tezduyar. Finite element methods for fluid dynamics with moving boundaries and interfaces. In E. Stein, R.D. Borst, and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, volume 3, chapter 17. John Wiley & Sons, 2004.

Authors:

Paulo H. de F. Meirelles

Jeferson W. D. Fernandes

Rodolfo A. K. Sanches

Wilson W. Wutzow