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--- ife-ps [2024/09/12 15:13] – paulo
+++ ife-ps [2024/09/12 20:15] (current) – paulo
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 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 ALE description. The generalized alpha method was used for temporal integration for both phases.
+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.
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  [**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 $\SolidVar^{k+1}=\SolidVar^{k}+\Delta \SolidVar^{k}$, 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.
+  * 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.
+{{:usy.png?nolink&400|}}
+Figure 4 - Two-dimensional saturated poroelastic domain under a single pulse: vertical displacement field with ρ<sub> ∞ </sub> = 0.3.
+{{:graficoushv.png?nolink&400|}}
+Figure 5 - Two-dimensional saturated poroelastic domain under a single pulse: time history of vertical vs. horizontal displacements in Point A.
+{{:pftime2d.png?nolink&400|}}
+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 ====