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fluid_structure_interaction/README.md

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 ## Validation Cases
 - [FSI lid driven cavity](https://github.com/KratosMultiphysics/Examples/blob/master/fluid_structure_interaction/validation/fsi_lid_driven_cavity/README.md)
+- [Flexible membrane airfoil (embedded)](https://github.com/KratosMultiphysics/Examples/blob/master/fluid_structure_interaction/validation/embedded_fsi_membrane_airfoil/README.md)
 - [Mixer with flexible blades (embedded)](https://github.com/KratosMultiphysics/Examples/blob/master/fluid_structure_interaction/validation/embedded_fsi_mixer_Y/README.md)
 - [Mok benchmark](https://github.com/KratosMultiphysics/Examples/blob/master/fluid_structure_interaction/validation/fsi_mok/README.md)
 - [Mok benchmark (embedded)](https://github.com/KratosMultiphysics/Examples/blob/master/fluid_structure_interaction/validation/embedded_fsi_mok/README.md)

fluid_structure_interaction/validation/embedded_fsi_membrane_airfoil/README.md

@@ -0,0 +1,45 @@
+# Flexible membrane airfoil
+
+**Author:** [Rubén Zorrilla](https://github.com/rubenzorrilla)
+
+**Kratos version:** 9.5
+
+**Source files:** [Flexible membrane airfoil](https://github.com/KratosMultiphysics/Examples/tree/master/fluid_structure_interaction/validation/embedded_fsi_membrane_airfoil/source)
+
+## Case Specification
+This example reproduces a reduced-scale experiment of a flexible membrane airfoil[1]. The embedded (i.e., CutFEM) framework for thin-walled bodies is applied in order to avoid the preprocessing and mesh entanglement issues appearing when using volumetric meshes around membrane-like bodies [2].
+
+The problem set up consists in a 2D idealization of the flexible membrane airfoil at an angle of attack of 4º. The lenght and thickness of the membrane are 0.15m and 2e-4m, respectively, while the material parameters are a Young modulus of 250e3Pa and zero Poisson ration. The fluid properties are set such that Re=2500. The structure and fluid density ratio is 441.75. The inlet characteristic velocity is 2.5833m/s.
+
+## Results
+The fluid domain is meshed with a 144k P1P1 elements unstructured mesh. The membrane is meshed with 128 line elements implementing a simplified 2D non-linear membrane formulation. The problem is run with a ramp up function of 1s and for an extra second so to ensure that the membrane reaches the steady state.
+
+The steady state velocity and pressure contour fields as well as the displacement vector field are shown below. The obtained results are in line with similar numerical experiments [3].
+
+<p align="center">
+<figure>
+  <img src="data/embedded_fsi_membrane_airfoil_fluid_v.gif" alt="Fluid velocity contour field." style="width: 600px;"/>
+  <figcaption>Velocity field and level set isosurface.</figcaption>
+</figure>
+</p>
+
+<p align="center">
+<figure>
+  <img src="data/embedded_fsi_membrane_airfoil_fluid_p.gif" alt="Fluid pressure contour field." style="width: 600px;"/>
+  <figcaption>Pressure field and level set isosurface.</figcaption>
+</figure>
+</p>
+
+<p align="center">
+<figure>
+  <img src="data/embedded_fsi_membrane_airfoil_structure_u.gif" alt="Membrane displacement vector field." style="width: 600px;"/>
+  <figcaption>Pressure field and level set isosurface.</figcaption>
+</figure>
+</p>
+
+## References
+[1] P. Rojratsirikul, Z. Wang and I. Gursul, I, Unsteady Aerodynamics of Membrane Airfoils. Paper presented at 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2008 [10.2514/6.2008-613](https://doi.org/10.2514/6.2008-613).
+
+[2] R. Zorrilla, R. Rossi, R. Wüchner and E. Oñate, An embedded Finite Element framework for the resolution of strongly coupled Fluid–Structure Interaction problems. Application to volumetric and membrane-like structures, Computer Methods in Applied Mechanics and Engineering (368), 2020 [10.1016/j.cma.2020.113179](https://doi.org/10.1016/j.cma.2020.113179).
+
+[3] R. E. Gordnier, High fidelity computational simulation of a membrane wing airfoil, Journal of Fluids and Structures (25), 2009, [10.1016/j.jfluidstructs.2009.03.004](https://doi.org/10.1016/j.jfluidstructs.2009.03.004).

fluid_structure_interaction/validation/embedded_fsi_membrane_airfoil/source/FluidMaterials.json

@@ -0,0 +1,20 @@
+{
+    "properties" : [{
+        "model_part_name" : "FluidModelPart",
+        "properties_id"   : 0,
+        "Material"        : null
+    },{
+        "model_part_name" : "FluidModelPart.FluidParts_Fluid",
+        "properties_id"   : 1,
+        "Material"        : {
+            "constitutive_law" : {
+                "name" : "Newtonian2DLaw"
+            },
+            "Variables"        : {
+                "DENSITY"           : 1.0,
+                "DYNAMIC_VISCOSITY" : 0.000155
+            },
+            "Tables"           : null
+        }
+    }]
+}

fluid_structure_interaction/validation/embedded_fsi_membrane_airfoil/source/MainKratos.py

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+import sys
+import math
+import time
+import importlib
+
+import KratosMultiphysics
+
+def CreateAnalysisStageWithFlushInstance(cls, global_model, parameters):
+    class AnalysisStageWithFlush(cls):
+
+        def __init__(self, model,project_parameters, flush_frequency=10.0):
+            super().__init__(model,project_parameters)
+            self.flush_frequency = flush_frequency
+            self.last_flush = time.time()
+            sys.stdout.flush()
+
+        def Initialize(self):
+            super().Initialize()
+            sys.stdout.flush()
+
+        def FinalizeSolutionStep(self):
+            super().FinalizeSolutionStep()
+
+            if self.parallel_type == "OpenMP":
+                now = time.time()
+                if now - self.last_flush > self.flush_frequency:
+                    sys.stdout.flush()
+                    self.last_flush = now
+
+    return AnalysisStageWithFlush(global_model, parameters)
+
+if __name__ == "__main__":
+
+    with open("ProjectParameters.json", 'r') as parameter_file:
+        parameters = KratosMultiphysics.Parameters(parameter_file.read())
+
+    analysis_stage_module_name = parameters["analysis_stage"].GetString()
+    analysis_stage_class_name = analysis_stage_module_name.split('.')[-1]
+    analysis_stage_class_name = ''.join(x.title() for x in analysis_stage_class_name.split('_'))
+
+    analysis_stage_module = importlib.import_module(analysis_stage_module_name)
+    analysis_stage_class = getattr(analysis_stage_module, analysis_stage_class_name)
+
+    global_model = KratosMultiphysics.Model()
+    simulation = CreateAnalysisStageWithFlushInstance(analysis_stage_class, global_model, parameters)
+    simulation.Run()