This tutorial demonstrates the use of unstructured meshes in 3D fluid-structure interaction problems. We combine two single-physics problems, namely
The two figures below show a sketch of the problem. An applied pressure drop drives fluid through an elastic, bifurcating tube whose branches have approximately rectangular cross-sections. We solve this problem as a fully-coupled fluid-structure interaction problem in which the fluid provides the traction onto the solid whose deformation changes the fluid domain.
As usual, we formulate the problem in non-dimensional form. For this purpose we non-dimensionalise all lengths on the half-width, , of the square inflow cross-section, and use the overall pressure drop, to define the (viscous) velocity scale
With this choice the Reynolds number becomes
and we choose to drive the flow by a dimensionless pressure drop of Using this non-dimensionalisation an increase in the Reynolds number may be interpreted as in increase in the applied (dimensional) pressure drop along the vessel. Note that all pressures are defined relative to an external pressure, which we set to zero.
We assume that the solid's constitutive equation is given by
oomph-lib's generalised Hookean constitutive law and non-dimensionalise the solid-mechanics stresses and tractions with the Young's modulus .
The FSI interaction parameter which represents the ratio of the (viscous) fluid stress scale to the reference stress used to non-dimensionalise the solid stresses, is therefore given by
The animation below illustrates the system's behaviour in a parameter study in which we keep the Reynolds number fixed at while increasing the FSI parameter in small increments. The increase in may be interpreted as a reduction in the tube's stiffness and the animation shows clearly how this increases the flow-induced deformation: the upstream end bulges and the downstream end is compressed.
The general procedure described below is essentially the same as that discussed in the two-dimensional unstructured FSI tutorial. The use of an unstructured mesh means that the most convenient node-update strategy for the fluid mesh is to treat it as a pseudo-elastic solid. Nonetheless, the majority of the steps described below are the same as for other fluid-structure-interaction problems.
We reiterate that an important prerequisite for the use of the automatic
FSI_functions is that each boundary between fluid and solid meshes must be parametrised by boundary coordinates. Moreover, the boundary-coordinate representations must be the same in both the fluid and solid meshes. This use of a continuous-coordinate representation of the boundaries means that the fluid and solid meshes do not need to match at the boundaries; see the section Fluid and solid meshes do not have to be matching. Unfortunately, the construction of a global surface parametrisation of a general surface is a non-trivial problem. We side-step this problem by making use of the fact that any third-party tetrahedral mesh generator represents domain boundaries using a surface triangulation, or planar surface facets. As the mesh is imported into
oomph-lib, each surface facet is treated as a separate boundary and given a unique boundary identifier. The surface parametrisation of each facet is then simply given by local coordinates of the plane; see How the boundary coordinates are generated. A consequence of this approach is that the physical boundaries will consist of several mesh "boundaries", as already discussed in the single-physics problems.
Since the driver code, discussed in detail below, is somewhat lengthy (partly because of the large number of self-tests and diagnostics included), we provide a brief overview of the main steps required to solve this problem:
tetgento generate the solid mesh, using the procedure discussed in another tutorial.
SolidMeshbase class to allow the use of pseudo-elasticity to update the nodal positions in response to the deformation of the domain boundary.
TetgenMeshclass, this may be done by calling the function
TetgenMesh::setup_boundary_coordinates(); see the section How the boundary coordinates are generated for details.
FSISolidTractionElementsto the faces of the "bulk" solid elements that are exposed to the fluid flow. These elements will apply the fluid traction to the solid.
FSISolidTractionElementsto a compound
GeomObjectthat provides a continuous representation of the solid's FSI boundary, required by the
ImposeDisplacementByLagrangeMultiplierElementsto the faces of the "bulk" fluid elements that are adjacent to the solid. These elements will employ Lagrange multipliers to deform the pseudo-solid fluid mesh so that its shape remains consistent with the motion of the solid's FSI boundary (as described by the compound
GeomObjectcreated in the previous step).
FSISolidTractionElements, using the function
In our experience, just as in two-dimensions, the most error-prone part of this procedure is the identification of the mesh boundaries. In particular, if the description of the FSI interface as viewed from the fluid and solid meshes is inconsistent, the automatic matching of the unstructured fluid and solid meshes will not work (see How the boundary coordinates are generated for details). For this reason, the driver code presented below generates a lot of output that can be used to identify and fix such problems. See also the section What can go wrong? at the end of this tutorial.
As usual we define the various problem parameters in a global namespace. We define the Reynolds number, , and the FSI interaction parameter .
We provide a pointer to the constitutive equation for the solid. For simplicity, the same constitutive equation will also be used for the (pseudo-)solid elements that determine the deformation of the fluid mesh. In general, of course, a different constitutive equation can (and probably should) be used to control the mesh motion.
Finally, we define the tractions that act on the fluid at the in- and outflow cross-sections. We enforce the non-dimensional pressure drop of (on the viscous scale) by setting , thus pressurising the tube's upstream end while applying an equal and opposite "suction" downstream. (Note that in the
corresponding single-physics fluids problem the actual pressure values were irrelevant because the vessel walls were rigid. In the FSI problem considered here it obviously makes a big difference if the fluid pressure is positive or negative because the external reference pressure is set to zero.)
Following the procedure discussed in the single-physics solid mechanics problem we create the mesh for the elastic tube using multiple inheritance from
TetgenMesh and the
SolidMesh base class.
As before, we set the Lagrangian coordinates to the current nodal positions to make the initial configuration stress-free. Next, we identify the elements next to the various boundaries (recall that the domain boundaries are specified in the tetgen
*.poly file), and set up the boundary coordinates. Adopting our usual state of continuous paranoia, we document these elements to facilitate debugging; see the section How the boundary coordinates are generated for details.
The creation of the fluid mesh follows the same process but uses the mesh created for the
single-physics fluids problem. The use of multiple inheritance from the
SolidMesh base classes is required to use pseudo-solid node-update techniques to move the fluid nodes in response to changes in the domain boundary. We refer to the fluids tutorial for a discussion of the
We create boundary coordinates along all mesh boundaries. To ensure that fluid and solid boundary coordinates are aligned properly, we use the flag
switch_normal to change the direction of the normal vector for the fluid mesh; see How the boundary coordinates are generated for details.
We specify an output directory and instantiate the constitutive equation (
oomph-lib's generalisation of Hooke's law), specifying a Poisson ratio of 0.3. As discussed above, this constitutive equation will be used for the "proper" solid mechanics that determines the deformation of the elastic vessel walls, and for the pseudo-solid that determines the deformation of the fluid mesh.
We create the
Problem object and output the initial guess for the solution.
Finally, we perform a parameter study in which we compute the solution of the fully-coupled FSI problem for increasing values of the FSI parameter – physically, an increase in can be interpreted as a reduction in the stiffness of the tube walls while keeping the fluid properties and the driving pressure drop constant.
Problem class has the usual members, with access functions to the fluid and solid meshes, and a post-processing routine.
We provide several helper functions to create the
FaceElements that (i) apply the applied traction on the fluid at the in- and outflow cross-sections; (ii) apply the fluid traction onto the solid, and (iii) create the Lagrange multipliers that apply the solid displacement onto the pseudo-solid fluid mesh.
We also provide a helper function that documents the boundary coordinates on the solid mesh.
Finally we provide a large number of additional helper functions that specify the various mesh boundaries (as defined in the tetgen
*.poly file) that make up the physical boundaries of interest: the in- and outflow boundaries in the fluid domain; the FSI boundaries of the fluid and solid domains and the boundaries along which the tube wall is held in a fixed position.
The private member data includes pointers to the various meshes and the
GeomObject representation of the FSI boundary (created from the
FaceElements attached to the solid mesh).
Finally, here are the vectors that store the mesh boundary IDs associated with the various domain boundaries of interest.
Next we associated the
tetgen boundary IDs with the various boundaries of interest: The inflow boundary is represented by
tetgen boundary 0, and the two outflow boundaries have IDs 1 and 2.
The FSI boundary (i.e. the boundary of the fluid mesh that is exposed to the elastic vessel wall) comprises 12 separate tetgen facets which were numbered 3 to 14 in the
*.poly that describes the fluid mesh.
Following the procedure used for the fluid mesh, we identify the mesh boundaries that make up the (pinned) ends of the tube (boundaries 0, 1 and 2, as defined in the tetgen
*.poly file) and the FSI boundary (boundaries 3 to 14 – note that this enumeration matches that in the fluid mesh; see How the boundary coordinates are generated for further details of how the fluid and solid meshes are matched).
We create the fluid traction elements that impose the applied pressures in the in- and outflow cross-sections.
Next, we create the
FaceElements that apply the fluid traction to the solid,
FaceElements that use Lagrange multipliers to deform the fluid mesh to keep it aligned with the FSI boundary.
We combine the various sub-meshes to a global mesh.
Next, we apply the boundary conditions for the fluid mesh: We impose parallel in- and outflow at the in- and outflow boundaries and apply boundary conditions for the pseudo-elastic deformation of the fluid mesh. Since the in- and outflow cross-sections of the elastic tube are held in place, we pin the position of the fluid nodes in these cross-sections too. To facilitate debugging we document the position of the fluid nodes whose (pseudo-solid) displacements we suppressed.
We apply the no-slip condition on the fluid nodes that are located on the FSI boundary. In addition, we apply boundary conditions for the Lagrange multipliers. We pin the Lagrange multipliers for nodes that are located on the in- and outflow boundaries where the nodal positions are pinned. Recall that the Lagrange multipliers are additional degrees of freedom that are added to the "bulk" degrees of freedom that were originally created by the "bulk" element. The storage for the Lagrange multipliers is added to the Nodes by the FaceElements and the values at which the Lagrange multipliers are stored are found using the function
BoundaryNodeBase::index_of_first_value_assigned_by_face_element(). The documentation of the position of the pinned Lagrange multiplier nodes in
RESLT/pinned_lagrange_multiplier_nodes.dat is here to facilitate the debugging of the code and is highly recommended.
We complete the build of the fluid elements by specifying the Reynolds number and the constitutive equation for the pseudo-solid equations; recall that we use the same constitutive equation as used for the tube wall.
We apply the "solid" boundary conditions by pinning the positions of the nodes that are located at the ends of the elastic tube, and, just to be on the safe side, document their positions to allow for debugging and sanity-checking.
We complete the build of the solid elements by passing the pointer to the constitutive equation.
Finally, we set up the fluid-structure interaction by determining which "bulk" fluid elements are located next to the FSI traction elements that apply the fluid load to the solid. This must be done separately for each of the mesh boundaries that make up the physical FSI boundary. To facilitate debugging, we document the boundary coordinates along the FSI interface (as seen by the fluid) by opening the
Multi_domain_functions::Doc_boundary_coordinate_file stream before calling
FSI_functions::setup_fluid_load_info_for_solid_elements(...). If this stream is open, the setup routine writes the Eulerian coordinates of the points on the FSI interface and their intrinsic surface coordinate to the specified file. This may be compared against the corresponding data for the solid's view of the FSI interface, documented in
All that's now left to do is to set up the equation numbering scheme and the problem is ready to be solved.
The creation of the FSI traction elements that apply the fluid traction to the solid elements that are adjacent to the FSI boundary follows the usual procedure: We loop over the relevant 3D "bulk" solid elements and attach the
FSISolidTractionElements to the appropriate faces.
Next we add the newly-created
FaceElement to the mesh of traction elements, specify which boundary of the bulk mesh it is attached to, and pass the FSI interaction parameter to the element.
The creation of the
FaceElements that use Lagrange multipliers to impose the boundary displacement of the pseudo-solid fluid mesh is again fairly straightforward (the use of Lagrange multipliers for the imposition of boundary displacements is explained in another tutorial). The only complication is that we must loop over the different parts of the FSI boundary. In each case we combine the
FSISolidTractionElements attached to the solid mesh into a compound
GeomObject provides a continuous representation of the relevant part of the FSI boundary, parametrised by the boundary coordinate assigned earlier while its shape is determined by the deformation of the 3D solid elements that the
FSISolidTractionElements are attached to.
Having represented the boundary by a (compound)
GeomObject, we now attach
ImposeDisplacementByLagrangeMultiplierElements to the appropriate faces of the "bulk" fluid elements that are adjacent to the FSI interface and add them to their own mesh:
Finally, we pass a pointer to the compound
GeomObject that defines the shape of the FSI interface and specify which boundary in the "bulk" fluid mesh the
ImposeDisplacementByLagrangeMultiplierElement is attached to.
The helper function
NavierStokesTractionElements to the in- and outflow cross-sections of the fluid mesh and thus imposes the prescribed pressure drop onto the fluid.
The post-processing routine simply executes the output functions for the fluid and solid meshes and documents their mesh boundaries. We also document the FSI traction that the fluid exerts onto the solid.
doc_solid_boundary_coordinates() documents the parametrisation of the solid's FSI boundary by writing into a file the solid's counterpart of the data that we created for the fluid side of the FSI interface when setting up the fluid-structure interaction with
FSI_functions::setup_fluid_load_info_for_solid_elements(...). The two parametrisations should be consistent; see What can go wrong? for more details. The implementation is straightforward and is suppressed for brevity; see the source code if you want to know how it works.
The use of pseudo-elasticity for the node update in the fluid mesh makes the solution of FSI problems extremely straightforward. They key feature that allows the "automatic" coupling of the unstructured fluid and solid meshes is the (consistent!) generation of the boundary coordinates along the FSI interface. For reasons already mentioned, the procedure is more involved than in the equivalent two-dimensional problem. The implementation of this functionality in
TetgenMesh::setup_boundary_coordinates(...) exploits the fact that
tetgenare bounded by planar facets, and
The figure below illustrates the generation of the boundary coordinates within a representative planar facet (shown in cyan) that defines a particular mesh boundary.
Here is how we generate the boundary coordinates:
FaceElementsto the appropriate faces of the "bulk" elements that are adjacent to the mesh boundary defined by the facet. These
FaceElementsprovide a surface triangulation of the boundary. In the sketch above, the surface triangulation is represented by the blue surface mesh.
This allows the assignment of boundary coordinates for each of the nodes on this boundary.
FaceElementsthat defined the surface triangulation.
Since the generation of the boundary coordinates by the above procedure only relies on the position of the "lower left" and "upper right" vertices (which are properties of the facet rather than the
tetgen mesh) the boundary coordinates are unique, regardless of the actual discretisation generated by
tetgen. Furthermore, the boundary coordinates created from the fluid and solid meshes will be consistent provided we reverse the direction of the outer unit normal in one of the meshes. This is exactly what the
switch_normal flag in
TetgenMesh::setup_boundary_coordinates(...) is for. When generating the boundary coordinates for our problem, we switched the direction of the normal in the fluid mesh; see the section The fluid mesh, above.
Here is an animation of the automatically-generated boundary coordinates along the 12 mesh boundaries that constitute the FSI boundary, when viewed from the fluid and the solid, respectively. In each of the frames the contours indicate the value of
within the respective boundary. Note that the boundary coordinates established in the two meshes match, even though the discretisations do not.
You should explore this procedure yourself as an exercise: For instance you may want to explore what happens if you don't switch the direction of the normal in the fluid mesh when generating the boundary coordinates.
The mesh shown in the problem sketch at the beginning of this tutorial already suggested that the fluid and solid meshes do not have to match across the FSI interface. To illustrate this point more clearly, here is the result of another computation for which we generated a much finer fluid mesh, using
The resulting tetgen
*.face files can be used with same driver code.
As an exercise, generate some finer fluid and solid meshes yourself and confirm that they may be used without having to change the driver code.
Here are a few things that can (and often do) go wrong in unstructured, three-dimensional FSI problems.
*.polyfiles describing the fluid and solid meshes do not use the same facets to describe the FSI interface. While we do not expect the fluid and solid meshes themselves to match, the representation of the FSI interface via facets
mustbe consistent. A simple way to ensure consistency is to use the following procedure:
*.polyfile for the fluid mesh. Make sure that each facet is given a separate boundary ID (at least for the facets describing the FSI interface).
*.polyfile for the fluid mesh and modify it to describe the solid mesh, leaving the facets that define the FSI interface unchanged. This automatically ensures that the representation of the FSI boundary from the fluid's and the solid's point of view is consistent.
*.polyfiles and the driver code so that it can handle the extended domain considered in the single-physics fluids problem.
tetgendoes not allow non-planar facets.
FSI_functions::setup_fluid_load_info_for_solid_elements(...)and attach the Lagrange multiplier elements but keep If the solid is not loaded by any forces/tractions other than those arising from the fluid, the solution should not change as the solid remains unloaded.
A pdf version of this document is available.