Simulation of pressurized bubbles

Repository: https://github.com/rprat-pro/mm-opera-hpc/tree/main/bubble

Problem description

The default example consists of a single spherical porosity in a quasi-infinite medium. The finite element solution can be compared with an analytical solution giving the elastic stress field as a function of the internal pressure, the bubble radius, and the distance from the bubble. As mentioned above, the boundary conditions for the problem are periodic, and we consider a null macroscopic displacement gradient, which in turn generates a uniform compressive hydrostatic pressure on the RVE. In this case with one porosity in a quasi-infinite medium, the compressive hydrostatic pressure is negligible, in agreement with the analytical solution mentioned above.

Modify the geometry for the single bubble case and mesh it

The geometry for the test case is contained in the file .geo stored in the mesh folder, and considers a sphere of radius equal to 400 nm at the centre of a (periodic) cube of 10 µm of size. For a more handy management of the geometry and of the mesh, the units in the geometry file are expressed in \(\mathrm{\mu m}\). One can modify it and use it as an input for gmsh to generate the computational mesh for the case by:

gmsh -3 single_sphere.geo

A file .msh is already provided in the folder mesh, generated based on the aforementioned geometry file. We have seen some slight differences in the final mesh based on the version of gmsh employed.

Note

If the bubble center, radius, or the surface label are modified, the corresponding data stored in single_bubble.txt must also be changed.

Note

single_bubble_ci.txt is used for GitHub continuous integration.

Set-up the physical problem

The simulation considers an empty (i.e., not meshed) cavity, on whose surface we impose an arbitrary uniform pressure (unitary by default). The medium is described by a purely elastic constitutive relationship, characterized by two elastic constants:

• \(E = 150\ \mathrm{N}\ \mu\mathrm{m}^{-3}\)

• \(\nu = 0.3\)

The elastic modulus is rescaled to coherently describe the geometry in micrometers, rather than in S.I. units. This choice is done to facilitate the creation of more complex geometries when using Mérope, given the characteristic length scale of the considered inclusions.

The geometry is meshed using quadratic elements, to better describe the spherical inclusions contained in the representative volume element (RVE). Despite MFEM allowing sub-, super-, and isoparametric analyses, we recommend sticking at least to the isoparametric choice (i.e., not subparametric) for the polynomial shape functions.

The boundary conditions for the problem are periodic, and we consider a null macroscopic displacement gradient, which in turn generates a uniform compressive hydrostatic pressure on the RVE.

Parameters

Command-line Usage:

Usage: ./test-bubble [options] ...

Option	Type	Default	Description
-h, --help	—	—	Print the help message and exit.
-m <string>, --mesh <string>	string	mesh/single_sphere.msh	Mesh file to use.
-l <string>, --library <string>	string	src/libBehaviour.so	Material behaviour library.
-f <string>, --bubble-file <string>	string	mesh/single_bubble.txt	File containing the bubble definitions.
-o <int>, --order <int>	int	2	Finite element order (polynomial degree).
-r <int>, --refinement <int>	int	0	Refinement level of the mesh (default = 0).
-p <int>, --post-processing <int>	int	1	Run the post-processing step.
-v <int>, --verbosity-level <int>	int	0	Verbosity level of the output.

The command to execute the test-case is:

mpirun -n 6 ./test-bubble

Below we show a contour plot of the \(YY\) component of the stress tensor (upper half of the cube) and of the first principal stress (bottom half of the cube).

Verification against the analytical solution

The problem of a pressurized spherical inclusion in an infinite elastic medium has a closed-form solution for the expressions of the hoop stress as a function of the distance from the sphere center:

\[\sigma_{\theta\theta}(r) \;=\; \dfrac{p_{in}\,R_b^3}{2\,r^3}\]

where \(p_{in}\) is the internal pressure, \(R_b\) the bubble radius, and the expression holds for \(r > R_b\).

The script available in verification/bubble can be used to compare the analytical solution to the MMM one:

python3 mmm_vs_analytical.py

The comparison between the computational results and the analytical solution is shown below.