MeshmerizeMe

IB2d and IBAMR are two software packages implementing the immersed boundary method (see below). These packages model fluid-structure interaction problems based on user given parameters and geometry. The manual creation of the initial geometry mesh can be difficult and time consuming, especially for the complex shapes encountered in biological applications. Oftentimes we have images of the geometry we wish to explore. I am developing software to help automate the creation of such CFD meshes for 2D simulations with a file-format suitable for use with IB2d and IBAMR from images. An initial prototype version is available on Github. A paper exploring the use of MeshmerizeMe in conjuction with IB2d for simulations is in preparation.

Usage

MeshmerizeMe needs two input files per experimental geometry: an SVG image file with the geometry of interest and an input2d file with the experiment parameters. When selecting an SVG for use with MeshmerizeMe it will automatically look for the input2d file in the same folder. It will then parse the paths, transform them into the correct coordinate system and appropriately sample the paths based on the size of the Cartesian grid set in the input2d file. The geometry will be exported as a vertex file. This file is readable by both IB2d and IBAMR.

SVGs were chosen as the image source as the are an open, text-based format making them very accesible to work with. They are standardized for web use and many tools exist for creating and manipulating SVG images. They can be created from source images such as photographs or scans by means of edge detection tools and by manually tracing the outline of a shape of interest Consider optimizing the SVG prior to processing to save time.

As the current version of MeshmerizeMe only handles a subset of SVG, tools that optimize the SVG files created by your editor are very useful. Examples of such software include SVGO, which also offers a webapp called SVGOMG. Another software is svgcleaner.

IBM Background

One aspect of computational fluid dynamics is the investigation of fluid-structure interactions. One method developed for the study of such interactions is the immersed boundary method (IBM) developed by Peskin¹. It is well known that fluids can be studied from both a Eulerian and a Lagrangian view. The IBM combines these - the domain of the problem is resolved as a Cartesian grid on which Eulerian equations are solved for fluid velocity and pressure. In the case of Newtonian fluids the incompressible Navier-Stokes equations comprising of

$$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla \mathbf{p} + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$

and

$$\nabla \cdot \mathbf{u} = 0$$

need to be solved.

The immersed structures are modeled as fibers in the form of parametric curves $X(s,t)$, where $s$ is a parameter and $t$ is time. The fiber experiences force distributions $F(s,t)$, and we can derive the force the fiber exerts on the fluid from the momentum equation. For the fibers we then solve

$$\mathbf{f} = \int_\Gamma \mathbf{F}(s,t),\delta\left(\mathbf{x}-\mathbf{X}(s,t)\right),ds$$

and

$$\frac{\partial \mathbf{X}}{\partial t} = \int_\Omega \mathbf{u}(\mathbf{x},t), \delta \left( \mathbf{x}-\mathbf{X}(s,t)\right),d\mathbf{x}.$$

Here, $\Gamma$ is the immersed structure and $\Omega$ is the fluid domain.

The immersed structures are discretized not on a Cartesian grid but on a separate Lagrangian grid on the fiber itself. Of import to CFD software users is that the initial discretization of the immersed structure has to be supplied by the user. While this is not too difficult for simple geometries, the often complex structures encountered in mathematical biology can present a significant time investment. This is the part where MeshmerizeMe comes in handy.