Eikonal solver

This repository uses the eikonal equation to compute the distance of each point in a 3D geometry from an inlet boundary and the walls boundary. It may be used (among many other applications) for fast approximations of mold filling simulations [1].

Theory

The solution of the eikonal equation $u(\mathbf{x})$ can be interpreted as the shortest path to travel from a boundary $\partial \Omega$ to an inner point $\mathbf{x} \in \Omega$. The boundary value problem is given by

$$ \begin{aligned} |\nabla \phi(\mathbf{x}) | &= \frac{1}{f(\mathbf{x})} \quad \mathbf{x} \in \Omega \\ \phi(\mathbf{x}) &= \phi^0(\mathbf{x}) \quad \mathbf{x} \in \partial \Omega , \end{aligned} $$

where $f(\mathbf{x})>0$ denotes the speed of travel. For $f \equiv 1$, it computes the geodesic distance to the boundary.

Methods

FEM solution (Fares & Schröder)

Unfortunately, the norm in the original formualtion is not friendly for the numerical solution. Hence, Fares and Schröder [2,3] derived the formulation

$$ \nabla G(\mathbf{x}) \cdot \nabla G(\mathbf{x}) + \sigma G(\mathbf{x}) \Delta G(\mathbf{x}) = (1 + 2\sigma)G^4(\mathbf{x}) \quad \mathbf{x} \in \Omega $$

for the inverse wall distance $G(\mathbf{x})$. The weak form of this equation is

$$ (1-\sigma) \int_\Omega \nabla G(\mathbf{x}) \cdot \nabla G(\mathbf{x}) H(\mathbf{x}) dV - \sigma \int_V G(\mathbf{x}) \nabla G(\mathbf{x}) \cdot \nabla H(\mathbf{x}) dV - (1+2\sigma) \int_\Omega G^4(\mathbf{x}) H(\mathbf{x}) dV = 0 \quad \forall H(\mathbf{x}) $$

assuming $\nabla G(\mathbf{x}) = 0$ on the Neumann boundary.

To use this method, edit the input mesh in the Notebook fares_schroeder.ipynb in this line

mesh_name = os.path.join("meshes", "control_arm_hole.msh")

and run all cells. The nutils [4] based solver computes the properties D_i and D_w representing the distance to the inlet surface and the next wall, respectively.

Fast iterative method for tetragonal meshes

An alternative to the FEM based solution approach are iterative methods, e.g. the Fast Marching Method [5] or Fast Sweeping Method [6]. For parallel execution on tetragonal domains, Fu et al. [7] suggest the tetragonal Fast Iterative Method, which is very suitable for the solution in this case.

To use this method, edit the input mesh in the Notebook fim.ipynb in this line

mesh_name = os.path.join("meshes", "control_arm_hole.msh")

and run all cells. The solver computes (among some other propeties) the properties D_i and D_w representing the distance to the inlet surface and the next wall, respectively.

Postprocessing

The results are exported as *.vtu file. These can be visualized e.g. with ParaView [9].

References

[1] Ospald, F., Herzog, R. (2018). SIMP based Topology Optimization for Injection Molding of SFRPs, 12th World Congress on Structural and Multidisciplinary Optimization, Braunschweig, Germany. https://doi.org/10.1007/978-3-319-67988-4_65

[2] Fares, E., Schröder, W. (2002). A differential equation for approximate wall distance. International Journal for Numerical Methods in Fluids, 39(8), 743–762. https://doi.org/10.1002/fld.348

[3] https://www.comsol.de/blogs/tips-using-wall-distance-interface/

[4] https://nutils.org

[5] Kimmel, R., Sethian, J. A. (1998). Computing geodesic paths on manifolds, Proceedings of the National Academy of Sciences of the United States of America 95(15) 8431–8435. https://doi.org/10.1073/pnas.95.15.8431.

[6] Zhao, H. (2004). A fast sweeping method for eikonal equations, Mathematics of Computation 74(250) 603–627. https://doi.org/10.1090/s0025-5718-04- 01678-3.

[7] Fu, Z., Kirby, R. M., Whitaker, R. T. (2013). A fast iterative method for solving the eikonal equation on tetrahedral domains, SIAM Journal on Scientific Computing 35(5) 473–494. https://doi.org/10.1137/120881956.

[8] Grandits, T. (2021) A fast iterative method python package, Journal of Open Source Software 6 (66) 3641. https://doi.org/10.21105/joss.03641.

[9] https://www.paraview.org

meyer-nils / eikonal_solver