CutFEM: Discretizing Partial Differential Equations and Geometry

by Andre’ Massing

Figure 1.  (Left) Finite element mesh of a blood vessel branch with an aneurysm on top. (Middle) Reconstruction of brain tissue revealing a complex network of neurons [1]. (Right) Even for simple geometries large motion can lead to highly distorted meshes which are not eligible for finite element-based simulation anymore.

The burden of mesh generation

The generation of meshes, as a fundamental prerequisite for the majority of computational methods for partial differential equations (PDEs), is a challenging task that can easily account for > 80% of the time in the overall simulation workflow. For instance, the simulation of blood flow dynamics in vessel geometries requires a series of highly non-trivial steps to generate a high-quality, full 3D finite element mesh from biomedical image data. Similar challenges arise in computational neuroscience, where current meshing capabilities are dramatically lacking behind available imaging technology [1]. The problem is even more pronounced if the geometry of the model domain changes substantially during the simulation, leading to highly distorted meshes, and ultimately an expensive remeshing.

The Cut Finite Element Approach As one potential remedy for the burden of mesh generation in modern simulation pipelines, the development of the Cut Finite Element Method (CutFEM) was initiated approximately a decade ago [2]. The CutFEM is a finite element-based framework for the numerical solution of PDEs which allows for a flexible representation of complicated and evolving geometries by decoupling the domain description from the underlying discretization, while maintaining the accuracy and robustness of a standard finite element method. This approach’s generality has enabled researchers worldwide to design more flexible solvers for a multitude of problems, and below we present a small selection to illustrate the versatility of the Cut Finite Element Method.

Figure 2. While a standard finite element approach can easily lead to highly distorted meshes in certain scenaorios (left), the Cut Finite Element method offers a more flexible description of the domain geometry, e.g.,   by means of implicit/level-set based description of interfaces (middle) or the possibility to composite independently mesh subdomains freely.

Flows in Complex Domains. A key step in the computer-assisted assessment of the risk of rupture for cerebral aneurysms is to compute the blood flow dynamics in the proximity of the aneurysm. Instead of generating a high-quality volume mesh for the blood vessel from biomedical image data, we developed a CutFEM approach [3, 4] where an easy-to-generate, low-quality mesh of the vessel surface can be embedded directly into a highly structured 3D mesh (Figure 3, left) shortening the simulation pipeline significantly. 

Figure 3. Numerical solution of complex flow problems. (Left) Simulation of blood flow passing an aneurysm. (Right) Flexible domain decomposition in fluid-structure interaction problems.

Fluid-structure interaction problems. To accommodate large deformations of the structure while assuring an accurate transfer of the stresses at the fluid-structure interface, a structure mesh (red) is surrounded by a boundary-fitted fluid mesh (green) which is then embedded into and coupled to a fluid background mesh (blue), see Figure 3 (right). While the body-fitted fluid mesh follows the structure under deformation, the background mesh stays fixed [5, 6].

Computational modeling of electrical signal formation in brain cell networks

The inherent temporal-spatial multiscale nature of brain cell networks and their electrical activity pose tremendous challenges when neuroscientists attempt to understand brain signaling formation and patterns through computational studies. A major challenge is the generation of high-quality 3D meshes from high-resolution neuronal network maps (Figure 1, middle) which conform to the complex boundary geometry of the neuronal networks. Last year, as part of PhD candidate Nanna Berre’s research project at the Department of Mathematical Sciences at NTNU, we started to collaborate with Research Professor Marie E. Rognes from Simula Research Laboratory within her RCN funded EMIx project on the computational modeling of excitable tissue. Nanna has successfully developed a CutFEM based computational model [7] which solves a complex mixed-dimensional PDE-ODE model coupling a Poisson-type interface equation for the potential in the intra/extracellular domains with a system of nonlinear ordinary differential equations modeling the electrical current over the cell membrane (the so-called EMI model).

Figure 4. Simulation of EMI model. (Left) Neuron surface geometry embedded into a structured 3D mesh. Only elements intersecting the neuron geometry are shown. (Middle/Right) Snapshots of the intracellular potential at different time-steps.

Solving PDEs on manifolds and coupled surface-bulk problems

The CutFEM approach can be used to easily solve PDEs on embedded manifolds [8] or even mixed-dimensional coupled surface-bulk problems. Important application examples include flow and transport problems in porous media perforated by large-scale fracture networks, two-phase flows with surfactants, and the mathematical modelling of cell motility.

Figure 5. (Left) Solution of diffusion-type problems on embedded surfaces and curves using a CutFEM approach. (Right). CutFEM allows for a seamless discretization of coupled diffusion-type surface-bulk problems.

Acknowledgements

We gratefully acknowledge the direct or indirect support of our research throughout the last decade  by the following research agencies:  Swedish Research Council, Swedish Foundation for Strategic Research,  Kempe Foundation (Sweden),  eSSENCE – The Swedish eScience collaboration, Research Council of Norway.

References

[1]: A. Motta et. al. Dense connectomic reconstruction in layer 4 of the somatosensory cortex. Science, 366(6469), 2019

[2]: Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.  CutFEM: discretizing geometry and partial differential equations.  Int. J. Numer. Meth. Engng. 105(4), pp. 472–501 (2015).

[3]: Massing  A., Larson, M.G., Logg, A., Rognes, M.E.  A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61(3), pp. 604-628 (2014).

[4]: Massing  A., Schott, B., Wall, W.A. A Nitsche cut finite element method for the Oseen problem. Comput. Methods Appl. Mech. Engrg. 328, pp. 262.-300 (2018)

[5]: Massing  A., Larson, M.G., Logg, A., Rognes, M.E. A partition of unity approach to fluid mechanics and fluid–structure interaction, Commun. Appl. Math. Comput. Sci. 10(2), pp. 97-120 (2015)

[6]: Balmus, M., Massing, A., Hoffman, J., Razavi, R., Nordsletten, D.A. A partition of unity approach to fluid mechanics and fluid–structure interaction, Comput. Methods Appl. Mech. Eng. 362, pp. 1-27 (2020)

[7]: Berre, N. Cut finite element methods for modeling excitable cells. Master thesis, NTNU, IMF (2022).

[8]: Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.   Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM: Math. Model. Numer. Anal. 52(6), pp. 2247–2282 (2018)

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