FEniCSx_ii (FEniCSx trace)

FEniCSx_ii is an extension of FEniCSx that allows users to work with non-conforming 3D-1D meshes. The core algorithm is based on the framework proposed by Kuchta 2021 {cite}intro-Kuchta2021trace and implemented in FEniCS_ii.

The new framework addresses the limitation of $\mathrm{FEniCS}_{\mathrm{ii}}$ not being MPI compatible. Currently, the new framework does not use {cite}intro-Mardal2012 cbc.block, and instead implements the matrix-matrix products in a non-lazy fashion, and uses PETSc Nest matrices to set up the blocked system.

Given a 3D domain with a function $u\in V(\Omega)$ and a $1D$ domain $\Gamma$. We define a restriction operator $\Pi:V\mapsto L^2(\Gamma)$, which becomes a central part of the variational formulation.

See for instance D'Angelo & Quarteroni, 2008 {cite}intro-dangelo20083d1d, Kuchta 2021 {cite}intro-Kuchta2021trace or Masri, Kuchta & Riviere, 2024 {cite}intro-masri2024coupled3d1d.

Several (non-local) operators are implemented in {py:mod}fenicsx_ii:

• {py:class}PointwiseTrace<fenicsx_ii.PointwiseTrace>, the operator: $\Pi(u)(\hat x)=u(\hat x)$, $\hat x \in \Gamma$ .
• {py:class}Circle<fenicsx_ii.Circle>, the operator $\Pi(u)(\hat x)=\frac{1}{\vert P_R \vert}\int_{P_{R}(\Gamma(\hat x))}u~\mathrm{d}s$, where $P_R(\Gamma(\hat x))$ is the perimeter of a disk with radius $R$, normal aligning with $\Gamma(\hat x)$ and origin at $\hat x$.
• {py:class}Disk<fenicsx_ii.Disk>, the operator $\Pi(u)(\hat x)=\frac{1}{\vert D_R \vert}\int_{D_R(\Gamma(\hat x))} u~\mathrm{d}x$, where $D_R(\Gamma(\hat x))$ is the disk with radius $R$, normal aligining with $\Gamma(\hat x)$ and origin at $\hat x$.

Any other operator can be implemented by following the {py:class}ReductionOperator<fenicsx_ii.ReductionOperator>-protocol.

Funding

The development of FEniCSx_ii has been funded by the Wellcome Trust, grant number: 313298/Z/24/Z

References

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