Extended Finite Elements for 3D-1D coupled problems via a PDE-constrained optimization approach

TL;DR

This work tackles diffusion-type 3D-1D coupled problems arising from dimensional reduction of thin tubular inclusions embedded in a 3D domain. It develops an XFEM-enhanced, PDE-constrained optimization approach that enables accurate coupling on non-conforming meshes using a log-like, globally continuous enrichment function and a specialized quadrature scheme for irregular enrichment terms. The method is discretized via a three-field domain-decomposition style formulation and enriched with partition-of-unity functions to capture singular behavior near inclusions, including crossing, bifurcated, and multi-branch geometries. Numerical results across five test cases show that the enrichment yields high accuracy on coarse meshes with substantial reductions in degrees of freedom compared to full equi-dimensional models, validating the approach for efficient simulations in applications with thin inclusions such as vascularized or porous media systems.

Abstract

In this work, we propose the application of the eXtended Finite Element Method (XFEM) in the context of the coupling between three-dimensional and one-dimensional elliptic problems. In particular, we consider the case in which the 3D-1D coupled problem arises from the geometrical model reduction of a fully three-dimensional problem, characterized by thin tubular inclusions embedded in a much wider domain. In the 3D-1D coupling framework, the use of non conforming meshes is widely adopted. However, since the inclusions typically behave as singular sinks or sources for the 3D problem, mesh adaptation near the embedded 1D domains may be necessary to enhance solution accuracy and recover optimal convergence rates. An alternative to mesh adaptation is represented by the XFEM, which we here propose to enhance the approximation capabilities of an optimization-based 3D-1D coupling approach. An effective quadrature strategy is devised to integrate the enrichment functions and numerical tests on single and multiple segments are proposed to demonstrate the effectiveness of the approach.

Figures (18)

• Figure 1: Domain with single inclusion and description of notation. The size of the inclusion is exaggerated for description purposes.
• Figure 2: Enrichment function $\zeta$ on a plane containing inclusion centreline. $\Omega=[0,1]^3$, $\Lambda$ aligned with the $z$-axis, $R=10^{-2}$. For $\zeta=\zeta^\flat$, $\Lambda$ extending from $z=0.2$ to $z=0.8$.
• Figure 3: Description of numerical integration strategy
• Figure 4: Description of the split strategy.
• Figure 5: Test \ref{['test1']}, trend of the relative errors under mesh refinement. Dashed lines: relative $H^1$-norm of the error; full lines: relative $L^2$-norm of the error.