Self-organized adaptive branching in frangible matter

TL;DR

This work addresses how flow can drive self-organized branching in frangible materials that abruptly switch from resisting to permitting flow when local stresses exceed a threshold. It introduces a unified framework coupling a conservation law to a threshold-driven mesoscopic state with a nonlocal field, capturing both fast dielectric breakdown and slow erosion as limiting cases; the dynamics involve a Rectified-Linear-Unit (ReLU) type activation on the gradient magnitude and a diffusion-relaxation nonlocal term, producing adaptive branching whose morphology is tunable via boundary conditions. Key findings include 2D and 3D demonstrations of boundary-controlled branching, the emergence of an eikonal constraint at steady state, and the ability to steer networks toward functional transport architectures, including self-healing behavior in curved and 3D domains. The work provides a general design principle for self-organized branched networks across diverse systems, with potential applications in engineered transport networks and biomedical perfusion platforms.

Abstract

Soft and frangible materials that remodel under flow can give rise to branched patterns shaped by material properties, boundary conditions, and the time scales of forcing. We present a general theoretical framework for emergent branching in these frangible (or threshold) materials that switch abruptly from resisting flow to permitting flow once local stresses exceed a threshold, relevant for examples as varied as dielectric breakdown of insulators and the erosion of soft materials. Simulations in 2D and 3D show that branching is adaptive and tunable via boundary conditions and domain geometry, offering a foundation for self-organized engineering of functional transport architectures.

Figures (4)

• Figure 1: Dynamics of frangible/threshold materials: (a) The local material state $\phi$ evolves under the combined influence of flow and material properties. Flow affects the material through the gradient field $\nabla u$ (Eq. \ref{['eq:flowmass']}), while non-local material interactions are described by the field $g$ (Eq. \ref{['eq:nonlocal']}). Together, these fields drive the threshold-like evolution of $\phi$ (Eq. \ref{['eq:ReLU']}), leading to the expansion of high-conductivity zones and the emergence of branched patterns in the presence of disorder. (b) Dielectric breakdown represents a fast, binary state limit of the general model. In our implementation on a network of capacitors and non-linear resistors (Eq. \ref{['eq:dielectric_closure']}), edges switch from insulating (yellow) to permanently conductive (purple) once the voltage threshold $\Gamma$ is exceeded. (c) Flow-driven erosion represents a slow, gradual limit, where the solid fraction $\phi$ of a porous medium is progressively degraded, enhancing local conductivity (Eq. \ref{['eq:erosion_closure']}).
• Figure 2: Boundary conditions affect branch shape: We visualize the steady-state $\phi$-field from numerical solutions of the dielectric breakdown model (End Matter, Appendix A) and erosion model (End Matter, Appendix B). For dielectric breakdown, $\phi_v$ is the vertex-averaged $\phi_e$ of connected edges. (a) Increasing the flux amplitude raises both the number of branches and their width. Parameters: dielectric breakdown $Q \in \{0.01,0.1,1.0\},\ T = 1.0$; erosion $Q \in \{0.05,0.5,5.0\},\ T=10, \ \xi = 0.05$. (b) Comparison between homogeneous flux sink (left) and homogeneous potential or pressure sink (right). Parameters: $Q = 0.5, \ T=1.0$ for dielectric system, $Q = 0.5, \ T = 10, \ \xi =0.025$ for eroding system. (c) Increasing the source-sink distance by a factor of 4 leads to more hierarchical branches that span a larger region. Parameters: dielectric $Q = 0.5, \ T = 1.0$, erosion $Q = 2, \ T = 10, \ \xi = 0.05$.
• Figure 3: Function of branched structures: (a) Scaled $A$, $E$, and $R$ from steady-state $\phi$-fields of the erosion model (End Matter, Appendix B) for $Q=[0.05,5]$, $T=[1,100]$ and $\xi=0.025$ (gray dots). Red circles mark the Pareto front. (b) Example structures with low (blue circle, $N_{\text{loops}}=12$) and high robustness (orange triangle, $N_{\text{loops}}=42$), see (a) for reference. (c) Robustness is probed by imposing random blockages ($\phi = 1$, white squares). The flux magnitude $\log_{10} \| \boldsymbol{j} \|$ shows the effect of these perturbations. (d) Normalized flux leaving the system at the sink versus the number of blockages, averaged over 5 random seeds. (e) Increased complexity of boundary control: Exponential flow-profile (Eq. S8, $Q=0.004$, $T=2.1$, $\alpha=18.5$) yields $N_{\text{loops}}=58$ (grey star). Side boundaries as a sink (polygon) at the start of invasion (Eq. S9, $Q=1.0$, $T=10.0$, $a=0.15$) yields $N_{\text{loops}}=56$ (polygon).
• Figure 4: Guided flow-driven branching by erosion in three dimensions. $\phi$-fields from numerical solutions of \ref{['eq:flowmass']} and \ref{['eq:General']} with $C=0$ and model parameters given in the Supplemental Material supp. (a) Dependence on $Q$ for a point source-planar sink geometry. Color shows $\phi$, with transparency increasing rapidly around $\phi=0.4$. (b) Flow network formation in long domains. Top: $\phi$-field as in (a). Bottom: skeleton (see Supplemental Material supp), revealing loops (blue cycles) and voids (blue blobs) toward the domain center. (c) Reconnection after insertion of a blockage demonstrates resilience. (d-e) Branching on curved surfaces. (f-g) Branching on curved surfaces with areal sink terms. Cardiac surface geometry from Finsberg_fenics-beat_2024PANKEWITZ2024103091.