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A 3D-1D Virtual Element Method for Modeling Root Water Uptake

Stefano Berrone, Stefano Ferraris, Denise Grappein, Gioana Teora, Fabio Vicini

TL;DR

The paper develops an optimization-based 3D-1D coupling framework for simulating root water uptake in complex soil geometries. It reduces a 3D-3D soil–root model to a decoupled 3D-1D system using cylindrical symmetry, with soil solved by a Virtual Element Method and xylem by a mixed FEM, while root growth is tracked by a discrete-tip RSA model. The coupling is enforced via a PDE-constrained optimization problem with auxiliary interface variables, yielding direct computation of interface fluxes and robust handling of obstacles like stones. Numerical experiments TP1–TP3 demonstrate accuracy, convergence, and scalability, including large-scale RSA growth in stony soils and effective preconditioning for the CG solver.

Abstract

An optimization-based strategy is proposed for coupling three-dimensional and one-dimensional problems (3D-1D coupling) in the context of soil-root interaction simulations. This strategy, originally designed to tackle generic 3D-1D coupled problems with discontinuous solutions, is here extended to the case of non-linear problems and applied, for the first time, along with a virtual element discretization of the 3D soil sample. This further enhances the capability of the method to handle geometrical complexities, allowing to easily mesh domains characterized, for instance, by the presence of stones and other impervious obstacles of arbitrary shape. A discrete-hybrid tip-tracking strategy is adopted to model both the root growth and the evolution in time of the water flux, the pressure head and the water content, both in the roots and in the surrounding soil sample. By choosing proper rules for the generation of branches, realistic root-network configurations are obtained. Several numerical examples are proposed, proving both the accuracy of the adopted method and its applicability in realistic and large scale simulations.

A 3D-1D Virtual Element Method for Modeling Root Water Uptake

TL;DR

The paper develops an optimization-based 3D-1D coupling framework for simulating root water uptake in complex soil geometries. It reduces a 3D-3D soil–root model to a decoupled 3D-1D system using cylindrical symmetry, with soil solved by a Virtual Element Method and xylem by a mixed FEM, while root growth is tracked by a discrete-tip RSA model. The coupling is enforced via a PDE-constrained optimization problem with auxiliary interface variables, yielding direct computation of interface fluxes and robust handling of obstacles like stones. Numerical experiments TP1–TP3 demonstrate accuracy, convergence, and scalability, including large-scale RSA growth in stony soils and effective preconditioning for the CG solver.

Abstract

An optimization-based strategy is proposed for coupling three-dimensional and one-dimensional problems (3D-1D coupling) in the context of soil-root interaction simulations. This strategy, originally designed to tackle generic 3D-1D coupled problems with discontinuous solutions, is here extended to the case of non-linear problems and applied, for the first time, along with a virtual element discretization of the 3D soil sample. This further enhances the capability of the method to handle geometrical complexities, allowing to easily mesh domains characterized, for instance, by the presence of stones and other impervious obstacles of arbitrary shape. A discrete-hybrid tip-tracking strategy is adopted to model both the root growth and the evolution in time of the water flux, the pressure head and the water content, both in the roots and in the surrounding soil sample. By choosing proper rules for the generation of branches, realistic root-network configurations are obtained. Several numerical examples are proposed, proving both the accuracy of the adopted method and its applicability in realistic and large scale simulations.

Paper Structure

This paper contains 22 sections, 119 equations, 14 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. The 3D-3D model
  4. Water flow in soil
  5. Water flow in roots
  6. The coupled problem
  7. The model reduction and the PDE-constrained optimization problem
  8. PDE-constrained optimization problem
  9. Modeling root growth
  10. Branching
  11. Managing the intersections between root segments
  12. The problem discretization and the solving strategy
  13. Discrete root growth
  14. Time discretization of the constraint equations
  15. Space discretization of the constraint equations
  16. ...and 7 more sections

Figures (14)

  • Figure 1: Details on notation (zoom-in on the right).
  • Figure 2: Nomenclature for root growth.
  • Figure 3: Branching angle and direction.
  • Figure 4: Nomenclature at branching points. The boundaries of the extended intersection volume are highlighted in red.
  • Figure 5: TP1: Numerical solution in the 3D soil sample and in the root segment at the final time $t=1$ on the finest mesh.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4: Numerical evaluation of the repulsion function