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.

翻译：本文提出了一种基于优化的策略，用于在土壤-根系相互作用模拟中耦合三维与一维问题（3D-1D耦合）。该策略最初设计用于处理具有不连续解的一般性3D-1D耦合问题，本文将其推广至非线性问题情形，并首次与三维土壤样本的虚拟单元离散化方法结合使用。这进一步增强了该方法处理几何复杂性的能力，使其能够轻松地对存在石块及其他任意形状不透水障碍物的区域进行网格划分。采用一种离散-混合的根尖追踪策略来模拟根系生长以及根系与周围土壤样本中水通量、压力水头和含水量的时间演化过程。通过选择适当的分支生成规则，可以获得逼真的根系网络构型。文中给出了若干数值算例，验证了所采用方法的准确性及其在真实和大规模模拟中的适用性。