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Comparison of two biomechanical models for the numerical analysis of tree growth
Thomas Guillon
Last modified: 2009-07-30
Abstract
The analysis of the shape evolution of growing trees requires modelling accurately the interaction between tree growth and its biomechanical response, including passive deformations due to external loads and active movements of the plant, e.g. tropisms. However, this coupling between tree growth and tree biomechanics exceeds the traditional framework of structural mechanics due to the changing in size of the studied domain and the resulting non-conservation of mass.
Previous works devoted to the modelling and simulation of plant biomechanics have been carried out using discrete time models at the stem and tree scale (Fourcaud et al. 2003), or based on a continuous formulation of growth at the level of a stem cross section (Alm'eras & Fournier, 2009) or applied to a branching structure (Senan et al., 2008).
The aim of this work is to compare the discrete approach developed by Fourcaud et al. (2003) and the continuous model given by Senan et al. (2008). This last model takes into account growth process inside the constitutive law of the material, using the configurational forces framework (Epstein & Maugin, 2000; Senan et al. 2008).
These biomechanical models have been integrated within the functional-structural model GreenLab (Yan et al., 2004). The architecture of the tree is described with substructures of the same physiological age in which growth process is applied recursively (Courn`ede et al., 2006). For the functional part of this model, biomass production is evaluated periodically using an empirical function of the leaf surface. This increment of biomass is then shared between the organs according to their sinks strength. Therefore, at the end of each growth cycle, the geometric features necessary to describe the evolving domain on which mechanical processes take place are fully provided.
In order to evaluate the differences between the two biomechanical models of growth, balance equations were solved numerically with the finite element method. As the balance equations of the Senan et al.'s model are expressed in continuous time, it is of great interest to compare the numerical convergence of the two models with respect to the time variable which will be illustrated by some examples.
These numerical analyses will provide a rigorous insight to decide the most relevant model to be applied for future studies in plant biomechanics.
Previous works devoted to the modelling and simulation of plant biomechanics have been carried out using discrete time models at the stem and tree scale (Fourcaud et al. 2003), or based on a continuous formulation of growth at the level of a stem cross section (Alm'eras & Fournier, 2009) or applied to a branching structure (Senan et al., 2008).
The aim of this work is to compare the discrete approach developed by Fourcaud et al. (2003) and the continuous model given by Senan et al. (2008). This last model takes into account growth process inside the constitutive law of the material, using the configurational forces framework (Epstein & Maugin, 2000; Senan et al. 2008).
These biomechanical models have been integrated within the functional-structural model GreenLab (Yan et al., 2004). The architecture of the tree is described with substructures of the same physiological age in which growth process is applied recursively (Courn`ede et al., 2006). For the functional part of this model, biomass production is evaluated periodically using an empirical function of the leaf surface. This increment of biomass is then shared between the organs according to their sinks strength. Therefore, at the end of each growth cycle, the geometric features necessary to describe the evolving domain on which mechanical processes take place are fully provided.
In order to evaluate the differences between the two biomechanical models of growth, balance equations were solved numerically with the finite element method. As the balance equations of the Senan et al.'s model are expressed in continuous time, it is of great interest to compare the numerical convergence of the two models with respect to the time variable which will be illustrated by some examples.
These numerical analyses will provide a rigorous insight to decide the most relevant model to be applied for future studies in plant biomechanics.