Plank, Michael

Wilkins, Andy

English

Mathematics in Industry Information Service Eprints Archive

 Page 19 of 25 
TREE GROWTH AND WOOD FORMATION – APPLICATIONS OF 
                           ANISOTROPIC SURFACE GROWTH 
 
                                          (Ensis Ltd) 
 
Industry Representatives:  
Dr Jennifer Grace,  Ensis Ltd, Rotorua, NZ 
Dr Jonathan Harrington, Ensis Ltd, Rotorua, NZ 
 
Moderators:  
Dr Michael Plank, University of Canterbury, Christchurch, NZ 
Dr Andy Wilkins, University of Queensland, Brisbane, Australia 
 
Student Moderator:  
Ms Joanne Mann, Massey University, Albany, NZ 
 
Ensis is a joint venture between Scion (New Zealand Forest Research 
Institute) and Australia’s CSIRO Forestry and Forestry Products. They deal 
with problems relating to: genetics; wood and fibre quality; wood 
processing and products; pulp, paper and packaging; forest bio security 
and protection; sustainable production forestry and integrated 
environmental forestry. Approximately 300 staff are located at eight sites 
in Australia and New Zealand working in trans-Tasman teams. 
 
The problem posed to MISG was to develop a model for growth of trees, 
with an emphasis on being able to provide a general framework for 
describing surface growth and to visualise the growth in three dimensions. 
The model had to be applied to a tree's surface, as well as cell wall 
thickening, and the formation of grain lines. Only growth of the girth of 
the tree's stem and branches (i.e. radial growth) needed to be modelled: 
the axial growth of the branch tips being considered outside the scope of 
the project. A numerically stable algorithm needed to be found, and 
implemented to solve the problems and produce the visualisation. 
Summarising, the main objectives were defined as: choose a suitable 
formulation for the surface velocity vector describing the tree's growth; 
determine grain directions; produce a stable numerical algorithm to 
evolve the anisotropic surface in time under any growth velocity; and then 
visualise the results. 
 
An initial literature search was undertaken to ascertain the methods used 
by previous models of tree growth. Simple models were found describing 
the growth in height and girth of trees, as well as a model concerning the 
formation of grain patterns (which approximated the grain pattern as a 
fluid flow field), but no papers were found producing a three-dimensional 
visualisation of the growth of the tree stem and branches. It was also 
realised that the problem of cell-wall thickening could not be considered 
satisfactorily due to a lack of experimental results. 
 
Literature and experimental samples provided by Ensis demonstrated that 
the velocity at which the stem and different branches grow is different and 
time dependent, so it was essential that the numerical algorithm was able 
to track the evolution of the trunk and branches separately – implying 
that there was more than one surface to track. The choice of the velocity 
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became a critical problem, as there was little literature concerning its 
dependence on hormones, environmental factors, stress, curvatures, etc. 
 
Some analytical work was undertaken to increase understanding of the 
effect of the velocity function, modelling the behaviour of a moving 
interface using partial differential equations for the position of the 
interface over time. Some theoretical results were obtained for simple 
curvature- and hormone-driven growth in two dimensions (although it was 
argued that curvature-driven growth is not likely to be a physically 
reasonable model for the growth velocity). These results reproduced some 
of the features observed in trees, notably the surface shapes seen at 
stem-branch junctions and when a tree is forced to grow around an 
external obstacle. 
 
The grain was modelled as streamlines of an incompressible fluid flowing 
on the tree's surface. The stem grain around a dead branch or obstacle 
was modelled as flow around a circular obstacle, whilst the grain pattern 
for a live branch was modelled as flow out of a cylinder (or source). 
Although this model was not a realistic representation of the physiological 
processes leading to grain formation, the results produced were 
qualitatively realistic. The physiologically-based problem of grain 
orientation evolution in time (straight grain becoming spiral grain over a 
period of years, for instance) was not considered. 
 
A fast marching method was implemented to track the surface evolution 
of a stem with multiple branches. This elegant method is based on the 
idea of representing the surface, at a given point in time, not explicitly 
using marker particles, but implicitly as the level set of some function. The 
method proved to be numerically efficient and stable, and was relatively 
straightforward to implement with simple velocity functions. Various 
scenarios of branch growth around objects and multiple branch growth 
were considered. Visualisation was then achieved using built-in tools in 
MATLAB, as shown in Fig. 1 where the fast marching method was 
implemented for a stem with five branches each with an independent 
growth rate.   
 
Figure 1 Growth simulation for stem and five branches, where the growth for the 
stem and each branch is independent. 
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Tree Growth and Wood Formation - Applications of Anisotropic Surface Growth

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