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Article published in
Journal of the Mechanical Behavior of Biomedical Materials, Vol. 123, No.104788, 2021
https://doi.org/10.1016/j.jmbbm.2021.104788
Topology of leaf veins: Experimental observation and computational
morphogenesis
Jiaming Ma a, Zi-Long Zhao a, Sen Lin b, Yi Min Xie a,*
a Centre for Innovative Structures and Materials, School of Engineering, RMIT University,
Melbourne 3001, Australia
b State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of
Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
ABSTRACT
The unique, hierarchical patterns of leaf veins have attracted extensive attention in recent
years. However, it remains unclear how biological and mechanical factors influence the
topology of leaf veins. In this paper, we investigate the optimization mechanisms of leaf veins
through a combination of experimental measurements and numerical simulations. The
topological details of three types of representative plant leaves are measured. The experimental
results show that the vein patterns are insensitive to leaf shapes and curvature. The numbers of
secondary veins are independent of the length of the main vein, and the total length of veins
increases linearly with the leaf perimeter. By integrating biomechanical mechanisms into the
topology optimization process, a transdisciplinary computational method is developed to
optimize leaf structures. The numerical results show that improving the efficiency of nutrient
transport plays a critical role in the morphogenesis of leaf veins. Contrary to the popular belief
in the literature, this study shows that the structural performance is not a key factor in
* Corresponding author. Tel.: +61 399253655
Email address: mike.xie@rmit.edu.au (Y.M. Xie).
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determining the venation patterns. The findings provide a deep understanding of the
optimization mechanism of leaf veins, which is useful for the design of high-performance shell
structures.
Keywords: Leaf veins; Topology; Computational morphogenesis; Nutrient transport; Structural
stiffness; Curved shell
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1. Introduction
Through a long history of evolution, biological materials have formed hierarchical
structures which are closely related to their functions (Allaire et al., 2002; Meyers et al., 2008;
Wang et al., 2016; Wright et al., 2004; Zhao et al., 2015, 2016). The fractal patterns in nature
such as the river networks, snowflakes, root systems, and venation systems, have attracted
considerable attention over the past few decades (Ball, 2009, 2016; Wolfram, 2002). Leaf veins
render gas exchange and fluid/nutrition transport, such that the mesophyll cells can achieve
sufficient supplies for living (Carvalho et al., 2017; Sack and Scoffoni, 2013). In return, the
photosynthesis from mesophyll provides organics and oxygen to support the plants’ lives and
flourishment (Dengler and Tsukaya, 2001; Efroni et al., 2008; Scarpella et al., 2010). To
maximize the sun lighted area for photosynthesis, a leaf surface requires sufficient stiffness to
support its expansion against weight and wind loads (Brodribb et al., 2007). Veins are stiffer
than the mesophyll, and they make a significant contribution to reinforce the mesophyll and
maintain the leaf shape (Ennos et al., 2000; Gibson et al., 1988; Niklas, 1999). The
biomechanical mechanisms underlying the beautiful vein patterns have attracted extensive
interests of scientists, engineers, and designers (Blonder et al., 2011; Gokmen, 2013; Md Rian
and Sassone, 2014; Runions et al., 2005).
Several numerical approaches have been established to investigate the branch-like vein
structures. The L-system method has been used to analyze the growth process of plants
(Prusinkiewicz and Lindenmayer, 1990). However, this mathematical method only models a
vein system morphologically, without considering the biomechanical requirements such as
nutrient transport and shape maintaining. An adaptive algorithm has been developed for vein
morphogenesis by generating stiffeners in leaf model in random directions (Liu et al., 2017).
Topology optimization has now become a powerful tool to explore the morphogenesis of
natural materials. In the past three decades, several optimization techniques have been
successfully established, including the solid isotropic material with penalization (SIMP)
method (Bendsøe, 1989; Bendsøe, 1995; Sigmund and Maute, 2013), the level-set method
(Allaire et al., 2002; Wang et al., 2003), the evolutionary structural optimization (ESO) method
(Xie and Steven, 1993; Xie and Steven, 1997) and the bi-directional evolutionary structural
optimization (BESO) method (Huang and Xie, 2007, 2009). Most recently, imposing
complicated constraints during the form-finding process has been realized (Chen et al., 2020;
He et al., 2020; Xiong et al., 2020; Zhao et al., 2020a). By establishing transdisciplinary
computational methods for biomechanical morphogenesis, Zhao et al. (2018, 2020b, 2020c)
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have revealed the optimization mechanisms of, e.g., plant leaves and animal stingers. Using
topology optimization, a golden ratio distribution rule is found in venation systems (Sun et al.,
2018). A multi-objective optimization approach is developed to explore the vein patterns on a
planar plate (Lin et al., 2020). However, there is still a lack of quantitative study on the
biomechanical mechanisms of vein distributions on the shell-like mesophyll.
In this work, we investigate, both experimentally and numerically, the topology of leaf
veins. The vein patterns of three representative plant leaves are measured, including the Syringa
vulgaris L., Rosa chinensis Jacq., and Cotoneaster submultiflorus Popov. Biomechanical
functions of veins such as nutrient transport and structural stiffness are integrated into the
computational morphogenesis. The influence of the curved shape of mesophyll on the vein
distribution is examined. This study reveals the optimization mechanisms underlying the
intriguing overall layout of venation systems. The presented methodology can be applied in the
design of high-performance shell structures such as aircraft skin ribs (Song et al., 2021).
2. Methodology
2.1. Experiments
Three types of fresh plant leaves with distinctly different shapes are collected in the same
area, including Syringa vulgaris L., Rosa chinensis Jacq., and Cotoneaster submultiflorus
Popov. Five samples of each species are randomly selected. Those samples have various
surface curvatures and their main veins have different angles of inclination. Images of the
samples are taken by a high-resolution digital camera immediately after collection. The
samples are flattened and put between a piece of A4-sized paper and a transparent thin film.
Rhinoceros (Rhino 6 SR28 version) was used for feature extractions and measurements.
2.2. Computational morphogenesis
An interdisciplinary topology optimization method is developed for investigating the
biomechanical morphogenesis of plant leaves. This method is capable of dealing with different
objective functions, e.g., the stiffness maximization and the enhancement of nutrition transport.
Veins have continuously varying thickness and material properties over the leaf surface. The
new method is developed under the framework of the SIMP technique which can produce
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results with transitional structural features (Bendsøe, 1989; Bendsøe, 1995; Sigmund and
Maute, 2013).
Denote the design domain of a leaf as Ω, which is divided into n elements for finite
element analysis (FEA). For each element i (
1, 2, 3, ...,in=
), it has a density value
i
. The
densities of vein and mesophyll elements are defined as
1
i
=
and
0
i
=
. In the beginning,
the densities for all elements are set to the target volume fraction
f
ˆ
v
. The FEA and density
update are carried out in each iteration during the optimization process. Elements with lower
sensitivities are gradually changed towards 1, while those with higher sensitivities are gradually
changed towards 0.
In the FEA, the curved leaves under self-weight is considered as shells under vertical
uniform surface traction. The leaf material is assumed to be linearly elastic. The governing
equation for maximizing the structural stiffness is expressed as below (Bendsøe and Sigmund,
2004a)
( )
dd
1
min: 2
C
=T
U K U
ρ
(1)
𝑠. 𝑡. ∶
d
=K U F
f
1
1
ˆ
n
ii
in
i
i
v
vv
=
=
=
0 1, 1, 2, ,
iin
=
(2)
where 𝑣𝑖 denotes the volume of element i. F, U and
d
K
are the uniformly distributed surface
traction, global displacement, and global stiffness matrix, respectively. Assume that the
Poisson’s ratios of the vein and mesophyll are the same. The Young’s modulus between two
neighboring phases is interpolated as (Bendsøe and Sigmund, 1999)
( ) ( )
m v m p
i
E E E E
= + −
(3)
where p is a penalization number (default as 3) (Bendsøe and Sigmund, 2004b), and
m
E
and
v
E
are the Young’s moduli of the mesophyll and vein, respectively. Thus, the objective
function can be rewritten as
( )
v
mm
dd
1vv
1
min: 1
2
np
i i i
i
EE
CEE
=
= + −
u k u
ρ
(4)
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where
ui
and
v
d
k
represent respectively the displacement vector and the stiffness matrix of
element i with E =
v
E
. The sensitivity is derived by the adjoint method as
( ) ( )
d1v
m
dd
v
C1
1
2p
i i i
i
E
pE
−
= = − −
u k u
(5)
To maximize the nutrient transport performance of a leaf, it can be considered as a
dissipation maximization problem. The objective function of minimizing the concentration
gradients of nutrient is written as
( )
tt
1
min: 2
C
=T
T K T
ρ
(6)
𝑠. 𝑡. ∶
t
=K T Q
f
1
1
ˆ
n
ii
in
i
i
v
vv
=
=
=
0 1, 1, 2, ,
iin
=
(7)
where Q, T, and
t
K
represent nutrient input, nutrient solution concentration, and global
conductivity matrix, respectively. The conductivity 𝜅 of each element is interpolated as
( ) ( )
m v m p
i
= + −
(8)
The sensitivity is derived by the adjoint method as
( ) ( )
t1 T v
mt
v
11 t t
2p
t i i i
i
Cp
−
= = − −
k
(9)
A weighting factor is introduced to control the effect of the two sensitivities on the final
design. The coupled sensitivity is calculated as
(1 )
i id it
= + −
(10)
where
( )
max id
id id
i
=
(11)
( )
max it
it it
i
=
(12)
The optimality criteria based optimizer is adopted for updating the design variables in
each iteration (Sigmund, 2001). The algorithm is implemented in the Python environment and
linked to Abaqus (Abaqus 6.20 version) for FEA. The finite element model of each leaf is
discretized into approximately 50,000 four-node S4R doubly curved shell elements. The
7
Poisson’s ratios of both veins and mesophylls are set as 0.3, and their Young’s moduli and
thickness are ascertained according to the previous study (Sun et al., 2018). For the force–
displacement analysis, the surface of the leaf is subjected to a uniformly distributed force in
the downward direction, and the petiole is fixed. For the steady conduction analysis, the surface
of the leaf is applied with uniformly distributed flux input, and the petiole is set to be zero
concentration. Adiabatic condition is applied on the leaf edges.
3. Results
Experimental and numerical results are presented and quantitatively analyzed in this
section. The geometries of real leaves are measured, including the leaf perimeter, and the
lengths, angles, and numbers of the main/secondary veins. The venation layouts optimized for
nutrition transport and stiffness maximization are presented, respectively. The experimental
and numerical results are compared in Section 3.2.3.
3.1. Experimental results
Three representative plant species, i.e., the Cotoneaster submultiflorus Popov, Rosa
chinensis Jacq., and Syringa vulgaris L., are selected for experiments. Their leaves have
distinctly different shapes, as shown in Figure 1. The leaf of Cotoneaster submultiflorus Popov
has a relative higher slenderness ratio with smooth edges, the leaf of Rosa chinensis Jacq. has
an approximately round shape with serrated edges, and the leaf of Syringa vulgaris L. is drop-
shaped.
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Figure 1. Three leaf species used in experiments: (a) Cotoneaster submultiflorus Popov, (b)
Rosa chinensis Jacq., and (c) Syringa vulgaris L.
Five samples of each species are measured for experiments. The samples for the three
plants are in different growth stages and hence have different sizes. Their surface curvatures
and inclination angles (Niklas, 1999) are also different, and therefore in the force–displacement
analysis, the loading conditions will be different. In most cases, the gravity direction is not
perpendicular to the leaf surfaces.
To conduct quantitative analysis on the vein patterns, the leaf samples are characterized
by their lengths and angles. The perimeters and main vein lengths are also measured. All the
three plant species are dicotyledons and have camptodromous pinnate vein. Accordingly, the
length measurement for the secondary veins starts from their growing point and ends at their
intersection with upper neighboring ones. For illustration, the secondary veins of a Rosa
chinensis Jacq. leaf are highlighted in Figure 2(b), and the method for measuring the lengths
and angles is shown in Figure 2(c).
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Figure 2. Measurements of the length and angle of secondary veins: (a) a leaf sample, (b) the
main vein (the blue line in the middle) and the secondary veins (separated in curved blue and
red lines), and (c) the method for measuring the lengths and angles.
The angles between the secondary and the main veins are measured. The rear-end of a
camptodromous pinnate vein bends and connects to a neighboring vein. Only the base segments
of these secondary veins are used to fit the straight lines for angle measurement. A base
segment starts from the main vein and ends at the intersection with its neighboring veins. The
base segments of the secondary veins are highlighted in blue in Figure 2(b). The method for
measuring the lengths and angles is shown in Figure 2(c).
Figure 3 shows the secondary vein lengths of the three plant species. Both left and right
secondary veins are observed and measured. The secondary vein lengths of each sample and
the position of each secondary vein are normalized by the main vein length. The normalized
length for each secondary vein is calculated as its measured length divided by the main vein
length. The relative position of each secondary vein is calculated as the distance from the root
of a secondary vein to the petiole of the main vein divided by the main vein length. It can be
seen that different species have different length distributions of secondary veins. For the same
species, the discrepancies of the normalized lengths of secondary veins at similar relative
positions are small.
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Figure 3. Measurements of the position–length relations of the secondary veins: (a)
Cotoneaster submultiflorus Popov, (b) Rosa chinensis Jacq., and (c) Syringa vulgaris L.
Figure 4 shows the angles of both left and right secondary veins. Similarly, different
species have different angle distributions of secondary veins, which for the same species,
samples have similar angle distributions.
Figure 4. Measurements of the position–angle relations of the secondary veins: (a) Cotoneaster
submultiflorus Popov, (b) Rosa chinensis Jacq., and (c) Syringa vulgaris L.
Apart from the length and angle of the secondary veins, their numbers are measured and
plotted in Figure 5(a). The result shows that, for each leaf species, the numbers of the secondary
veins are almost the same, although the five samples have significantly different main vein
lengths.
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Figure 5(b) shows the relation between the leaf perimeter and the total length of the main
and the secondary veins. For each plant, the total length of veins increases linearly with the
increasing leaf perimeter. It is found that the three straight fitting lines have similar slopes. For
the three species, the ratios of the total length of veins to the leaf perimeter are 2.17, 2.63, and
2.93, respectively. The biomechanical mechanisms underlying the similar ratios require further
research.
Figure 5. Comparison of vein features across different species and samples: (a) number of the
secondary veins against the main vein length, (b) variation of the total length of veins with
respect to the leaf perimeter.
3.2. Numerical results
Topology optimizations are performed to investigate the biomechanical mechanisms of
the distribution of leaf veins. Both flat and curved leaf models are considered. Two
biomechanical functions are considered, including the stiffness maximization and the
dissipation optimization (corresponding to nutrient transport performance). A representative
leaf profile is chosen for the leaf model. Curvatures for the leaf models are determined based
on typical shapes of real leaves.
Figure 6 shows four leaf models for topology optimization. The model in Figure 6(a) is
flat. The model in Figure 6(b) is folded along its middle axis. The models in Figure 6(c) and
(d) are folded perpendicular to their middle axes, where the one in Figure 6(d) has ruffles on
its edges. All leaf models have the same surface area and the objective functions of the four
cases can be compared.
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Figure 6. Four leaf models for topology optimization: (a) a flat model, (b) a model folded along
its middle axis, (c) a model folded perpendicular to its middle axis, and (d) a model folded
perpendicular to its middle axis and with ruffles on its edges.
3.2.1. Nutrient transport
The optimized topologies of the leaves for maximizing the efficiency of nutrient transport
are shown in Figure 7. Here and in the following, the vein material with the highest conductivity
is highlighted in light-yellow, and the mesophyll materials are marked in dark green.
Intermediate colors on the models indicate the transitional change of material properties
between the two phases. Figure 7 shows that the veins have hierarchical and fractal patterns,
regardless of the curvatures of the leaf models. The optimized results are very similar to the
real leaves. The quantitative comparison between the numerical and the experimental results
are presented in Section 3.2.3. It can be seen that veins with the highest conductivity are located
near the petiole. The secondary and tertiary veins have lower conductivity and thickness. These
results are in good agreement with real leaves.
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Figure 7. Optimized topologies for nutrient transport maximization: (a) a flat leaf, (b) a leaf
folded along the middle axis, (c) a leaf folded perpendicular to the middle axis, and (d) a leaf
folded perpendicular to the middle axis and with wrinkled edges.
3.2.2. Structural stiffness
The optimized topologies of the leaves for stiffness maximization are shown in Figure 8.
Similarly, the light-yellow color indicates the highest stiffness vein material, and the dark green
one reveals where the soft mesophylls are. The obtained topologies depend strongly on the
curvature of the models. This phenomenon is on the contrary to that of the nutrition
transportation optimization results. Although the flat model results in a branch-like pattern of
veins, the curved models result in truss-like patterns. Such difference could be attributed to the
membrane stiffness (i.e., the in-plane stiffness) of shell elements. For a curved shell, the
external load induces in-plane stresses, while for a flat plate, there exists only bending and
shear stresses. The structural compliance of the structures in Figure 7(a-d) are 7.21 N·mm, 5.22
N·mm, 7.59 N·mm and 6.90 N·mm, respectively. The compliances of the structures in Figure
8(a-d) are significantly smaller, which are 3.95 N·mm, 1.72 N·mm, 4.36 N·mm and 3.65 N·mm,
respectively. In Figure 8, veins near the petioles are very thick, which contribute to supporting
the leaf structure.
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Figure 8. Optimized topologies for stiffness maximization: (a) a flat leaf, (b) a leaf folded along
the middle axis, (c) a leaf folded perpendicular to the middle axis, and (d) a leaf folded
perpendicular to the middle axis and with wrinkled edges.
3.2.3. Quantitative analysis
The numerical and experimental results are compared quantitatively. For the optimization
results of nutrition transportation, the skeleton lines of the main and the secondary veins are
extracted and shown in Figure 9. The lengths of the secondary veins are measured and
normalized by the length of the main vein. Figure 10(a) shows the position–length relations of
the secondary veins.
Figure 9. Skeleton (red lines) of the main vein and the secondary veins.
15
Figure 10. (a) Position–length and (b) position–angle relations of the secondary veins
optimized for nutrient transport.
Figure 10 (b) shows the position–angle relations of the secondary veins optimized for
nutrient transport. Despite the shape difference of the four leaf models, their veins exhibit
similar position–length relations and position–angle relations.
The topologies optimized for stiffness maximization are distinctly different from the
patterns of real veins, which are here not quantitatively analyzed.
To investigate the influence of the main vein length on the numbers of secondary veins
through topology optimization, the leaf models should have different sizes but the same shape.
The element sizes need to be scaled to keep the element number and layout unchanged. Since
the optimized results will be unchanged if the model and its elements are enlarged
simultaneously, the computational method is not used to check the relationship between main
vein length and the numbers of secondary veins.
4. Discussion
It is seen that different leaf species exhibit different feature sizes, edge profiles, and
material properties. However, the topologies of their veins are similar. Our experimental results
reveal that the number of secondary veins is independent of the length of the main vein. The
total length of veins increases linearly with the leaf perimeter. Although the leaves have
distinctly different shapes, curvatures, inclination angles, and living environments, the
topological differences of their veins are limited.
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From the results in Section 3.2, the vein patterns optimized for nutrition transport are
insensitive to the leaf shapes, while the patterns optimized for stiffness depend strongly on the
leaf curvature. For stiffness optimization, truss-like veins are obtained in curved models, while
feather-like veins are generated only in the flat model. It is well known that real leaves could
have various curved shapes and inclination angles (loading conditions). Therefore, the
structural stiffness is less likely to play a dominated role in determining the topology of leaf
veins.
Figure 11. Optimization results from coupling stiffness and dissipation: (a) 𝜆Stiffness = 0.0
and 𝜆Dissipation = 1.0 , (b) 𝜆Stiffness = 0.2 and 𝜆Dissipation = 0.8 , (c) 𝜆Stiffness = 0.6 and
𝜆Dissipation = 0.4, (d) 𝜆Stiffness = 1.0 and 𝜆Dissipation = 0.0.
Veins, consisting of parenchyma, sclerenchyma, and sheath, usually have higher Young’s
modulus than the mesophyll (Gibson et al., 1988; Sun et al., 2018). Vein is usually thicker than
mesophyll. Therefore, veins can be considered as reinforcing ribs of leaves. However,
according to our results in Section 3.2.2, the stiffness enhancement is not a dominated factor
that determines the vein topologies. Figure 11 shows the coupled optimization results of vein
morphologies on a curved leaf model with different weight factors, combining stiffness
enhancement with nutrient transport enhancement. It can be seen that the nutrient transport,
which is of significant importance for photosynthesis, governs the topology of veins.
This study reveals that the enhancement of nutrient transport plays a predominant role in
determining the vein patterns. The main features of the topologies optimized for nutrient
transport agree well with the real patterns. Some other factors such as biological constraints,
17
the intercellular stress during growth, and the microstructures of plant tissues, can also affect
the topology of veins, which require further research.
5. Conclusion
In this study, intricate topologies of leaf veins have been investigated through both
experimental observation and computational morphogenesis. It is found that the enhancement
of nutrient transport plays a predominant role in determining the form of venation patterns.
Contrary to popular belief in the literature, this research reveals that the structural performance
is not the key factor of leaf vein patterns. Furthermore, the experimental measurements show
that the numbers of secondary veins are independent of the length of main veins, and the total
length of veins has a linear relationship with the leaf perimeter. The numerical results of
nutrient transport show that the vein patterns are insensitive to the variation of leaf shapes. This
research provides a deep understanding of the biomechanical mechanisms underlying the
intriguing layout of leaf veins. The presented computational method can be used for designing
efficient and innovative free-form shell structures.
Acknowledgement
The authors received financial support from the Australian Research Council
(FL190100014 and DE200100887).
Conflict of interest
The authors declare that they have no conflict of interest.
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Reference
Allaire, G., Jouve, F., Toader, A. M., 2002. A level-set method for shape optimization. Comptes rendus.
Mathématique 334, 1125-1130.
Ball, P., 2009. Nature's Patterns: A Tapestry in Three Parts. Oxford University Press, New York.
Ball, P., 2016. Patterns In Nature: Why The Natural World Looks The Way It Does. The University of
Chicago Press, Chicago.
Bendsøe, M.P., 1989. Optimal shape design as a material distribution problem. Struct. Multidiscip.
Optim. 1, 193-202.
Bendsøe, M.P., 1995. Optimization of Structural Topology, Shape, and Material. Springer-Verlag
Berlin Heidelberg, Berlin.
Bendsøe, M.P., Sigmund, O., 1999. Material interpolation schemes in topology optimization. Arch.
Appl. Mech. (1991) 69, 635-654.
Bendsøe, M.P., Sigmund, O., 2004a. Topology optimization by distribution of isotropic material,
Topology Optimization: Theory, Methods, And Applications. Springer-Verlag Berlin Heidelberg,
Berlin, pp. 1-69.
Bendsøe, M.P., Sigmund, O., 2004b. Topology Optimization: Theory, Methods, and Applications,
Second Edition, Corrected Printing. ed.: Springer-Verlag Berlin Heidelberg, Berlin.
Blonder, B., Violle, C., Bentley, L.P., Enquist, B.J., 2011. Venation networks and the origin of the leaf
economics spectrum. Ecol. Lett. 14, 91-100.
Brodribb, T.J., Feild, T.S., Jordan, G.J., 2007. Leaf maximum photosynthetic rate and venation are
linked by hydraulics. Plant Physiol. 144, 1890-1898.
Carvalho, M.R., Turgeon, R., Owens, T., Niklas, K.J., 2017. The scaling of the hydraulic architecture
in poplar leaves. New Phytol. 214, 145-157.
Chen, A.B., Cai, K., Zhao, Z.L., Zhou, Y.Y., Xia, L., Xie, Y.M., 2021. Controlling the maximum first
principal stress in topology optimization. Struct. Multidiscip. Optim. 63, 327-339
Dengler, N.G., Tsukaya, H., 2001. Leaf morphogenesis in dicotyledons: Current issues. Int. J. Plant Sci.
162, 459-464.
Efroni, I., Blum, E., Goldshmidt, A., Eshed, Y., 2008. A protracted and dynamic maturation schedule
underlies arabidopsis leaf development. Plant Cell 20, 2293-2306.
Ennos, A.R., Spatz, H.C., Speck, T., 2000. The functional morphology of the petioles of the banana,
Musa textilis. J. Exp. Bot. 51, 2085-2093.
Gibson, L.J., Ashby, M.F., Easterling, K.E., 1988. Structure and mechanics of the iris leaf. J. Mater.
Sci. 23, 3041-3048.
Gokmen, S., 2013. A morphogenetic approach for performative building envelope systems using leaf
venetian patterns. eCAADe 2013: Computation and Performance, Proceedings of the 31st
International Conference on Education and research in Computer Aided Architectural Design in
Europe, Delft University of Technology 1, 497-506.
He, Y.Z., Cai, K., Zhao, Z.L., Xie, Y.M., 2020. Stochastic approaches to generating diverse and
competitive structural designs in topology optimization. Finite Elem. Anal. Des. 173, 103399.
Huang, X., Xie, Y.M., 2007. Convergent and mesh-independent solutions for the bi-directional
evolutionary structural optimization method. Finite Elem. Anal. Des. 43, 1039-1049.
Huang, X., Xie, Y.M., 2009. Bi-directional evolutionary topology optimization of continuum structures
with one or multiple materials. Comput. Mech. 43, 393-401.
Lin, S., Chen, L., Zhang, M., Xie, Y.M., Huang, X.D., Zhou, S.W., 2020. On the interaction of
biological and mechanical factors in leaf vein formation. Adv. Eng. Softw. 149, 1-8.
Liu, H.L., Li, B.T., Yang, Z.H., Hong, J., 2017. Topology optimization of stiffened plate/shell structures
based on adaptive morphogenesis algorithm. J. Manuf. Syst. 43, 375-384.
Md Rian, I., Sassone, M., 2014. Tree-inspired dendriforms and fractal-like branching structures in
architecture: A brief historical overview. Front. Archit. Res. 3, 298-323.
Meyers, M.A., Chen, P.Y., Lin, A.Y.M., Seki, Y., 2008. Biological materials: structure and mechanical
properties. Prog. Mater. Sci. 53, 1-206.
Niklas, K.J., 1999. A mechanical perspective on foliage leaf form and function. New Phytol. 143, 19-
31.
19
Prusinkiewicz, P., Lindenmayer, A., 1990. The Algorithmic Beauty of Plants. Springer-Verlag, New
York.
Runions, A., Fuhrer, M., Lane, B., Federl, P., Rolland-Lagan, A.G., Prusinkiewicz, P., 2005. Modeling
and visualization of leaf venation patterns. Acm. T. Graphic. 24, 702-711.
Sack, L., Scoffoni, C., 2013. Leaf venation: structure, function, development, evolution, ecology and
applications in the past, present and future. New Phytol. 198, 983-1000.
Scarpella, E., Barkoulas, M., Tsiantis, M., 2010. Control of leaf and vein development by auxin. Cold
Spring Harb. Perspect. Biol. 2, a001511.
Sigmund, O., 2001. A 99 line topology optimization code written in Matlab. Struct. Multidiscip. Optim.
21, 120-127.
Sigmund, O., Maute, K., 2013. Topology optimization approaches: A comparative review. Struct.
Multidiscip. Optim. 48, 1031-1055.
Song, L., Gao, T., Tang, L., Du, X., Zhu, J., Lin, Y., Shi, G., Liu, H., Zhou, G., Zhang, W., 2021. An
all-movable rudder designed by thermo-elastic topology optimization and manufactured by
additive manufacturing. Comput. Struct. 243, 106405.
Sun, Z., Cui, T.C., Zhu, Y.C., Zhang, W.S., Shi, S.S., Tang, S., Du, Z.L., Liu, C., Cui, R.H., Chen, H.J.,
Guo, X., 2018. The mechanical principles behind the golden ratio distribution of veins in plant
leaves. Sci. Rep. 8, 13859.
Wang, M.Y., Wang, X.M., Guo, D.M., 2003. A level set method for structural topology optimization.
Comput. Methods in Appl. Mech. Eng. 192, 227-246.
Wang, X.J., Xu, S.Q., Zhou, S.W., Xu, W., Leary, M., Choong, P., Qian, M., Brandt, M., Xie, Y.M.,
2016. Topological design and additive manufacturing of porous metals for bone scaffolds and
orthopaedic implants: A review. Biomaterials 83, 127-141.
Wolfram, S., 2002. A New Kind of Science. Wolfram Media, Champaign, Illinois.
Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J.,
Chapin, T., Cornelissen, J.H.C., Diemer, M., Flexas, J., Garnier, E., Groom, P.K., Gulias, J.,
Hikosaka, K., Lamont, B.B., Lee, T., Lee, W., Lusk, C., Midgley, J.J., Navas, M.L., Niinemets,
U., Oleksyn, J., Osada, N., Poorter, H., Poot, P., Prior, L., Pyankov, V.I., Roumet, C., Thomas,
S.C., Tjoelker, M.G., Veneklaas, E.J., Villar, R., 2004. The worldwide leaf economics spectrum.
Nature 428, 821-827.
Xie, Y.M., Steven, G.P., 1993. A simple evolutionary procedure for structural optimization. Comput.
Struct. 49, 885-896.
Xie, Y.M., Steven, G.P., 1997. Evolutionary Structural Optimization, Springer, London.
Xiong, Y.L., Yao, S., Zhao, Z.L., Xie, Y.M., 2020. A new approach to eliminating enclosed voids in
topology optimization for additive manufacturing. Addit. Manuf. 32, 101006.
Yang, K., Zhao, Z.L., He, Y.Z., Zhou, S.W., Zhou, Q., Huang, W.X., Xie, Y.M., 2019. Simple and
effective strategies for achieving diverse and competitive structural designs. Extreme Mech. Lett.
30, 100481.
Zhao, Z.L., Shu, T., Feng, X.Q., 2016. Study of biomechanical, anatomical, and physiological
properties of scorpion stingers for developing biomimetic materials. Mat Sci Eng C-Mater 58,
1112–1121.
Zhao, Z.L., Zhao, H.P., Ma, G.J., Wu, C.W., Yang, K., Feng, X.Q., 2015. Structures, properties, and
functions of the stings of honey bees and paper wasps: a comparative study. Biol. Open. 4, 921-
928.
Zhao, Z.L., Zhou, S.W., Feng, X.Q., Xie, Y.M., 2018. On the internal architecture of emergent plants.
J. Mech. Phys. Solids. 119, 224-239.
Zhao, Z.L., Zhou, S.W., Cai, K., Xie, Y.M., 2020a. A direct approach to controlling the topology in
structural optimization. Comput. Struct. 227, 106141.
Zhao, Z.L., Zhou, S.W., Feng, X.Q., Xie, Y.M., 2020b. Morphological optimization of scorpion telson.
J. Mech. Phys. Solids. 135, 103773.
Zhao, Z.L., Zhou, S.W., Feng, X.Q., Xie, Y.M., 2020c. Static and dynamic properties of pre-twisted
leaves and stalks with varying chiral morphologies. Extreme Mech. Lett. 34, 100612.
