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Abstract
This paper presents a study of the morphogenesis of brain corals based on an
experimental investigation and a topological optimization method. The re-
sistance to matter interchange was employed to allocate the optimal space for
the growth of polyp colonies from the perspective of topological optimization,
where the optimized structures are those of natural brain corals. Computa-
tional fluid dynamics simulations revealed that these complicated structures
can provide shelter to protect polyps from ocean currents. A reverse mold
was prepared from silica gel and used to cast models from mixtures of ce-
ment and calcium carbonate, where the mixture ratio was determined based
on compressive strength and biocompatibility. Based on an acid corrosion
experiment, the matter interchange capability was verified. This study also
proved that the many folds in the structure of brain corals contribute to the
circulation of seawater, thus maintaining the concentration of nutrients and
hindering the deposition of harmful substances. This paper establishes an
Preprint submitted to COMPUTERS & STRUCTURES December 17, 2022
Article published in
Computers & Structures, 276 (2023) 106955.
https://doi.org/10.1016/j.compstruc.2022.106955
Design and fabrication of artificial brain coral:
Evolution principle, turbulent hydrodynamics and
matter interchange
S. Lin1*, N.Z. Chou1, D.W. Bao2,3, G.B. Zhang4, C.W. Xiong5, J. Fang6,
Y.M. Xie2 and G.Y.Li7∗
1State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College
of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
2Centre for Innovative Structures and Materials, School of Engineering, RMIT
University, GPO Box 2476, Melbourne 3001, Australia
3School of Architecture and Urban Design, RMIT University, GPO Box 2476, Melbourne
3001, Australia
4College of Civil Engineering, Hunan City University, Yiyang, Hunan 413000, China
5College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001,
China
6Beijing Institute of Control and Electronic Technology, Beijing 102308, China
7Shenzhen Automotive Research Institute (Shenzhen Research Institute of National
Engineering Laboratory for Electric Vehicles), Beijing Institute of Technology, Shenzhen
518118, Guangdong, China
innovative methodology for the creation of artificial brain corals, which is
important for environmental restoration.
Keywords: Brain corals, Topological optimization, Turbulent
hydrodynamics, Matter interchange
1. Introduction1
In recent decades, coral aquaculture techniques have been under develop-2
ment in many countries to support the conservation of endangered natural3
coral reefs in the face of an increasing live coral market [1]. Such techniques4
could provide an effective means of restoring coral reefs [2], which are de-5
clining worldwide due to human impacts such as pollution [3], fishing [4],6
coastal development [5], and climate change [6]. Industries that specialize in7
the production of artificial reefs often use reinforced concrete structures to8
assist in coral reproduction [7]. Accordingly, these artificial reefs can gener-9
ally be defined as man-made structures placed underwater to mimic certain10
characteristics of natural reefs [8].11
The potential of community-based coral aquaculture has been evaluated12
from various perspectives in recent years. In Madagascar, researchers in-13
vestigated suitable farming techniques using the species Acropora nasuta14
and Seriatopora caliendrum [9]. In their studies, the survival and growth15
rates of the nubbins during wet/warm and dry/cold seasons were compared.16
Similarly, a significant variation in growth rate has been observed between17
naturally recruited and transplanted corals during the monsoon period [10].18
As an emerging class of techniques, advanced manufacturing [11] has en-19
abled researchers to produce man-made experimental models that can serve20
as structural replicates of or specifically manipulated alternatives to natural21
habitats in order to study the behaviors and habitat preferences of aquatic22
species [12]. For instance, the interstitial space of a 3D printed oyster reef23
has been quantitatively manipulated to gain an understanding of the struc-24
tural properties that mediate the foraging success of Callinectes sapidus [13].25
Moreover, 3D-printed reefs have natural surface pH values that are favorable26
for the growth of coral larvae. With the assistance of 3D printing techniques,27
it is possible to mimic the natural complexity (caves and connecting tunnels)28
of coral reefs and enhance the diversity of artificial reefs [14]. Although 3D29
printing has the potential to advance efforts to protect coral reefs, the im-30
maturity of some relevant technologies may seriously limit the application31
2
prospects of this advanced concept [15]. One important reason is that en-32
gineers tend to simply imitate the shape of corals [16] or use engineering33
methods to design artificial coral reefs [17] that are not consistent with the34
growth principle of corals [18]. Another aspect of additive manufacturing35
that is of concern to oceanologists is that the printing materials used must36
meet certain manufacturability requirements. Consequently, most objects37
are fabricated in plastic, which is incompatible with colonization by marine38
organisms [19]. Instead, nontoxic composites such as cementitious mixtures39
of biological bone would be more suitable for printing artificial marine con-40
structions [20].41
Interdisciplinary efforts offer further possibilities; in particular, many re-42
searchers have attempted to employ topological optimization methods to43
analyze the features of biological structures [21] for subsequent application44
[22]. The precondition for such application is that scholars in topological45
optimization have proven that the dependence of the objective function on46
the design variables can be incorporated into structural designs using the47
adjoint technique [23]. Furthermore, methods for solving partial differen-48
tial equations using artificial neural networks [24] or deep neural networks49
(DNNs) [25] have been proposed. Optimal heat conduction paths have been50
designed by using a novel topological optimization method [26]. The com-51
plexity of a structure is usually characterized in terms of the geometries of52
the interior holes, which can be well controlled based on graph theory [27].53
Additionally, the bidirectional evolutionary structural optimization (BESO)54
algorithm has been verified as an effective means of explaining the hollow55
sectional structures of aquatic plants [28]. Recently, a new topological op-56
timization method for designing workpieces with lightweight structures and57
acceptable mechanical strength has been proposed for generating bioinspired58
porosities based on bones of various shapes, sizes and orientations [29]. More59
recently, human-made corals have been developed by minimizing a carbon-60
solidification-related objective function and considering the morphological61
characteristics of staghorn corals [30].62
Regarding the turbulent hydrodynamics of corals, the complex structures63
of these marine habitats influence wave activity, which appears to enhance64
the rates of mass transport in the water surrounding corals [31]. For instance,65
strong bottom currents play a key role in cold-water coral environments by66
shaping their morphology and providing the necessary food for the corals67
to thrive [32]. This interesting phenomenon has motivated researchers to68
employ fluid mechanics methods to investigate it [33]. The effects of reef69
3
permeability on the spatial distributions of the wave-induced currents in-70
side and outside an inhomogeneous coral reef body have been studied using71
an improved weakly compressible smoothed particle hydrodynamic mixture72
model [34]. Such research can serve as a crucial reference for differential sur-73
vivorship during heat-induced coral bleaching, particularly as thermal stress74
events become increasingly common due to global climate change [35].75
The rest of this paper is organized as follows. Section 2 introduces how76
the resistance to matter interchange in the projected surface domain was77
set as the objective for topological optimization under volume and length78
constraints. As described in section 3, computational fluid dynamics (CFD)79
simulations were then implemented to investigate the characteristics of the80
flow field around these human-made corals. Section 4 reports an experimen-81
tal investigation in which a silica gel reverse mold was prepared for casting82
models in cement and calcium carbonate, where the mixture ratio was de-83
termined based on tests of workability and compressive strength. An acid84
corrosion test was conducted to verify the matter interchange capability of the85
optimized morphology. The proposed methodology is intended to facilitate86
the creation of artificial brain corals, which is important for environmental87
restoration.88
4
2. Methodology89
2.1. Aquatic observation90
a
10 m
-50 m
Lizard Island
N
145°20’ E
14°50’ S
b
c d e
Figure 1: The sample collection site and the appearances of samples: (a) the elevation
data (from the National Geophysical Data Center (NGDC)) near the site; (b) a panoramic
view of the mariculture chamber; (c) the appearance of Platygyra; (d) the appearance of
Goniastrea; (e) the appearance of Lobophyllia (b–e were photographed by N.Z. Chou).
As a kind of reef-building coral, brain corals live in tropical sea water91
at latitudes below 19◦. They tend to reproduce in shallow water with suf-92
ficient illumination. The samples of Platygyra,Goniastrea and Lobophyllia93
(approximately 0.03–0.10 m in dimension) studied here were collected from94
the seabed near Lizard Island, Australia (14◦40’37” S, 145◦26’17” E; shown95
in Fig. 1a), and were subsequently observed for 10 months (Fig. 1b,c&d).96
In the sea water used to cultivate brain corals, sufficient calcium is crucial97
to ensure the healthy growth of their internal skeletons. In contrast, it is98
better to limit the levels of NO−
3and PO3−
4, as high concentrations of these99
chemicals can cause polyps to become necrotic. Brain corals are well suited100
for living in moderate water flows. When exposed to swift currents, however,101
5
they often contract, which is likely to cause tissue damage. Another key102
factor in brain coral feeding is light; specifically, medium or bright light is103
appropriate. Nevertheless, a freshly transplanted coral should not be exposed104
to strong light immediately. It is suggested that it be placed in a deep and105
dark location, where its position can be gradually raised to permit adaptation106
to stronger light.107
In this study, an experimental chamber (0.80 m×0.50 m×0.80 m) made108
of polymethyl methacrylate was employed for mariculture purposes, in which109
the current speed was controlled between 1500 and 2000 gph by an MP10wQD110
propeller pump (EcoTech Marine Inc., United States) and a Seio 620 circu-111
lation propeller pump (Zhongshan SOBO Electric Co., Ltd., China). The112
salinity was controlled between 3.45% and 3.55% (mean=3.50%). The water113
temperature ranged from 24.1◦C to 26.5◦C and was regulated by an HC-114
2200BH water temperature control machine (Shenzhen HaiLea Tech. Co.,115
Ltd., China). The stability of the primary nutrients was maintained within116
reasonable limits by means of a smart dosing pump (Wavereef Aquarium117
Co., Ltd., China), with the concentrations of Ca2+, NO−
3and PO3−
4being118
400, 12.5 and 0.02, respectively. The pH value was stabilized in the range119
of 8.1–8.4, as measured by an Orion Star A121 portable pH meter (Thermo120
Fisher Scientific Inc., United States). Artificial light was maintained for 10–121
18 hours every day with a Radion XR30w G4 light fixture (EcoTech Marine122
Inc., United States). For water purification, an AE-DF130 protein skimmer123
(Guangzhou Kas Aquarium Equipment Manufacturing, Germany) was used.124
2.2. Topological optimization125
Like most reef-building corals, once the larvae of brain corals have cho-126
sen a place to settle, they remain stationary throughout their lives and grow127
on reef ridges. Being symbiotic with zooxanthellae, these animals use pho-128
tosynthesis from symbiotic algae to create organic matter for growth, and129
they also use their stinging tentacles to capture plankton from the water.130
Therefore, the living spaces of brain corals, which are home to competition131
among polyps on the same individual, are significant for investigating the132
formation of their shapes. In this case, the problem can be regarded as the133
optimal design of a space-occupying configuration. The distribution of the134
polyps on a domed colony can be simplified to a design ‘density’ variable135
ρon a 2-D domain with an irregular shape, where low values represent the136
‘valleys’ of the polyps, which serve a predation function, while high values137
represent the ‘ridges’ of the polyps, which play a protective or partitioning138
6
role. The coefficient of matter interchange, k(ρ), corresponds to the living139
spaces of the polyps. The solid isotropic material with penalization (SIMP)140
method was employed here for topological optimization to ensure robust141
topological development and convincing physical validation. Notably, the142
SIMP method comprises abundant subalgorithms involving size constraints143
and mesh independence. The interchange of matter between brain coral and144
nearby seawater can be regarded as the distribution and exchange of organics145
between polyps with strong metabolic activity and essential nutrients in the146
seawater. In particular, the interchange activity associated with convective147
water flow occurs mainly at the interface between the coral and the seawater.148
The formulation of a suitable objective function for such a matter (or heat)149
transfer optimization problem has been a topic of discussion for a long time150
[36]. It has been found that minimizing the compliance of matter (heat) can151
usually lead to a desirable structure with better comprehensive properties152
compared with directly minimizing the matter concentration (temperature)153
in the design domain. Accordingly, the objective function for matter inter-154
change, fm, can be defined to minimize the matter interchange resistance of155
the design domain Ω by means of an optimized material distribution ρ:156
min :fm=ZΩ
(∇[k(ρ)∇T]−C)T dΩ
s.t.
RΩρdΩ≤V∗
ρmin ≤ρ≤1
∇[k(ρ)∇T]−C=C
(1)
where T(x, y) is the growing matter content in the coral, xand yare two-157
dimensional coordinates, and V∗is an upper-bound limit on the allowable158
solid part volume fraction. In a confined domain, the mouth part of a coral159
structure can increase the efficiency of matter interchange, while the ridges160
can influence the direction of water flow to carry away waste. Therefore, the161
polyps must be properly proportionately allocated between the mouth and162
ridges. A corresponding volume constraint can be set based on biological163
characteristics (V∗≈0.5, obtained from aquatic observations). ρmin ≈0.1164
is the minimum density of the coral, which is generally measured from the165
density ratio between the valleys and ridges. Cis matter interchange on the166
surface of the coral, and Cis the production of matters in the polyps. Re-167
garding numerical implementation, in this work, the above 2-D conduction168
topology optimization problem has been extended to consider side-surface169
7
convection, with the added benefit of design in two dimensions. The in-170
corporation of side-surface convection into the design problem leads to the171
following governing equation for the objective function:172
C=s(ρ)(T−T0)(2)
where s(ρ) represents the design-dependent side-surface convection coefficient173
at a reference matter content of T0. In addition, the coefficient of matter174
interchange can be interpolated as follows:175
k(ρ) = [σ+ (1 −σ)ρp]k0(3)
where p≥3 is a penalty parameter, k0is a reference value for the matter176
interchange coefficient of the polyps (including their zooxanthellae), and σ177
represents the ratio of matter interchange between the ridges/seawater and178
the valleys of the polyps, which should be very small (10−3). Moreover, the179
side-surface convection coefficient s(ρ) in Eq. 2 is specified by a simplified180
hat function:181
s(ρ) =
0.01s0, ρl≤ρ<ρu
s0, ρmin ≤ρ<ρl
0, ρu≤ρ≤1
(4)
where s0is the reference full-magnitude convective matter interchange coeffi-182
cient defined at the external surface of the coral and ρland ρuare lower and183
upper cutoff values, respectively, for the side-surface convection coefficient.184
At the valley-to-seawater interfaces (ρmin ≤ρ<ρl), where the side-surface185
convection coefficient takes its full-magnitude value, matter interchange ac-186
tivities occur most frequently. Meanwhile, because of the many cavities, the187
density value here is relatively low (ρ≈0.1, based on specimen estimation).188
On the ridges (ρl≤ρ < ρu), the side-surface convection coefficient turns out189
to be a small fraction of s0due to relative metabolic inactivity. Furthermore,190
the inner parts of the coral (ρu≤ρ≤1) are skeletons, where the side-surface191
convection coefficient is 0. Finally, the lower cutoff value should be defined192
as the maximum valley density, ρl≈0.5, while the upper cutoff value should193
be set to the surface density on ridges, ρu≈0.9. Predictably, because of the194
high matter exchange efficiency in the valleys, the corals will evolve toward a195
more sinuous mouth. In the case of ridges, a large size or a concentrated dis-196
8
tribution would reduce metabolic activity; thus, slender ridges are preferred197
for brain corals.198
In the described example and most of the literature referenced above,199
a primary challenge facing the implementation of brain-like patterns is the200
thickness of the ridges and valleys. This is because wrinkles that are too201
large or small are rare due to their biological nature. Therefore, a length-202
scale constraint for controlling the thickness of the wrinkles is introduced203
here [37]. Suppose that the average density in the vicinity of element iis204
expressed as205
eρi= (X
j∈Ωi
vj)−1X
j∈Ωi
vjρξ
j(5)
where Ωidenotes the element design domain for elements located in a circle206
centered at element i,vjrepresents the area of each element included in the207
domain Ωi, and the exponent ξ= 0.5 acts as a density amplifier. A smaller208
value of the parameter ξindicates that an element with an intermediate209
density value (0 < ρj<1) is more likely to be counted as a solid element210
(e.g., for ρj= 0.7, ρξ
j= 0.84 with ξ= 0.5, while ρξ
j= 0.90 with ξ= 0.3). In211
this case, the length-scale constraint that describes the biological features of212
corals is imposed by minimizing the objective function213
min :fl=X
j∈Ωi
max(0,eρi−γ)(6)
where 0 < γ < 1 is the upper area fraction ratio in the length-scale domain214
Ωiof element i. When γis given a certain value, e.g., 0.95, the volume215
fraction in Ωiwill not exceed that value. The sensitivity of the length-scale216
constraint can be expressed as ∂fl/∂ρ =Pj∈Ωimax(0,eρi/∂ρ). By removing217
the constant, the sensitivity expression can be further simplified to218
∂fl
∂ρ =1
ξvi
∂fl/∂ρ =ρξ−1
iX
j∈Ωi
(βj/vj)(7)
where βis related to the Heaviside function in topological optimization [37]219
and can be simplified to220
βj=(1,eρi> γ
0,eρi≤γ(8)
9
The derivation of the sensitivity of the objective function to the design221
variable is given in the Appendix. Finally, the overall normalized sensitivity222
of element iis defined as223
∂f
∂ρ =λ1αm+λ2αl(9)
where λ1,2(λ1+λ2= 1, 0 ≤λ1,2≤1) are weighting factors and αm=224
∂fm
∂ρ (max
i
∂fmi
∂ρ )−1and αl=∂ fl
∂ρ (max
i
∂fli
∂ρ )−1represent the corresponding nor-225
malized sensitivities. Specifically, the normalized length constraint sensitivity226
αlshould satisfy227
αl=
1−∂fl
∂ρ (max
i
∂fli
∂ρ )−1,max
i
∂fli
∂ρ >0
0,max
i
∂fli
∂ρ <0(10)
In this study, the optimization formulation and procedure were imple-228
mented in MATLAB. The weight function λidepends on the particular coral229
species but is independent of the element sensitivity and thus could be de-230
termined before the finite element analysis. The design domain was dis-231
cretized into 104∼105(approximately 120 ×120, depending on the shape232
of the coral) four-node planar elements. Both the ridges and valleys were233
assumed to be homogeneous, and Neumann boundary conditions were ap-234
plied to the external elements. The ratio of matter interchange between the235
ridges and valleys, σ, was set to 10−3. In addition, a sensitivity filtering236
technique was applied as follows: ∂ˆ
f
∂ρ =Pi∈ΩeHei ρi∂f
∂ρ /(ρePi∈ΩeHei ), where237
Hei =max(0, rmin −∆(e, i)), with ∆(e, i) being the distance between ele-238
ments eand iand rmin being the filter radius. Ωe(the domain within the239
filter radius) was employed in the optimization procedure to avoid problems240
such as checkerboarding and mesh dependency. Notably, the filter radius was241
chosen to be rmin = 2.0∼5.0, corresponding to the minimum dimensions of242
the wrinkles. The area fraction of valleys, χ= 0.2∼0.6, was determined243
based on the experimentally observed appearance of the reference corals. The244
upper area fraction ratio, γ= 0.90 ∼0.99, was set to satisfy the length-scale245
control requirements. For the original polyps settled on the domain, their246
distribution was assumed to follow247
(PΩi=p0
ρi(xi, yi)ρj(xj, yj)≥d0, ρi, ρj∈Ω(11)
10
where PΩiis the survival possibility of larva iin a specific part of the domain,248
Ωi. The possibility p0can vary in accordance with the nature of the species249
(some larvae prefer to congregate in specific locations, such as the center or250
edge of the domain.). ρi=ρj= 1 represents the first group of settled polyps,251
in which each pair is separated by a distance larger than d0. This is because252
sufficient living space should be ensured for the polyp larvae, as polyps that253
are too close to each other will not survive.254
0 5 10 15 20 25 30
Iteration
30
40
50
60
70
80
90
100
110
Normalized objective (%)
Valley
Ridge
It: 4
It: 12
It: 20
It: 27
Figure 2: Normalized objective value (%) versus the number of iterations of optimization.
The corresponding evolution of Platygyra is also illustrated, where light colors represent
valley regions and dark colors represent ridge regions. Iteration 4–Settling of the larvae.
Iteration 12–Accelerated reproduction. Iteration 20–Bifurcation. Iteration 27–Stabilized
layout.
Regular Platygyra structures can be obtained via the above optimization255
algorithm. The evolution history of the objective value and corresponding256
coral appearances are shown in Fig. 2. When the larvae have just settled in a257
limited space, the coral starts to grow in a disorderly manner around the first258
11
group of colonists. During this time, the normalized objective value sharply259
decreases during the early stage of evolution (from 100% to 40% during260
iterations 1–5). Afterward, the decrease in the objective value is retarded261
because the breeding of the polyps is accelerating, until the polyps occupy the262
entire domain (iterations 5–15). In iterations 15–22, the polyps are fighting263
for the last remaining space. They gradually evolve into winding shapes with264
a small number of bifurcations. The optimization process converges after 27265
iterations, at which time a small number of closed loops appear in the polyps.266
These features cause water to circulate in the loops, resulting a slow flow rate267
and better efficiency of nutrient capture. The evolution of the optimization268
process indicates that the matter interchange characteristics of coral depend269
on the maze pattern of the polyps.270
2.3. Shape, expansion and curvature271
Most brain corals do not have perfectly rounded outlines. In contrast, due272
to space constraints (the shape of the base or the presence of other neigh-273
bors), they usually have irregular shapes. On the other hand, brain corals274
grow very slowly as each generation adds to the limestone skeleton. Benefit-275
ing from their massive, sturdy structures, brain corals form the foundation276
of coral reefs and can live nearly 1000 years. In other words, the slow growth277
process of brain corals might affect their groove patterns. To explore the278
influence of shape and the growth process on gully morphology, related pro-279
grams were implemented in MATLAB, and the corresponding results were280
analyzed. In the early stage of an individual’s growth process, fertilized lar-281
vae attach themselves to the seafloor or stones. The places where they settle282
determine the future living space of the coral. As seen in Fig. 3a and b, the283
early expansion is generally homogeneous, with the rough outline remaining284
essentially a convex polygon. The polyps prefer to grow along preformed285
contours, forming moat-like gullies.286
12
a b c d
Figure 3: Sketches of the evolution of an irregularly shaped Platygyra pattern: (a) the
original state of the coral; (b) the state of the coral after homogeneous expansion; (c)
the final appearance of the gully morphology with an anisotropic growth rate; (d) the
corresponding dome-like 3D structure.
Notably, the shapes of the previously formed structures will not signifi-287
cantly change in the next phase of expansion. To ensure adherence to this288
rule in the evolutionary optimization process related to coral expansion, the289
design domain Ωpwas divided into several parts, one corresponding to each290
growth stage (p= 2,3, ...). In each evolutionary step, the amplitude of the291
change in the density of each element was limited by ρchmax, defined as fol-292
lows:293
ρchmax =(δpρch, ρ ∈Ωp−1
ρch, ρ ∈Ωp
(12)
where the parameter ρch = 0.1 determines the amplitude of the change in294
the density of each element between consecutive growth stages. As observed295
from mariculture, the further evolution of the region formed in the previous296
stage is quite slow. In quantitative terms, the change in the density was297
restricted by the parameter δp= 0.05.298
As the coral becomes stationary after the period of early expansion, the299
polyps will form buds at the bottom. Thus, expansion becomes much eas-300
ier; however, surrounding obstacles such as stones or other colonies must be301
avoided. In this stage (Fig. 3c), a mature individual may grow in arbitrary302
directions depending on environmental conditions such as illumination, wa-303
ter flow and nutrients. Consequently, the directions of the gullies are no304
longer regular; they may sometimes even grow perpendicular to the previous305
contour or form closed loops.306
Brain corals growing in high-flow environments exhibit higher growth307
rates than those growing in lower-speed currents because the water flow308
13
speeds up the transportation of nutrients. Therefore, the morphology of309
brain corals is elaborately formed to adapt to the current, especially to ensure310
stability under flowing water. Accordingly, a hemispherical shape is highly311
favorable. Limited by the available growth space, however, not all brain312
corals have perfectly spherical skeletons. In general, deeper water causes313
corals to have flatter structures, which could be better able to resist strong314
water pressure. According to the literature review and aquatic observations,315
most corals preferentially expand from the attachment surface and slightly316
swell vertically, forming a hill-like skeleton that can be mathematically rep-317
resented as318
(e=min(AB)
za= 1 −ζ(e−emax )2(13)
where A(xa, ya) is an element in the design domain Ω and B(xb, yb) is an319
arbitrary element on its outline. The dome-like skeleton can be defined in320
terms of the 3D coordinates (xa, ya, za) along with a slope governing param-321
eter ζ. In addition, a significant correlation between the surface roughness322
of coral and its matter interchange rate has been confirmed. In terms of323
biological species, corals with smooth surfaces have more space for larger324
individual polyps. Conversely, a wrinkled surface is able to increase nutrient325
capture from a fast water flow. On the other hand, corals with deep ridges326
have the ability to establish diverse environments with varying light and flow327
conditions, making them attractive to small creatures. Therefore, numerical328
simulation of the 3D morphology of the gullies is crucial. For this purpose, a329
formulation based on the local coordinate system can be written as follows:330
A0(x0
a, y0
a, z0
a) = A(xa, ya, za) + gρa~ma(14)
where A0(x0
a, y0
a, z0
a) denotes the revised coordinates of the element A(xa, ya, za).331
Corresponding to the aforementioned side-surface convention, the density at332
this location determines whether it belongs to a valley (ρa≤ρl) or a ridge333
(ρa>ρl) in the design domain. The gully depth is proportional to the op-334
timized density of aand is amplified by g>0 (the value of gis determined335
based on statistical data from aquatic observations), while the direction is336
determined by the external normal vector ~maon the surface. It is noted337
that the gully depth affects the optimization results through the efficiency of338
matter interchange. Finally, the 3D morphology corresponding to Fig. 3c is339
as shown in Fig. 3d.340
14
2.4. Species and wrinkles341
Generally, the wrinkle patterns of different brain coral species are di-342
verse and correspond to their polyp growth characteristics. For instance, the343
various shapes formed by the polyps are strongly influenced by the water344
flow. Individuals in deep water exhibit lower calcification ratios, with wider345
gaps between neighboring polyps. In contrast, heavier calcification occurs in346
colonies living under rapid water flows, where dense cellular structures ap-347
pear. Stony corals (Scleractinia) can also control the shapes of their colonies348
through budding and different relative growth rates in different locations.349
Buds can be classified into internal and external tentacle buds. In an inter-350
nal tentacle bud, the polyps simply divide in the orifice; the bud retains part351
of the original orifice, while the remaining portion undergoes development.352
External tentacle buds appear outside the tentacle ring of the main body353
and cannot maintain the function of the main body. To reveal the princi-354
ple of this interesting phenomenon, the evolution of coral wrinkle patterns355
was investigated. For Platygyra (Fig. 4a), the larvae have a relatively high356
survival rate according to the expression given in Eq. 11, with thin valley re-357
gions occupying nearly half of the area of the domain (rmin = 3.0, p0= 80%,358
d0= 3.0). On the other hand, the polyps of Goniastrea are arranged in a359
completely different way (Fig. 4b). The structure is usually composed of360
nubby skeletons, where the calices (0.003–0.005 m in the lateral dimension361
and 0.001–0.002 m in depth) are irregular polygons. Two neighboring cal-362
ices are separated by the coenosteum, where a columella is connected to 12363
septa and an equal number of phanic spindle-like paliform lobes. Another364
group of brain corals has a folded rather than spherical shape and lives in365
a free-standing manner rather than as part of the structure of a larger reef.366
These species are called open brain corals (rmin = 3.0, p0= 5%, d0= 1.0).367
Lobophyllia, as a genus of open brain corals commonly found in aquariums,368
is distinguished by tenacious vitality and a fluorescent protein that emits vi-369
brant colors when exposed to blue light (Fig. 4c). Such corals usually exhibit370
a variety of brilliant fluorescent colors. They are composed of spiral-petal-371
like flaps that grow upward or inward at the edges. The reticular structure372
of the coenosteum of the polyps buckles into wrinkles. The wrinkled lobes373
are auricular or fan-shaped. Sometimes, however, other shapes also develop,374
such as more regular oval shapes or shapes with more protruding skeletons in375
the middle (rmin = 5.0; p0= 70% for the spiral-petal-like flaps and p0= 2%376
for the rest of the domain; d0= 1.0).377
15
a b c
Figure 4: The appearances of different brain coral species, represented by sketches of
actual specimens (top row), simulated results (middle row) and 3D structures (bottom
row): (a) Platygyra; (b) Goniastrea; (c) Lobophyllia.
3. Numerical simulation378
3.1. Vorticity analysis379
Sufficient light allows symbiotic algae to grow normally and provide en-380
ergy for the corals. A good flow environment provides not only plankton,381
which are carried by the water, but also protection to the corals, helping382
them to breathe and carrying away excreta and secretions. In such an envi-383
ronment, Acropora grows faster and appears denser, giving these corals an384
advantage in coral reefs because they can obtain more sunlight more eas-385
ily. In addition, they can cut off the flow of water from other corals around386
them. Conversely, Colpophyllia does not exhibit aggressivity due to having387
no sharp edges, yet these corals can still gain competitive superiority. When388
a neighbor is within striking range, the sweeper tentacles of brain corals can389
stretch 10 times in length to attack the invader with the current. In addi-390
tion, Colpophyllia, like Platygyra sinensis, not only extends its tentacles on391
16
the side facing the invading corals but also randomly radiates a small num-392
ber of tentacles in different directions for detection. Similar sweeper tentacles393
have been found in many genera, including Goniastrea,Acanthastrea,Favia,394
Favites and Galaxea.395
The fluid dynamics, especially the turbulence, around brain corals is a396
crucial basis for studying the matter interchange efficiency of artificial corals.397
The seawater movement considered in this research can be regarded as the398
horizontal sloshing of a homogeneous, isotropic and viscous Newtonian fluid399
described in a fixed Cartesian coordinate system with reference to an oscil-400
lating rectangular chamber. In the numerical model, the spatial and tempo-401
ral discretization scheme of the lattice Boltzmann method (LBM) [38] was402
adopted. For the Smagorinsky eddy viscosity, the filtered particle distri-403
bution function fv(n, t) was employed, where the cap represents a filtering404
operator, ndenotes the coordinate tensor of the particles, tis the time vari-405
able and vrepresents a velocity orientation. The iterative temporal process406
followed the forced Bhatnagar–Gross–Krook (BGK) model [39], consisting of407
a collision step and a streaming step, as follows:408
˜
fv(n, t) = ¯
fv(n, t)−∆t
κ∗[¯
fv(n, t)−¯
feq
v(n, t)] + ¯
Fv∆t(15)
¯
fv(n+Dv∆t, t + ∆t) = ˜
fv(n, t)(16)
where ˜
fvis the postcollision distribution function, Dvis the fluid speed ten-409
sor, and κ∗is the relaxation time corresponding to the total viscosity ν∗. The410
external force ¯
Fvis fixed on the oscillating chamber from the noninertial ref-411
erence frame. The formula for the equilibrium distribution function ¯
feq
vis as412
follows:413
¯
feq
v(n, t) = ωvρf[1 + uf·cv
c2
s
+(uf·cv)2
2c4
s
−uf·uf
2c2
s
](17)
where csis the speed of sound propagation in the lattice and ρfand ufare414
the macroscopic density and velocity, respectively, of the fluid, which can be415
calculated from the distribution function as follows:416
ρf=
18
X
v=0
¯
fv,uf=1
ρf
18
X
v=0
¯
fvcv+¯
F∆t
2(18)
According to a related study [40], the force term ¯
Fvcan be expressed as417
17
¯
Fv=ωv(1 −1
2κ)[cv−uf
c2
s
+cv(cv·uf)
c4
s
]·¯
F(19)
Here, ωvis a weighting factor related to the lattice velocity, which can be418
derived from the D3Q19 model [38].419
Numerical CFD simulations were conducted in a domain of 1.0 m ×420
0.3 m ×0.3 m. For both the upper and lower boundaries, slip boundary con-421
ditions were employed, while periodic boundary conditions were adopted on422
both lateral sides. At the inlet, a uniform velocity of 0.1 m/s was employed,423
and free outlet boundary conditions were used. Additionally, octree element424
structures with a minimum mesh size of 0.001 m were used for local encryp-425
tion surrounding the coral structure. A convergence test was performed to426
verify the mesh independence.427
18
ab
0.02 m 0.01 m
100Vorticity (s-1) : 0
(Pa)
c
0.02 m
Figure 5: The vorticity and surface-stress contours of artificial brain coral: (a) one wave
cycle from the global perspective; (b) the corresponding local perspective on the reverse
side of the coral; (c) the corresponding stress contours under the worst-case conditions.
19
In Fig. 5, the periodic changes in the surface and the surrounding vorticity428
of artificial coral under unidirectional turbulence are represented in sequence,429
and some characteristics of the flow field affecting matter exchange are noted.430
Generally, the variation in vorticity is entirely caused by periodic vortex431
shedding. The maximum vorticity occurs at the ridges on the side facing the432
current, causing a relatively smooth laminar flow to envelop the back side433
of the coral. Under these circumstances, the polyps in the valleys are well434
shielded from the dangers presented by the current, including sharp objects435
and fierce predators. On the other hand, flow reversal occurs on the back436
side of the coral when vortex shedding begins. The periodic change in the437
flow line spectrum could cause a change in the pressure distribution, which438
would lead to a change in the magnitude and direction of the fluid pressure439
acting on the coral and ultimately cause vibration of the coral.440
Table 1: Summary of the influence of the morphological characteristics of brain coral on
the surrounding vorticity.
Case ζ g (mm) Mean vorticity near the surface (s−1)
1 0.0001 1.50 15.72
2 0.0001 2.00 18.68
3 0.0001 2.50 21.49
4 0.0002 1.50 17.16
5 0.0002 2.00 21.52
6 0.0002 2.50 24.33
7 0.0003 1.50 19.41
8 0.0003 2.00 23.27
9 0.0003 2.50 26.55
To analyze the influence of the morphological characteristics of brain coral441
on the surrounding seawater flow field, models were generated using various442
combinations of the slope (ζ) and fold depth (g) parameters. Through CFD443
simulations, vorticity distributions similar to those in Fig. 5 were obtained,444
and the corresponding results are shown in Table 1. For a better quantita-445
tive analysis of the water flow, the mean vorticity in the space close to the446
coral surface (0–3 cm) was computed for comparison. The results show that447
when the slope is small ζ≤0.0001 (corresponding to a flat shape), rapid448
vortices cannot be generated on the side facing the current. Such vortices449
20
could increase the amount of time nutrients would remain in contact with450
the polyps, which would increase the efficiency of matter interchange. As a451
result, most brain corals form hemispherical structures. However, a large ζ452
leads to a great impact from the seawater that is detrimental to the coral’s453
stability. Therefore, the value of ζusually should not exceed 0.0003. On454
the other hand, as the most distinct morphological characteristics of brain455
corals, the grooves can collect microorganisms that fall from vortices, pro-456
viding the polyps with nutrients other than those obtained through photo-457
synthesis. Rougher surface features (g≥2.00) create strong vortices that458
allow polyps to capture abundant food through their tentacles. However,459
this also results in low efficiency for seawater to carry the polyps’ fertilized460
eggs and excreted metabolites. Over time, such matter will be deposited in461
the gullies, preventing the population from reproducing. In conclusion, vor-462
ticity analysis can be applied to more reasonably determine the values of 3D463
morphological parameters for coral evolution. The flow chart of the whole464
optimization process is shown in Fig. 6.465
21
Aquatic observation
Objective function
Vorticity analysis
Length constraint
Shape correction
Expansion Slope gradient
Evolutionary results
Matter interchange
Platygyra
Goniastrea
Lobophyllia
Figure 6: Flow chart showing the step-by-step process of evolutionary optimization.
At the moment representing the worst conditions, for a hemispherical466
structure with a fixed bottom, a maximum stress of 2.71×102Pa occurs on467
the side facing the current (Fig. 5(c)). This is a safe value for solid structures468
predominantly consisting of calcium carbonate (which has a strength limit of469
approximately 1×106–2×106Pa). Moreover, a relatively large stress is also470
found at the pole facing away from the current, which is a result of the flow471
reversal. In fact, the stress resulting from hydrostatic pressure is often much472
greater than the stress caused by ocean currents. Due to its bulbous shape,473
Platygyra has less to fear from pressure and currents than staghorn coral and474
thus tends to live on deeper seabeds (approximately 30–50 m).475
22
3.2. Matter interchange simulation476
To study the influence of the gully structure of brain coral on matter477
exchange, the Fluent module of the commercial finite element software AN-478
SYS (ANSYS, Inc., United States) was employed to simulate the matter479
interchange process in a static water environment. The numerical model was480
imported into the software and placed in a chamber filled with solving liquid.481
The pressure-based solver was employed, using the transient solution process482
and a given gravitational field. Regarding the material-related assumptions,483
the liquid domain and the valleys of the brain coral were set as liquid (water)484
materials, whereas it was assumed that the ridges could not be dissolved. For485
the boundary conditions, the concentration of the liquid against the cham-486
ber was set to 0. Both the solid and liquid domains were discretized into487
tetrahedral elements, with the coral and liquid containing 78938 and 161765488
elements, respectively. The process could be interpreted as the valleys of the489
brain coral gradually being dissolved into the surrounding liquid until they490
were completely dissolved. The simulation was conducted using the SIMPLE491
algorithm, and the results are presented in the figure below.492
(a)15.24% (b) 35.63% (c) 47.24%
(d) 60.51% (e)79.77% (f )100%
a100%
0%
bc
d e f
Figure 7: Simulation of the matter interchange process for Platygyra: (a) a state in which
15.24% of the polyps (cyan parts) have completed matter interchange; (b-f ) similarly,
states of (b) 35.63%, (c) 47.24%, (d) 60.51%, (e) 79.77%, and (f) 100% completion. The
color bar represents the percentage of polyps that have completed matter interchange.
23
The dissolution process began in the upper right region of the man-made493
coral; this relative location was subject to some randomness (Fig. 7a). How-494
ever, the matter-interchange-sensitive region (valleys) occupied a relatively495
large area proportion in the upper right corner, which was one of the reasons496
for the active matter interchange here. From another point of view, dissolu-497
tion generally occurred at the edges of the valleys. This be explained by the498
fact that the dissolution process started at random spots, from which it then499
spread. However, at the junctions of valleys and ridges, the surface curvature500
of the model varies greatly over a short distance. For dissolution spots at501
these junctions, the solution containing the dissolved matter tended to be502
trapped, making it difficult to enter outside circulation. Consequently, the503
dissolution along the valley edges was accelerated (Fig. 7b&c). Then, the504
centers of the valleys also started to dissolve (Fig. 7d&e), and the dissolution505
process gradually spread to the entire dissoluble domain (Fig. 7f).506
24
4. Experimental justification507
4.1. Mold prefabrication508
0.02m
ab
c d
0.02m
0.02m 0.02m
Figure 8: The fabrication procedure for man-made Platygyra: (a) the polished geometric
model; (b) the UV-curable resin model fabricated using the DLP technique; (c) the reverse
mold made of platinum AB silica gel; (d) the cast Platygyra model made of a cement and
calcium carbonate mixture.
Because of the complex geometry and slump of coral structures, it would509
be difficult to use direct printing techniques such as fused deposition modeling510
(FDM) for fabrication. Therefore, a soft material with good demolding per-511
formance was used for the manufacture of casting molds. First, we imported512
the optimized structure into the commercial 3D modeling application ZBrush513
(version 2020, Pixologic Inc.) (Fig. 8a). Then, the numerical coral model was514
printed using UV-curable resin (SP-RH; Soonsolid Co., Ltd., China; density:515
1.14 ×103kg/m3; curing wavelength: 4.05 ×10−7m; viscosity (25◦C): 1.06516
×108Pa·s) by a desktop digital light processing (DLP) printer (SprintRay517
25
Pro; Soonsolid Co., Ltd., China) (Fig. 8b). We used this printing method518
because of its high printing resolution, which could help to maintain the ge-519
ometrical configuration of the numerical model. The printer had a 10−4m520
printing precision, a 6 ×10−5m layer thickness, and a printing time of 10521
seconds per layer. It took approximately 30 min to fabricate a 0.0708 m522
×0.0644 m ×0.0190 m sample. Afterward, a reverse mold was fabricated523
using platinum AB silica gel (SJ3212 Beijing Sanjing Xinde Technology Co.,524
Ltd., China; Shore A hardness: 15) (Fig. 8c). After the mold had solidified,525
artificial corals were fabricated via a casting process (Fig. 8d).526
4.2. Casting process527
Table 2: Mixture proportions and compressive strength of the materials used for the
fabrication of man-made corals
Sample Cement Calcium carbonate Water Compressive strength
label (g) (g) (g) (MPa)
S1 33.3 33.3 33.3 11.390 ±0.826
S2 30.3 36.3 33.3 7.197 ±0.534
S3 27.8 38.8 33.3 5.537 ±0.311
S4 25.6 41.0 33.3 4.384 ±0.285
The specimens were composed of a material that provided strength (Port-528
land cement PO42.5) and a material that satisfied the necessary requirements529
in terms of biological properties (calcium carbonate powder, fineness 800).530
To study the strength of the artificial coral composites, the proportions of531
these two solid matter materials were employed as the influencing factors for532
orthogonal experiments. The various environmentally friendly compositions533
employed for manufacturing [41], all of which are similar to the composition of534
stony corals, are shown in Table 2. To prepare the cementitious composites,535
the two powders were first poured into a mixer for dry mixing. Afterward,536
the mixed concrete was poured into the mold, and the top surface was lev-537
eled. The mold was removed after three days of curing at a temperature of538
20.0±2◦C and a relative humidity of 95±5%. Then, the sample was covered539
with plastic film and further cured for 25 days. Additionally, to compare the540
strengths of the various recipes, we prepared corresponding standard cubic541
blocks (0.15 m ×0.15 m ×0.15 m) with the same mixture proportions. For542
26
compression tests, a compression testing machine (CONCRETO 2000; Shi-543
madzu Co., Ltd., Japan) was employed, whose loading speed was controlled544
at 0.3 MPa/s. Each group of samples was subjected to compression three545
times, and the average results were recorded. Generally, the compressive546
strength decreased as the proportion of calcium carbonate in the composites547
increased. According to the CFD simulations, the compressive strength of548
the S4 material was still more than sufficient to withstand the current impact549
and hydrostatic pressure at a depth of 50 m. Furthermore, such a mixture550
with high calcium carbonate content is closer to the composition of natural551
coral than plain cement is. However, a high calcium carbonate content will552
also lead to a decrease in workability, especially in terms of the cohesion of553
the concrete. In summary, the mixture proportions of the S4 material were554
found to be the most suitable for the manufacture of this man-made coral.555
27
4.3. Matter interchange test556
0.02m 0.02m
0.02m 0.02m
ab
c d
Figure 9: Control experiment to validate the matter interchange capability: (a) the original
man-made coral with concentric ring-shaped gullies; (b) the original man-made Platygyra;
(c) the coral with concentric ring-shaped gullies after the matter interchange test; (d) the
man-made Platygyra after the matter interchange test.
To verify the superiority of the optimized brain-shaped grooves in terms557
of their matter interchange capabilities, a pair of control samples were fabri-558
cated for calcium carbonate–acetic acid corrosion experiments. The test pro-559
cess was performed as follows: First, a sample with concentric ring-shaped560
gullies was cast, whose ridges and valleys each occupied half the surface area561
(Fig. 9a). For comparison, a man-made Platygyra sample was cast using562
28
the topological optimization method described above, where the valleys also563
occupied 50% of the sample area (Fig. 9b). The samples were photographed564
with a digital camera (12 million pixels) (Fig. 9a,b) and were each subse-565
quently placed at the bottom of a chamber of the same size as the seawater566
tank for coral cultivation (0.80 m ×0.50 m ×0.80 m), with a propeller pump567
placed 0.4 m away from the sample. Because the predation behavior of the568
polyps occurs within the valleys, the observations needed to focus on the569
valleys rather than the ridges. It was therefore necessary to preprocess the570
models by pasting plastic film on all ridge surfaces in Fig. 9a,b. In this way,571
it was ensured that the matter interchange reaction would occur only in the572
valleys, as these were the only parts of the samples that were in contact with573
the solution. After the chamber was filled with pure water, the flow speed574
was controlled by the pump at 2000 gph. Every 60 seconds, 10 ml of 80%575
acetic acid solution was dropped into the chamber from a burette beside the576
pump. One hour later, the sample was removed, and its surface was rinsed577
with deionized water before drying. According to the observation of the con-578
centric ringed structure after corrosion (Fig. 9c), the calcium carbonate in579
the structure reacted with the acetic acid to form calcium acetate, which580
then dissolved in the water, leaving a few tiny holes in the valleys. These581
holes were fairly evenly distributed throughout all of the gullies. In contrast,582
the valleys of the man-made Platygyra exhibited many holes of different sizes583
(Fig. 9d). Generally, the matter interchange reaction was more severe on the584
man-made Platygyra, especially in the valleys at an intermediate height from585
the bottom. This difference can be explained by the flow field simulation.586
Normally, the grooves tend to block water from flowing over the ridges and587
force it to run along the valleys. Thus, circular water flows form in the con-588
centric grooves. The flow carrying the acetate ions reacts with the calcium589
carbonate, and the products of this reaction gradually precipitate from the590
valley surfaces and accumulate. This prevents the remaining calcium carbon-591
ate from coming in contact with acetic acid, resulting in insufficient material592
interchange. In contrast, after entering the man-made Platygyra, the wa-593
ter flow travels through the crisscross valleys and finally escapes from the594
gullies, carrying away the calcium acetate produced by the reaction. This595
allows the subsequent replacement reaction to continue, resulting in more596
extensive material interchange between the coral surface and the water. In597
other words, the brain-like surface maintains the concentration of nutrients598
while hindering the deposition of harmful substances.599
29
250 μm250 μm
250 μm250 μm
a b
cd
Figure 10: The micromorphology on the valley surfaces of (a) the original man-made coral
with concentric ring-shaped gullies; (b) the original man-made Platygyra; (c) the coral
with concentric ring-shaped gullies after the matter interchange test; (d) the man-made
Platygyra after the matter interchange test.
To observe the micromorphology on the surfaces after the matter in-600
terchange treatment, the samples were rinsed and observed using a digital601
microscope capable of showing details as small as 4 ×10−7m (DVM6, Le-602
ica, Germany). As seen from the optical images, the valley surfaces of both603
the coral with the concentric ring-shaped gullies (Fig. 10a) and the man-604
made Platygyra (Fig. 10b) were flat and nonporous. However, after the605
corrosion treatment, the microstructures of the two models were completely606
different. Calcium acetate was deposited on the surface of the concentrically607
ringed sample and prevented further corrosion. Under these conditions, only608
a small proportion of the calcium carbonate particles could react with acetic609
acid to produce carbon dioxide. Pores of approximately 10–70 microns in610
diameter were observed on the surface of the sample, with an average spac-611
30
ing of approximately 100 microns (Fig. 10c). On the other hand, matter612
interchange in the valleys of the man-made Platygyra was unhindered. A613
large number of pores with a spacing of approximately 80 microns are visible614
in Fig. 10d.615
5. Conclusion616
For the topological optimization of artificial coral structures, their re-617
sistance to matter interchange was minimized in the projected surface do-618
main under length constraints. This optimization algorithm was proven to619
be correct by comparison with the morphological characteristics and growth620
processes of Platygyra,Goniastrea and Lobophyllia. In addition, CFD simula-621
tions showed that these artificial corals could provide shelter to shield polyps622
from swift currents and predators. A reverse mold was prepared from silica623
gel and used to cast models from mixtures of cement and calcium carbon-624
ate, where the mixture ratio was determined based on tests of compressive625
strength and biological adaptation. Finally, a numerical simulation and an626
acid corrosion test were conducted to verify the matter interchange capability627
of the optimized morphology with many folds.628
The numerical and experimental validations showed that the brain-like629
folds contribute to the circulation of seawater, thus maintaining the concen-630
tration of nutrients and hindering the deposition of harmful substances. Mul-631
tiobjective optimization based on a linear weighted sum can address some632
of the challenges of biological evolution mechanisms, especially those that633
are difficult to express mathematically. The presented optimization method-634
ology is able to generate 3D topologies with promising matter interchange635
performance. Additionally, based on data from aquatic observations, the636
morphological characteristics of different species of brain corals could be suc-637
cessfully numerically reproduced by suitably arranging the settlement of the638
larvae.639
However, only the matter interchange capability was considered in this640
study to evaluate the performance of the optimized structures. In fact, the641
biological process of coral growth is lengthy and influenced by a combina-642
tion of factors, such as temperature and light intensity, for which accurate643
mathematical descriptions are still lacking. Strongly universal methods such644
as evolutionary optimization algorithms might be applied to analyze these645
problems. In subsequent work, the colonization and reproduction of polyps646
on artificial coral surfaces will be investigated. Regarding industrial applica-647
31
tions, the results of this study might assist in the design of structures that can648
adsorb hazardous substances, particularly in oscillatory flows, which could649
help to effectively improve water and air quality in various environments.650
6. Acknowledgments651
The authors gratefully acknowledge financial support from the National652
Natural Science Foundation of China (No. 51908201), the Shenzhen Science653
and Technology Program (KQTD20200820113110016), the Natural Science654
Foundation of Hunan Province (No. 2020JJ4945).655
Conflicts of interest656
The authors declare that they have no conflicts of interest.657
Appendix A. Sensitivity analysis658
With the goal of solving the governing equation of fm, the elemental-based659
matter transport matrix Kis given by660
K=k(ρ)BTB(A.1)
where Bdetermines the relation between the growing matter content and its661
spatial derivative. Accordingly, the discretized Lagrangian function can be662
rewritten as663
L=TTKT+TT(KT −F)(A.2)
where KT =Cand Fis independent of the design. The sensitivity of the664
density can be expressed as665
∂fm
∂ρ =∂S
∂ρ (TT∂K
∂S+TT∂K
∂S)T(A.3)
where S=σ+ (1 −σ)ρp. The adjoint problem of solving Tcan be expressed666
as667
KTT=−(K+KT)T(A.4)
Both Kand Kare symmetric. Therefore, the adjoint problem can be668
solved through back substitution.669
32
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