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SUMMARY:Modelling the force-dependent morphogenesis of fish vertebra with
topology optimization
DTSTART;VALUE=DATE-TIME:20180709T050000Z
DTEND;VALUE=DATE-TIME:20180709T052000Z
DTSTAMP;VALUE=DATE-TIME:20241111T084601Z
UID:indico-contribution-346@conferences.maths.unsw.edu.au
DESCRIPTION:Speakers: Misaki Sakashita (Osaka University)\nWolffâ€™s law s
tates that bone morphology evolves according to their external mechanical
loading. Following this law\, researchers have tried to simulate bone shap
e formation\, especially for trabecular bone\, using topology optimization
[1]. Less attention has been given to the bone outer shape\, composed of
cortical bone. However\, trabecular bone and cortical bone are both mainly
formed by osteoblasts and osteoclasts. Therefore\, we hypothesize that th
e bone outer shape also adapts to the external forces. The aim of this res
earch is to understand the mechanism that generates the bone outer shape b
y reproducing the latter using topology optimization. \n\nThe fish vertebr
a can be divided into two parts: an inner hourglass-like structure and an
outer lateral structure. Based on our observations\, it turns out that num
erous species present a similar hourglass-like structure but that the late
ral structure strongly depends on the fish species. Lateral structures can
be classified into two types. The first type exhibits a ridge structure w
ith one or more thick plate-like bones and the second type exhibits a netl
ike structure in which thin plate-like bones are randomly oriented. These
differences seem related to the fish motion\, i.e. the swimming type of th
e fish\, and therefore\, it is assumed that lateral structures also evolve
based on external loading conditions.\n\nIn standard topology optimizatio
n\, the density at each material point is only constrained by the total vo
lume of material. However\, the activity of osteoblast and osteoclast is m
ore a local phenomenon. Hence\, the standard topology optimization problem
is supplemented with a local density penalization to mimic this local phe
nomenon. To solve the optimization problem\, a method based on a time depe
ndent reaction-diffusion equation is employed [2]. The equation is driven
by the sensitivity $S$\, in which the Lagrangian multiplier $\\lambda$ is
introduced. Solving this optimization problem gives an optimized structure
with respect to the imposed boundary and loading conditions. The penaliza
tion term enables to control locally the feature size. \n\nUsing this equa
tion\, we developed a 3-D mathematical model to generate fish vertebra. Wi
thout local density penalization\, thick beams appear similarly to the rid
ge structure. Penalizing locally the density\, thinner beams are promoted
and they tend to form a netlike structure. Numerical results show that the
proposed model can produce both types of lateral structures\, i.e. ridge
structure and netlike structure\, which are similar to the actual fish bon
e. In the future\, it would be interesting to be able to produce various f
orms of fish vertebra by only adjusting a few parameters of the penalizati
on law.\n\n[1] Jang *et al.*\, 2009\n[2] Kawamoto *et al.*\, 2013\n\nhttps
://conferences.maths.unsw.edu.au/event/2/contributions/346/
LOCATION:University of Sydney New Law School/--106
URL:https://conferences.maths.unsw.edu.au/event/2/contributions/346/
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