14 Flocking Rules Governing Swarm Robot as Tool to Describe …
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where dis(t, k, j) is the Euclidean distance between points k and j at time t. This is
the formula for the point j at time t. It is an attempt to simulate the potential energy
of a material point obeying Hooke’s law.
To compare time contiguous configuration C t and C t − 1 ,we define for each point
j and each time t
PE2(t, j) = ||C t − C t−1 ||
where || is the norm of the vector defined by the point j at time t and t − 1.
It must be underlined that this artifice has no direct connection with the usual
energy definition (this is the reason we use the term pseudo-energy). Actually, it is
only a graphic tool for a better understanding of the deformation. Extended use of
them can be found in (dell’Erba 2018b).
14.3 Results from Previous Works
In this section, we briefly resume some results obtained by this tool in the preceding
works (dell’Erba 2018a, c) where different strains of the leaders, with different
choices, have been investigated. In preceding papers, we also have described the
behaviour of some ASTM sample and the respect of Saint Venant principle; but now
we want to focus our attention on other simple simulation to outline what we have
to improve and what we need to understand better.
So far consider a case of simple strain and release in tensile test of a rectangular
shape specimen. We are considering a square sample undergoing strain from one
side (the other side is clamped) at constant velocity in the x-direction (speed 0.6
unit length/step time), with a square lattice of 10 × 10 units. At a certain time,
the pull is released and the leaders return to original configuration (this means that
the leaders have changed category and are now followers) attracted by the other
points. The simple rule governing the followers’ motion is that every point must be
placed in the barycentre of its neighbours (Eq. 14.1); the neighbours are determined
by the coordination number of the lattice; therefore, the leader’s motion implies a
displacement of the first layer that propagates in successive time steps to the other
particles. This means the displacements, at each time step, involve a larger shell of
points until all the lattice points are moving. In second gradient, (dell’Erba 2018c),
we have considered also the neighbours of the first neighbours as second shell. In
Fig. 14.1, we can see the configuration of the lattice over different time steps together
with the PE1 contour plot. Red points are the leaders; blue are the followers; and
orange are the frame. From the figure, we can outline that the x displacement of the
points seems do not depend on the y-coordinate; however, looking at the PE1 picture
we can note a light convexity that does mean this is not true.
A deeper examination of the points’ displacements confirms that, close to the
frame, the displacements along x are lower than the central points. This can be
explained as an edge effect. In fact if we consider points on the same vertical lines
227
where dis(t, k, j) is the Euclidean distance between points k and j at time t. This is
the formula for the point j at time t. It is an attempt to simulate the potential energy
of a material point obeying Hooke’s law.
To compare time contiguous configuration C t and C t − 1 ,we define for each point
j and each time t
PE2(t, j) = ||C t − C t−1 ||
where || is the norm of the vector defined by the point j at time t and t − 1.
It must be underlined that this artifice has no direct connection with the usual
energy definition (this is the reason we use the term pseudo-energy). Actually, it is
only a graphic tool for a better understanding of the deformation. Extended use of
them can be found in (dell’Erba 2018b).
14.3 Results from Previous Works
In this section, we briefly resume some results obtained by this tool in the preceding
works (dell’Erba 2018a, c) where different strains of the leaders, with different
choices, have been investigated. In preceding papers, we also have described the
behaviour of some ASTM sample and the respect of Saint Venant principle; but now
we want to focus our attention on other simple simulation to outline what we have
to improve and what we need to understand better.
So far consider a case of simple strain and release in tensile test of a rectangular
shape specimen. We are considering a square sample undergoing strain from one
side (the other side is clamped) at constant velocity in the x-direction (speed 0.6
unit length/step time), with a square lattice of 10 × 10 units. At a certain time,
the pull is released and the leaders return to original configuration (this means that
the leaders have changed category and are now followers) attracted by the other
points. The simple rule governing the followers’ motion is that every point must be
placed in the barycentre of its neighbours (Eq. 14.1); the neighbours are determined
by the coordination number of the lattice; therefore, the leader’s motion implies a
displacement of the first layer that propagates in successive time steps to the other
particles. This means the displacements, at each time step, involve a larger shell of
points until all the lattice points are moving. In second gradient, (dell’Erba 2018c),
we have considered also the neighbours of the first neighbours as second shell. In
Fig. 14.1, we can see the configuration of the lattice over different time steps together
with the PE1 contour plot. Red points are the leaders; blue are the followers; and
orange are the frame. From the figure, we can outline that the x displacement of the
points seems do not depend on the y-coordinate; however, looking at the PE1 picture
we can note a light convexity that does mean this is not true.
A deeper examination of the points’ displacements confirms that, close to the
frame, the displacements along x are lower than the central points. This can be
explained as an edge effect. In fact if we consider points on the same vertical lines