Magnetic particles are either positively charged or negatively charged. Particles of the same charge repel each other, while particles of differing charges are attracted to each other. Figure 2 shows this: (a) and (b) show repulsion, while (c) shows attraction.

Coulomb’s Law quantifies the force between the particles according to the Coulomb’s Law, which mathematically is as follows:

F i j = 1 4 π ε 0 q i q j d i j 2 (1)

The first term of this (1/4πe₀) is essentially a constant, used to convert the result into units of Newtons (N). This constant is not used here, as we are only interested in relative forces. The value of q_i and q_j are the charges of two particles, measured in Coulombs (C). The value of d_ij is the Euclidean distance between particles i and j. As such, the force between two charged particles reduces to the following:

F i j = q i q j d i j 2 (2)

We can state that the force between two charged particles is the product of their charges and inversely proportional to the squared distance between them, where the squared distance is as follows:

d i j 2 = ( x i − x j ) 2 + ( y i − y j ) 2 (3)

With a basic understanding of Coulomb’s Law, we now need to understand how the forces of several charged particles affect a single particle. For this research, only positive particles move, while negative particles remain stationary – the forces imposed by the negative particles do, however, affect the movement of the positively-charged particles. Table 1 provides some relevant terms:

It is intended to determine the net force exerted on all positive particles. To do this, we combine the forces exerted on all positive particles in terms of horizontal and vertical components. Consider Figure 3, detailing a specific particle at the point (7, 11). It is our intent to sum all forces imposed on this specific particle to determine the net force acting on this particle.

The find the net force exerted on some specific positive particle i⁺, the following two equations are used:

( F x ) i + = ( ∑ j + = 1 n + ( x i + − x j + ) d i + j + ) + ( ∑ j − = 1 n − ( n + / n − ) ( x j − − x i + ) d i + j − ) , ∀ i + , ∀ j ≠ i + (4)

( F y ) i + = ( ∑ j + = 1 n + ( y i + − y j + ) d i + j + ) + ( ∑ j − = 1 n − ( n + / n − ) ( y j − − y i + ) d i + j − ) ∀ i + , ∀ j ≠ i + (5)

It is important to note that the first component of the above two formulae relates to the force between the particle i⁺ and other positively-charged particles, while the second component relates to the force between positively-charged particle i⁺ and the negatively-charged particles. It is also important to note that the (n⁺/n⁻) component is used to make sure the total positive force (in Coulombs) is the same as the total negative force (in Coulombs). For this example, there are four positively-charged particles, and one negatively-charged particle. As such, the (n⁺/n⁻) multiplier assures the negatively-charged particle has a charge of four Coulombs, the same as the sum of the positively-charged particles.

The magnitude of the resultant net force is as follows:

F i + = ( F x 2 ) i + + ( F y 2 ) i + , ∀ i + (6)

The direction, or heading, of this resultant net force is as follows:

h i + = tan − 1 ( ( F y ) i + ( F x ) i + ) , ∀ i + (7)

Table 2 shows how the net resultant force acting on the point (7, 11) is determined:

Using Equations (4) and (5) above, and summing the two right-most columns of Table 2, we have ( F x ) i + of −0.3377, and ( F y ) i + of −0.6263. This results in a net force magnitude of 0.7116 ( F i + ) , and a heading of −151.67 degrees ( h i + ) . Figure 4 details the resultant force.

The importance of Figure 4 is that the resultant force (both magnitude and direction) presents the location to where the particle is inclined to move. Hereafter, this type of particle movement is referred to as “polar movement.”

Again, it is important to note that only positively-charged particles are subject to movement, so resultant forces are only determined for the positively-charged particles.

Unfortunately, polar movement is not always possible, because resultant forces may take us outside the realm of feasible solutions (outside the unit square). Because in polar movement, only the positively-charged particles move, and their nature to repel each other will motivate these particles outside the realm of feasible solutions. Figure 5 shows an example of this.

Figure 5 shows us that the positively-charged particle desires to move outside the search space, rendering an infeasible solution. Furthermore, the heading of this particle will never change, essentially putting our search at a “dead end.”

In order to combat this problem, all movable particles are treated as “neutral” particles, and they are moved a very small amount according to randomly-selected magnitudes and headings. This approach is referred to as random movement, and is illustrated in Figure 6.

This research effort utilizes both polar movement and random movement in our attempt to find optimal spacing.

Prior to describing the search process, Table 3 provides definitions, some of which are a slight repeat of the definitions from Table 1.

Values for the following search parameters are chosen: Iter, k_B, CR (< 1.0), T₁ and T_F (where T_F < T₁), and T = T₁. The iteration index, q, is set to 1. The number of movable particles for spacing (n) is chosen, and for each particle i, x and y coordinates are assigned the following:

x i ~ U ( 0 , 1 ) , ∀ i (8)

y i ~ U ( 0 , 1 ) , ∀ i (9)

This is the initial solution. The objective function of this initial solution is as follows:

max ( min ( d i j ) ) , ∀ i , j ≠ i , w h e r e (10)

d i j = ( x i − x j ) 2 + ( y i − y j ) 2 (11)

Subject to the constraints:

0 ≤ x i ≤ 1 , ∀ i (12)

0 ≤ y i ≤ 1 , ∀ i (13)

The values of Z_C, Z_T and Z_B are set equal to the objective function value.

In layman’s terms, we wish to arrange n particles in the unit square such that the distance between the closest two particles is maximized. The mathematics of this model shows a scenario where the objective function is both nonlinear and discontinuous, meaning conventional solvers are unlikely to guarantee an optimal solution, especially for larger problems. This is the reason a search heuristic is pursued.

Prior to movement, the current solution is copied to the temporary solution. Movement is either in the form of polar movement or random movement, both described in Section 3. The probability of polar-movement is τ, implying that the probability of random movement is (1 − τ).

In polar movement, only the positively-charged particles move, while the negatively-charged particles remain stationary. All positively charged particles simultaneously move in the direction of their heading ( h i + ) at a distance of ( F i + ) . In random movement, the n-particles move to a nearby, randomly-selected location. Regardless of the type of move, infeasible moves are not executed. The result of all movements is the temporary solution.

Once the (previously-described) move is complete, the objective function value (Z_T) is determined via Equation (10). A determination needs to be made as to whether or not to accept the temporary solution as the current solution. If the objective function value of the temporary solution is better than the objective function value of the current solution—that is, if Z_T > Z_C, the current solution is replaced by the temporary solution.

Otherwise, further evaluation is needed before accepting the temporary solution as the current solution. The inferiority of the temporary solution compared to the current solution is represented as δ, and is determined as follows:

δ = Z T − Z C Z C (14)

The probability of accepting the inferior temporary solution as the current solution occurs with the following probability:

P ( A ) = exp ( δ k B T ) (15)

A uniformly-distributed random number of the interval (0, 1) is generated. If this value is less than P(A), the inferior temporary solution is accepted as the current solution. Otherwise, the inferior temporary solution does not replace the current solution. The content provided by Equations (14) and (15) cuts to the uniqueness of Simulated Annealing. Inferior solutions are occasionally accepted as current solutions so as to avoid being trapped at local optima. The probability of accepting inferior solutions decreases as the search progresses. The Boltzmann Constant (k_B) controls the probability of accepting inferior solutions [10].

A test step compares the temporary solution to the best solution. If the temporary solution objective function value exceeds the best solution objective function value (that is, if Z_T > Z_B), the temporary solution replaces the best solution.

If q is less than the specified number of iterations per temperature level (Iter), the value of q is incremented by one (q = q + 1), and control if the search returns to that described in Section 4.2. Otherwise, the current Temperature (T) is multiplied by the cooling rate to set the new current temperature (T = T * CR), and q is reset to 1. If the new resultant temperature is greater than T_F, control of the search returns to that described by Section 4.2, and q is reset to 1. Otherwise, the search is over and the best solution is returned.

For a better understanding of the search process, details are provided in pseudocode form.

initialize();

while (T > T_F){

for(q = 1; q <= Iter; q++){

set temp-solution = current-solution();

if(~U(0,1)<τ)polar-movement();

else random-movement();

evaluate temp-solution();

if (Z_T > Z_C) set current-solution = temp-solution();

if (Z_T > Z_B) set best-solution = temp-solution();

else {if(~U(0,1)

set current-solution = temp-solution();}

} //end else

} //end q

T = T * CR

} //end while

report-best-solution();

When the best solution is returned, the x and y-coordinates of all particles associated with the best solution are reported, along with the objective function value.

The methodology can be used for any value of n. As stated earlier, NetLogo was the programming software employed. The images provided via Figure 7 and Figure 8 are images from the NetLogo environment. Figure 7 details the randomly-generated initial solution for n = 18. The particles to be arranged are positively-charged, and shown via a blue “+” symbol. Additionally, the size of the particle is proportional to the net force acting on the particle, and the heading of the particle is also shown via the arrow at the end of the “+” symbol. The negatively-charged particles are shown in red via “−” symbols. These particles are stationary—they do not move, and are only present for their influence on the mobile, positively-charged particles.

After careful inspection of Figure 7, one will notice that the positive particles close to the negative particles have larger magnitudes and are pointed toward the negative particles. This is affirmation of Coulomb’s Law. Similarly, positive particle further from other particles have smaller magnitudes, again affirming Coulomb’s Law. Finally, positively-charged particles somewhat near each other are pointing in opposing directions.

After the Simulated Annealing was performed, the final solution was obtained and detailed via Figure 8. The is the “best” arrangement of the positively-charged particles is shown the maximized distance between the two closest

particles was 0.3000, where the proven-optimal distance was found to be 0.3005 [14].

Figure 8 shows what the NetLogo program found to be the best overall solution. Here, the positively-charged particles are shown via black circles, and the negatively-charged particles are not shown. The maximized distance between the two closest particles is also shown.

In finding the best solutions for values of n ≥ 2, the search settings shown in Table 3 had to be adjusted for each unique value of n. For the trivial problems involving optimal spacing of 2, 3 and 4 particles (n = 2, 3, 4), the search best responded by eliminating the negative charge in the unit square. This makes sense, because for these values of n, the positively-charged particles are motivated to get as far away from each other as possible, regardless of other circumstances. Staring with five or more particles (n ≥ 5), the negative force in the central portion of the unit square becomes more important. This is because for larger quantities of particles, some of them need to be attracted toward the central part of the unit square for optimal spacing. Adjusting the negative force from problem to problem was required to obtain desirable results.

For all problems, T₁ was set to 40, while T_F was set to 1. The cooling ratio, CR was set to 0.9995, and the number of Iter increased in proportion to n. For all problems, the Boltzmann Constant, k_B, was set such that a temporary solution 5% inferior to the current solution had a 5% probability of acceptance when T = T₁.

Table 4 shows the results of the methodology presented here as compared to results proven to be optimal for several problems.

For the problems pursued, the presented methodology averaged 99.89% as compared to proven optimal. The worst-performance was for n = 15, which was 99.27% as compared to optimal.