^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

The process of high gradient magnetic separation (HGMS) using a microferromagnetic wire for capturing weakly magnetic nanoparticles in the irrotational flow of inviscid fluid is simulated by using parallel algorithm developed based on openMP. The two-dimensional problem of particle transport under the influences of magnetic force and fluid flow is considered in an annular domain surrounding the wire with inner radius equal to that of the wire and outer radius equal to various multiples of wire radius. The differential equations governing particle transport are solved numerically as an initial and boundary values problem by using the finite-difference method. Concentration distribution of the particles around the wire is investigated and compared with some previously reported results and shows the good agreement between them. The results show the feasibility of accumulating weakly magnetic nanoparticles in specific regions on the wire surface which is useful for applications in biomedical and environmental works. The speedup of parallel simulation ranges from 1.8 to 21 depending on the number of threads and the domain problem size as well as the number of iterations. With the nature of computing in the application and current multicore technology, it is observed that 4–8 threads are sufficient to obtain the optimized speedup.

High gradient magnetic separation (HGMS) is a separation technique which has been proven as a powerful one for the capture of weakly magnetic particles from suspension which conventional magnetic separation techniques using only permanent magnet cannot achieve. Some examples of potential applications of HGMS technique in the fields of biomedical and environmental science are the separation of red and white blood cells from small amount of blood sample in microfluidic device [

In this work, we study the problem of HGMS of weakly magnetic particle using a microferromagnetic wire as capture center in irrotational flow of inviscid fluid. The effect of diffusion becomes significant for nanoparticle motion so dynamic of capture process will be described statistically in terms of particle volume concentration. The problem is considered in an annular domain surrounding the wire. The continuity equation describing time rate of change of volume concentration in any part of the domain is formulated and is solved numerically as an initial and boundary value problem by using the explicit finite-difference method. A parallel algorithm for updating concentration value at each time step is developed based on openMP (

Figure

The simulation domain, direction of incoming fluid flow (

By solving a magnetostatic boundary value problem in the domain using the boundary condition of uniform magnetic field (

The magnetic force acting of any weakly magnetic particle of susceptibility

In this work, we follow the work of Davies and Gerber [

Dynamic of nanoparticle concentration in the domain is described by the continuity equation which states that, in any small elements of the domain, time rate of change of particle volume concentration is proportional to net volume flux passing through the elements and can be expressed as

We partition the annular domain in Figure

Simulation domain partitioning.

In Figure

In the simulation, two two-dimensional arrays of equal size named

The concentration array.

The steps of simulation can be summarized as shown in Algorithm

(1) Assign initial concentration

(2) while (time < end_time) do

{

Loop1: FOR (

{

Check the value

IF (

{

ELSE IF (

{ Apply (

/* Assign downstream boundary condition */

/* Assign array

computing at next discrete time */

Loop2: FOR (

Loop3: FOR (

end:} /* End of simulation */

(3) Save concentration results in all elements.

The size of normalized time step

In this work, we use openMP platform to parallelize the above algorithm. To simplify this, the challenge of the algorithm is twofold.

The domain size can be large due to the domain size of

The number of iterations is more due to

Due to the first issue, there is no dependency in computing each element in array

Depending on the openMP schedule used, the work rows may be distributed to threads statically or dynamically. For example, we may try with the clause

The algorithm for computing the simulation is similar to the above one except that the line marked by Loop1 is the

Threads are created based on each angle.

We present the results in two parts. The first part is the results of the correctness of the simulation approach and the second part is the performance of the parallel computation on the varying factors such as the domain size and the iteration numbers.

In this work, we simulate concentration dynamics of weakly magnetic particle which has known magnetic susceptibility value. Simulation parameters are obtained from the previously reported work of Davies and Gerber [_{2}P_{2}O_{7}·3H_{2}O) particles of 24 nm diameter dispersed in water that provides _{2}P_{2}O_{7}·3H_{2}O particle flows traverse to the wire with incoming velocity of

Figure _{2}P_{2}O_{7}·3H_{2}O in this work so the concentration of gold nanodrug carriers may not be increased 100 times. However, increasing of drug carriers concentration from only 2 to 4 times is sufficient for therapy process so HGMS is very a interesting technique to increase the effectiveness of cancers or tumors therapies. It is also observed that the region of accumulation of the particles,

The concentration contours of Mn_{2}P_{2}O_{7}·3H_{2}O nanoparticles of 24 nm diameter at

Figure

The concentration contours of Mn_{2}P_{2}O_{7}·3H_{2}O nanoparticles of 24 nm diameter at

In the experiments, we test the simulation time on Intel Xeon Phi 7110P located at Kasetsart University, Thailand. It has 61 cores at 1.1 GHz, with 64-bit addressing running Linux

Figure

The speedup for the varying number of columns, 201, 401, and 801, and varying number of iterations, 5,000 and 10,000 iterations, varying schedule types.

For 5,000 iterations, 1 thread, 98,585 milliseconds is used while for 10,000 iterations, 1 thread, 195,396 milliseconds is used. From Figure

The speedup for the case number of columns, 201, 401, and 801 with the dynamic schedule varying number of iterations, 5,000 and 10,000 iterations.

Figure

The speedup for 5,000 and 10,000 iterations, dynamic schedule varying number of columns.

Figure

The computation time for the case numbers of columns, 201, 401, and 801, dynamic schedule varying number of iterations.

The computation time for 5,000 and 10,000 iterations, dynamic schedule varying number of columns.

For further analysis, we study the workload for each thread at each iteration. We observe that the number of points calculated for each thread is very much stable for every 1,000 iterations. For the first 1,000 iterations, all threads have about the same amount of points calculated. As the iteration goes, the concentration values become steady at some points. Thus, the thread responsible for the rows containing more of these points has less work load. Therefore the static schedule is not best performed. With the dynamic schedule, the thread that finishes fast due to this reason will be able to perform computation in other rows. Figures

Total points calculated for every 1,000 iterations for the case of 201, 401, 801 column size.

The workload of each thread for 4 threads using static schedule

The workload of each thread for 4 threads using dynamic schedule

Figure

Figure

In this work, we present the process of high gradient magnetic separation (HGMS) using microferromagnetic wire for capturing the weakly magnetic nanoparticles in the irrotational flow of inviscid fluid. The parallel simulation using openMP is proposed. The domain problem is viewed in two-dimensional domains of wire radius and annular step and with the magnetic fore and fluid flow. The timing factors are the size of granularity and the iterations performed. Both also affect the correctness of the simulation process.

To parallelize it, each concentration element can be computed independently. Each thread computes rows of each concentration. The varying schedule types are considered. From the experiments, the dynamic schedule is very suitable since, along the iterations, the number of computed elements is less when the simulation moves gradually to the stable state. Also, with the parallelization by angle, we obtain double speedup when the number of threads goes double. This is consistent for the case when we double the domain size and double number of iterations. For the study of 720 rows, the number of threads should be 16 for the optimum speedup.

In the future, if the domain is very large, we can distribute the domain using MPI approach, and within each MPI node, the proposed openMP method can be used. With the MPI, the concentration array is divided with overlapped rows [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported in part by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program, Contract no. PHD/0275/2551, and Silpakorn University Research and Development Institute for the funding support, Contracts nos. SURDI 55/01/18 and SURDI 55/01/01. The authors are grateful to Intel Many Core Testing Lab under the collaboration of Intel Co. Ltd. and HPCNC, Kasetsart University, Thailand, for the testing and the experiments.