### From nanoparticles to plasma modeling

Electrostatic modeling, although a mature academic area, was a source of vivid activity for a past few years. Industrial design of many electronic devices, such as capacitive sensing (touch screen technology), electrical capacitance tomography or microelectromechanical systems (MEMS) design, relied on the ability to calculate and post-process electrostatic charge distribution, potential and field as part of a detailed analysis and optimization process.

[Study how Robin Hood Solver is used to optimize capacitive sensors.]

Besides industrial design, electrostatic interaction dominates many physical, chemical or biological systems some of which are of major interest to Tier 1 research institutions. Graphene, electrically conducting allotrope of carbon, present in various configurations (monolayer and bilayer graphene, graphene flakes and superlattice) promises an enormous amount of use in industry, electronics, biology and biochemistry. Components like graphene transistor (through the use of graphene logic gates), graphene ultracapacitors or inexpensive graphene DNA sequencing are within the reach. Electrostatic properties of nanoparticles in various configurations and surroundings are successfully used in nanowires, nanotubes and nanobatteries modeling. Also, a nanoparticle interaction and dynamics as a component of colloid solution, such as metal polymer nanocomposite colloid, is of wide academic and industrial interest. There are also many other areas with the need for reliable and efficient electrostatic calculation and modeling such as plasma dynamics, hard sphere models, quantum dot modeling and simulation, charged particle and ion dynamics and beam control and focusing (cone traps, quadrupole traps, electrostatic lens systems) and many others.

### Robin Hood nature of the Robin Hood method

The functioning of an algorithm is best illustrated by a transparent example. Let us consider an insulated metallic sphere and a fixed point charge placed close to it. Under the influence of the electric field of the fixed charge the free charge in the metal redistributes until the surface of the sphere becomes equipotential. The solution of the electrostatic problem amounts to finding the distribution of the surface charge at the metallic surface of the sphere.

[See more details about this example calculation.]

How does the Robin Hood method do it? The surface of the sphere is divided into triangles. An initial surface charge distribution is chosen. The electric potential at every triangle is calculated. Then the triangles with the maximal and the minimal value of the potential are identified. An amount of charge is transferred from the triangle with the maximal potential to the triangle with the minimal potential so that the potentials at these two triangles become the same after the transfer. The potentials at other triangles are updated to account for the transferred charge. The entire procedure of finding minimum/maximum of the potential, transferring the charge and updating the potential is iterated until the predefined precision is achieved.

The main steps of the method explain the choice of the name: taking from the rich (high potential) and giving to the poor (low potential) thus making them equal, just as the hero of the Sherwood forest did.

### Proprietary implementation

The principle of the Robin Hood method, illustrated above, consists in achieving the equipotentiality of conducting surfaces by iterative non-local charge transfers. It could be used in many areas of numerical computation and modeling. Our proprietary design, numerical procedures and implementation ensure the most efficient exploitation of the principle depending on the specifics of the problem.

The calculation procedure of the Robin Hood method for electrostatics consists of several steps. Initial surface charge distributions at all conducting and dielectric surfaces are chosen. The surface charge distribution at surfaces of homogeneous dielectrics is the charge distribution induced by the polarization of the dielectric. The potential is calculated at all triangles of conducting surfaces and the electric field is calculated at all triangles of the dielectric surfaces. Special procedures of calculation of potential and electric field of a charged triangle are developed and incorporated into the Robin Hood method and implemented in the Robin Hood Solver. The procedure of identifying the triangles which deviate the most from equipotentiality, performing the charge transfers and updating the potential/electric field is iterated until the precision condition is satisfied. The rules of non-local charge transfers differ for insulated conducting objects, conducting objects kept at fixed potential and dielectric objects, but they are all aligned with the Robin Hood principle described in the preceding section. For a system of multiple objects the charge transfers are performed separately within each object.

Metal electrode charge distribution can be of a major interest in electron and ion optics modeling, nanocapacitors and supercapacitors design, palladium gates modeling, nanoparticles in aqueous (water) solution or even microtubule studies.

[See how rough surfaces are modeled using Robin Hood Solver.]

Special care is devoted to post-processing capabilities. Charge distribution sometimes is, but very often is not, a desired quantity user is aiming for, but derivative quantities have to be calculated. In that respect, using the most appropriate numerical integrators, potential and field calculation procedures result in a superior analysis and simulation features compared to other solutions available on the market.

An overview of the method and a detailed description of the implementation and technical properties discussed below can be found in the References.

### Optimal memory requirements

The implementation of the Robin Hood method results in minimal memory requirements: the required memory scales linearly with the total number of surface elements N, thus achieving absolute theoretical limit in that aspect. This important feature of the Robin Hood method allows the Robin Hood Solver to work with very large number of triangles on a single processor.

This fact makes the Robin Hood Solver an ideal tool for high-precision calculations in complex geometries such as spectrometers or accelerometers design, but also a protein folding and docking and capsid analysis.

### Highly efficient convergence

The Robin Hood method is endowed with excellent convergence properties. The convergence is geometric through many orders of magnitude of error reduction. The Robin Hood method reduces error very efficiently at error level of 0.01 or at error level of 10-4 or at any error level whatsoever with great computational complexity regarding the number of calculating elements. A phase of very fast initial error reduction is a particularly appealing property of the Robin Hood method. It indicates that a solution characterized with a potential accurate to 3 decimal places can be obtained in a very short time. This is especially a case using our proprietary customized implementations and procedures we developed over the years, optimally combining advanced analytical and numerical approaches.

### References

Predrag Lazić, Hrvoje Štefančić, Hrvoje Abraham, “The Robin Hood method – A novel numerical method for electrostatic problems based on a non-local charge transfer”, Journal of Computational Physics 213 (2006) 117-140. doi:10.1016/j.jcp.2005.08.006

Predrag Lazić, Hrvoje Štefančić, Hrvoje Abraham, “The Robin Hood method—A new view on differential equations”, Engineering Analysis with Boundary Elements 32 (2008) 76-89.

doi:10.1016/j.enganabound.2007.06.004

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**Keywords:** plasma, nanoparticles, graphene, protein, capsid, quantum dots, colloid solution, palladium, logic gates, microtubule