Elegant Mathematics was found in 1991 in the USA, (Washington State), for development and manufacture of linear systems and eigenvalue solvers for vector-pipelined and massively-parallel algorithms, that was requested by the industry of the nineties in the last century, for the solution of mathematical, physical, chemical, aerodynamic and other tasks.

Our experts worked on the newest computer facilities of that time: 32 processor vector-pipelined system Cray C90, massively-parallel computers Cray T3D-T3E with 1,024 processors installed in NASA, Cray Research Inc, and the University of Pittsburgh, on many massively-parallel clusters with IRIX, DEC, RS6000, HP, Convex processors in the Hawaiian High-Performance Center, and on a huge amount of Linux clusters worldwide.

In the beginning of the 21st century our company underwent significant changes. There came a new generation of employees, we adjusted to new industrial problems and entered the European market.

In 2006 Elegant Mathematics moved its activity to Germany where its head office is situated at the present. The staff of our company are highly qualified specialists, who received Master's and PhD degrees on graduating from the worldwide recognized universities. Our mathematicians, physicists, chemists and programmers search out new technologies and directions in the industry. The results of their research and development activities are regularly published in scientific journals, such as Nature, JACS, LAA, etc. Hence, you are assured to find the high standard of scientific knowledge at your disposal.

All our products result from the search of the optimal task solution for the customers and application of our know-how in software development. We work out the detailed task description for each customer, and then proceed to solve the challenges in the most effective and fast way. We help the customer to find the solution that best fits the available computer facilities, and to optimize the price/performance ratio for these particular needs.

RADIANT (RADar Innovation ANtenna Technology) is our new ground penetrating radar with synthetic aperture antenna that can achieve very deep (up to 1 km) geologic exploration. Recent our publications about RADIANT technology:

Transmitting antenna:

  • up to 24 J energy of each pulse,
  • up to 800 Mega Watt peak performance,
  • short period of pulse: 3-300 ns depending on application,
  • 5-1500 Watt sustained power (depending on used area and local country regulations),

Receiving antenna:

  • digitize signals with up to 2.3 GHz,
  • up to 48 numerical phased array channels,
  • very bright voltage range from 3~mkV to 1 V with 14-16 bits,
  • CE confirm,
  • achieve 200 dB total dynamic range,
  • work with many emitting antennas to save the acquisition time,
  • can be situated up to 1 km from transmitters,
  • equipped with high performance computing FPGA module and GPU accelerator that achieves up to 2 TFlop/s equivalent performance (like a small supercomputer with 100-200 CPU cores),
  • compute and display 3D inverse wave propagation results.
More details about our RADIANT Technology is available on www.sar-saar.de


Our company is continuously researching Nuclear Magnetic Resonance (NMR) physics, with applications for spectrometry and tomography. Our achievements in this fields are published in highly cited scientific journals.

We have incorporated NMR tomography into our
ground penetrating radar RADIANT so that we can distinguish different geological layers and predict minerals' element composition during exploration.

We successfully constructed a pocked size NMR spectrometer based on our high tech mathematics and are currently participating in many NMR related projects. More details on our NMR applications could be found on our satellite projects:

Do the same with EMBoltzmann solver and complete this 3D simulation in one minute on modern PC!

Our achievements in CFD allow to simulate sub-, super- and hypersonic aerodynamics and aeroacoustics solving deterministic Boltzmann equation with the complexity of modern Navier-Stocks solver.

Fluid Dynamics

Elegant Mathematics Ltd researches and develops the software for numerical simulation of the most physical effects connected with transport of matter. The spectrum of our mathematical solutions ranges from construction of solid state models, which are simulated with the Lame equation, covering the majority of convection-diffusion models, up to the solution of under- and supersonic computational fluid dynamics models, which we successfully solve using the Boltzmann equation approach. All our models are based on the advanced scientific development in the field of finite elements and adaptive grids.

Elegant Solvers are exceptionally robust and are able to solve highly ill-conditioned problems, even those, which can not be solved by any other reasonable cost method.

Our new algorithm for the deterministic solution of the Boltzmann equation is based on the multilinear decomposition of velocity space. It enables us to compute the results with the same computational efforts like with the Navier-Stokes approach. Moreover, the simulation of the Boltzmann equation enhances considerably the accuracy of received results.

Solving problems of numerical subsonic gas dynamics for the given shape of a plane, stream speed and an attack angle, you can compute turbulence and all forces acting on the plane. Based on these results, you learn a behavior of a plane on various sites of the flight. If you provide a set of possible geometries of a plane and run the simulations for them, it is possible to foretell flight characteristics for each of them and to find the most appropriate one.

Moreover, if optimum values of force distribution, acting on a plane, are given, you can optimize a form of the wing, without constructing a large set of different wings, and carrying out tests with them.

Our algorithms are used at all stages of development of marine power plant turbines and helicopter propellers.

It is often difficult – and sometimes impossible – to foresee the right form of turbine blades. At the same time, designing and checking on the efficiency of different shapes of blades turn out to be unprofitable, since this method demands great expense of time and resources.

Based on the blade's geometry, temperature, liquid pressure, fluid stream velocity, and other provided physical characteristics, our algorithms will compute both the efficiency of accumulation, in case of power plant turbines, and the efficiency of feedback, in case of marine turbines. Thus, for each possible form of the blade you can learn its characteristics without the actual construction of expensive test experiments to find the optimum parameters.
Using our adaptive grid algorithms with moving boundaries, you will obtain not only a full picture of fluid motion, but also of all forces and energy transfer that act to the boundaries. We also suggest to automate the process of search for the optimal shape of blades. Thus, it is enough to specify the criterion function (for example, the efficiency of a turbine), possible restrictions (for example, a condition for a subsonic fluid flow), and varied parameters (for example, some points of spline interpolation for all surface of a turbine blade). In this case the search for the optimum design of your turbine is carried out automatically with the help of our minimizer package.

Suppose that you have the best shape of a turbine blade, it is powerful and optimal in respect to computational fluid dynamics (CFD). Then you need to have it forged with the maximal durability. Here we will help you again, solving Lame equation for non-elastic deformation. It will allow you to choose the most effective strategy of forging the blade, reducing probability of occurrence of shifts and material defects, and to construct the turbine with the maximal durability and reliability.

Our new project deals with numerical simulations of hypersonic flows with deterministic and stochastic models for the Boltzmann equation. We solve the 3-dimensional, real-time Boltzmann equation with an adaptive grid and billion particles for the stochastic approach, and with million values of average velocities on each physical space cell for deterministic approaches. It gives a chance to carefully predict hypersonic gas flows at the speed close to space flights and strongly turbulent flows. Thus, these programs make it possible to compute:
  • atmospheric entry heating of lander surfaces,
  • supersonic flow in propulsion compressors,
  • flow separation phenomenon,
and many other important physical phenomena. The mathematical model, included in the Boltzmann equations, allows simulating most of shock waves and does not distort results even in high Mach numbers.
We suggest to optimize your Ram-, and Scramjets by means of simulating combustion process based on the three-dimensional Boltzmann equation.

For the accurate simulation we offer you a wide range of mesh generators and proper simulation algorithms, as follows:
  • tensor uniform grids with Toplitz matrices and fast Fourier transformation,
  • tensor nonuniform grids with Kronecker matrices,
for velocity space simulations;
  • adaptive grids with tetrahedral finite elements of first and higher orders,
  • mesh-free and dual mesh approaches based on the Delaunay tessellation,
for the discretization of physical space;
  • 3D high-order finite elements
for the best approximation of boundary elements.

Our Contacts

Elegant Mathematics Ltd,
Hanauer Muehle 2,
66564, Ottweiler-Fuerth,
Tel: +49 6858 79 79 858
Email: info at elegant - math dot de

Elegant Mathematics LLC,
82834, WY, USA
Tel: +1(631)490-0521
Email: info at elegant - mathematics dot com

Legal registration numbers:
Cheyenne 746213
Cardiff 05975337
HRB 16570
Active and registered since 23 Oct 2006
German tax payer's account number 030/146/00565
EU VAT account number DE 257663693
Customs number (EORI) DE 1753525

Our technical support and information office is always available for you. You can contact us at any time from any point of the world by our contact phones, and receive competitive guidance and consulting about our products and services in English and German languages.

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