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
(software package EMLibHMatrix), 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 (software package EMBoltzmann).
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 (software packages EMLibSparse and
EMLibIter).
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, (software package EMLibMinimize), 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.
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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.
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Using our adaptive grid algorithms with moving boundaries, (software package
EMLibGrid), 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 – software package
EMLibSmooth). In this case the search for the optimum design of your
turbine is carried out automatically with the help of our EMLibMinimize 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, (software package EMLibHMatrix). 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.
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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, (software packages EMBoltzmann,
EMParBoltzmann and EMGPUBoltzmann). Thus, these programs make
it possible to compute:
- atmospheric entry heating of lander surfaces,
- supersonic flow in propulsion compressors,
- flow separation phenomenon,
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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 (software package {\bf EMLibMDD});
- 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 (software package {\bf EMLibGrid});
- 3D high-order finite elements
for the best approximation of boundary elements (software package {\bf EMLibGrid}).
Multiphysics
To find a successful solution of a real problem, it is often necessary to simulate several physical phenomena at a time, which are described by different mathematical models.
For example, you are going to construct an injector for a fuel engine and it is required to optimize its functioning in real time on different operating points, you need to solve the CFD task for fuel propargation together with magnet wave propargation.
While constructing a sea jet engine it is important to consider not only hydrodynamics of liquids, but also gas dynamics equations, moving boundary problems, and gas-liquid equilibrium.
A wide range of physical phenomena successfully simulated by our company, and broad experience in their numerical implementation are at your disposal. We are ready to research your problem and find the accurate solution of it.
Wave Simulation
Direct Wave Simulation Problems:
I. Numerical Simulation of Magnetostatics for Prediction of Electro- and Magnetic Field Around Solenoids
This modeling allows to predict parameters of a magnetic field of any form of
the magnetic coil or superconducting solenoid without gathering the test stand.
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In case you make the challenge with different model parameters, for example,
proportions of dimensions of the solenoid, number of coils, etc., – you
can find optimum values of these parameters concerning the efficiency or other
required characteristics. The test stand assembly is often rather labor-consuming
and financially unprofitable. In some cases such a stand is impossible to be
constructed because the customer can afford to construct only one device with
expected properties. For such cases numerical modeling and optimization of
parameters of the device are carried out before this device is designed.
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We also provide you with self-acting searching algorithm for the best device.s
Thus, it is enough to specify the objective function (for example, the efficiency),
varied parameters (for example, thickness of a wire, number of coils in the
solenoid), and ranges of their values. In this case, the search for the optimum
design of your device will be made automatically for you by our package EMLibMinimize.
We use modern BEM-FEM coupling algorithms for simulation. The boundary element
method (BEM) is used for the simulation of magnetic parts, and the finite element
method (FEM) for isolators. As we need to move magnetic parts, we do not recalculate
the BEM matrix, thus, it reduces numerical errors that occur in FEM during remeshing.
It allows us to solve most effectively and precisely the systems with moving parts,
for example, rotation of the core of an electromagnet, movement of an electromagnetic
valve, and many other similar problems.
II. Numerical Modeling of Radars and Radar Invisibility
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For the successful simulation of stealth properties, or optimization of radar antennas,
one need to discretize large volumes of physical space with a very fine grid, so the
grid step size would be considerably smaller than the wave length. It leads to huge
systems of linear equations. The system matrix in these equations is often so large,
that it does not fit the main memory of the modern computer. Therefore we have developed
for you parallel implementations of these algorithms (software packages with labels
MPP, GPU and Cell), as well as the algorithms with effective disk
memory usage (software packages with labels Out-of-Core).
Here we are using the last achievements in the numerical linear algebra developed by
our company. These methods allow to solve sparse unstructured linear equation systems
(software package EMLibSparse), and structured linear equation systems
(software package EMLibMDD). So, systems of linear equations with several
millions of unknowns are solved in a few seconds on a notebook; and systems of linear
equations with several billions of unknowns are solved within a reasonable time in
a small Linux cluster.
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The software package LRA_CDENSE of the predecessor company Elegant Mathematics Inc
was used by Lockheed Martin Concern for radar invisibility of stealth aircrafts. The
relative wave number in these problems exceeded 500. Now you can make this simulation
using our new improved packages EMLibIter, EMLibSparse, EMLibHMatrix
on workstations, and with the EMParLibIter, EMParLibSparse packages on
massively-parallel computers.
The experience obtained over the last 15 years in software development for the solution
of ill-conditioned linear systems allows us to take part in the radar antennas
development for the most aerospace applications.
3D Inverse Maxwell Solutions
I. Ultrasound Non-Destructive Diagnostics, Tomography, Acoustic Geological Exploration and Upstreaming
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We will help you to find a three-dimensional structure of the mineral distribution
if only a few emitters and targets of sound waves on the surface are available, and
the studied object has considerably larger dimensions than the wave length.
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The current problems can be divided into two basic classes:
- The studied object is between sources and detectors, and wave reflection from heterogeneity of the object is not considered. Common applications here are those of medical tomography and, partially, ultrasonic non-destructive diagnostics of materials.
- Sources and receivers are situated in such a way, that waves from sources are reflected on heterogeneity of the studied object and registered by receivers. Common applications are problems of seismic prospecting of oil- and gas-fields, and some applications of ultrasonic not-destructive diagnostics of materials.
As this method does not describe wave interference, its use is limited by the condition, that the studied object should have much greater sizes than the characteristic wave length of radiation (software package {\bf EMInverse}).
The basic complexity of the solution lies in the correct discretization and
solution of huge ill-conditioned linear system. The discretization is carried
out by finite elements of first order, so the size of each element corresponds
approximately to the average size of heterogeneity in the position where the
corresponding finite element is situated. In case we have no a priory
information about heterogeneity distribution, we can start calculations with
a regular finite element grid, and later, try to improve the grid according
to computed properties using the Voronoi-Delaunay approach (software package
EMLibGrid). It allows to improve the accuracy of computation, to reduce
the total amount of unknown parameters, to save computational time and reduce
memory requirements (software packages EMLibIter and EMLibSparse).
Our company has 15 years experience of development of iterative methods for the solution of linear systems of equations that allows us to find the most suitable and steady method of the solution of linear systems and, if necessary, to apply a correct regularization to a singular matrix (software package EMLibIter).
II. Ground Penetrating Radars and Upstreaming
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We will help you to find the structure of a three-dimensional object using
interference, nonstationary time distribution of sound and electromagnetic waves.
Solving this mathematical problem, you can predict a distribution of petroliferous
stratum on the basis of seismic prospecting, and simultaneously conduct correction
of drilling. Our algorithms were successfully applied for numerical modeling of
distribution of petroliferous stratum, and for correct direction of oil drilling.
Sources and receivers of electromagnetic signals were placed on the drill and allowed
"to see" the end of a petroliferous stratum already 5-7 meters before, without
drilling a dead rock.
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For the solution of this problem we use a new method with two different discretization
grids. The first grid is used for distribution of electromagnetic waves, it is
structured and fine, to make better approximation for physical phenomena of the
Maxwell equations. The second grid is used for dielectric permeability approximation,
and this grid is based on the 3D adaptive Voronoi-Delaunay tessellation. This dual
grid approach allows describing magnetic and electric fields with minimum complexity.
At the same time, adaptive grid, on which dielectric permeability is calculated, does
not overload the system of equations with excessive unknown variables.
The software packages of Elegant Mathematics run on supercomputers in the High-Performance
Supercomputing Center in Maui of Hawaii (USA) on behalf of Mobil Oil Corporation for the
solution of ill-conditioned sparse problems with billion unknowns (software package
A_SPARSE_T3D – the predecessor of our EMParLibIter
and EMParLibSparse software packages).
Nowadays, our algorithms, (software package EMMaxwell}), solve 3D inverse ground
penetrating radar problems for Timer Ltd,
the leading supplier of georadar technology in Russia.
Signal Processing
Elegant Mathematics Ltd has several successful approaches for image and video
stream compression based the modern Multilinear Adaptive Cross Approximation
method. It can increase the compression ratio for video streams, and it is very
effective for noise reduction. Using the new numerical method together with the
ultra-broadband lenses, we can construct a 3D surface of a pictured object and
give the information about materials over its surface.
If there are some pictures of the object, which were made from different points,
a new picture of this object at an arbitrary point can be attained, provided that
this point is not far from an effective central distance. This method allows
substantial simplification of air and space admission: while flying past the
object, it is sufficient to make a photo of the desired object only from a few
points. Then, using our algorithm based on the EMLibMDD software package,
you can restore the picture of the object from any other desired point, and even
its 3D surface. These algorithms can be used also for recognition of flying
objects by fragmentary picturing from the ground and comparing them with the
picture in the data base. In this case, we do not need to have all possible
pictures of the flying object from all possible points in the data base.
We have a wide experience of work with various computer architectures from
Embedded Systems up to TOP 1 supercomputers. If necessary, we are ready to
transfer our algorithms on your built-in microprocessors, or to offer you a
complete solution using the microprocessors of our partners. Successful results
in this area are shown with our software package EMGPULibMDD, which
was ported to NVIDIA 2xx and Tesla GPU processors.
Engine Optimization
Modeling of the fuel engine and optimization of its functioning in real time
with different operating points. Due to the data evolutions per minute, quantity,
temperature and pressure of inlet air and fuel, detonation time and properties
of mixtures in the exhaust, one can build a mathematical model that makes
regulation of the fuel force engine possible. It allows the optimization of
several parameters of the engine, for example, to tune for fuel consumption
or increasing of engine's power.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibIter
The software library is developed for iterative solutions of linear systems and
eigenvalue problems. This package contains CG, GMRES, GMRESF, NGMRES, BiCGStab
algorithms for the linear system solutions, and Lanczos, Arnoldy and Davidson
methods for eigenvalue problems. All these implementations allow you to apply
any external preconditioner and Newton acceleration technique.
Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
Optimized for Multi-Core, Vector-Pipeline, Out-of-Core, GPU and MPP;
complex numbers support.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibSparse
The software library is developed for solutions of large sparse linear systems and
eigenvalue problems. This package contains all iterative methods of the EMLibIter
package. The wide range of preconditioners allows to considerably reduce computational
costs. The package contains several incomplete LU and Cholesky preconditioners with
optimal thresholds for neglecting preconditioner entries. To reduce computational
expenses such algorithms can be implemented as of maximum degree, approximate maximum
degree, nested dissection, minimal Schur complement norm and optimal permutation.
Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
Optimized for Multi-Core, Vector-Pipeline, Out-of-Core
and partially for MPP, complex numbers support.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibHMatrix
The software library is developed for solutions of dense linear systems and eigenvalue
problems. This package contains all iterative methods of EMLibIter package together
with the complete set of H-Matrix arithmetic. It allows to create, multiply, add up,
solve systems with hierarchical matrixes, using direct and iterative methods with
hierarchical preconditioners.
Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
Optimized for Multi-Core and Vector-Pipeline, complex numbers support.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibGrid
The software library is developed for construction and recalculation of
the adaptive Voronoi-Delaunay grids in three-dimensional space, and for
generation of piece-wise constant and linear finite elements on these grids.
Optimized for Multi-Core and Vector-Pipeline.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibSmooth
The software library is developed for construction of spline approximations
of any multivariate partially defined functions, or data sets. The package
also includes smoothing and noise filtration algorithms.
Optimized for Multi-Core and Vector-Pipeline, complex numbers support.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibMinimize
The software library is developed for unconditional minimization of piecewise
smooth function, or for the solution of nonlinear system of equations. This
software package automatically tunes to minimization problem choosing between
conjugated directions, Broyden, BFGS, and Quasi-Newton methods. The gradient
can be very efficiently computed using a very powerful method of Baur-Strassen.
Optimized for Multi-Core and Vector-Pipeline.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMLibIter
EMLibSparse
EMLibHMatrix
EMLibGrid
EMLibSmooth
EMLibMinimize
EMLibMDD
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EMLibMDD
The software library is developed for multilinear decompositions of all sorts:
Parallel Factor (PARAFAC), Alternate Least Squares (ALS), Parallel Decomposition (PARDEC),
Multilinear Adaptive Cross Approximation (MACA). Using this library you can treat both
fully populated and partially defined data sets.
Optimized for Multi-Core, Vector-Pipeline, partially for
Out-of-Core, MPP and GPU, complex numbers support.
Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.
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EMBoltzmann
EMPartBoltzmann
EMBoltzmannModules
EMHydroBoltzmann
EMMaxwell
EMInverse
EMLame
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EMBoltzmann, EMParBoltzmann, EMGPUBoltzmann
Packages for CFD calculations of sub-, super- and hypersonic aerodynamics and
aeroacoustics, turbine compressors and combustion chambers.
The main difference of these packages from EMPartBoltzmann series of
packages is based on the grid discretised deterministic approach. This method
is heavier for computations, however it guaranties the quadratic convergence of
physical variables on the amount of discretized velocity space finite elements.
The software package is designed for the solution of computational fluid dynamics
problems with a special emphasis on the solution of super- and hypersonic challenges.
Our recent scientific results allow to solve Boltzmann equations with the complexity
comparable to the Navier-Stokes algorithms. It vanishes a lot of unstable empirical
models and gives us a chance to simulate ram- and scramjets, atmospheric entry
heating of lander surfaces, supersonic turbulence, boundary layer problems,
flow-separation phenomenon, and many others. Velocity subspace is discretized
with an adaptive tensor grid and piece-wise linear finite elements, and physical
subspace is discretized with adaptive Voronoi-Delaunay grids.
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EMBoltzmann
EMPartBoltzmann
EMBoltzmannModules
EMHydroBoltzmann
EMMaxwell
EMInverse
EMLame
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EMPartBoltzmann
Packages for CFD calculations of sub-, super- and hypersonic aerodynamics and
aeroacoustics, turbine compressors and combustion chambers.
These software packages solve the Boltzmann equation with the mesh free particle
method for arbitrary gas or gas mixture with possible chemical reactions.
The key idea of these packages is based on the mesh free approach, i.e. you need
to provide only a 2D grid with boundary conditions, and this grid can evolve
according to your conditions during the computational time. It gives a possibility
to make simple and fast estimations of your properties for a computing task,
positioning of shock waves and their forms, separation phenomena, and others.
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EMBoltzmann
EMPartBoltzmann
EMBoltzmannModules
EMHydroBoltzmann
EMMaxwell
EMInverse
EMLame
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EMHydroBoltzmann
This programm package solves the equation similar to the Boltzmann approach and
allows us to have a multivelocity model for incompressible fluid dynamics. It is
applied to power plant turbines and sea propellers.
Together with the gas dynamical approach it can be used for complicated multiphase
problems for optimizing sea jet turbines, simulating a cavitation, and optimizing
multiphase high-speed flow for water missiles and submarines.
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EMBoltzmann
EMPartBoltzmann
EMBoltzmannModules
EMHydroBoltzmann
EMMaxwell
EMInverse
EMLame
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EMMaxwell
This software package is designed for direct 3D Maxwell problem solutions.
It allows to simulate eddy current and other electromagnetic effects.
Several modern approaches are implemented in this package to solve
frequencies from Hz up to GHz range. Is allows us to simulate properties
of low frequency electromotors and high frequency radar cells.
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EMBoltzmann
EMPartBoltzmann
EMBoltzmannModules
EMHydroBoltzmann
EMMaxwell
EMInverse
EMLame
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EMInverseTomo, EMInverseMaxvell
These software packages are designed for inverse the 3D Helmholz and Maxwell
problem solutions. The software package EMInverseTomo solves simplification
of the Maxwell problem to the case of no diffraction useful for tomography and
similar tasks. The software package EMInverseMaxvell solves complete
3D vector problems.
These packages are designed for the solution of inverse aeroacoustics of geo-
and oil-radar improvements, antenna design and fully featured stealth optimization.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Iterative Linear System Solvers
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An average performance of CG, BiCGStab, Lanczos with real arithmetic equals to 7 GFlop/s
on a single NVIDIA 260. This performance is achieved from ca 100,000 unknowns. A complex
version of these iterative methods increases twice, 14 GFlop/s. It is almost 50 times
faster than on the Quad-Core Xeon 2.6 GHz processor with 666 FSB, which can deliver only
100 MFlop/s.
The main reason of our achievements lies in the comprehensive usage of fast GPU memory.
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Block versions of CG [1,2], BiCGStab, GMRES/FGMRES/NGMRES [3], Arnoldy and Davidson
algorithms provide even faster performance. In particular zGMRES with 10 simultaneous
right hand sides achieves 70 GFlop/s on NVIDIA 260; so it is almost the peak performance
of double-precision arithmetic. The similar algorithm on the Quad Core Xeon 2.6 GHz
processor with 666 FSB produces only 3 GFlop/s.
Our Kronecker Preconditioner and Kronecker sparse matrix multiplication algorithm [4,5]
show the incredible 250 GFlop/s on one NVIDIA GPU 260!
Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
- Nikishin A., Yeremin A. Variable block CG algorithms for solving large
sparse symmetric positive definite linear systems on parallel computers. I. General
iterative scheme. SIAM J. Matrix Anal. Appl. 16(4), 1995, 1135-1153.
- Nikishin A., Yeremin A. An automatic scheme for regulating the block size
in the block conjugate gradient method for solving linear systems.
Zap. Nauchn. Sem. POMI, 2000, J. Math. Sci., 114(6), 2003, 1844-1853.
- Kharchenko S., Yeremin A. Multiplicative correction of a matrix on a
sequence of subspaces. I. Basic algorithms and theory for the general non symmetric
sign-indefinite case. Zap. Nauchn. Sem. POMI, 2002, J. Math. Sci., 121(4), 2004, 2546--2575.
- Ibraghimov I. Application of the three-way decomposition for matrix compression.
Numer. Lin. Alg. Appl. 2002; 9:551--565.
- Ibraghimov I., Sublinear Complexity of Krylov Subspace Method for the
Kronecker Product Matrices. In press in Numer. Lin. Alg. Appl, 2009.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Dense Compressed/Hierarchical Linear System Solvers
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The subject of discussion is preliminary benchmarks with the NVIDIA 260 GTX
hardware for the solution of large dense linear systems with hierarchical structures.
A linear system with 81,920 unknowns was generated and solved in GPU with reasonable
speedup in regards to Quad Core Xeon 2.66 HGz.
Matrix generation shows 15 times speedup and delivers the peak performance of 60 GFlop/s.
The iterative solver and compressed matrix multiplication algorithm produce up to
50 times speedup with the peak performance of 6 GFlop/s, equal to 45 GB/s of memory bandwidth.
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Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Particle Simulation Boltzmann Solver with GPU Kernel
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This software package solves the Boltzmann equation with a particle method.
It allows making mixtures of particles and particle transformation, i.e.
chemical reactions. The collision part is almost 30 times faster than on
the modern Quad Core Xeon CPUs. The free flow part is even faster –
up to 120 times.
The "simple" problem with 15,000,000 particles can be solved within several
minutes without out-of-core algorithms.
The true out-of-core GPU memory to the CPU memory allows to solve huge problems
that fits only the main CPU memory. So it gives a possibility to solve real
simulations with up to 1,000,000,000 particles during one computation week on
one GPU together with the large main CPU memory of 64 Gb.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Deterministic Boltzmann Solver with GPU Acceleration
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The main core part of the EMBoltzmann software package was tuned for NVIDIA GPU
processors. This part is always about 90-95% of the total computation time of the
EMBoltzmann package. The core is now almost 30 times faster than on CPU, so
the total computation time was 10 times reduced.
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- I. Ibragimov, S. Rjasanow. Numerical solution of the Boltzmann equation on the uniform grid.
Computing, 69(2):163--186, 2002.
- I. Ibragimov, S. Rjasanow. Three Way Decomposition for the Boltzmann Equation.
J. Comp. Math., 27(2-3):184--195, 2009.
- I. Ibragimov. Fast numerical solution of Boltzmann equation.
ZAMM, 7(1):1110101-1110102, 2007.
- E. Ibragimova. Solution of industrial CFD problems with structured Boltzmann approach.
ZAMM, 7(1):1110105-1110106, 2007.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Fast GPU 3D Inverse Maxwell Solver for Georadars
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The recent results with the solution of 3D inverse Maxwell equation and their application
to georadar signal processing move the solver from supercomputers to modern PCs. However,
the solution time can take hours on a modern workstation. To speed up this computation
the core part was tuned to the NVIDIA 2xx series of GPU. That made it possible to solve
the most computation 50 times faster; so the total solution time on NVIDIA 280 proved to
be 40 times shorter than on modern workstations.
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EMGPUIter
EMGPUHMatrix
EMGPUPartBoltzmann
EMGPUBoltzmann
EMGPUGeo
EMGPUNMR
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Fast Multilinear NMR Deconvolution
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The multilinear decomposition has been recently approved as a new robust
method for data processing of multidimensional Nuclear Magnetic Resonance.
These results were published in "Nature". The application problem has so
huge sizes, that modern workstations need several days to find a solution.
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A new high performance implementation of this algorithm for the GPU NVIDIA 2xx
stream processors is presented in what follows.
The algorithm is based on sparse implementation of parallel factor decomposition
algorithm (PARAFAC) that performs alternate sparsely defined least squares minimization.
The nuclear magnetic resonance (NMR) data are usually huge and have a large
amount of data entries. To handle them one needs to solve several (often hundreds)
almost nonoverlapping regions with a considerably small rank, and then to make a
final tune of the total large rank system.
This package allows to compute large regions independently using all power of NVIDIA GPU.
This part often shows significant slow down in the CPU architecture since it requires
solving a lot of small problems where CPU cannot show all power.
Since these parts are multithreaded over NVIDIA multiprocessors, it gives us high
performance improvement. The only bottleneck in this part is load balancing,
however, it is not very important with usage of large data sets. Hence, with
data sets published in the articles below, we reached 50-60 times speedup.
The final solution of the joined multidimensional problem with a large rank
(often about 1,000 components) was again efficiently implemented, and showed
us improvement 30-40 times as much as before compared with Quad Core Xeon workstations.
- Jaravine V., Ibraghimov I., Orekhov V. Removal of a Time Barrier for High-Resolution
Multidimensional NMR Spectroscopy. Nature Methods 3(8):605--607, 2006.
- Jaravine V., Zhuravleva A., Permi P., Ibraghimov I., Orekhov V. Hyperdimensional NMR
Spectroscopy with Nonlinear Sampling. JACS 130(12):3927-3936, 2008.
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Consulting
Our company develops software packages for the solution of industrial problems, thereto we often need to use a huge computer power: massively-parallel, vector-pipeline processors, and out-of-core solutions.
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. In case you are deciding what computer hardware to buy, we can help you to make this choice and optimize the price/performance ratio for these particular needs.
When our programs are used, we shall always accompany you and solve together your problems. You will never encounter problems with our software as we shall provide training and advise you on using our products.
We keep abreast of the quickly developing computer industry --- many our algorithms are ported on such special hardware platforms like Tesla and 2xx GPU of NVIDIA and IBM Cell. Based on broad experience on all possible massively-parallel and vector-pipelined platforms that have existed in the world since the end of the last century, we are ready to develop and optimize our algorithms for your actual computer resources.
Our algorithms allow solving problems by means of massively-parallel computers. If you have a computer with 1,000 processors, we solve your problem practically 1,000 times faster.
If your problem is so complex, that your computing facilities do not manage to solve it in a reasonable time, we will try to solve it on our facilities or to place an order for computational time in the leading supercomputer centers.
In case usage of the main memory is the most critical part of modeling, we offer our out-of-core solutions, which almost do not slow down our algorithms, allowing you to expand your main memory to the size of a hard disk.
Elegant Mathematics Ltd hopes for a successful co-operation with you!
Into the High-Speed Century – with High-Performance Algorithms!
In our hi-tech century many industrial branches are confronted by difficulties
with production, which requires precisely defined properties. It is often
necessary to simulate these characteristics to improve the quality and reduce
production costs. Most often it occurs in industries with intensive power- and
labor-consumption where the development overhead makes up the main part of the
total production costs. Sometimes it is impossible to predict the properties
of a product before it leaves the assembly line. In this case, the only
solution is to make numerical simulations using algorithms, which are precisely
developed for specific industrial problems.
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Development of the high-performance software is the main duty of our team. The core
of our software products is based on our high-performance matrix algorithms that are
unique with respect to their accuracy and efficiency.
Based on the wide experience of work with industrial customers, we apply the newest
scientific methods and the latest technical achievements for the solution of industrial
tasks. We elaborate specific software products individualized for each of our customers,
and we are always ready to work with highly complicated requirements. We shall satisfy
needs of the most particular customer, and base our cooperation on mutual understanding
and decency.
Elena Ibragimova, Managing Director
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Elegant Mathematics was found in 1992 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 taks.
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.
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Here you can download booklets about our company
24 pages, in English, 4Mb
24 Blatt, auf Deutsch, 4Mb
24 pages, en Francaise, 4Mb
24 pages, in Russian, 4Mb
To make our pricing easy to understand, please, check your requirements online.
Software Prices
Prices include all updates, new releases and support during validity of licences.
Consulting/Software development
50,00 Euro per Hour + VAT
Prices don't include travel and accommodation expencies in case if your company is more than 300 km away from our main office.
Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.
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