MA-1's High-Performance Computing Clusters

MA-1's High-Performance Computing Clusters

Index

Description

Eixam and Maya are the high-performance parallel computing clusters of the Departament de Matemàtica Aplicada 1 at UPC.

Eixam (the Catalan word for "swarm") is composed of 26 Dell PowerEdge SC1425 servers, each with two Intel Xeon 3.2 GHz processors and 2 GB of RAM, for a total of 52 CPU.

Maya (introduced November 2007) is composed of 11 Acer Altos R520 servers, each with two quad-core Intel Xeon processors at 2.0 GHz and 8 GB of RAM, for a total of 88 CPU.

The servers are interconnected through a Gigabit Ethernet dedicated network.

eixam.jpg (click to enlarge)

A picture of Eixam : 26 "pizzas" enclosed in a single rack. On the right there is the console. Mounted on the ceiling, two large A/C units.

maya.jpg (click to enlarge)

A close-up of Maya. It is much smaller in size than Eixam (it only occupies 11 U's), but it is more powerful. Notice two ethernet cables per server to achieve higher communication rates.

Eixam and Maya are used for scientific computing by the EGSA research group from our Department. In particular, the following research projects are involved:

An essential point is the detailed study of invariant (stable and unstable) manifolds of various invariant objects (such as quasi-periodic orbits or Normally Hyperbolic Invariant Manifolds) and their relative position, as well as their dynamical implications.
This is accomplished by means of both symbolic and numerical computations. All the software utilized for these applications is developed, analyzed and implemented by members of our research group.
One of our main goals is to implement parallel numerical methods for our special-purpose computations.
The whole group has also a large experience in applying these techniques to real life problems, like transfer of orbits and station keeping maneuvers in the analysis of several missions to ESA and NASA.
  • Partial Differential Equations
    • Nonlinear PDEs, fluid dynamics.
  • Geometry and Topology
    • Algebraic geometry computations.
Moreover, we host a number of "invited" scientific research computations by people from outside our Department. This includes research projects in areas as diverse as
  • Astrodynamics
  • Neuroscience
  • Modeling and visualization in virtual reality
  • Soft computing
  • Numerical modeling and optimization
  • Parallel algorithms
  • Nonlinear fluid dynamics
  • Statistics and applications
Researchers from outside our Department may submit a proposal to request an allocation of cycles on our clusters. To submit a proposal to request an allocation, please contact eixam.ma1 arrobillaupc.edu
Finally, our clusters are catering not only basic research projects, but also strategic research projects from INTAS, NASA-JPL, ICREA and PROFIT.

Sample research projects

Libration point trajectories

These are natural trajectories of a spacecraft in a gravitational vector-field governed mainly by two primaries (e.g. Sun and Earth, Earth and Moon, Sun and Jupier,...) where the relative position of the spacecraft with respect to the primaries is almost constant.

These trajectories are located around five fixed points with respect to the primaries. Three of them are placed in the line joining the primaries, and the remaining two form an equilateral triangle with them. In a rotating reference frame which keeps the primaries in fixed positions, libration points are seen as places where the gravitational forces of the primaries compensates the centrifugal force of the non-inertial frame.

Due to its privileged location, libration point trajectories, among other applications, are suitable places to put satellites devoted to solar science, space telescopes and possible future gateway stations to the Solar System. See http://sohowww.estec.esa.nl/ for a current libration point mission.

In order to illustrate libration point orbits, consider the following figure.

frame0024.gif

The white sphere on the left represents the Moon, and the blue sphere on the right represents the Earth. The moving spheres (click here for an animation of the previous figure) stand for a cluster of artificial satellites, flying in formation. Their trajectories describe a geometrical object known as "invariant torus" in Dynamical Systems theory. The trajectories of these satellites are examples of quasi-periodic motion with two frequencies: the one corresponding to the rotation of the cluster, and the other corresponding to the translation of the cluster along the invariant torus.

If we plot a dot every time a satellite crosses the horizontal plane (which is the plane of rotation of the Moon around the Earth) we get a closed curve, which is the intersection of the invariant torus with the horizontal plane. This curve is surrounded by the green curve, which is a closed trajectory, known as Lyapunov planar periodic orbit. For a given energy level, the traces on the horizontal plane of all the bounded trajectories of the vicinity of the libration point L1 (which is the point where the gravitational forces of the Earth and the Moon compensate) are inside the Lyapunov periodic orbit of this level of energy.

These trajectories cannot be obtained by direct numerical integration because they are highly unstable. One approach to compute them is to use a technique known as "reduction to the central manifold", which consists in computing, using symbolic manipulation, the expansions of a change of coordinates that allows to remove the unstable directions. The resulting system of differential equations can be integrated numerically for long time spans. In the third demo, we do this for a representative set of all the trajectories of a fixed energy level, and plot their intersections with the horizontal plane in the figure below. The trajectories are integrated in parallel.

barrsem.gif

The drawback of the above approach is limited to the regions where the expansions computed are accurate. Out of this regions, one has to use a different approach, that is more expensive in terms of computing time. It consists in computing these trajectories by direct numerical methods (in opposition to the above approach, which is semi-analytical). In the fourth demo, we start the parallel computation of several of these trajectories. We do not actually compute the trajectories, but the invariant tori in which they are contained.

barrnum.gif

In the above figure, each '+' symbol represents an invariant torus, similar to the one represented in the animation above. The horizontal coordinate is the energy level, and the vertical coordinate is the quotient of the frequencies of the trajectories on each invariant torus (this is known as "rotation number").

Conclusions

  • Eixam and Maya are parallel computers, specially well-suited for massive computations.
  • They are fully scalable, updatable and recyclable.
  • The software for parallel computations is developed by members of the group.
  • There are several projects, involving different kind of people (PQS, Ph.D., post-Doc), for development of parallel algorithms for Eixam and Maya, covering several fields like the computation of normal forms, invariant manifolds, finite element methods and transport in fluid mechanics.

Acknowledgements

Eixam and Maya have been funded with several PEIR grants from the Generalitat de Catalunya. The clusters were only truly implemented after our Engineering school ETSEIB built a room for them and installed two industrial-grade air conditioning units.

Moreover, a research assistant position (PQS) was awarded to our research group by the CUR (Comissionat d'Universitats i Recerca) and co-funded by the UPC and our Department. This position is currently held by Pau Roldán (eixam.ma1 arrobillaupc.edu), who looks after the clusters.

Special thanks go to J.M. Mondelo for putting together the original Eixam.

Last but not least, Eixam is a 'son' of Hidra, another cluster of the UB-UPC Dynamical Systems Group, located at the University of Barcelona. We are indebted to Joaquim Font, Àngel Jorba, Carles Simó and Jaume Timoneda for their help and support.





RMC 03/08