Effective Grid Algorithm for Calculating Deformations of Load Carrying Beams
Török János Dr. <>
BME, IK
Construction industry uses pre-stressed concrete beams for assembling
large edifices. These big beams can deform during transportation
and naturally at their destination under different loads (rest of the
building, wind, heat, etc.). The precise calculation of these
deformations requires the solution of highly non-linear set of
equations the number of which increases with the degrees of freedom
and the complexity of the setup. What is common in all these problems
that they are boundary value problems: we know the stresses at some
points (generally at the end of the beam) and want to calculate the
shape of the beam.
The high number of non-linear equations make the problem really
difficult to handle. It is not trivial that intuition delivers all
possible deformation modes. Therefore we must rely on algorithms that
scan the parameter space for such solutions. Different versions
of this scanning algorithm (parallel, solution following) were
developed at our institute. The common of all these solutions were
that the space was divided into small hypercubes which in turn were
divided into simplexes upon which the solutions were searched for. As
the most CPU expensive point is the calculation of the set of
equations in the corners of the hypercubes all the above features
contributed a lot to the effectiveness of the algorithm which resulted
in a number of publications.
The above solution has its limits. As the number of equations gets
high enough the scanning of the whole space gets impossible. Therefore
we chose a completely new approach: an artificial potential is
generated from the equations with minima at the solutions. We try to
find a minimum and then follow the solution by solving the problems
on the simplexes of a small hypercube. Since the solution space is
one-dimensional we can choose very small hypercubes thus be very
accurate and find no fake solutions. The algorithm uses a
sophisticated bit encoding for the visited cubes which can be tuned
between memory or CPU efficiency depending on whether the code is run
an a GRID or a cluster. The parallelization of the original code was
master-slave type. This new realization can be run in parallel without
interprocess communication with small overlap between two different
runs. This will allow the algorithm to run in GRID environment easier
accessible than clusters.