16,445 research outputs found
Jacobi fields, conjugate points and cut points on timelike geodesics in special spacetimes
Several physical problems such as the `twin paradox' in curved spacetimes
have purely geometrical nature and may be reduced to studying properties of
bundles of timelike geodesics. The paper is a general introduction to
systematic investigations of the geodesic structure of physically relevant
spacetimes. The investigations are focussed on the search of locally and
globally maximal timelike geodesics. The method of dealing with the local
problem is in a sense algorithmic and is based on the geodesic deviation
equation. Yet the search for globally maximal geodesics is non-algorithmic and
cannot be treated analytically by solving a differential equation. Here one
must apply a mixture of methods: spacetime symmetries (we have effectively
employed the spherical symmetry), the use of the comoving coordinates adapted
to the given congruence of timelike geodesics and the conjugate points on these
geodesics. All these methods have been effectively applied in both the local
and global problems in a number of simple and important spacetimes and their
outcomes have already been published in three papers. Our approach shows that
even in Schwarzschild spacetime (as well as in other static spherically
symetric ones) one can find a new unexpected geometrical feature: instead of
one there are three different infinite sets of conjugate points on each stable
circular timelike geodesic curve. Due to problems with solving differential
equations we are dealing solely with radial and circular geodesics.Comment: A revised and expanded version, self-contained and written in an
expository style. 36 pages, 0 figures. A substantially abridged version
appeared in Acta Physica Polonica
Boundary-Conforming Finite Element Methods for Twin-Screw Extruders: Unsteady - Temperature-Dependent - Non-Newtonian Simulations
We present a boundary-conforming space-time finite element method to compute
the flow inside co-rotating, self-wiping twin-screw extruders. The mesh update
is carried out using the newly developed Snapping Reference Mesh Update Method
(SRMUM). It allows to compute time-dependent flow solutions inside twin-screw
extruders equipped with conveying screw elements without any need for
re-meshing and projections of solutions - making it a very efficient method. We
provide cases for Newtonian and non-Newtonian fluids in 2D and 3D, that show
mesh convergence of the solution as well as agreement to experimental results.
Furthermore, a complex, unsteady and temperature-dependent 3D test case with
multiple screw elements illustrates the potential of the method also for
industrial applications
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