VECTOR FINITE DIFFERENCE METHOD

by NICHOLAS M. BESSONOV

Institute of the Problems of Mechanical Engineering 
Acad of Sci. of Russia, Bolshoj pr.61, St.Petersburg, 199178, Russia 
bessonov@bess.ipme.ru


Abstract

The vector finite difference (VFD) method is a coordinate-free generalization
of a traditional (``scalar") technology of approximation used in finite
difference (FD), volume (FV) or elements (FE) methods.  The VFD method
conserves compact and invariant vector-tensor notation (the natural language of
continuum mechanics) from mathematical formulation, through numerical scheme,
and computer program nomenclature and implementation.  As a result (in
comparison with traditional FD, and other methods) the two last steps are
simplified drastically, especially for multi-dimensional problems in the
regions with irregular geometry, where nonorthogonal meshes are applied.  So
more attention can be focused on physical part of problems.  Apart from the
cleaner syntax, VFD allows to conserve the structure of traditional numerical
algorithms (resolution methods) and to solve the high-dimensional problems with
little additional programming efforts also.  Partial applications of this
technique are described as well (simulation of elastic-plastics problems,
simulation of hyperelastic 4D pyramid twisting)