System Stochasticity: Discrete Formulation with Mathematica
Professor, Civil Engineering and Engineering Mechanics
600 Seeley W. Mudd
New York, NY 10027, USA
Material and geometrical
randomness are depicted by variability of numerical solutions.
This second order effect in response cannot be faithfully captured
unless the equilibrium equation is exactly satisfied. Hence
engineering approximations of system stochasticity demand higher
accuracy than their deterministic counter parts.
Consequently, the contamination in numerical responses
cannot be removed by selecting large Monte Carlo samples.
Stochastic shape functions and stochastic Green's functions constitute
the bases for finite and boundary elements, respectively.
These fundamental solutions need to be modified distinctly for each Monte Carlo
representative via symbolic manipulation of the governing algebraic equation.
The subsequent closed form analytical integration of the energy density
function is illustrated here for a beam problem with geometrical stochasticity
after zero shear locking is met in a patch-test.