System Stochasticity: Discrete Formulation with Mathematica

Gautam Dasgupta
Professor, Civil Engineering and Engineering Mechanics
Columbia University
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.