Spanos, Pol D.
||dc.creator||Evangelatos, Georgios I.
Evangelatos, Georgios I.. "Propagation via a Peridynamics Formulation: A
Stochastic\Deterministic Perspective." (2011) Diss., Rice University. https://hdl.handle.net/1911/64430.
Novel numerical methods for treating fractional differential and
integrodifferential equations arising in non local mechanics formulations are
proposed. For fractional differential equations arising in modeling oscillatory
systems incorporating viscoelastic elements governed by fractional derivatives,
the devised scheme is based on the Grunwald-Letnikov fractional derivative
representation, dual time meshing technique and Taylor expansion. The proposed
algorithm transforms the governing fractional differential equation into a second
order differential equation with appropriate effective coefficients. The enhanced
efficiency of the scheme hinges upon circumventing the calculation of the non
local fractional derivative operator. Several examples of application are provided.
Further, the concept of non locality, specifically viscoelasticity, governed
by fractional derivatives is utilized to accurately model polyester materials.
Specifically, the linear standard solid (Zener model) is extended to capture non
linear viscoelastic behavior. Then, experimental data of polyester ropes are
utilized using the Gauss Newton and Levenberg-Marquart minimization algorithm
to determine the model parameters.
Next, for integrodifferential equations arising in peridynamics theory of
mechanics, an approach is formulated based on the inverse multi-quadric (IMQ)
radial basis function (RBF) expansion and the Kansa collocation method. The
devised scheme utilizes interpolation functions and basis function expansion for
the spatial discretization of the peri dynamics equation. This significantly reduces
the computational effort required to numerically treat the peri dynamics equations.
Further, the proposed method is extended to account for mechanical systems with
random material properties operating under random excitation. For this, the
stochastic peridynamics governing equation of motion is solved using the
benchmark Monte Carlo analysis and tools of stochastic analysis. The stochastic
analysis is done by numerical evaluation of the requisite Neumann expansion
using pertinent Monte Carlo simulations.
Further, the usefulness of the radial basis function (RBF) collocation
method in conjunction with a polynomial chaos expansion (PCE) is explored in
stochastic mechanics problems. It is shown that the proposed approach renders
further solution improvements in solving stochastic mechanics problems vis-a-vis
the stochastic finite element method and the element free Galerkin method.
Propagation via a Peridynamics Formulation: A
Civil and Environmental Engineering
Doctor of Philosophy