Propagation via a Peridynamics Formulation: AStochasticDeterministic Perspective

[摘要] Novel numerical methods for treating fractional differential andintegrodifferential equations arising in non local mechanics formulations areproposed. For fractional differential equations arising in modeling oscillatorysystems incorporating viscoelastic elements governed by fractional derivatives,the devised scheme is based on the Grunwald-Letnikov fractional derivativerepresentation, dual time meshing technique and Taylor expansion. The proposedalgorithm transforms the governing fractional differential equation into a secondorder differential equation with appropriate effective coefficients. The enhancedefficiency of the scheme hinges upon circumventing the calculation of the nonlocal fractional derivative operator. Several examples of application are provided.Further, the concept of non locality, specifically viscoelasticity, governedby fractional derivatives is utilized to accurately model polyester materials.Specifically, the linear standard solid (Zener model) is extended to capture nonlinear viscoelastic behavior. Then, experimental data of polyester ropes areutilized using the Gauss Newton and Levenberg-Marquart minimization algorithmto determine the model parameters.Next, for integrodifferential equations arising in peridynamics theory ofmechanics, an approach is formulated based on the inverse multi-quadric (IMQ)radial basis function (RBF) expansion and the Kansa collocation method. Thedevised scheme utilizes interpolation functions and basis function expansion forthe spatial discretization of the peri dynamics equation. This significantly reducesthe computational effort required to numerically treat the peri dynamics equations.Further, the proposed method is extended to account for mechanical systems withrandom material properties operating under random excitation. For this, thestochastic peridynamics governing equation of motion is solved using thebenchmark Monte Carlo analysis and tools of stochastic analysis. The stochasticanalysis is done by numerical evaluation of the requisite Neumann expansionusing pertinent Monte Carlo simulations.Further, the usefulness of the radial basis function (RBF) collocationmethod in conjunction with a polynomial chaos expansion (PCE) is explored instochastic mechanics problems. It is shown that the proposed approach rendersfurther solution improvements in solving stochastic mechanics problems vis-a-visthe stochastic finite element method and the element free Galerkin method.

[发布机构] Rice University

[效力级别] engineering