Nonlinear hierarchical substructural parallelism and computer architecture

Computer architecture is investigated in conjunction with the algorithmic structures of nonlinear finite-element analysis. To help set the stage for this goal, the development is undertaken by considering the wide-ranging needs associated with the analysis of rolling tires which possess the full range of kinematic, material and boundary condition induced nonlinearity in addition to gross and local cord-matrix material properties

Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure. Part 1: Theory

In a three part series of papers, a generalized finite element analysis scheme is developed to handle the steady and transient response of moving/rolling nonlinear viscoelastic structure. This paper considers the development of the moving/rolling element strategy, including the effects of large deformation kinematics and viscoelasticity modelled by fractional integro-differential operators. To improve the solution strategy, a special hierarchical constraint procedure is developed for the case of steady rolling/translating as well as a transient scheme involving the use of a Grunwaldian representation of the fractional operator. In the second and third parts of the paper, 3-D extensions are developed along with transient contact strategies enabling the handling of impacts with obstructions. Overall, the various developments are benchmarked via comprehensive 2- and 3-D simulations. These are correlated with experimental data to define modelling capabilities

Parallelized modelling and solution scheme for hierarchically scaled simulations

This two-part paper presents the results of a benchmarked analytical-numerical investigation into the operational characteristics of a unified parallel processing strategy for implicit fluid mechanics formulations. This hierarchical poly tree (HPT) strategy is based on multilevel substructural decomposition. The Tree morphology is chosen to minimize memory, communications and computational effort. The methodology is general enough to apply to existing finite difference (FD), finite element (FEM), finite volume (FV) or spectral element (SE) based computer programs without an extensive rewrite of code. In addition to finding large reductions in memory, communications, and computational effort associated with a parallel computing environment, substantial reductions are generated in the sequential mode of application. Such improvements grow with increasing problem size. Along with a theoretical development of general 2-D and 3-D HPT, several techniques for expanding the problem size that the current generation of computers are capable of solving, are presented and discussed. Among these techniques are several interpolative reduction methods. It was found that by combining several of these techniques that a relatively small interpolative reduction resulted in substantial performance gains. Several other unique features/benefits are discussed in this paper. Along with Part 1's theoretical development, Part 2 presents a numerical approach to the HPT along with four prototype CFD applications. These demonstrate the potential of the HPT strategy

Diophantinized Fractional Representations for Nonlinear Elastomeric Media

This paper explores the use of diophantinized fractional fits for the nonlinear constitutive representation of elastomeric media. These are caste in terms of either the principal stretches or the strain invariants. Both polynomial and rational forms are considered. To construct the requisite complete and physically admissible basis space, the diophantinized set of fractional powers is bound by the curvature properties of the experimental data set. This set is then employed in conjunction with a remezed least square scheme to obtain an optimal fit. To verify the scheme a sample application case is presented

Frequency Driven Phasic Shifting and Elastic-Hysteretic Partitioning Properties of Fractional Mechanical System Representation Schemes

Based on the Louiville–Riemann fractional formulation of lumped hysteretic mechanical system simulations, asymptotic-type relationships are derived. These are employed to determine how such operators, which act as viscoelastic elements, partition system energy into conservative and nonconservative components. Special emphasis is given to: (a) determine how operator order serves to weigh such a splitting, (b) determine how partitioning affects system phasing and amplitude response, and (c) to establish how conservative and nonconservative effects modulate during a given system cycle. The generality of the undertaken approach is such that multi-element fractional Kelvin Voigt formulations subject to spectrally rich inputs can be handled, i.e., the multi-modal splitting of energies. As a result of the insights derived, improved frequency dependent simulations of system amplitude, phasing and energetics will be possible

Hierarchically Parallelized Constrained Nonlinear Solvers with Automated Substructuring

This paper develops a parallelizable multilevel multiple constrained nonlinear equation solver. The substructuring process is automated to yield appropriately balanced partitioning of each succeeding level. Due to the generality of the procedure,_sequential, as well as partially and fully parallel environments can be handled. This includes both single and multiprocessor assignment per individual partition. Several benchmark examples are presented. These illustrate the robustness of the procedure as well as its capability to yield significant reductions in memory utilization and calculational effort due both to updating and inversion

Nonlinear Vibrations of Fractionally Damped Systems

This paper deals with the harmonic oscillations of periodically excited nonlinear systems where hysteresis is simulated via fractional operator representations. Employing a diophantine version of the fractional operational powers, the energy constrained Lindstedt–Poincaré perturbation procedure is utilized to establish the harmonic solution. The constrained perturbation procedure was employed since it allows for the handling of strong damping and exciting forces over the full span of the driving frequency range. Based on the approach taken, the long time behavior of the fractionally damped Duffing\u27s equation is studied in detail. Of special interest is the determination of the influence of fractional order on the frequency amplitude response behavior

Diophantine Type Fractional Derivative Representation of Structural Hysteresis, Part I: Formulation

Based on a diophantine representation of the operational powers, a fractional derivative modelling scheme is developed to simulate frequency dependent structural damping. The diophantine set of powers is established by employing the curvature properties of the defining empirical data set. These together with a remezed least square scheme are employed to construct a Chebyschev like optimal differintegro simulation. Based on the use of the rational form resulting from the diophantine representation, a composition rule is introduced to reduce the differintegro simulation to first order form. The associated eigenvalue/vector properties are then explored. To verify the robustness-stability accuracy of the overall modelling procedure, correlation studies are also presented. Part I of this series focuses on the diophantine representation, its use in formulating a numerically more workable first order form as well as formal representations of its transient and steady state solutions. This will include investigations of the asymptotic properties of the various formulations. Part II will introduce the model fitting scheme along with a look at eigen properties and fitting effectiveness

Diophantine Type Fractional Derivative Representation of Structural Hysteresis, Part II: Fitting, Computational Mechanics

Part I of this series introduced the diophantinized fractional model and the decompositional formulation. The various important properties of fractional continuum formulation and its decomposed version were developed. In Part II the dynamic properties of the diophantine representation are investigated. The model fitting scheme will be developed to handle an arbitrary frequency dependent structural hysteriesis. This is followed up with the results of benchmark studies which demonstrate the effectiveness of fitting

Hierarchial parallel computer architecture defined by computational multidisciplinary mechanics

The goal is to develop an architecture for parallel processors enabling optimal handling of multi-disciplinary computation of fluid-solid simulations employing finite element and difference schemes. The goals, philosphical and modeling directions, static and dynamic poly trees, example problems, interpolative reduction, the impact on solvers are shown in viewgraph form