Gregory M. Hulbert

A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method

A generalized-a method is developed and analyzed for linear, first-order systems. The method is then extended to the filtered Navier-Stokes equations within the context of a stabilized finite element method. The formulation is studied through the application to laminar flow past a circular cylinder and turbulent flow past a long, transverse groove. The method is formulated to obtain a second-order accurate family of time integrators whose high frequency amplification factor is the sole free parameter. Such an approach allows the replication of midpoint rule (zero damping), Gears method (maximal damping), or anything in between.

Space-time finite element methods for second-order hyperbolic equations

Space-time finite element methods are presented to accurately solve elastodynamics problems that include sharp gradients due to propagating waves. The new methodology involves finite element discretization of the time domain as well as the usual finite element discretization of the spatial domain. Linear stabilizing mechanisms are included which do not degrade the accuracy of the space-time finite element formulation. Nonlinear discontinuity-capturing operators are used which result in more accurate capturing of steep fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. The space-time finite element method possesses a firm mathematical foundation in that stability and convergence of the method have been proved. In addition, the formulation has been extended to structural dynamics problems and can be extended to higher-order hyperbolic systems.

Explicit time integration algorithms for structural dynamics with optimal numerical dissipation

Abstract A new predictor-corrector explicit time integration algorithm is presented for solving structural dynamics problems. The basis of the algorithm is the implicit generalized-α method developed by the authors. Like its implicit parent, the explicit generalized-α method is a one-parameter family of algorithms in which the parameter defines the high-frequency numerical dissipation. For a given value of high-frequency dissipation, the explicit generalized-α method minimizes low-frequency dissipation. The algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations. Parameter values are given that enable physical damping to be treated explicitly while maintaining second-order accuracy without the need for an expensive second corrector pass.

Dispersive elastodynamics of 1D banded materials and structures : analysis

Abstract The elastodynamics of 1D periodic materials and finite structures comprising these materials are studied with particular emphasis on correlating their frequency-dependent characteristics and on elucidating their pass-band and stop-band behaviors. Dispersion relations are derived for periodic materials and are employed in a novel manner for computing both pass-band and stop-band complex mode shapes. Through simulations of harmonically induced wave motion within a finite number of unit cells, conformity of the frequency band structure between infinite and finite periodic systems is shown. In particular, only one or two unit cells of a periodic material could be sufficient for “frequency bandedness” to carry over from the infinite periodic case, and only three to four unit cells are necessary for the decay in normalized transmission within a stop band to practically saturate with an increase in the number of cells. Dominant speeds in the scattered wave field within the same finite set of unit cells are observed to match those of phase and group velocities of the infinite periodic material within the most active pass band. Dynamic response due to impulse excitation also is shown to capture the infinite periodic material dynamical characteristics. Finally, steady-state vibration analyses are conducted on a finite fully periodic structure revealing a conformity in the natural frequency spread to the frequency band layout of the infinite periodic material. The steady-state forced response is observed to exhibit mode localization patterns that resemble those of the infinite periodic medium, and it is shown that the maximum localized response under stop-band conditions could be significantly less than in an equivalent homogenous structure and the converse is true for pass-band conditions.

Optimal synthesis of 2D phononic crystals for broadband frequency isolation

The spatial distribution of material phases within a periodic composite can be engineered to produce band gaps in its frequency spectrum. Applications for such composite materials include vibration and sound isolation. Previous research focused on utilizing topology optimization techniques to design two-dimensional (2D) periodic materials with a maximized band gap around a particular frequency or between two particular dispersion branches. While sizable band gaps can be realized, the possibility remains that the frequency bandwidth of the load that is to be isolated might exceed the size of the band gap. In this paper, genetic algorithms are used to design squared bi-material unit cells with a maximized sum of band-gap widths, with or without normalization relative to the central frequency of each band gap, over a prescribed total frequency range of interest. The optimized unit cells therefore exhibit broadband frequency isolation characteristics. The effects of the ratios of contrasting material properties are also studied. The designed cells are subsequently used, with varying levels of material damping, to form a finite vibration isolation structure, which is subjected to broadband loading conditions. Excellent isolation properties of the synthesized material are demonstrated for this structure.

FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY

The frictional heat generated during braking causes thermoelastic distortion that modifies the contact pressure distribution. If the sliding speed is sufficiently high, this can lead to frictionalfy excited thermoelastic instability, characterized by major nonuniformi-ties in pressure and temperature. In automotive applications, a particular area of concern is the relation between thermoelasticalfy induced hot spots in the brake disks and noise and vibration in the brake system. The critical sliding speed can be found by examining the conditions under which a perturbation in the temperature and stress fields can grow in time. The growth has exponential character, and subject to certain restrictions, the growth rate b is found to be real. The critical speed then corresponds to a condition at which b = 0 and hence at which there is a steady-state solution involuing nonuniform contact pressure. We first treat the heat sources Q at the contact nodes as given and use standard finite element analysis (FEA) to d...

Phonon band structure and thermal transport correlation in a layered diatomic crystal

To elucidate the three-way relationship among a crystal’s structure, its phonon dispersion characteristics, and its thermal conductivity, an analysis is conducted on layered diatomic Lennard-Jones crystals with various mass ratios. Lattice dynamics theory and molecular dynamics simulations are used to predict the phonon dispersion curves and the thermal conductivity. The layered structure generates directionally dependent thermal conductivities lower than those predicted by density trends alone. The dispersion characteristics are quantified using a set of band diagram metrics, which are used to assess the contributions of acoustic phonons and optical phonons to the thermal conductivity. The thermal conductivity increases as the extent of the acoustic modes increases, and it decreases as the extent of the stop bands increases. The sensitivity of the thermal conductivity to the band diagram metrics is highest at low temperatures, where there is less anharmonic scattering, indicating that dispersion plays a more prominent role in thermal transport in that regime. We propose that the dispersion metrics i provide an indirect measure of the relative contributions of dispersion and anharmonic scattering to the thermal transport, and ii uncouple the standard thermal conductivity structure-property relation to that of structure-dispersion and dispersion-property relations, providing opportunities for better understanding of the underlying physical mechanisms and a potential tool for material design.

Automatic time step control algorithms for structural dynamics

Abstract An adaptive time step size control strategy is presented for use with step-by-step time integration methods to solve structural dynamics problems. A simple and efficient local error estimate is derived for the newly developed generalized-α time integration method. By judicious choice of normalization, the local error estimate is related to a user-specified tolerance of desired number of time steps per period of effective system response frequency. Step size control algorithms are given that control the undesirable, frequent changes in time step size. Numerical results are presented that demonstrate the performance of the local error estimate and the time step size control strategy.

A new load-dependent Ritz vector method for structural dynamics analyses: quasi-static Ritz vectors

Abstract Existing load-dependent Ritz vector (LDRV) methods employ static recurrence procedures to generate the Ritz vectors. As such, these vector methods are best suited for low-frequency problems. For higher-frequency problems, the existing methods may engender large sets of Ritz vectors, which significantly reduces the methods’ efficiency. A new algorithm is presented for LDRV generation using a quasi-static recurrence procedure, denoted as the quasi-static Ritz vector (QSRV) method. A tuning parameter, designated as the centering frequency, controls the behavior of the QSRV approach, enabling the new method to improve upon existing LDRV methods for particular frequency ranges of interest. Compared with existing LDRV methods, the QSRV method is more efficient (in terms of the number of Ritz vectors), more accurate (in terms of response errors), and more stable (in terms of orthogonality). Numerical examples are provided to illustrate the accuracy, efficiency and generality of the proposed method.

A new component mode synthesis method: quasi-static mode compensation

A new component mode synthesis method is presented in this paper that combines the computational efficiency of the well-known constraint mode approach with the dynamic compensation accuracy obtained by higher-order expansion methods. Instead of employing static constraint modes, quasi-static modes are used to capture inertial effects of the truncated modes. The method is ideally suited for mid-band frequency analysis in which both high-frequency and low-frequency modes may be omitted. A tuning parameter, designated as the centering frequency, controls the dynamic range of the quasi-static modes. Numerical examples are provided which demonstrate the improved accuracy of the proposed method.