Nathan L. Mundis

Quasi-periodic Time Spectral Method for Aeroelastic Flutter Analysis

A recently developed quasi-periodic time spectral method is applied to the demanding problem of aeroelastic flutter. Both a standard time-implicit method and a quasi-periodic time spectral method are developed that take into account the coupling among the three fundamental aspects of computational aeroelastic calculations: unsteady flow equations, time dependent structural response to aerodynamics loads, and dynamically moving meshes. These two methods are then compared in order to demonstrate the capability of the quasi-periodic time spectral method to solve aeroelastic flutter problems. Finally, it is demonstrated that the quasi-periodic time spectral method can be used to solve aeroelastic flutter problems.

Extensions of Time Spectral Methods for Practical Rotorcraft Problems

For ows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit methods which operate directly in the time domain. In their original form, time spectral methods are applicable only to purely periodic problems and formulated for single grid systems. A wide class of problems involve quasi-periodic ows, such as maneuvering rotorcraft problems, where a slow transient is superimposed over a more rapid periodic motion. Additionally, the most common approach for simulating combined rotor-fuselage interactions is through the use of a dynamically overlapping mesh system. Thus, in order to represent a practical approach for rotorcraft simulations, time spectral methods that are applicable to quasi-periodic problems and capable of operating on overlapping mesh systems need to be formulated. In this paper, we propose separately an extension of time spectral methods to quasi-periodic problems, and an extension for overlapping mesh congurations. In both cases, the basic implementation allows for two levels of parallelism, one in the spatial dimension, and another in the time-spectral dimension, and is implemented in a modular fashion that minimizes the modi cations required to an existing steady-state solver. Results are given for three-dimensional quasi-periodic problems on a single mesh, and for two-dimensional periodic overlapping mesh systems.

GMRES applied to the Time Spectral and Quasi-periodic Time Spectral Methods

The time-spectral method applied to the Euler equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. A recently developed quasi-periodic time-spectral (BDFTS) method extends the time-spectral method to problems with fast periodic content and slow mean flow transients, which should lead to faster solution of these types of problems as well. However, attaining superior efficiency with TS or BDFTS methods over traditional time-implicit methods hinges on the ability to rapidly solve the large non-linear system resulting from TS discretizations which become larger and stiffer as more time instances are employed. In order to increase the efficiency of these solvers, and to improve robustness, particularly for large numbers of time instances, the TS and BDFTS methods are reworked such that the Generalized Minimal Residual Method (GMRES) is used to solve the implicit linear system over all coupled time instances. The use of GMRES as the linear solver makes these methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods.

Highly-Accurate Filter-Based Artificial-Dissipation Schemes for Stiff Unsteady Fluid Systems

It is well known that the unmodified application of central difference schemes to the Euler equations produces numerically unstable results because such schemes do not naturally damp the high-frequency modes involved in odd-even decoupling. This shortcoming is usually overcome by adding artificial-dissipation terms, thereby producing stable schemes at the price of potential loss of solution accuracy. For unsteady fluid dynamics, solution filtering schemes have been proposed as a more accurate alternative to artificial dissipation especially when explicit physical-time integration is utilized. However, to solve computationally stiff problems efficiently, it is necessary to use a dual-time stepping approach, to which the application of solution filtering is not straightforward. Restricting the solution filtering only to the physical-time level does not guarantee numerical stability, as errors can accumulate in pseudo-time, causing divergence, while including it at the pseudo-time level introduces inconsistencies that lead to convergence problems. In the present work, Shapiroand Purser-type explicit solution filters are used to derive a new class of filter-equivalent artificial-dissipation operators that can be applied in pseudo time to produce stable, convergent, low-dissipation solutions. These novel artificial-dissipation operators are shown to be indistinguishable from their corresponding filtering procedure for explicit, single-time schemes and are just as effective within a dual-time framework. In addition, the filter-based artificial-dissipation schemes are formulated for use with local preconditioning methods and applied to stiff problems such as low-Mach unsteady flows.

An Efficient Flexible GMRES Solver for the Fully-coupled Time-spectral Aeroelastic System

The time-spectral (TS) method applied to the coupled aeroelastic equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. However, attaining superior efficiency with TS methods over traditional time-implicit methods hinges on the ability to rapidly solve the large non-linear system resulting from TS discretizations which become larger and stiffer as more time instances are employed. An ideal time-spectral solver would scale linearly, such that a doubling of the number of time-instances used would double the wall-clock time needed to converge the solution on the same computational hardware. The present work is focused on achieving an optimal solver for large numbers of time instances. In order to increase the efficiency of the solver, and to improve robustness particularly for large numbers of time instances and fluid/structure coupling, TS methods are reworked such that the Generalized Minimal Residual Method (GMRES) is used to solve the implicitly coupled, time-spectral, fluid/structure linear system over all time instances at each non-linear iteration. The use of GMRES as the linear solver makes these methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods.

Wave-number Independent Preconditioning for GMRES Time-spectral Solvers

The time-spectral method applied to the Euler equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. However, attaining superior efficiency with time-spectral methods over traditional time-implicit methods hinges on the ability rapidly to solve the large non-linear system resulting from time-spectral discretizations which become larger and stiffer as more time instances are employed. In order to increase the efficiency of these solvers, and to improve robustness, particularly for large numbers of time instances, the Generalized Minimal Residual Method (GMRES) is used to solve the implicit linear system over all coupled time instances. The use of GMRES as the linear solver makes the time-spectral methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods. However, it has been shown in previous work that when the number of time instances and/or the reduced frequency of motion increases (i.e. the maximum resolvable wave-number increases), the convergence degrades rapidly, requiring many more total preconditioning iterations to reach a converged solution. To alleviate this convergence degradation, this work formulates a wave-number independent preconditioner by inverting the spatial-temporal diagonal blocks in the preconditioner instead of the spatial diagonal blocks individually, as has been done previously. The solver utilizing this wave-number independent preconditioner is shown to be more efficient than past solvers under all conditions, but especially for high reduced frequencies and large number of time instances, i.e. conditions with high maximum wave numbers.

Optimal Runge-Kutta Schemes for High-order Spatial and Temporal Discretizations

Abstract : Numerical discretization for unsteady flow simulations can be broken down into spatial and temporal parts which interplay in complex and sometimes unexpected ways. This paper attempts to address how the properties of the spatial discretization help drive the choice of temporal discretization. In addition, it examines methods for higher than second-order accurate time integration using L-stable singly-diagonally-implicit (ESDIRK) Runge-Kutta methods. Von Neumann analysis is used to examine the theoretical effects of different spatial/temporal discretization combinations. The predictive nature of the von Neumann analysis is then validated through the exploration of the convection of acoustic waves in one dimension and an isentropic vortex in three dimensions. Is is shown that the computational results follow the expected trends taking the von Neumann analysis of the schemes into account. This work highlights that, for unsteady problems, both dissipation and dispersion errors must be accounted for when selecting optimal Runge-Kutta time integrators.

Isolating Flow-field Discontinuities while Preserving Monotonicity and High-order Accuracy on Cartesian Meshes

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Toward an optimal solver for time-spectral fluid-dynamic and aeroelastic solutions on unstructured meshes

Abstract The time-spectral method applied to the Euler and coupled aeroelastic equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. However, attaining superior efficiency with time-spectral methods over traditional time-implicit methods hinges on the ability rapidly to solve the large non-linear system resulting from time-spectral discretizations which become larger and stiffer as more time instances are employed or the period of the flow becomes especially short (i.e. the maximum resolvable wave-number increases). In order to increase the efficiency of these solvers, and to improve robustness, particularly for large numbers of time instances, the Generalized Minimal Residual Method (GMRES) is used to solve the implicit linear system over all coupled time instances. The use of GMRES as the linear solver makes time-spectral methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods. In previous work, a wave-number independent preconditioner that mitigates the increased stiffness of the time-spectral method when applied to problems with large resolvable wave numbers has been developed. This preconditioner, however, directly inverts a large matrix whose size increases in proportion to the number of time instances. As a result, the computational time of this method scales as the cube of the number of time instances. In the present work, this preconditioner has been reworked to take advantage of an approximate-factorization approach that effectively decouples the spatial and temporal systems. Once decoupled, the time-spectral matrix can be inverted in frequency space, where it has entries only on the main diagonal and therefore can be inverted quite efficiently. This new GMRES/preconditioner combination is shown to be over an order of magnitude more efficient than the previous wave-number independent preconditioner for problems with large numbers of time instances and/or large reduced frequencies.

Finite-element Time Discretizations for the Unsteady Euler Equations

This work examines finite-element time discretizations for the Euler equations and methods for the robust and efficient solution of these discretizations. Specifically, the spectralelement (SEMT) and discontinuous-Galerkin (DGMT) methods in time are derived and examined in detail. To solve the SEMT and DGMT Euler equations, the flexible variant of the Generalized Minimal Residual method (FGMRES), utilizing the full second-order accurate spatial Jacobian and complete temporal coupling of the chosen time discretization, is implemented. The FGMRES solver developed utilizes a block-colored Gauss-Seidel (BCGS) preconditioner augmented by a defect-correction process to increase its effectiveness. This preliminary examination of the SEMT and DGMT shows that these time discretization have promise for the solution of the unsteady Euler equations, but also that the behavior of the methods around discontinuities must be improved to make them broadly applicable to unsteady flow problems. The investigation conducted in this work indicates that SEMT, rather than DGMT, is likely the better time discretization for use in solution of the Euler equations.