Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

Abstract

Abstract. For flows 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. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace engineering problems. This paper presents an extension of the time-spectral approach for problems that include a slow transient in addition to strong periodic behavior, suitable for applications such as transient turbofan simulation or maneuvering rotorcraft calculations. The formulation is based on a collocation method which makes use of a combination of spectral and polynomial basis functions and results in the requirement of solving coupled time instances within a period, similar to the time spectral approach, although multiple successive periods must be solved to capture the transient behavior.The 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 which minimizes the modifications required to an existing steady-state solver. For dynamically deforming mesh cases, a formulation which preserves discrete conservation as determined by the Geometric Conservation Law is derived and implemented. A fully implicit approach which takes into account the coupling between the various time instances is implemented and shown to preserve the baseline steady-state multigrid convergence rate as the number of time instances is increased. Accuracy and efficiency are demonstrated for periodic and non-periodic problems by comparing the performance of the method with a traditional time-stepping approach using a simple two-dimensional pitching airfoil problem, a three-dimensional pitching wing problem, and a more realistic transitioning rotor problem.

Development of a High-Fidelity Time-Dependent Aero-Structural Capability for Analysis and Design

“…For periodic and quasi-periodic problems, parallel time-spectral (TS) and BDF-time-spectral (BDFTS) time discretizations have been implemented and demonstrated for rotorcraft applications. 23 NSU3D has also been coupled with a finite-element beam structural model, both for steady state 21 and time dependent problems 18 in analysis and adjoint mode.…”

Jacobian‐free Newton–Krylov method for implicit time‐spectral solution of the compressible Navier‐Stokes equations

“…Also note that in Equation (9) we have omitted, showing the explicit dependence of Q and R on X and J is now a matrix that consists of 2N h C 1 blocks of the diagonal matrices containing the transformation Jacobian at each time level (which are the same in this paper as the grid is not time dependent). From Equation (8), the Jacobian @Q @X in Equation (9) can be written as @Q @X D J 1 C @R @X ; (11) and Equation (9) can now be written as Ä…”

“…In Equation (15), ı .2/ A, ı .2/ Á B, and ı .2/ C are matrices that consist of the combination, over all time instances, of the second-order (central) spatial discretizations of the flux Jacobians at each grid point. This second-order discretization gives rise to an additional form of approximation in the factorization of Equation (11). A diagonalized form of the approximate factorization [38] is used that, when considered along with the aforementioned approximations (e.g., approximate factorization, second-order spatial discretization of flux Jacobians and ignoring of time-spectral coupling terms), results in a block, tridiagonal matrix.…”

“…A solution for X in Equation (12) is found using the (non-restarted) flexible generalized minimal residual method (FGMRES) [29], a variant of the original (Arnoldi-based) GMRES solver proposed by Saad and Schultz [30]. The use of a GMRES-based solver is due to the weak, or lack of, diagonal dominance of the Jacobian matrix given in Equation (11), and thus, a solver that does not require diagonal dominance, such as GMRES, is preferred [10]. If the solution to Equation (12) is only approximate, the resulting method is called an inexact Newton method [31] with a convergence criteria, which can be written as Ä…”

Algorithmic Advanced for the Adaptive Non-Linear Frequency Domain Method.

“…Therefore, the effect of nonlinear harmonic coupling would not be considered. On the other hand, in the nonlinear harmonic method, known as Harmonic Balance , 13,15,16 Non-Linear Frequency Domain (NLFD), 29,30,34,35 and Time Spectral, 5,6,11,12,26,27,44,53 all the harmonic equations are solved coupled. Besides, there is no restriction on the amplitude of the perturbation terms.…”