Computation of frequency responses for linear time-invariant PDEs on a compact interval

Abstract

We develop mathematical framework and computational tools for calculating frequency responses of linear time-invariant PDEs in which an independent spatial variable belongs to a compact interval. In conventional studies this computation is done numerically using spatial discretization of differential operators in the evolution equation. In this paper, we introduce an alternative method that avoids the need for finitedimensional approximation of the underlying operators in the evolution model. This method recasts the frequency response operator as a two point boundary value problem and uses state-of-the-art automatic spectral collocation techniques for solving integral representations of the resulting boundary value problems with accuracy comparable to machine precision. Our approach has two advantages over currently available schemes: first, it avoids numerical instabilities encountered in systems with differential operators of high order and, second, it alleviates difficulty in implementing boundary conditions. We provide examples from Newtonian and viscoelastic fluid dynamics to illustrate utility of the proposed method.

Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids

“…This is accomplished by rewriting the evolution equations (3.7) into an equivalent two-point boundary value problem and then reformulating it into a system of integral equations. The procedure for achieving this along with easy-to-use Matlab source codes is provided in Lieu & Jovanović (2011). This new paradigm for computing frequency responses utilizes the Chebfun computing environment (Trefethen et al 2011) and it exhibits superior numerical accuracy compared to conventional numerical schemes.…”

Reconstructed drill-bit motion for sonic drillstring dynamics

“…Using the obtained analytical form of the sonic drill PDE model and integrating the boundaries, at first, we have computing the different model resonant frequencies. For this, we are referred to the work of Lieu and Jovanovic [9] and the Matlab Chebfun'tool in order to solve the analytical form in a frequency domain. The detailed frequency analysis was addressed in [12] where the obtained ξ = 68 Hz matches the value requested in practice in tunnel reinforcing domain [7] .…”

Well-conditioned ultraspherical and spectral integration methods for resolvent analysis of channel flows of Newtonian and viscoelastic fluids

“…In the presence of purely harmonic inputs, this analysis is colloquially referred to as resolvent analysis and the largest singular value of the frequency response operator determines the worst-case amplification of inputs with a particular temporal frequency [2,5,6]. This quantity determines the so-called "resolvent norm" and it has been used to address nonmodal amplification and robustness to modeling imperfections in channel flows of Newtonian [2,3,7] and viscoelastic fluids [8][9][10].…”

“…In [10], the frequency response operator and its adjoint were cast as two-point boundary value problems (TPBVPs) which take the form of high-order differential equations in a spatially-independent variable. Reference [10] converted these TPBVPs into a system of integral equations and used Chebfun [16] to compute the eigenvalue decomposition of a cascade connection of the frequency response operator with its adjoint. This approach avoids numerical ill-conditioning of pseudo-spectral collocation techniques in resolvent analysis and facilitates straightforward implementation of boundary conditions.…”

“…We build on the formulation provided in [10] and develop a robust method for resolvent analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint, as done in [10].…”