Application of Gegenbauer polynomial expansions to mitigate Gibbs phenomenon in Fourier–Bessel series solutions of a dynamic sphere problem

Abstract

SUMMARYWe utilize the inverse polynomial reconstruction (IPR) method to mitigate the Gibbs phenomenon observed in Fourier-Bessel (FB) series. Gibbs phenomenon is the oscillatory behavior that occurs near discontinuities when evaluating series solutions for Sturm-Liouville eigenvalue problems. We employ an approach that uses expansions of the solution in terms of Gegenbauer polynomials on each side of solution discontinuities, the location of which must be known in advance. The IPR solutions provide pointwise values that are more accurate than the truncated series solution, which are polluted by Gibbs phenomenon. We apply this method to discontinuous solutions of a time dependent, linear elastic spherical shell problem, for which a series solution is derived in terms of a FB expansion. For the loading conditions and material properties, we consider the Gibbs phenomenon in the FB solution for a perfectly elastic shell renders the numerically evaluated results unusable as an 'exact' solution for code verification analysis. We quantify the degree to which the IPR method eliminates the Gibbs phenomenon in the computed solution.

Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

“…Since the generalised reconstruction procedure is well-conditioned, there may also be a benefit in this regard. Outside of PDEs, the Gegenbauer reconstruction technique has also been extended to other types of series, including radial basis functions [39], Fourier-Bessel series [40] and spherical harmonics [22]. Future work will also consider generalisation of the method of this paper along these lines.…”

The jump-off velocity of an impulsively loaded spherical shell

“…−p(0) = α o u (r o , 0) + β o u(r o , 0)), so there is a discontinuity that propagates along the characteristics. We compute the velocity as t approaches the jump-off time R/c both for η ∈ {1}, ξ ∈ [1] and for η ∈ {1}, ξ ∈ [2], which correspond to the velocities obtain by the pressure wave traveling along the characteristic line η = r o and ξ = r i − R, respectively. These velocities correspond to what [3] calls the particle velocity and the material surface velocity, respectively.…”

“…Therefore, these velocities apply to the case where a constant pressure p(t) = P 0 is applied to the outer surface for all time. Second, we notice that the material surface velocityu {1} [2] (r i , R c ) is twice the particle velocityu {1} [1] (r i , R c ). This difference is explained by the velocity doubling rule described in [4, pg.…”

“…This is not physically realistic, but it is sometimes used as a simplifying assumption [1,2,3]. For such a spherical shell, the equation of motion (1) and the initial conditions (4) are the same as in the compressible case, but the boundary conditions take a slightly different form.…”

“…is the speed of sound through the material [1,2,3]. We want to compute the jump-off time and velocity of the pressure wave, which are the first time after t = 0 at which the pressure wave from the outer surface reaches the inner surface.…”