Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

Abstract

SummaryTo increase the robustness of a Padé-based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed.The resulting original approach is shown to allow for a straightforward, efficient, subsequent Padé-based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Padé approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component-wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Padé approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural-acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed.

An assessment of two popular Padé-based approaches for fast frequency sweeps of time-harmonic finite element problems

“…While the increase of the numerator polynomial degree from nL = 8 to nL = 12 extends the range of convergence, these approximations do not cover the range of 7 poles as theoretically expected. This known limitation was recently improved with a procedure proposed by the authors 15 for conservative problems. The component-wise approach remains however limited in range by the inherent ill-conditioned nature of the procedure associated with the calculation of the solution derivatives, thus requiring a specific multi-interval strategy.…”

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

“…Since the models are described by standard secondorder transfer functions, sp yields 3 columns for each interpolation point. Using n s = 80 shifts linearly distributed in the same range and augmented by shifts at 2πi [46,47,48,50] yields the intermediate reduction basis with q = 250. The additional shifts are introduced to capture the local behavior near the tuning frequency of the TVAs.…”

“…A local order of 10 is chosen for soa presampling. The intermediate basis of order q = 250 is computed considering n s = 21 shifts linearly distributed in 2πi [1,250] and four additional shifts at 2πi [46,47,48,50]. The expansion point sampling for equi is modified similarly to the presampling methods to account for the high impact of the TVAs on the transfer function near their tuning frequency.…”

“…The expansion point sampling for equi is modified similarly to the presampling methods to account for the high impact of the TVAs on the transfer function near their tuning frequency. The shifts at 2πi [46,47,48,49,50] are always considered, the location of the remaining shifts are linearly distributed in the frequency range of interest. For orders r < 5 only the first r extra shifts were considered.…”

“…For efficient numerical computations, ideas for Arnoldi-like algorithms have been extended to the second-order system case [6,39] for application to structural and vibro-acoustic systems [5,54,59]. Further related methods for vibro-acoustic systems with poroelastic materials can be found in, e.g., [25,48,49]. An important aspect in interpolatory model order reduction is the choice of appropriate interpolation points.…”