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.
Highlights
1.1 BackgroundMost engineering problems to be solved by numerical methods for engineering design purposes are parameter-dependent, often requiring multiple analyses to be conducted with increasingly complex models
In the context of frequency-dependent problems solved by the finite element (FE) method, several approaches have been developed to limit the impact of their computational burden
The main conceptual difference between the 2 procedures can be clearly seen in Figure 1A,B as the original procedure relies on the scalar-component data to determine the pole of the Padé approximant when the updated approach relies on the eigenfrequencies of the full-size problem
Summary
Most engineering problems to be solved by numerical methods for engineering design purposes are parameter-dependent, often requiring multiple analyses to be conducted with increasingly complex models. Lanczos-based interpolatory methods, such as the Padé-via-Lanczos algorithm[14] or its subsequent development for more general partial fields and unsymmetric systems with the matrix-valued Padé-via-Lanczos approach, or Krylov subspace methods,[15,16,17] have been extended to second-order problems, being limited to constant non-proportional damping models These extensions are made by a linearization of the second-order problem which involves doubling the dimension of the state-space vector, substantially increasing the memory requirements. Further works have focused on such iterative approaches targeting projection-based interpolatory methods, eg, using residuals compared between 2 consecutive steps of increasing size of the reduced system.[26,29] In Rumpler et al,[24] the authors presented an adaptive frequency windowing for the component-wise class of Padé approximant methods This approach consists in a combination of sequential, a priori estimates of the intervals of convergence of the Padé approximants, with a posteriori refinements using an error estimator. This lack of robustness, obviously strongly dependent on the problem of interest, may hamper the efficiency and elegance of Padé-based approximations
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