Parametric finite elements, exact sequences and perfectly matched layers

Abstract

The paper establishes a relation between exact sequences, parametric finite elements, and perfectly matched layer (PML) techniques. We illuminate the analogy between the Piola-like maps used to define parametric H 1 -, H(curl)-, H(div)-, and L 2 -conforming elements, and the corresponding PML complex coordinates stretching for the same energy spaces. We deliver a method for obtaining PML-stretched bilinear forms (constituting the new weak formulation for the original problem with PML absorbing boundary layers) directly from their classical counterparts.

Influence of borehole‐eccentred tools on wireline and logging‐while‐drilling sonic logging measurements

“…The method also incorporates a Perfectly Matched Layer (PML) for truncation of the computational domain, as first proposed by Bérenger (1994). Our understanding and implementation of the PML is consistent with the one provided by Chew and Weedon (1994), which is based on a change of coordinates of the governing equations into the complex plane within the PML layer (Matuszyk et al 2012; Matuszyk and Demkowicz 2012). The coordinates are transformed according to the formula where x j and X j represent unstretched and stretched coordinates, respectively, k = 2π/λ min (λ min is the shortest anticipated wavelength for a given frequency and all the material properties) and The coordinates and define the PML region for the coordinate x j and δ j denotes its thickness.…”

On perfectly matched layers for discontinuous Petrov–Galerkin methods

“…However, our methods can very easily be applied to any other conceivable variational formulation. Complex coordinate stretching has been discussed in an earlier work for different equations using Bubnov-Galerkin methods [74]. Inspired by this and to elaborate the differences, we present two possible categories of PML-consistent ultraweak variational formulations.…”

“…(2) Alternatively, first multiply with test functions and integrate by parts to construct a variational formulation in the stretched coordinates, then pull back the corresponding stretched variational formulation to spatial coordinates. In classical functional settings, it is well-known that these two approaches are identical in that they return the same ultimate variational formulation [74]. However, there are subtle differences to acknowledge in alternative functional settings.…”

Frequency-domain simulation of logging-while-drilling borehole sonic waveforms

“…The formulation also includes a PML to impose radiation (Sommerfeld) conditions, thereby enabling a finite truncation of the computational domain. We introduce the PML into the weak form given by equation 1 and following the method developed by Matuszyk and Demkowicz (2013), namely, by introduction of the stretching JacobianJ ¼ detðJÞ and matrixà ¼JJ −1J−T , where the complex coordinate stretching JacobianJ and its determinant are given bỹ…”