Stability of perfectly matched layers, group velocities and anisotropic waves

Bécache, Éliane; Fauqueux, Sandrine; Joly, Patrick

doi:10.1016/s0021-9991(03)00184-0

Cited by 304 publications

(364 citation statements)

References 39 publications

(47 reference statements)

Supporting

Mentioning

355

Contrasting

Unclassified

Order By: Relevance

“…elastodynamics), due to the presence or absence of specific symmetries in the physical fields in the PML introduced medium, the interpretation of the PML in terms of an anisotropic material seems to be impossible. A rigorous proof of stability for such systems remains open [9].…”

mentioning

confidence: 99%

Parametric finite elements, exact sequences and perfectly matched layers

Matuszyk

Demkowicz

2012

Comput Mech

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

Parametric finite elements, exact sequences and perfectly matched layers

Matuszyk

Demkowicz

2012

Comput Mech

View full text Add to dashboard Cite

show abstract

“…We therefore observe that the error manifests itself in the existence of reflected modes in the approximate solution 3 . The method will be deemed accurate if these reflections are small or, more properly, if the reflection coefficients R n (α, L), ∀n ∈ N, are small, the values of the parameters α and L being fixed.…”

Section: Truncation Of the Layermentioning

confidence: 89%

“…For linear elastic systems in which the propagative medium presents particular anisotropy properties, numerical instabilities can be observed in time-domain simulations [3]. In the context of waveguides, anisotropy of the material is not even a necessary feature for this phenomenon to occur.…”

Section: Introductionmentioning

confidence: 99%

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

Dhia¹,

Chambeyron²,

Legendre³

2014

Wave Motion

View full text Add to dashboard Cite

An efficient method to compute the scattering of a guided wave by a localized defect, in an elastic waveguide of infinite extent and bounded cross section, is considered. It relies on the use of perfectly matched layers (PML) to reduce the problem to a bounded portion of the guide, allowing for a classical finite element discretization. The difficulty here comes from the existence of backward propagating modes, which are not correctly handled by the PML. We propose a simple strategy, based on finite-dimensional linear algebra arguments and using the knowledge of the modes, to recover a correct approximation to the solution with a low additional cost compared to the standard PML approach. Numerical experiments are presented in the two-dimensional case involving Rayleigh-Lamb modes.

show abstract

“…Efficient and reliable domain truncation becomes essential, since it enables more accurate numerical simulations. More than thirty years of extensive research in this area has resulted in two standard, competing approaches for artificial boundary closures: high order local non-reflecting boundary condition (NRBC) [19,20], and damping layers such as the perfectly matched layer (PML) [1][2][3][4][5][6][7][8][9][10][11][12], grid stretching techniques [14] and others [13,15]. An NRBC is a boundary condition defined on an artificial boundary such that little or no spurious reflections occur as a wave passes the boundary.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, if the original governing equation and its numerical approximations do not support growth, neither should the augmented system nor a corresponding numerical approximation. For hyperbolic systems, the temporal stability of the PML Cauchy problem is well known; see for example [3,2]. When the domain is bounded or semi-bounded more care is required, since a PML which is stable in an unbounded domain, can support growth when boundaries are introduced [10,6].…”

Section: Introductionmentioning

confidence: 99%

Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form

Duru

Kozdon

Kreiss

2015

Journal of Computational Physics

View full text Add to dashboard Cite

In computations, it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer (PML) of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation. Unfortunately, the PML does not give us an indication about appropriate boundary conditions needed to close the edges of the PML, or how those boundary conditions should be enforced in a numerical setting. Terminating the PML with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the PML which will eventually destroy the accuracy of a numerical simulation everywhere. In this paper, we analyze the stability and the well-posedness of boundary conditions terminating the PML for the elastic wave equation in first order form. First, we consider a vertical modal PML truncating a two space dimensional computational domain in the horizontal direction. We freeze all coefficients and consider a left half-plane problem with linear boundary conditions terminating the PML. The normal mode analysis is used to study the stability and well-posedness of the resulting initial boundary value problem (IBVP). The result is that any linear well-posed boundary condition yielding an energy estimate for the elastic wave equation, without the PML, will also lead to a well-posed IBVP for the PML. Second, we extend the analysis to the PML corner region where both a horizontal and vertical PML are simultaneously active. The challenge lies in constructing accurate and stable numerical approximations for the PML and the boundary conditions. Third, we develop a high order accurate finite difference approximation of the PML subject to the boundary conditions. To enable accurate and stable numerical boundary treatments for the PML we construct continuous energy estimates in the Laplace space for a one space dimensional problem and two space dimensional PML corner problem. We use summation-by-parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. In order to ensure numerical stability of the discrete PML, it is necessary to extend the numerical boundary procedure to the auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical experiments are presented corroborating the theoretical results. Moreover, in order to ensure longtime numerical stability, the boundary condition closing the PML, or its corresponding discrete implementation, must be dissipative. Furthermore, the numerical experiments demonstrate the stable and robust treatment of PML corners.

show abstract

Stability of perfectly matched layers, group velocities and anisotropic waves

Cited by 304 publications

References 39 publications

Parametric finite elements, exact sequences and perfectly matched layers

Parametric finite elements, exact sequences and perfectly matched layers

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form

Contact Info

Product

Resources

About