Nonreflecting Boundary Conditions for Elastodynamic Scattering

Grote, Marcus J.

doi:10.1006/jcph.2000.6509

Cited by 28 publications

(9 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As conjectured in (26), this apparent contradiction is at the heart of the unstable behavior of the OE and FE algorithms in high-order calculations. Indeed, as argued there, substantial cancellations occur in (13) and (21) so that the overall sums in their respective right-hand sides give rise to finite quantities in spite of possible singularities in the individual terms.…”

Section: Cancellations and Ill-conditioningmentioning

confidence: 89%

“…These relations, in turn, can be realized by simply introducing the notion of a Dirichlet-Neumann operator (DNO), and its higher order analogues, associated with the governing differential operator, and which is defined precisely so as to produce normal derivatives from boundary values. In this manner, the DNO has been brought to bear on problems (direct and inverse) relating to a wide variety of applications that include electromagnetic and acoustic scattering, nondestructive evaluation, and boundary value and free boundary problems from solid and fluid mechanics; see (13,21), (16,28), and (9,12), respectively, and the references therein. Within this framework then, a successful treatment of the corresponding models hinges on a thorough understanding of the mathematical properties of DNO and on the design of accurate and efficient numerical algorithms for their evaluation; these issues are the subject of the present discussion.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Nicholls¹,

Reitich²

2001

Journal of Computational Physics

117

View full text Add to dashboard Cite

In this paper we present results on the stability of perturbation methods for the evaluation of Dirichlet-Neumann operators (DNO) defined on domains that are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition, have long been recognized to constitute efficient and accurate means for the approximation of DNO and, in fact, several numerical implementations have been previously proposed. Inspired by our recent analytical work, here we demonstrate that the convergence of these algorithms is, quite generally, limited by numerical instability. Indeed, we show that these standard perturbative methods for the calculation of DNO suffer from significant ill-conditioning which is manifest even for very smooth boundaries, and whose severity increases with boundary roughness. Moreover, and again motivated by our previous work, we introduce an alternative perturbative approach that we show to be numerically stable. This approach can be interpreted as a reformulation of classical perturbative algorithms (in suitable independent variables), and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series. It can also be related to classical (preconditioned) spectral approaches and, as such, it retains, in finite arithmetic, the spectral convergence properties of classical perturbative methods, albeit at a higher computational cost (as it does not take advantage of possible dimensional reductions). Still, as we show, an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high-order harmonics and /or perturbation coefficients of the solution.

show abstract

Section: Cancellations and Ill-conditioningmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Nicholls¹,

Reitich²

2001

Journal of Computational Physics

117

View full text Add to dashboard Cite

show abstract

“…Peng & Töksoz 1995), exact absorbing conditions on a spherical contour (e.g. Grote 2000), or asymptotic local or non‐local operators (e.g. Givoli 1991; Hagstrom & Hariharan 1998).…”

Section: Introductionmentioning

confidence: 99%

A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation

Komatitsch¹,

Tromp²

2003

Geophysical Journal International

349

159

View full text Add to dashboard Cite

SUMMARY The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first‐order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second‐order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite‐element method, the spectral‐element method and some finite‐difference methods. We illustrate the efficiency of this second‐order perfectly matched layer based upon 2‐D benchmarks with body and surface waves.

show abstract

“…The same approach can be used to derive exact NBCs for multiple scattering for other wave equations or geometries, such as ellipsoids or wave guides, for which the NBC with a single (convex) artificial boundary is explicitly known. In particular, the derivation presented here extends to time-dependent electromagnetic and elastic wave scattering, where similar boundary conditions are known [11,15,16].…”

Section: Resultsmentioning

confidence: 98%