The thin-shape breakdown (TSB) of the Helmholtz integral equation

Martínez, R.

doi:10.1121/1.2028750

Cited by 20 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If, however, it is required to evaluate first, or higher, derivatives of I, the integral is improper and more advanced approaches are required. Such integrals arise in the calculation of acoustic scattering by thin bodies [16], due to the "thin-shape breakdown" of the Helmholtz equation, when the Burton and Miller approach is used to avoid spurious resonances [5] and when gradients of the potential are required as, for example, in computing velocities in potential problems in fluid dynamics. It should be noted that in many problems, such as those in axisymmetric domains, f (x, t) will contain logarithmic terms of the form (x − t) log |x − t| which are exposed by differentiation so that the integrand will contain singularities of more than one order.…”

Section: Introduction Boundary Integral Methods Employing Hypersingumentioning

confidence: 99%

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Carley¹

2007

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract.A method is developed for the computation of the weights and nodes of a numerical quadrature which integrates functions containing singularities up to order 1/x 2 , without the requirement to know the coefficients of the singularities exactly. The work is motivated by the need to evaluate such integrals on boundary elements in potential problems and is a simplification of a previously published method, but with the advantage of handling singularities at the endpoints of the integral. The numerical performance of the method is demonstrated by application to an integral containing logarithmic, first, and second order singularities, characteristic of the problems encountered in integrating a Green's function in boundary element problems. It is found that the quadrature is accurate to 11-12 decimal places when computed in double precision. 1. Introduction. Boundary integral methods employing hypersingular integrals have become increasingly popular over recent decades with applications in potential problems such as acoustics [6] and fracture mechanics [13]. Hypersingular integrals arise naturally when it is required to compute the field quantities in such problems, for example, the potential and its gradients, and when specialized techniques for the avoidance of "interior resonance" are used, such as that of Burton and Miller [5]. When such integrals arise, they require special numerical treatment and cannot be evaluated using the standard tools of Gaussian quadrature. Instead, approaches tailored to the problem must be used, which can add considerable complexity to the code. This paper introduces a method for the design of numerical quadrature rules which evaluate hypersingular integrals without requiring a detailed analysis of the integrand. The method is based on that of Kolm and Rokhlin [12], who developed a procedure for the design of such rules, but is considerably simplified by the use of Brandão's approach to finite part integrals [4] and is extended to the case of integrals with endpoint singularities-essential if such a scheme is to be used with boundary elements.To fix ideas, we assume that we are dealing with a two-dimensional potential problem, such as the Laplace or Helmholtz equation:

show abstract

Section: Introduction Boundary Integral Methods Employing Hypersingumentioning

confidence: 99%

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Carley¹

2007

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…A serious shortcoming is the failure when it is applied to bodies of thin shape or regular bodies with thin appendages. This thin-shape breakdown of the direct formulation was investigated extensively by Martinez 15 and recently by Cutanda et al, 16 and remedies ensuring a meaningful formulation for thin shapes were proposed.…”

Section: Boundary Integral Formulations Of the Forward Problemmentioning

confidence: 99%

Sound source reconstruction using inverse boundary element calculations

Schuhmacher¹,

Hald²,

Rasmussen

et al. 2003

The Journal of the Acoustical Society of America

130

View full text Add to dashboard Cite

Whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure, the aim in this work is to reverse the process, i.e., to determine vibration from acoustic field data. This inverse problem is brought on a form suited for solution by means of an inverse boundary element method. Since the numerical treatment of the inverse source reconstruction results in a discrete ill-posed problem, regularization is imposed to avoid unstable solutions dominated by errors. In the present work the emphasis is on Tikhonov regularization and parameter-choice methods not requiring an error-norm estimate for choosing the right amount of regularization. Several parameter-choice strategies have been presented lately, but it still remains to be seen how well these can handle industrial applications with real measurement data. In the present work it is demonstrated that the L-curve criterion is robust with respect to the errors in a real measurement situation. In particular, it is shown that the L-curve criterion is superior to the more conventional generalized cross-validation ͑GCV͒ approach for the present tire noise studies.

show abstract

“…In particular, the complementary interior problem is now an extremely thin cavity, so areas of the cavity walls opposite one another interact very strongly, dominating over their interactions with other parts of the body and making the problem ill-posed. This phenomenon is known as Thin-Shape Breakdown (TSB) and has been found to cause frequency-independent illconditioning in the solution stage of frequency domain BEM models 16 . The TSB also affects the time domain BEM, where it is most likely to manifest as solver instability.…”

Section: Obstacles With Thin Appendagesmentioning

confidence: 99%