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Weighted integral solvers for elastic scattering by open arcs in two dimensions

Bruno, Oscar P. and Xu, Liwei and Yin, Tao (2021) Weighted integral solvers for elastic scattering by open arcs in two dimensions. International Journal for Numerical Methods in Engineering, 122 (11). pp. 2733-2750. ISSN 0029-5981. doi:10.1002/nme.6639.

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We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral‐equation solutions at the open‐arc endpoints. Crucially, the method also incorporates a certain “open‐arc elastic Calderón relation” introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open‐arc elastic Calderón relation generalizes a corresponding elastic Calderón relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open‐surface Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov‐subspace linear algebra solver GMRES, the proposed overall open‐arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2021 John Wiley & Sons, Ltd. Issue Online: 05 May 2021; Version of Record online: 25 February 2021; Accepted manuscript online: 25 January 2021; Manuscript accepted: 21 January 2021; Manuscript revised: 21 August 2020; Manuscript received: 21 February 2020. Funding information: Air Force Office of Scientific Research, FA9550‐15‐1‐0043; Defense Advanced Research Projects Agency, HR00111720035; National Science Foundation, DMS‐1714169; Office of Naval Research, N00014‐16‐1‐2808. Data Availability Statement: The authors confirm that the data supporting the findings of this study are available within the article.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-15-1-0043
Defense Advanced Research Projects Agency (DARPA)HR00111720035
Office of Naval Research (ONR)N00014‐16‐1‐2808
Subject Keywords:Calderón relation; elasticity; open arc; second‐kind integral solver
Issue or Number:11
Record Number:CaltechAUTHORS:20190906-093843706
Persistent URL:
Official Citation:Bruno, OP, Xu, L, Yin, T. Weighted integral solvers for elastic scattering by open arcs in two dimensions. Int J Numer Methods Eng. 2021; 122: 2733–2750.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98477
Deposited By: George Porter
Deposited On:06 Sep 2019 16:53
Last Modified:07 May 2021 18:00

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