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Algorithms 2012, 5(4), 529-544; doi:10.3390/a5040529

Finite Element Quadrature of Regularized Discontinuous and Singular Level Set Functions in 3D Problems

1,* , 2
 and 1
1 Engineering Department, University of Ferrara, via Saragat 1, 44122 Ferrara, Italy 2 Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
* Author to whom correspondence should be addressed.
Received: 10 September 2012 / Revised: 22 October 2012 / Accepted: 29 October 2012 / Published: 7 November 2012
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Regularized Heaviside and Dirac delta function are used in several fields of computational physics and mechanics. Hence the issue of the quadrature of integrals of discontinuous and singular functions arises. In order to avoid ad-hoc quadrature procedures, regularization of the discontinuous and the singular fields is often carried out. In particular, weight functions of the signed distance with respect to the discontinuity interface are exploited. Tornberg and Engquist (Journal of Scientific Computing, 2003, 19: 527–552) proved that the use of compact support weight function is not suitable because it leads to errors that do not vanish for decreasing mesh size. They proposed the adoption of non-compact support weight functions. In the present contribution, the relationship between the Fourier transform of the weight functions and the accuracy of the regularization procedure is exploited. The proposed regularized approach was implemented in the eXtended Finite Element Method. As a three-dimensional example, we study a slender solid characterized by an inclined interface across which the displacement is discontinuous. The accuracy is evaluated for varying position of the discontinuity interfaces with respect to the underlying mesh. A procedure for the choice of the regularization parameters is proposed.
Keywords: 3D-quadrature; regularization; XFEM; interface 3D-quadrature; regularization; XFEM; interface
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Benvenuti, E.; Ventura, G.; Ponara, N. Finite Element Quadrature of Regularized Discontinuous and Singular Level Set Functions in 3D Problems. Algorithms 2012, 5, 529-544.

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