##
**Accuracy of three-dimensional analysis of regularized singularities.**
*(English)*
Zbl 1352.65080

Summary: In computational mechanics, the quadrature of discontinuous and singular functions is often required. To avoid specialized quadrature procedures, discontinuous and singular fields can be regularized. However, regularization changes the algebraic structure of the solving equations, and this can lead to high errors. We show how to acquire accurate and consistent results when regularization is carried out. A three-dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method. The accuracy obtained via Gaussian quadrature is compared with that obtained by means of CUBPACK adaptive quadrature FORTRAN tool. The use of regularized functions with non-compact and compact support is investigated through an error evaluation procedure based on the use of their Fourier transform. The proposed procedure leads to the remarkable conclusion that regularized delta functions with non-compact support exhibit superior performance.

### MSC:

65D30 | Numerical integration |

### Software:

CUBPACK
PDF
BibTeX
XML
Cite

\textit{E. Benvenuti} et al., Int. J. Numer. Methods Eng. 101, No. 1, 29--53 (2015; Zbl 1352.65080)

Full Text:
DOI

### References:

[1] | Chorin, Numerical study of slightly viscous flow, Journal of Fluid Mechanics 57 pp 785– (1973) |

[2] | Peskin, Numerical analysis of blood flow in the heart, Journal of Computational Physics 25 pp 220– (1977) · Zbl 0403.76100 |

[3] | Beyer, Analysis of a one-dimensional model for the immersed boundary method, SIAM Journal of Numerical Analysis 29 pp 332– (1992) · Zbl 0762.65052 |

[4] | Chen, Phase field models for microstructure evolution, Annual Review of Materials Research 32 pp 113– (2002) |

[5] | Lee, Regularized Dirac delta functions for phase field models, International Journal for Numerical Methods in Engineering 91 pp 269– (2012) · Zbl 1246.76148 |

[6] | Leveque, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal of Numerical Analysis 31 pp 1019– (1994) · Zbl 0811.65083 |

[7] | Abbas, The XFEM for high-gradient solutions in convection-dominated problems, International Journal for Numerical Methods in Engineering 82 pp 1044– (2010) · Zbl 1188.76224 |

[8] | Mousavi, Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds, International Journal for Numerical Methods in Engineering 91 pp 343– (2012) · Zbl 1253.65034 |

[9] | Xu, A two-dimensional co-rotational Timoshenko beam element with XFEM formulation, Computational Mechanics 49 pp 667– (2012) · Zbl 1398.74423 |

[10] | Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 |

[11] | Löhnert, 3D corrected XFEM approach and extension to finite deformation theory, International Journal for Numerical Methods in Engineering 86 pp 431– (2011) · Zbl 1216.74026 |

[12] | Müller, Simple multidimensional integration of discontinuous functions with application to level set methods, International Journal for Numerical Methods in Engineering 92 pp 637– (2012) · Zbl 1352.65084 |

[13] | Dooren, An adaptive algorithm for numerical integration over an n-dimensional cube, Journal of Computational and Applied Mathematics 2 pp 207– (1976) · Zbl 0334.65024 |

[14] | Mousavi, Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 199 pp 3237– (2012) · Zbl 1225.74099 |

[15] | Ventura, On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, International Journal for Numerical Methods in Engineering 66 pp 761– (2006) · Zbl 1110.74858 |

[16] | Tornberg, Multi-dimensional quadrature of singular and discontinuous functions, BIT Numerical Mathematics 42 pp 644– (2002) · Zbl 1021.65010 |

[17] | Tornberg, Regularization techniques for numerical approximation of PDEs with singularities, Journal of Scientific Computing 19 pp 527– (2003) · Zbl 1035.65085 |

[18] | Brandt, Multilevel matrix multiplication and fast solution of integral equations, Journal of Computational Physics 90 pp 348– (1990) · Zbl 0707.65025 |

[19] | Benvenuti, The fast Gauss transform for non-local integral FE models, Communications in Numerical Methods in Engineering 22 pp 505– (2006) · Zbl 1105.65345 |

[20] | Osher, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics 79 pp 12– (1988) · Zbl 0659.65132 |

[21] | Zahedi, Delta function approximations in level set methods by distance function extension, Journal of Computational Physics 229 pp 2199– (2010) · Zbl 1186.65018 |

[22] | Belytschko, A review of the extended/generalized finite element methods for material modelling, Modelling and Simulation in Materials Science and Engineering 17 (; 2009) |

[23] | Fries, The extended/generalized finite element method: an overview of the method and its applications, International Journal for Numerical Methods in Engineering 84 pp 253– (2010) · Zbl 1202.74169 |

[24] | Gravouil, Stabilized globallocal X-FEM for 3D non-planar frictional crack using relevant meshes, International Journal for Numerical Methods in Engineering 88 pp 1449– (2011) · Zbl 1242.74121 |

[25] | Benvenuti, A regularized XFEM model for the transition from continuous to discontinuous displacements, International Journal for Numerical Methods in Engineering 74 pp 911– (2008) · Zbl 1158.74479 |

[26] | Benvenuti, Mesh-size-objective XFEM for regularized continuous discontinuous transition, Finite Elements in Analysis and Design 47 pp 1326– (2011) |

[27] | Benvenuti, Variationally consistent extended FE model for 3D planar and curved imperfect interfaces, Computer Methods in Applied Mechanics and Engineering 267 pp 434– (2013) · Zbl 1286.74095 |

[28] | Benvenuti, XFEM with equivalent eigenstrain for matrix-inclusion interfaces, Computational Mechanics 53 pp 893– (2014) · Zbl 1398.74301 |

[29] | Benvenuti, A regularized XFEM framework for embedded cohesive interfaces, Computer Methods in Applied Mechanics and Engineering 197 pp 4367– (2008) · Zbl 1194.74364 |

[30] | Benvenuti, Simulation of finite-width process zone for concrete-like materials, Computational Mechanics 50 pp 479– (2012) · Zbl 1398.74302 |

[31] | Benvenuti, Delamination of FRP-reinforced concrete by means of an extended finite element formulation, Composite Part B: Engineering 43 pp 3258– (2012) |

[32] | Ventura, Fast integration and weight function blending in the extended finite element method, International Journal for Numerical Methods in Engineering 77 pp 1– (2009) · Zbl 1195.74201 |

[33] | Fries, A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering 75 pp 503– (2007) · Zbl 1195.74173 |

[34] | Patzák, Process zone resolution by extended finite elements, Engineering Fracture Mechanics 70 pp 957– (2003) |

[35] | Iarve, Mesh-independent matrix cracking and delamination modeling in laminated composites, International Journal for Numerical Methods in Engineering 88 pp 749– (2011) · Zbl 1242.74126 |

[36] | Kolmogorov, Introductory Real Analysis (1975) |

[37] | Areias, Two-scale method for shear bands: thermal effects and variable bandwidth, International Journal for Numerical Methods in Engineering 72 pp 658– (2007) · Zbl 1194.74355 |

[38] | Tornberg, Numerical approximations of singular source terms in differential equations, Journal of Computational Physics 200 pp 462– (2004) · Zbl 1115.76392 |

[39] | Benvenuti, Finite element quadrature of regularized discontinuous and singular level set functions in 3D problems, Algorithms 5 pp 529– (2012) |

[40] | Akisanya, Interfacial cracking from the free-edge of a long bi-material strip, International Journal of Solids and Structures 34 pp 1645– (1997) · Zbl 0944.74626 |

[41] | Rao, Stress concentrations and singularities at interface corners, Zeitschrift fur Angewandte Mathematik und Mechanik 51 pp 395– (1971) · Zbl 0228.73092 |

[42] | Sinclair, Stress singularities in classical elasticity-I: removal, interpretation, and analysis, Applied Mechanics Reviews 57 pp 251– (2004) |

[43] | Sinclair, Stress singularities in classical elasticity-II: asymptotic identification, Applied Mechanics Reviews 57 pp 385– (2004) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.