Improving the Modified XFEM for Optimal High-Order ApproximationCitation formats

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Improving the Modified XFEM for Optimal High-Order Approximation. / Saxby, Ben; Hazel, Andrew.

In: International Journal for Numerical Methods in Engineering, 03.09.2019.

Research output: Contribution to journalArticle

Harvard

Saxby, B & Hazel, A 2019, 'Improving the Modified XFEM for Optimal High-Order Approximation', International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.6214

APA

Saxby, B., & Hazel, A. (2019). Improving the Modified XFEM for Optimal High-Order Approximation. International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.6214

Vancouver

Saxby B, Hazel A. Improving the Modified XFEM for Optimal High-Order Approximation. International Journal for Numerical Methods in Engineering. 2019 Sep 3. https://doi.org/10.1002/nme.6214

Author

Saxby, Ben ; Hazel, Andrew. / Improving the Modified XFEM for Optimal High-Order Approximation. In: International Journal for Numerical Methods in Engineering. 2019.

Bibtex

@article{785cb7b564f943fba196de3559f7608d,
title = "Improving the Modified XFEM for Optimal High-Order Approximation",
abstract = "This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method (SGFEM), is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher order contributions to the approximation space. An improved modified XFEM is presented, with basis functions ‘corrected’ by projecting out higher order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order 푝≤4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity, and also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order 푝 > 1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order 푝≤4.",
keywords = "extended finite element method, high-order accuracy, curved weak discontinuity, optimal convergence",
author = "Ben Saxby and Andrew Hazel",
year = "2019",
month = "9",
day = "3",
doi = "10.1002/nme.6214",
language = "English",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley & Sons Ltd",

}

RIS

TY - JOUR

T1 - Improving the Modified XFEM for Optimal High-Order Approximation

AU - Saxby, Ben

AU - Hazel, Andrew

PY - 2019/9/3

Y1 - 2019/9/3

N2 - This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method (SGFEM), is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher order contributions to the approximation space. An improved modified XFEM is presented, with basis functions ‘corrected’ by projecting out higher order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order 푝≤4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity, and also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order 푝 > 1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order 푝≤4.

AB - This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method (SGFEM), is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher order contributions to the approximation space. An improved modified XFEM is presented, with basis functions ‘corrected’ by projecting out higher order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order 푝≤4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity, and also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order 푝 > 1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order 푝≤4.

KW - extended finite element method

KW - high-order accuracy

KW - curved weak discontinuity

KW - optimal convergence

U2 - 10.1002/nme.6214

DO - 10.1002/nme.6214

M3 - Article

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

ER -