Premium
Interface problems with quadratic X‐FEM: design of a stable multiplier space and error analysis
Author(s) -
Ferté G.,
Massin P.,
Moës N.
Publication year - 2014
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4787
Subject(s) - lagrange multiplier , quadratic equation , finite element method , mathematics , multiplier (economics) , a priori and a posteriori , convergence (economics) , interpolation (computer graphics) , lagrange polynomial , space (punctuation) , mathematical analysis , quadratic programming , mathematical optimization , computer science , geometry , physics , polynomial , animation , philosophy , computer graphics (images) , epistemology , economics , economic growth , operating system , thermodynamics , macroeconomics
SUMMARY The aim of this paper is to propose a procedure to accurately compute curved interfaces problems within the extended finite element method and with quadratic elements. It is dedicated to gradient discontinuous problems, which cover the case of bimaterials as the main application. We focus on the use of Lagrange multipliers to enforce adherence at the interface, which makes this strategy applicable to cohesive laws or unilateral contact. Convergence then occurs under the condition that a discrete inf‐sup condition is passed. A dedicated P1 multiplier space intended for use with P2 displacements is introduced. Analytical proof that it passes the inf‐sup condition is presented in the two‐dimensional case. Under the assumption that this inf‐sup condition holds, a priori error estimates are derived for linear or quadratic elements as functions of the curved interface resolution and of the interpolation properties of the discrete Lagrange multipliers space. The estimates are successfully checked against several numerical experiments: disparities, when they occur, are explained in the literature. Besides, the new multiplier space is able to produce quadratic convergence from P2 displacements and quadratic geometry resolution. Copyright © 2014 John Wiley & Sons, Ltd.