Element-Independent Implementation for Method of Lagrange Multipliers

Authors: Gil-Eon Jeong, Sung-Kie Youn, K. C. Park

Abstract:

Treatment for the non-matching interface is an important computational issue. To handle this problem, the method of Lagrange multipliers including classical and localized versions are the most popular technique. It essentially imposes the interface compatibility conditions by introducing Lagrange multipliers. However, the numerical system becomes unstable and inefficient due to the Lagrange multipliers. The interface element-independent formulation that does not include the Lagrange multipliers can be obtained by modifying the independent variables mathematically. Through this modification, more efficient and stable system can be achieved while involving equivalent accuracy comparing with the conventional method. A numerical example is conducted to verify the validity of the presented method.

Keywords: Element-independent formulation, non-matching interface, interface coupling, methods of Lagrange multipliers.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1128799

References:

[1] F. Ben Belgacem, “The mortar finite element method with Lagrange multipliers,” Numer. Math., vol. 84, pp. 173–197, 1999.
[2] B. I. Wohlmuth, “A mortar finite element method using dual spaces for the Lagrange multiplier,” Soc. Ind. Appl. Math., vol. 38, pp. 989–1012, 2000.
[3] A. Popp, “Mortar Methods for Computational Contact Mechanics and General Interface Problems,” 2012.
[4] J. A. González and K. C. Park, “A simple explicit – implicit finite element tearing and interconnecting transient analysis algorithm,” Int. J. Numer. Methods Eng., vol. 89, pp. 1203–1226, 2012.
[5] M. R. Ross, M. A. Sprague, C. A. Felippa, and K. C. Park, “Treatment of acoustic fluid – structure interaction by localized Lagrange multipliers and comparison to alternative interface-coupling methods,” Comput. Methods Appl. Mech. Eng., vol. 198, pp. 986–1005, 2009.
[6] Y.-U. Song, S.-K. Youn, and K. C. Park, “A gap element for treating non-matching discrete interfaces,” Comput. Mech., vol. 56, no. 3, pp. 551–563, 2015.
[7] K. C. Park, C. a. Felippa, and G. Rebel, “A simple algorithm for localized construction of non-matching structural interfaces,” Int. J. Numer. Methods Eng., vol. 53, pp. 2117–2142, 2002.
[8] M. a. Puso, “A 3D mortar method for solid mechanics,” Int. J. Numer. Methods Eng., vol. 59, no. January 2003, pp. 315–336, 2004.
[9] G. S. Abdoulaev, Y. Achdou, Y. A. Kuznetsov, and C. Prud, “On a parallel implementation of the mortar element method,” Math. Model. Numer. Anal., vol. 33, no. 2, pp. 245–259, 1999.
[10] T. Matsuo, Y. Ohtsuki, and M. Shimasaki, “Efficient linear solvers for mortar finite-element method,” IEEE Trans. Ind. Electron., vol. 43, pp. 1469–1472, 2007.
[11] M. a. Crisfield, “Re-visiting the contact patch test,” Int. J. Numer. Methods Eng., vol. 48, pp. 435–449, 2000.