A Reduced Lagrange Multiplier Method for Dirichlet Boundary Conditions in Isogeometric Analysis

Volume 1 - Issue 1

Shuohui Yin¹*, Tiantang Yu² and Tinh Quoc Bui³

Received: January 22, 2018;   Published: January 29, 2018

DOI: 10.32474/TCEIA.2018.01.000102

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Abstract

Although the well-known standard Lagrange multiplier method (LMM) can even produce higher accuracy and easier implementation than other conventional schemes (e.g., the modified variational principle, the Nitsche’s method), however it inherently owns many difficulties in solving the system of discretized equations, mainly caused by new unknown Lagrange multipliers. The LMM naturally increases the problem size and leads to a poorly conditioned matrix equation. The singularity is also often encountered because of inappropriate selection of interpolation space for the Lagrange multiplier. In this paper, we propose an improved method, called reduced Lagrange multiplier method, which can overcome such drawbacks raised by the LMM in treating the Dirichlet-type boundary conditions in terms of Isogeometric Analysis. By simply splitting the system equations into boundaries and interior groups, the size of system equations derived from the LMM is reduced; no additional unknowns have been added into the resulting system of equations; the Lagrange multiplier is hence disappeared; and more importantly the singular problem mentioned is avoided. The accuracy and convergence rates of the proposed method are studied through three numerical examples, exhibiting all the desirable features of the method. Optimal convergence rate and high accuracy for the present method is found.

Keywords: NURBS; Isogeometric analysis; Dirichlet boundary conditions; Lagrange multiplier method; Finite element method

Abbrevations: LMM: Lagrange Multiplier Method; IGA: Iso-Geometric Analysis; CAD: Computer-Aided DBCs: Design Boundary Conditions; DM: Direct Method; LSCM: Least-Squares Collocation Method; RLMM: Reduced Lagrange Multiplier Method

Abstract| Introduction| NURBS-Based Isogeometric Analysis| Reduced Lagrange Multiplier Method For Dirichlet Boundary Conditions| Numerical Examples and Discussion| Conclusion| Acknowledgement| References|