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

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

Journal Title: Trends in Civil Engineering and its Architecture - Year 2018, Vol 1, Issue 1

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. The Isogeometric analysis (IGA) [1] employs the same computer- aided design (CAD) spline basis functions (e.g., NURBS, T-splines) to describe exact geometries and approximate physical responses. The IGA offers possibilities of integrating finite element analysis into CAD design tools and avoids mesh generation process and subsequent communication with a CAD model during refinement. The IGA has shown to be a highly accurate and robust approach for numerical simulation with exact geometry representation even with coarse mesh [2]. Moreover, the high-order smoothness of the NURBS basis functions allows for a direct construction of rotation- free plate/shell formulations [3,4] and is attractive for solving PDEs that incorporate fourth-order (or higher) derivatives of the field variable, such as the Hill-Cahnard equation [Gomez et al., 2008] [5] and gradient damage models [6].

Authors and Affiliations

Shuohui Yin, Tiantang Yu, Tinh Quoc Bui

Keywords

Lupine Publishers Lupine Publishers Group lupine publishers indexing Journals lupine publishers indexing list lupine publishers indexing sites Indexed Journals indexed journals list List of Indexed Journals lupine publishers impact factor lupine publishers open access journals lupine publishers indexing Lupine group Lupine online open access open access journals open access journals publishers 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 lupinepublishers Civil engineering Shuohui Yin

EP ID EP604402
DOI 10.32474/TCEIA.2018.01.000102
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