Abstract
In recent years, isogeometric analysis (IGA) has received great attention in many fields of computational mechanics research. Especially for computational contact mechanics, an exact and smooth surface representation is highly desirable. As a consequence, many well-known finite element methods and algorithms for contact mechanics have been transferred to IGA. In the present contribution, the so-called dual mortar method is investigated for both contact mechanics and classical domain decomposition using NURBS basis functions. In contrast to standard mortar methods, the use of dual basis functions for the Lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional Lagrange multiplier degrees of freedom from the global system. This condensed system is smaller in size, and no longer of saddle point type but positive definite. A very simple and commonly used element-wise construction of the dual basis functions is directly transferred to the IGA case. The resulting Lagrange multiplier interpolation satisfies discrete inf–sup stability and biorthogonality, however, the reproduction order is limited to one. In the domain decomposition case, this results in a limitation of the spatial convergence order to O(h32) in the energy norm, whereas for unilateral contact, due to the lower regularity of the solution, optimal convergence rates are still met. Numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order Lagrange polynomials.
Highlights
Robust and accurate contact discretizations for nonlinear finite element analysis have been an active field of research in the past decade and a new class of formulations emerged with the introduction of isogeometric analysis (IGA) [1]
IGA is intended to bridge the gap between computer aided design (CAD) and finite element analysis (FEA) by using the smooth non-uniform rational B-splines (NURBS) or T-splines common in CAD as a basis for the numerical analysis
This high continuity results, amongst others, in a smooth surface representation which makes the application to computational contact mechanics appealing, which has already been anticipated in the original proposition of IGA by Hughes et al [1]
Summary
Robust and accurate contact discretizations for nonlinear finite element analysis have been an active field of research in the past decade and a new class of formulations emerged with the introduction of isogeometric analysis (IGA) [1]. In the context of domain decomposition in IGA, optimality and stability of standard mortar methods have only very recently been investigated in [30,31,32,33], where the construction of dual B-spline basis functions has been outlined theoretically [33] In this contribution, we present a mortar contact formulation for IGA using a dual Lagrange multiplier method where the inequality constraints are reformulated using nonlinear complementarity (NCP) functions. The newly developed isogeometric dual mortar method is applied to both domain decomposition and (unilateral) frictional contact problems with finite deformations In both applications, spatial convergence orders for the discretization error using uniform mesh refinement are studied numerically.
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