There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Contact treatment in isogeometric analysis with NURBS

A primal–dual active set strategy for non-linear multibody contact problems

We give a review of Nitsche’s method applied to interface problems, involving real or artificial interfaces Applications to unfitted meshes, Chimera meshes, cut meshes, fictitious domain methods, 

/pdf/nitsche-s-method-for-interface-problems-in-computational-3yhpl55tlg.pdf

Nitsche's method for interface problems in computational mechanics

This paper focuses on the application of NURBS-based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar-based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C0-continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.

/pdf/a-large-deformation-frictional-contact-formulation-using-2hpgnm5xgl.pdf

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

In this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth complementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples.

/pdf/a-primal-dual-active-set-algorithm-for-three-dimensional-4mx7l8mp2w.pdf

A Primal-Dual Active Set Algorithm for Three-Dimensional Contact Problems with Coulomb Friction

Nonconforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations. For the discretization of a nonlinear multibody contact problem, we use linear mortar finite elements based on dual Lagrange multipliers. Under some regularity assumptions on the solution, an optimal convergence order of $h^{0.5+\nu}$, $0<\nu\leq 0.5$, can be established in two dimensions (2D) and three dimensions (3D). Compared with a standard linear saddle point formulation, two additional terms which provide a measure for the nonconformity and the nonlinearity of the approach have to be taken in account. Numerical examples illustrating the performance of the nonconforming method and confirming our theoretical result are presented.

An Optimal A Priori Error Estimate for Nonlinear Multibody Contact Problems

http://www-m2.ma.tum.de/download-publications/hueeber_wohlmuth_07.pdf

Thermo-mechanical contact problems on non-matching meshes

A common approach for the numerical simulation of non-linear multi-body contact problems is the use of Lagrange multipliers to model the contact conditions. The stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes. By this, the spurious algorithmic oscillations in the multiplier do not occur any more, which facilitates the application of the primal–dual active set strategy to dynamical contact problems. The new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost. In addition the conservation properties of the underlying algorithm are transferred to the modified mass version. Different numerical examples for frictional two-body contact problems illustrate the improvement in the results for the contact stresses. Copyright © 2007 John Wiley & Sons, Ltd.

S. Hüeber

Papers

A primal–dual active set strategy for non-linear multibody contact problems

A Primal-Dual Active Set Algorithm for Three-Dimensional Contact Problems with Coulomb Friction

An Optimal A Priori Error Estimate for Nonlinear Multibody Contact Problems

Thermo-mechanical contact problems on non-matching meshes

A stable energy-conserving approach for frictional contact problems based on quadrature formulas