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 e lement m ethods a nd a lgorithms f or c ontact m echanics h ave b een t ransferred t o I GA. I n t he 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(h 3 /2 ) 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.

No abstract

Summary This work introduces a novel, mortar‐based coupling scheme for electrode‐electrolyte interfaces in 3‐dimensional finite element models for lithium‐ion cells and similar electrochemical systems. The coupling scheme incorporates the widely applied Butler‐Volmer charge transfer kinetics, but conceptually also works for other interface equations. Unlike conventional approaches, the coupling scheme allows flexible mesh generation for the electrode and electrolyte phases with nonmatching meshes at electrode‐electrolyte interfaces. As a result, the desired spatial mesh resolution in each phase and the resulting computational effort can be easily controlled, leading to improved efficiency. All governing equations are solved in a monolithic fashion as a holistic, unified system of linear equations for computational robustness and performance reasons. Consistency and optimal convergence behavior of the coupling scheme are demonstrated in elementary numerical tests, and the discharge of two different realistic lithium‐ion cells, each consisting of an anode, a cathode, and an electrolyte, is also simulated. One of the two cells involves about 1.35 million degrees of freedom and very complex microstructural geometries obtained from X‐ray tomography data. For validation purposes, characteristic numerical results from the literature are reproduced, and the coupling scheme is shown to require considerably fewer degrees of freedom than a standard discretization with matching interface meshes to achieve a similar level of accuracy.

Summary Finite deformation contact problems with frictional effects and finite shape changes due to wear are investigated. To capture the finite shape changes, a third configuration besides the well‐known reference and spatial configurations is introduced, which represents the time‐dependent worn state. Consistent interconnections between these states are realized by an arbitrary Lagrangean–Eulerian formulation. The newly developed partitioned and fully implicit algorithm is based on a Lagrangean step and a shape evolution step. Within the Lagrangean step, contact constraints as well as the wear equations are weakly enforced following the well‐established mortar framework. Additional unknowns due to the employed Lagrange multiplier method for contact constraint enforcement and due to wear itself are eliminated by condensation procedures based on the concept of biorthogonality and the so‐called dual shape functions. Several numerical examples in both 2D and 3D are provided to demonstrate the performance and accuracy of the proposed numerical algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

A finite element framework based on dual mortar methods is presented for simulating fretting wear effects in the finite deformation regime. The mortar finite element discretization is realized with Lagrangean shape functions as well as isogeometric elements based on non-uniform rational B-splines (NURBS) in two and three dimensions. Fretting wear effects are modeled in an incremental scheme with the help of Archard’s law and the worn material is considered as additional contribution to the gap function. Numerical examples demonstrate the robustness and accuracy of the presented algorithm.