The methods developed in this thesis are aiming to consistently extend the well-established mor-tar methods in the context of the FEM towards contact of non-smooth geometries, wear modeling and general volume coupled problems. For all three aspects, dual Lagrange multipliers (i.e. dual mortar methods) are very promising since they have been successfully employed in various ap-plications outlined in the previous section. Thus, the general research objective of this thesis is specified in the following. In addition, detailed specifications of requirements are given for each of the three mentioned topics individually at the beginning of the corresponding chapters.
1.3.1. General specification of requirements
Based upon the explanations stated in the previous section, the most important requirements for the improvements of mortar methods are listed and illustrated in the following.
Mortar formulations for point, line and surface contact General interactions of two arbi-trarily formed bodies could not only lead to contacting surfaces, but also contact of edges and vertices. For these scenarios, the existing mortar methods are not desirable, since well-known drawbacks such as large penetrations at vertices and edges would occur. The mentioned large penetrations could be avoided by a node-to-segment scheme at vertices and edges. However, contact stress oscillations for edge (line) contact are inevitable. Thus, it would be desirable to create suitable mortar contact formulations which directly address the interaction of vertices, edges and surface in one comprehensive model. Up to now, there is no such a model available in the existing literature on computational contact mechanics and mortar methods. In addition,
1.3. Research objective dual Lagrange multipliers are the preferred discretization approach for the development of a combined point, line and surface contact formulation since they naturally allow for an easy and efficient elimination of the arising Lagrange multiplier unknowns.
Fretting wear modeling within mortar contact frameworkThe fretting wear phenomenon is usually modeled as an additional contribution to the gap function, which leads to slightly overlapping bodies up to an extent which is equivalent to the wear depth. For 3D mortar contact, such an approach for wear modeling is published in Cavalieri and Cardona [36] and Cavalieri et al. [37]. But, this model does not consider frictional effects and no beneficial dual Lagrange multipliers are employed. The only frictional contact formulation based on dual mortar methods for fretting wear has been published in Gitterle [87], being restricted to 2D problems. The ac-curate and efficiently calculated results from Gitterle [87] motivate to realize a general 3D wear formulation based on dual mortar methods, which will be presented in this thesis.
Finite wear modeling within mortar contact framework The effect of material loss at the contact interface due to wear resulting into finite shape changes is only rarely addressed in the existing literature on mortar methods for computational contact mechanics. The only known pub-lications which address this topic are Doca [61] and Gitterle [87], but only for 2D models and explicit time integration of the wear phenomena. In Stupkiewicz [263], it has been demonstrated that implicit treatment of wear effects is necessary in order to obtain stable results, especially when performing simulations based on a steady-state assumption with large time step sizes.
However, the construction of an implicit finite wear framework based on dual mortar methods is a hitherto unanswered question.
Dual mortar methods for 3D projection operatorFor a large variety of applications, projec-tion of nodal informaprojec-tion between two different meshes is required. Despite classical collocaprojec-tion methods being well-established for this purpose, nodal information transfer methods based on weak conservation properties are often beneficial to satisfy overall conservation demands. Mor-tar projection operators for nodal information transfer of 2D problems have been developed and applied in Dureisseix and Bavestrello [64] and Néron and Dureisseix [181]. In addition, the mor-tar method as basis of a 3D projection operator has already been employed in Bussetta et al. [35].
But, all mentioned publications are based on the standard mortar approach. Consequently, their construction becomes very costly. A volume projection operator based on dual mortar methods has not been developed in the existing literature, although numerical efficiency of dual mor-tar methods has been demonstrated by several authors for DD and contact applications, see for example Gitterle et al. [88], Popp et al. [212] and Wohlmuth [289]. Consequently, it is very promising in terms of computational efficiency to develop such a projection operator with help of dual mortar methods.
Volume coupled multiphysics on non-matching meshes Classical mortar methods are de-signed in order to allow for flexible discretization via interface coupling of subdomains with non-matching meshes. Transferring this idea towards the context of coupled multiphysics, it would be desirable to allow for different bulk discretizations of the involved fields. Such an ap-proach has been published in Dureisseix and Bavestrello [64] and Néron and Dureisseix [181], but it is restricted to 2D problems, which are solved within a partitioned solution scheme.
Mono-lithic solution schemes are often proven to be of superior robustness compared to partitioned counterparts. This causes the need for having a general methodology, which allows for solving volume coupled multiphysics on non-matching meshes within a monolithic approach. In ad-dition, contact interaction of two bodies with multiphysics effects and non-matching interface and volume discretizations has never been considered in the existing literature. Aiming for high-est possible flexibility with respect to spatial discretization, these topics need further exploration.
1.3.2. Proposal for novel mortar approaches
This thesis describes the consistent extension of the mortar finite element method to computa-tional contact mechanics of complex geometries, wear simulations and general volume coupled problems. The most important ingredients and new scientific contributions of the presented ap-proaches are given in the following:
• the first consistent extension of the dual mortar contact formulation to point, line and sur-face contact scenarios of vertices, edges and sursur-faces being involved in a finite deformation regime, see also Farah et al. [71].
• development of the first dual mortar formulation for the calculation of fretting wear prob-lems in 3D, see also Farah et al. [69].
• implementation of the first fully implicit finite wear algorithm in a non-steady-state regime based on dual mortar methods with an Arbitrary-Lagrangian-Eulerian approach and first extensions towards thermo-structure interaction problems, see also Farah et al. [74].
• successful extension of the dual mortar method to a 3D information transfer scheme with application to monolithic volume coupled multi-field problems, e.g. porous media, thermo-structure interaction, fluid-structure interaction and thermo-structure-contact in-teraction, see also Farah et al. [70] and La Spina et al. [145].
All methods and models devised as part of this work have been implemented in the in-house C++ code BACI (cf. Wall et al. [282]) of the Institute for Computational Mechanics at the Tech-nical University of Munich. This code integrated open-source libraries of the Trilinos Project conducted by Sandia National Laboratories, see Heroux et al. [102]. The basic data structures and existing features like time integration schemes or iterative solution techniques were reused within this thesis. Other modules have been adapted or were written completely from scratch.