6. Mortar Methods for Volume Coupled Problems 153
6.1.1. Fundamental approaches and applications
In the following, the fundamental approaches on nodal information transfer methods for volume coupled problems are reviewed. In addition, their strongly related counterparts in the field of computational contact mechanics are mentioned. Although contact mechanics and general inter-face mechanics are based on constraint enforcement techniques such as penalty regularization and Lagrange multiplier approaches, their type of constraint discretization is crucial for the per-formance of these methods. Therein, interface projection operators for nodal information trans-fer implicitly arise, which share a lot of conceptual similarities with abstract volume operators.
Thus, the following explanations provide a state-of-the-art overview of general methodologies for nodal information transfer. Moreover, their various fields of application which are beyond the scope of computational contact mechanics are reviewed.
The easiest way of transferring nodal information is a node-to-node(NTN) approach, where each coupling pair consists of two nodes. It is also commonly known as matching grid dis-cretization. Here, it is assumed that the nodes initially match and thus the coupling constraints or the information transfer can be enforced individually for each node pair. This approach was implicitly employed in many multiphysics problems, such as for poroelastic media simulations in Vuong et al. [278], thermo-structure interaction in Danowski et al. [52] and fluid-structure interaction problems with moving grid approach in Ramm and Wall [225] and Wall [280]. In the context of computational contact mechanics, such an approach has been employed in Fran-cavilla and Zienkiewicz [84], Hughes et al. [120] and Ireman et al. [125]. In a regime of in-finitesimal deformation this approach is valid since no significant relative movements between the domains are allowed. However, for classical structural applications in a finite deformation regime together with a pure Lagrangian description the NTN approach obviously fails. To over-come this problem, an Arbitrary-Lagrangian-Eulerian (ALE) approach was developed in Benson [24] and Haber [93], which split physical material point motion and mesh motion to enforce a matching grid setting at the contact interface. Such an approach was also applied for finite wear simulations in Chapter 5 and the interested reader is also referred to the explanations therein.
However, this approach is still restricted to nearly equally discretized contact interfaces and thus it is not the approach of choice for contact simulations. For general domain decomposition ap-plications with non-matching meshes the NTN approach is obviously not an adequate choice.
In order to treat the problem of non-matching discretizations that naturally restricts the NTN approach, a scheme based on coupling pairs consisting of a node and an element was developed, which is denoted asnode-to-segment(NTS) scheme. A NTS scheme for nodal information trans-fer enforces the coupling point-wise (strong) at each node. It was employed for example in Bus-setta et al. [35], Ortiz and Quigley [190], Ramm et al. [226] and Saksono and Peri´c [239]. The more general form of the NTS approach is denoted as collocation scheme, where the coupling pair consists of a general collocation point and an element. Then, the nodal information is inter-polated at these collocation points and after the projection is completed, the nodal values from the target mesh can be recovered. In the literature on information transfer methods, these ap-proaches are often termed collocation schemes although the information transfer is enforced at the nodes, see Dureisseix and Bavestrello [64]. Since this chapter focuses on information transfer schemes, the naming collocation method is maintained in the following of this thesis. In
addi-6.1. Fundamental approaches and research objective tion, it should be noted that among various other declarations, the NTS type of discretization is also usually calledelement transfer methodin some publications, see for example Bussetta et al.
[35]. In the context of computational contact mechanics, such node-wise enforcement is indeed commonly termed NTS discretization and has been proposed in Hallquist [97] and Hughes et al.
[120]. It has successfully been extended to more general contact cases in Bathe and Chaudhary [14], Hallquist et al. [96], Papadopoulos and Taylor [195], Simo et al. [253] and Wriggers et al.
[296].
Since the collocation approach enforces the nodal information transfer in a point-wise (strong) sense, it cannot guarantee for weak conservation properties and does not naturally fit in the fi-nite element philosophy. To be consistent in a fifi-nite element sense, weak conservation methods have been developed, see for example Orlando and Peri´c [188] and Orlando [189]. These types of nodal information transfer schemes are denoted assegment-to-segment(STS) methods in the following. Due to their characteristic of weak information conservation, they naturally require for numerical integration procedures, which cause increased computational costs compared to the previously introduced methods. For computational contact methods, first investigations re-garding the weak enforcement of contact constraints can be found in Papadopoulos and Taylor [195] and Simo et al. [253]. Themortar methodcan be identified as special type of STS methods.
As already introduced in this thesis, it is based on a separation of a so-called slave (target) and master (source) mesh and the nodal information transfer is based on weak conservation assump-tions over the slave side, see Bernardi et al. [25, 26]. The mortar method has been employed for the creation of general volume coupling operators in Dureisseix and Bavestrello [64] and Néron and Dureisseix [181]. Therein, the computation of the mortar operators is based on the construc-tion of a so-called "super mesh", from which each element is included in only one element of the source and target mesh. This basically describes the integration cells/segments for the segment-based integration procedure from Section 3.4.2. This was also employed in Farrell and Maddison [76] and Farrell et al. [77] for the construction of projection operators, which are strongly related to mortar operators. Mortar methods for computational contact mechanics have been already in-troduced in Chapter 3 and the interested reader is referred to the explanations therein and the literature Gitterle [87], Gitterle et al. [88], Popp [210], Popp et al. [211, 212], Puso and Laursen [218] and Yang et al. [301]. For the sake of completeness, it should be noticed that operators for nodal information transfer based on weak conservation properties have also been developed based on finite volume (FV) schemes, see Alauzet and Mehrenberger [4] and Rashid [227].
Finally, besides the already mentioned field of computational contact mechanics, the various types of applications for nodal information transfer schemes are briefly given. One of the most obvious type of application is the remeshing operation, which might becomes necessary for structural mechanics where extremely large deformations occur. Remeshing procedures can be found in Bussetta et al. [35], Fernandes and Martins [78] and Peri´c et al. [203]. Other application types are global/local schemes in Gould and Hara [91], Mote [176] and Voleti et al. [276], the Arlequin method for structural problems in Dhia [58] and Dhia and Rateau [59], the Chimera scheme for fluid problems in Meakin [169], Renaud et al. [229], Steger and Benek [258] and Wall et al. [281] and geometrical multigrid methods in Adams [2] and Biotteau et al. [28]. Nodal in-formation transfer schemes have also been applied for the simulation of volume coupled mul-tiphysics problems on different meshes for the individual fields, see Dureisseix and Bavestrello [64] and Néron and Dureisseix [181]. Yet, in these publications only 2D multiphysics problems solved within a partitioned scheme are considered. However, it has been proven, that monolithic
solution schemes lead to superior robustness for coupled multiphysics problems, see for exam-ple Danowski et al. [52], Gee et al. [85] and Verdugo and Wall [275]. Hence, it seems promising to develop methodologies to allow for monolithic multiphysics solvers on non-matching meshes.
6.1.2. Specification of requirements
Based on the previous explanations on already existing coupling approaches, the most important requirements for the development of accurate and efficient mortar approaches for nodal infor-mation transfer and their application to multiphysics problems are listed in the following.
Computational efficiency due to dual shape functions Compared to collocation methods, mortar approaches require a much higher evaluation effort due to its characteristic weak enforce-ment of the information transfer and the consequently arising need for numerical integration. In addition, a standard mortar approach leads naturally to a fully coupled system for information transfer, meaning that one node of the target mesh is usually subjected to information of all nodes from the source mesh. In the algebraic form, this can be identified by a dense projection operator which inevitably causes high numerical effort for matrix-matrix and matrix-vector multiplica-tions. Also the creation of the mortar projection operator itself is very costly since it requires the inversion of a non-diagonal, sparse matrix that is of global problem size. The use of dual shape functions based on biorthogonality conditions would significantly increase the computa-tional efficiency of a mortar based projection operator, since they localize the coupling effect and lead to a NTS like, decoupled algebraic structure of the information transfer. Furthermore, the mentioned inversion of a global matrix becomes very efficient because it can be created in a way that it becomes of diagonal form. Up to now, the use of dual shape functions for coupling operators in a completely 3D setting for first- and second-order elements cannot be found in existing literature.
Application to general monolithic multiphysics problems In the existing literature, multi-physics problems are predominantly discretized in a NTN manner, meaning that nodes of the discretized single-fields match. In contrast, volume coupled multiphysics with different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and com-putational costs. A first approach for multiphysics coupling on different meshes was outlined in Dureisseix and Bavestrello [64], but with restrictions to partitioned coupling schemes in a 2D setting. A comprehensive methodology for multiphysics simulations on non-matching and non-conforming meshes at the boundaries cannot be found in the existing literature so far.
Extension of the general coupling approach for monolithic multiphysics problems to-wards contact mechanicsWhile the simulation of volume coupled problems on non-matching meshes is a rarely focused topic, the extension of such approaches towards contact mechanics problems, like thermo-structure-contact interaction, is a completely unanswered question. When considering a body with non-matching volume meshes in the most general case, usually its inter-face discretizations are also non-matching. Thus, a further coupling of the interinter-face quantities, e.g. possibly employed Lagrange multipliers, is required in this case. Furthermore, the compu-tational contact interaction with a second body in a finite deformation regime generally leads to non-matching interface meshes between these two bodies, see Chapter 3. Consequently, a very
6.2. Fundamentals on volume projection of nodal information complex interface and volume coupling scheme arises for such scenarios, which is presented throughout this thesis.
Extendable implementation regarding general volume coupled problemsDespite the men-tioned application to volume coupled multiphysics problems, the consideration of overlapping volumes with different meshes is also required for many other applications, such as remeshing schemes and so-called zooming procedures. The implementation of the presented algorithms can be realized in very general manner to allow for employing similarities for a bunch of numerical application such as a Hu-Washizu approach for first-order tetrahedral elements (cf. Lamichhane et al. [149]), a novel and computationally efficient algorithm for fluid-structure interaction prob-lems and many more methods.