The idea of this kind of application is to compose independent meshes also re- ferred to as dissimilar meshes. The issue of connecting them is one of the major problems in FEM. The meshes could be non-matching and the discrete interface between the meshes may present gaps or overlaps. Coupling non-conforming meshes help us to piece together the parts that are modelled independently, without sacrificing accuracy or efficiency, such as wing and fuselage structures that may have been modelled by different analysts in different groups or or- ganizations. Moreover, one can also impose selective refinement only on those components where it is required. Hence, in a variety of industrial applications, it is important to employ an efficient method that uses the existing meshes to solve the global system.The origins were in aerospace engineering and was also extended to offshore and shipbuilding industries or even in the electronic field. Another advantage of this technique is found in problems with repeated structural components like blades in an wind turbine.
names. In the NASA Langley Research Center, a method has been developed for analysing structures composed of two or more independent substrucutures, based on a hybrid variational formulation with Lagrange multipliers and in- terfaces elements, Aminpour, Ransom, & McCleary (1995). The software de- veloped by them is MSC Patran/Nastran, see Schiermeier, Housner, Ransom, Aminpour, & Stroud (1996), which offers the possibility to connect dissimilar meshes and they refer to this option as glue. They apply the technique mostly for permitting a high level of refinement in the local region and a coarser level of refinement in the global region, that is, as a local refinement technique. The software Abaqus also offers the possibilty of mesh gluing, known by the developers as tie, via constraint equations and using a master-slave formula- tion. The constraint prevents slave nodes from separating or sliding in relation to the master surface. See this manual for further details: ABAQUS (2014). In both softwares, the union between meshes could be for a shell and solid or two solids. In other commercial softwares like Altair (2014) only the contact between surfaces is valid.
Mesh gluing is also necessary to solve multiphysic problems. As an example, in fluid-structure interaction (FSI) computations, a natural decomposition is carried out by using non-overlapping meshes for the flow and the structure, also referred to as partitioned coupling. See Piperno, Farhat, & Larrouturou (1995); Farhat, Lesoinne, & Tallec (1998); Dureisseix (2008). In FSI realistic applications, the fluid and structure meshes are non-matching either because they have been designed by different analysts or because the fluid and structure problems have different resolution requirements, as we have explained before. In the following reference, Felippa, Park, & Farhat (2001) we find an interesting review of the use of partitioned strategies in coupled dynamical systems and a particular summary of the terminology related to this issue. Among many other definitions, the authors describe two kinds of partitioning which we want to mention:
Partitioning may be algebraic or differential. In algebraic par- titioning the complete coupled system is spatially discretized first, and then decomposed. In differential partitioning the decomposition is done first and each field then discretized separately.
Differential partitioning often leads to non-matched meshes, as a typical of fluid-structure interaction and handles those naturally. Algebraic partitioning was originally developed for matched meshes and substructuring, but recent work has aimed at simplifying the treatment of non-matched meshes through frames
As the authors note, since the problem that we are considering can be studied from many angles, the terminology is far from standard. We want to refer to the general problem of coupling systems with independent meshes, whether it be coupling different structural domains, multiphysical problems or even con- tact problems, such as mesh gluing.
The Arlequin method is another tool for the multimodel and multiscale analy- sis of complex mechanical problems, see Dhia & Rateau (2005). This technique locally allows refinement to link structure models (also called substructuring or external junctionby the authors) or to introduce an essential local modification in the models themselves (which is referred to by the authors as internal junc- tion). The coupling operator is based on the Lagrange multiplier field belonging to the dual space of admissible displacement fields restricted to the gluing zone. Different ways can be found to transfer data between non-matching meshes in FSI and could be divided in a direct way or dual way. The former is formed by techniques such as nearest neighbour interpolation, projection methods and methods based on interpolation by splines. A good reference where these meth- ods are compared is Boer, Zuijlen, & Bijl (2007). A dual way to impose bound- ary conditions is by Lagrange multipliers, explained before. Another example of multiphysical problem is the work of Dureisseix & Bavestrello (2006), for thermo-viscoelasticity coupled problems where an extension of mortar tech- nique is applied.
Techniques in contact computational finite element analysis involve tying or gluing several non-matching computational domains. In this problem, inter- faces are physical ones and present the boundary between two components and could present gaps or overlapping between them. Many of the mortar meth- ods with non-matching grids applied to this problems fail the so-called patch test, meaning that they do not exactly recover any globally linear solution of the governing equations (see Dohrmann, Key, & Heinstein (2000) for fur- ther discussion of this issue). In Parks, Romero, & Bochev (2007) this test is guaranteed using a novel Lagrange multipliers technique joining finite element models on two independently meshed subdomains in two dimensions that share a curved interface solving the Poisson equation. Another example of mortar method with curved surfaces is Flemisch, Melenk, & Wohlmuth (2005) or Puso (2004) in three dimensional problems for solid mechanics.
We find also composing meshes in aeroacoustic computation. This is the case of the work of Lee, Vouvakis, & Lee (2005) for modelling large finite an- tenna arrays, or the work of Peers, Zhang, & Kim (2010) in high-order finite difference schemes for multiblock computational aeroacoustic in complex ge- ometries.
To unite the rotor and stator independent meshes to compose an electric machine we find the work of Tsukerman (1992). In this reference, the authors propose an overlapping elements method which is valid only if the surfaces to be connected are regular. This drawback is overcome in Krebs, Clénet, & Tsuk- erman (2010) where the method is extended to non-planar surfaces introducing modified shape functions in the overlapping area. The method is valid for mov- ing meshes without any distortion in it. The method has been developed for
three-dimensional meshes with hexahedral, tetrahedral or prism elements. The idea is to project the nodes of one surface to the other and these projections create virtual nodes and virtual elements. The virtual nodes do not entail new degrees of freedom and are just used to define the nodal shape function for the scalar potential in the overlapping area with the result that it is continuous in the whole domain. In the framework of Generalized Finite Element Methods (GFEM) we find the work of Duarte, Liszka, & Tworzydlo (2007) to unite non- matching meshes. In particular, the authors present a GFEM with clustering since they cluster a set of nodes and elements which define a modified finite element partition of unity that is constant over part of the clusters.
In the solid mechanics framework, the meshless technique could be useful for problems which present cracks or large deformations. See, for example, Cho & Im (2006) where different couplings are compared in such situations. Other references are Cho et al. (2005); Tian & Yagawa (2007); Kim (2008). Fluid- structure interaction is another example of applicability in the meshless and finite element combination for mesh joining application. In fact, one of the first works related to this issue is found in this application, see Attaway, Heinstein, & Swegle (1994).
Mesh-based techniques like the work of Lo & Wang (2004) or Cebral et al. (2001) also have also been applied to solve mesh union problems.