The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis

Abstract

In this contribution, a mortar-type method for the coupling of non-conforming NURBS (Non-Uniform Rational B-spline) surface patches is proposed. The connection of non-conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor-product structure of NURBS surfaces. Thus, methods to handle non-conforming meshes are essential in NURBSbased isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master-slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation, the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given.The combination of these two issues requires methods to handle non-conforming patches in order to establish an efficient mechanical response analysis framework for the computation of complex NURBS surface models.Domain decomposition methods decompose the domain of a given problem into subdomains, of which the parametrization does not have to match. The situation at hand is an ideal field of application for domain decomposition methods, where each patch can be regarded as one subdomain. A general overview over domain decomposition methods is given in [6]. The Penalty method [7], the Lagrange multiplier method [8], and Nitsche's method [9] were proposed in the 1970s to enforce Dirichlet boundary conditions in a weak manner. In recent engineering literature, these methods are used as a basis for domain decomposition techniques. Mortar methods are the prevailing domain decomposition method in mathematical literature. Basically, two standard types of mortar formulations exist, which are referred to as the non-conforming positive definite problem and the saddle point problem based on unconstrained product spaces in [6]. In all cases, a Lagrange multiplier space is used for the matching conditions along the interface. The mortar method leading to the nonconforming positive definite problem was initially proposed by Bernardi et al. [10]. The function spaces are constrained to fulfill the matching condition at the interface in a weak manner. Basically, this means that the basis functions of one side are related to those of the other side in a way that the matching condition is fulfilled. The discretization of the interface condition with Lagrange multiplier functions yields an equation...

Isogeometric dual mortar methods for computational contact mechanics

“…dual shape functions with an extended support [33,50]. Nonetheless, the presented method provides an easy and efficient way of coupling patches without increasing the global system size (compared to [33]) and gaining a localization of the coupling (compared to [32]). …”

“…A comprehensive review on dual mortar methods for contact mechanics can be found in [28,29]. In the context of domain decomposition in IGA, optimality and stability of standard mortar methods have only very recently been investigated in [30][31][32][33], where also the construction of dual B-spline basis functions has been outlined theoretically [33].…”

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

“…Although it appears interesting because of the strong mathematics behind it and the absence of additional degrees of freedom, the Nitsche method leads to considerable implementation work and an increased cost of computation, since an additional eigenvalue problem has to be solved for the stabilizing term. As a result, Dornisch et al [26] has recently developed a weak substitution method that can be interpreted as a mortar method. Although the combination of the FCM with Nitsche coupling may be promising, the drawback of such a strategy, if directly applied to the local enrichment of a NURBS patch, is that it suffers from some intrusiveness.…”

Development of a new, more regular, mortar method for the coupling of NURBS subdomains within a NURBS patch: Application to a non-intrusive local enrichment of NURBS patches

“…This is in contrast with the more usual situation of the coupling of IGA patches which is achieved along C 0 interfaces (see, e.g., [11,25,26,29,30]). …”

“…Even if it may appear interesting due to the absence of additional degrees of freedom, the Nitsche method leads to a comparatively high computational effort since an additional eigenvalue problem has to be solved for the stabilizing term. As a result, Dornisch et al [30] developed a weak substitution method to simplify the implementation and reduce the computational cost. From this overview, it may be noticed that most of the coupling techniques elaborated for NURBS nowadays are dedicated to the connection of NURBS patches, i.e., the coupling along a C 0 interface.…”