The Nitsche method applied to a class of mixed-dimensional coupling problems Daniel Rabinovich a , Yoav Ofir b , Dan Givoli a,⇑ a Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa 32000, Israel b Interdepartmental Program of Applied Mathematics, Technion — Israel Institute of Technology, Haifa 32000, Israel article info Article history: Received 8 September 2013 Received in revised form 2 February 2014 Accepted 9 February 2014 Available online 25 February 2014 Keywords: Mixed-dimensional Nitsche 2D–1D 1D–2D Time-harmonic Hybrid model abstract A computational approach for the mixed-dimensional modeling of time-harmonic waves in elastic structures is proposed. A two-dimensional (2D) structure is considered, that includes a part which is assumed to behave in a one-dimensional (1D) way. The 2D and 1D structural regions are discretized using 2D and 1D finite element formulations. The cou- pling of the 2D and 1D regions is performed weakly, by using the Nitsche method. The advantage of using the Nitsche method to impose boundary and interface conditions has been demonstrated by various authors; here this advantage is shown in the context of mixed-dimensional coupling. The computational aspects of the method are discussed, and it is compared to the slightly simpler penalty method, both theoretically and numer- ically. Numerical examples are presented in various configurations: where the 1D model is either confined laterally or laterally free, and where the 2D part is either simply connected or doubly connected. The performance is investigated for various wave numbers and various extents of the 1D region. Varying material properties and distributed loads in the 1D and 2D parts are also considered. It is concluded that the Nitsche method is a viable technique for mixed-dimensional coupling of elliptic problems of this type. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction The need for coupling structural models (or other types of models) of a high dimension to models of a low dimension arises from the practical objective of reducing the size of the models used, by replacing certain portions of the model by simpler structural components involving a much lower computational effort. This need is typical to structures dealt with in various applications, and in many cases the use of members of reduced dimensionality is quite natural. One important case in point is aircraft structures, where most of the components are one-dimensional (1D) in nature, namely beams that undergo bending, torsion and tension/compression, some parts are regarded as two-dimensional (2D), namely plates and shells that undergo in-plane and out-of-plane deformation, and small but important regions behave in a fully three-dimen- sional (3D) way, e.g., joints that connect these structural members [1]. In such hybrid modeling, the location of the interface between the high-dimensional (high-D) and low-dimensional (low-D) parts has to be determined by the modeler. This choice has to be made in such a manner that the displacements and stresses in the part that is to be idealized will indeed behave in an approximately low-D manner; otherwise a large ide- alization error might arise. This issue is an important part of model validation practice; see, e.g., [2]. Efforts have also been http://dx.doi.org/10.1016/j.cma.2014.02.006 0045-7825/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +972 829 3814; fax: +972 829 2030. E-mail addresses: aedaniel@technion.ac.il (D. Rabinovich), joav@technion.ac.il (Y. Ofir), givolid@technion.ac.il (D. Givoli). Comput. Methods Appl. Mech. Engrg. 274 (2014) 125–147 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma