The Nitsche method applied to a class of mixed-dimensional coupling problems

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 coupling 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 numerically. 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.