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X-FEM and Lagrange multipliers method for complex interface problems

Initial Situation

There are a lot of problems, where the certain interface conditions between adjacent subdomains has to be taken into account. The motivation for such coupling and its efficient numerical implementation is originated from both applied problems, e.g., coupling between different models in different subdomains in acoustics, fluid-structure interaction, coupled elctro-mechanical problems, etc., - and numerical analysis, e.g., mortar and domain decomposition methods. In both cases, the problem under consideration can be described as a set of equations in subdomains and a set of corresponding interface conditions.

In this work we consider a class of more complex problems, where the set of governing equations, described above, is extended by a certain equations, which are defined and has to be solved at the interface. As a result, the full system of equation consist of, generally, (i) Equations in subdomains of the dimension n; (ii) Equations at the interface (of the dimension n-1 or n-2), and (iii) Interface conditions, which couple interface and subdomains problems.

Despite of generality of the description presented above, there are lot of examples which fit such description. Examples are low-dimesional inclusions (of the dimesion n-1 or n-2) in n-dimensional elastic body, dynamics of thin shells or membranes in fluid or magnetic field, etc. The common features of all such problems is a presence of some "active" interface, which is efficiently of the lower dimension and which can not be, practically, resolved by the underlying finite element mesh.

Current Results

As a first example, we consider several elastic strings, immersed in 2d elastic solid. The solid and the strings have different elastic properties and are described by 2d Lame equations for solid and 1d equations for strings respectively. Coupling interface conditions in that case describe continuity of displacements of both solid and strings and force balance at the interface, which describes the strings. Such problem introduce very good approximation for many enginering applications, e.g., metal armature in mortar constructions or fiber-reinforced composite media.

Based on the mathematical framework of X-FEM method and and Lagrange multipliers approach, the new numerical approximations were developed and implemented in Matlab and first numerical results were obtained. X-FEM approach, originally presented for the solution of the crack problems in elasticity, allows to consider interfaces with the geometry complitely independent on the underlying finite element mesh. Therefore, no adaptivity of this mesh to the interface is needed, which is especially important in dynamical problems, where interface can change its shape and position in time. An examples are crack propagation in elasticity or thin shell or mebrane dynamics in fluidi-structure interation problems. At the same moment, Lagrange multipliers provides a very flexible tool to deal with interface conditions (or constraints) of, basically, an arbitrary nature. It allows, among others, to consider more complex models for the interfaces and coupling conditions, e.g., different contact problems at the interface, etc.

Work Plan

The future activity will be focused on the extensive numerical and theoretical studies of the new method. As a primary further extension we consider an important class of fluid-structure interaction problems, precisely, dynamics of thin shells and membrains in incompressible fluid flow.

cassini_geom.gif cassini_mesh.gif cassini_vital.gif


Author(s) Title Appeared In Year Type Download
Christian Waluga, Barbara Wohlmuth Quasi-optimal A Priori Interface Error Bounds and A Posteriori Estimates for the Interior Penalty Method SIAM Journal on Numerical Analysis 2013 Article in Journal Abstract
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Eric Bechet, Nicolas Moes, Barbara Wohlmuth A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method Internat. J. Numer. Methods Engrg., Vol. 78, 931--954 2009 Article in Journal Abstract
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Christian Wieners, Barbara Wohlmuth The coupling of mixed and conforming finite element discretizations Domain Decomposition Methods 10, Boulder, August 1997 (J. Mandel, C. Farhat, and X.-C. Cai, eds.), pp. 453-459, American Mathematical Society 218 1998 Article in Conference Proceedings Abstract