Imposing Constraints on Surfaces with Embedded Finite Element Methods

Abstract eng:
The computational mechanics community is increasingly seeking numerical methods suitable for treating problems with complex, often evolving, geometry. Examples span applications from fracture mechanics and the dynamics of fault networks to fluid-structure interaction. Recently, methods have emerged that permit some relaxation between the surface of interest and the underlying finite element mesh. This includes the eXtended Finite Element Method and the Generalized Finite Element Method. These methods embed the geometry within the mesh and locally enhance the approximation to capture the kinematics and/or asymptotic solution. While such an embedding facilitates the representation of complex surfaces and their evolution, it complicates the imposition of constraints, particularly for three-dimensional realizations. Accordingly, we describe both stable and stabilized methods for imposing constraints on embedded, three-dimensional surfaces in finite elements. The stable method makes use of the vital vertex algorithm of Bechet et al. [1], albeit with a modified set of discontinuous basis functions for the Lagrange multiplier field on the embedded surface. The stabilized method extends the work of Dolbow and Harari [2] to three-dimensional surfaces. We present results from several threedimensional benchmark problems designed to compare and contrast the accuracy of the two approaches. We show that both methods yield optimal rates of convergence in quantities of interest, with the primary differences being in the accuracy of the surface flux. References [1] E. Bechet, N. Moes, B. Wohlmuth, A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method. International Journal for Numerical Methods in Engineering, 78, 931-954, 2009 [2] J.E. Dolbow, I. Harari, An efficient finite element method for embedded interface problems. International Journal for Numerical Methods in Engineering, 78, 229-252, 2009

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