Publications

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Interface-conforming elements generated by the conformal decomposition finite element method can have arbitrarily poor quality due to the arbitrary intersection of the base triangular or tetrahedral mesh with material interfaces. This can have severe consequences for both the solvability of linear systems and for the interpolation error of fields represented on these meshes. The present work demonstrates that snapping the base mesh nodes to the interface whenever the interface cuts close to a node results in conforming meshes of good quality. Theoretical limits on the snapping tolerance are derived, and even conservative tolerance choices result in limiting the stiffness matrix condition number to within a small multiple of that of the base mesh. Interpolation errors are also well controlled in the norms of interest. In 3D, use of node-to-interface snapping also permits a simpler and more robust vertex ID-based element decomposition algorithm to be used with no serious detriment to mesh quality.

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This report briefly outlines an algorithm for dividing a tetrahedron intersected by a planar interface into conforming sub-tetrahedra. The problem of conformal decomposition of tetrahedral meshes arises in enriched finite element methods; in particular, we are concerned with the Conformal Decomposition Finite Element Method (CDFEM) and variants of the eXtended Finite Element Method (XFEM). The algorithm presented is based on the paper How to Subdivide Pyramids, Prisms and Hexahedra into Tetrahedra by Dompierre, Labbe, Vallet, and Camarero (1999), and here is applied and extended to the problem of fully defining and tracking all geometric features of the sub-tetrahedra generated when a tetrahedron is cut by a planar surface.

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Surface effects are critical to the accurate simulation of electromagnetics (EM) as current tends to concentrate near material surfaces. Sandia EM applications, which include exploding bridge wires for detonator design, electromagnetic launch of flyer plates for material testing and gun design, lightning blast-through for weapon safety, electromagnetic armor, and magnetic flux compression generators, all require accurate resolution of surface effects. These applications operate in a large deformation regime, where body-fitted meshes are impractical and multimaterial elements are the only feasible option. State-of-the-art methods use various mixture models to approximate the multi-physics of these elements. The empirical nature of these models can significantly compromise the accuracy of the simulation in this very important surface region. We propose to substantially improve the predictive capability of electromagnetic simulations by removing the need for empirical mixture models at material surfaces. We do this by developing an eXtended Finite Element Method (XFEM) and an associated Conformal Decomposition Finite Element Method (CDFEM) which satisfy the physically required compatibility conditions at material interfaces. We demonstrate the effectiveness of these methods for diffusion and diffusion-like problems on node, edge and face elements in 2D and 3D. We also present preliminary work on h -hierarchical elements and remap algorithms.

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We present a new extended finite element method with algebraic constraints (XFEM-AC) for recovering weakly discontinuous solutions across internal element interfaces. If necessary, cut elements are further partitioned by a local secondary cut into body-fitting subelements. Each resulting subelement contributes an enrichment of the parent element. The enriched solutions are then tied using algebraic constraints, which enforce C0 continuity across both cuts. These constraints impose equivalence of the enriched and body-fitted finite element solutions, and are the key differentiating feature of the XFEM-AC. In so doing, a stable mixed formulation is obtained without having to explicitly construct a compatible Lagrange multiplier space and prove a formal inf-sup condition. Likewise, convergence of the XFEM-AC solution follows from its equivalence to the interface-fitted finite element solution. This relationship is further exploited to improve the numerical solution of the resulting XFEM-AC linear system. Examples are shown demonstrating the new approach for both steady-state and transient diffusion problems. © 2013 Elsevier B.V.

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