An extended finite element method with algebraic constraints (XFEM-AC) for problems with weak discontinuities

Kramer, Richard Michael Jack; Bochev, Pavel B.; Siefert, Christopher; Voth, Thomas Eugene

doi:10.1016/j.cma.2013.07.013

Cited by 13 publications

(20 citation statements)

References 19 publications

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“…The linear system corresponding to eXFEM-AC can be solved much in the same way as the system corresponding to the nodal XFEM-AC [13]. We begin by solving the conforming problem for c and use (33) to get c, and then solve for the Lagrange multipliers, λ = (B B T ) −1 C (f − A c).…”

Section: Solution Of the Resulting Linear Systemmentioning

confidence: 99%

“…We recall that the key junctures in the XFEM-AC are: (i) the subdivision of every element intersected by the interface (the "cut" elements) into subelements of the same shape; (ii) the enrichment of the local element space on cut elements by the products of the nodal basis and the indicators of the subelements; and (iii) the restoration of C 0 continuity along the cuts by means of algebraic constraints. The first step is similar to the CDFEM [24,12], the second step follows traditional XFEM [16], while the third step is the original contribution of [13].…”

Section: Algebraically Constrained Extended Edge Element Methodsmentioning

confidence: 99%

“…1(b) shows an example of such a mesh. Our approach extends the nodal XFEM-AC method [13] to the two-dimensional eddy-current problem. We recall that the key junctures in the XFEM-AC are: (i) the subdivision of every element intersected by the interface (the "cut" elements) into subelements of the same shape; (ii) the enrichment of the local element space on cut elements by the products of the nodal basis and the indicators of the subelements; and (iii) the restoration of C 0 continuity along the cuts by means of algebraic constraints.…”

Section: Algebraically Constrained Extended Edge Element Methodsmentioning

confidence: 99%

“…The third condition requires us to enforce the jump condition across edges e 25 and e 53 . Doing so is simpler than the inter-element connection in the nodal XFEM-AC, since we do not have to worry about potentially overconstraining the field [13]. Thus we can specify the final condition as (see Fig.…”

Section: Formulation Of Exfem-acmentioning

confidence: 98%

“…We use this observation to construct a matrix of algebraic constraints, following the basic approach of the XFEM-AC method [13].…”

Section: The Mixed Enriched Formulationmentioning

confidence: 99%

See 4 more Smart Citations

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

Kramer¹,

Bochev²,

Siefert³

et al. 2014

Journal of Computational Physics

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Section: Solution Of the Resulting Linear Systemmentioning

confidence: 99%

Section: Algebraically Constrained Extended Edge Element Methodsmentioning

confidence: 99%

Section: Algebraically Constrained Extended Edge Element Methodsmentioning

confidence: 99%

Section: Formulation Of Exfem-acmentioning

confidence: 98%

“…We use this observation to construct a matrix of algebraic constraints, following the basic approach of the XFEM-AC method [13].…”

Section: The Mixed Enriched Formulationmentioning

confidence: 99%

See 3 more Smart Citations

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

Kramer¹,

Bochev²,

Siefert³

et al. 2014

Journal of Computational Physics

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A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Kramer

Noble

2014

Numerical Meth Engineering

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SUMMARYInterface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities. The conformal decomposition FEM decomposes multimaterial elements of a non-conforming background mesh into sub-elements that conform to material interfaces captured using a level set method. As the interface evolves, interfacial nodes move, and background nodes may change material. The present work describes approaches for handling moving interfaces in the context of the conformal decomposition FEM for both weakly and strongly discontinuous fields. Dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first-order and second-order time integration methods. The moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one-dimensional and two-dimensional test problems, while achieving the expected second-order error convergence rate in space and time.

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Extended quasicontinuum methodology for highly heterogeneous discrete systems

Werner,

Zeman,

Rokoš

2023

Numerical Meth Engineering

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SummaryLattice networks are indispensable to study heterogeneous materials such as concrete or rock as well as textiles and woven fabrics. Due to the discrete character of lattices, they quickly become computationally intensive. The QuasiContinuum (QC) Method resolves this challenge by interpolating the displacement of the underlying lattice with a coarser finite element mesh and sampling strategies to accelerate the assembly of the resulting system of governing equations. In lattices with complex heterogeneous microstructures with a high number of randomly shaped inclusions the QC leads to an almost fully‐resolved system due to the many interfaces. In the present study the QC Method is expanded with enrichment strategies from the eXtended Finite Element Method (XFEM) to resolve material interfaces using nonconforming meshes. The goal of this contribution is to bridge this gap and improve the computational efficiency of the method. To this end, four different enrichment strategies are compared in terms of their accuracy and convergence behavior. These include the Heaviside, absolute value, modified absolute value and the corrected XFEM enrichment. It is shown that the Heaviside enrichment is the most accurate and straightforward to implement. A first‐order interaction based summation rule is applied and adapted for the extended QC for elements intersected by a material interface to complement the Heaviside enrichment. The developed methodology is demonstrated by three numerical examples in comparison with the standard QC and the full solution. The extended QC is also able to predict the results with 5% error compared to the full solution, while employing almost one order of magnitude fewer degrees of freedom than the standard QC and even more compared to the fully‐resolved system.

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An extended finite element method with algebraic constraints (XFEM-AC) for problems with weak discontinuities

Cited by 13 publications

References 19 publications

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Extended quasicontinuum methodology for highly heterogeneous discrete systems

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