An embedded Dirichlet formulation for 3D continua

Gerstenberger, Axel; Wall, Wolfgang A.

doi:10.1002/nme.2755

Cited by 97 publications

(107 citation statements)

References 36 publications

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“…A recent effort has been done by Gerstenberger and Wall Gerstenberger and Wall (2010) to eliminate this obstacle and facilitate the use of Lagrange multipliers. They have developed a Lagrange multiplier formulation for the solution of problems involving voids in the geometry.…”

Section: Introductionmentioning

confidence: 99%

A XFEM Lagrange Multiplier Technique for Stefan Problems

Martin¹,

Chaouki²,

Robert³

et al. 2016

Frontiers in Heat and Mass Transfer

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The two dimensional phase change problem was solved using the extended finite element method with a Lagrange formulation to apply the interface boundary condition. The Lagrange multiplier space is identical to the solution space and does not require stabilization. The solid-liquid interface velocity is determined by the jump in heat flux across the i nterface. Two methods to calculate the jump are used and c ompared. The first is based on an averaged temperature gradient near the interface. The second uses the Lagrange multiplier values to evaluate the jump. The Lagrange multiplier based approach was shown to be more robust and precise.

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Section: Introductionmentioning

confidence: 99%

A XFEM Lagrange Multiplier Technique for Stefan Problems

Martin¹,

Chaouki²,

Robert³

et al. 2016

Frontiers in Heat and Mass Transfer

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“…Families of fixed mesh methods are often classified precisely depending, firstly, on the way each method deals with the imposition of boundary conditions, and secondly, on the way they deal with time and space integration over the fixed mesh [55]. Regarding the imposition of Dirichlet boundary conditions, several approaches exist such as the use of penalty terms as in the original immersed boundary method [50,59], the use of Lagrange multipliers [15,42,43,13,33,20,16,3] which may require the use of additional unknowns accounting for the fluxes on the Dirichlet boundary, or the well-known Nitsche's method [57,47,44,23], which yields symmetric, stable variational formulations through the use of a limited penalty term whose value needs to be estimated.…”

Section: Introductionmentioning

confidence: 99%

An adaptive Fixed-Mesh ALE method for free surface flows

Baiges

Codina

Pont

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian-Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

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“…On the embedded domain side, the importance of geometrically faithful quadrature of trimmed elements and corresponding techniques have been discussed in a series of recent papers [28,27,[29][30][31][32][33][34][35]. For the weak enforcement of boundary and interface conditions at trimming curves and surfaces, variational methods such as Lagrange multiplier [36][37][38] or Nitsche techniques [39][40][41][42][43][44] have been successfully developed.…”

Section: Introductionmentioning

confidence: 99%

A parameter-free variational coupling approach for trimmed isogeometric thin shells

2016

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The non-symmetric variant of Nitsche's method was recently applied successfully for variationally enforcing boundary and interface conditions in non-boundary-fitted discretizations. In contrast to its symmetric variant, it does not require stabilization terms and therefore does not depend on the appropriate estimation of stabilization parameters. In this paper, we further consolidate the non-symmetric Nitsche approach by establishing its application in isogeometric thin shell analysis, where variational coupling techniques are of particular interest for enforcing interface conditions along trimming curves. To this end, we extend its variational formulation within Kirchhoff-Love shell theory, combine it with the finite cell method, and apply the resulting framework to a range of representative shell problems based on trimmed NURBS surfaces. We demonstrate that the non-symmetric variant applied in this context is stable and can lead to the same accuracy in terms of displacements and stresses as its symmetric counterpart. Based on our numerical evidence, the non-symmetric Nitsche method is a viable parameter-free alternative to the symmetric variant in elastostatic shell analysis.

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An embedded Dirichlet formulation for 3D continua

Cited by 97 publications

References 36 publications

A XFEM Lagrange Multiplier Technique for Stefan Problems

A XFEM Lagrange Multiplier Technique for Stefan Problems

An adaptive Fixed-Mesh ALE method for free surface flows

A parameter-free variational coupling approach for trimmed isogeometric thin shells

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