Higher‐order accurate integration of implicit geometries

Fries, Thomas‐Peter; Omerovic, Samir

doi:10.1002/nme.5121

Cited by 135 publications

(151 citation statements)

References 51 publications

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“…We note that the symmetric Nitsche method with local stabilization parameters seems to be more sensitive to quadrature error, which additionally affects the evaluation of integrals (52) and (53). For a more comprehensive account on the importance of geometrically faithful surface and volume quadrature in immersed finite element methods, we refer the interested reader to a series of of recent papers [28,41,45,47,49,71,74,76]. It is also worthwhile to mention in this context that with the quadrature scheme described, the A matrix of the eigenvalue problem (51) may be nonzero while the B matrix is zero (or vice versa).…”

Section: Laplace Problem: Boundary Conditionsmentioning

confidence: 99%

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The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Schillinger

Harari

Hsu

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2" role="presentation"> error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest. KeywordsImmersed finite element methods, Weak boundary and coupling conditions, Non-symmetric Nitsche method AbstractWe explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L 2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

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Section: Laplace Problem: Boundary Conditionsmentioning

confidence: 99%

“…First, they need to be able to evaluate integrals in cut elements. In this context, the importance of geometrically faithful quadrature has been recently highlighted [28,41,45,47,49,71,74,76]. The second component is the enforcement of Dirichlet boundary and interface conditions at immersed boundaries.…”

Section: Introductionmentioning

confidence: 99%

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Schillinger

Harari

Hsu

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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“…zero level sets is discussed in more detail in [36], including the extension to the three-dimensional case.…”

Section: Reconstruction Of the Zero Level Setsmentioning

confidence: 99%

Conformal higher‐order remeshing schemes for implicitly defined interface problems

Omerovic

Fries

2016

Numerical Meth Engineering

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SUMMARYA new higher-order accurate method is proposed that combines the advantages of the classical p-version of the FEM on body-fitted meshes with embedded domain methods. A background mesh composed by higherorder Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher-order accurate sub-elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher-order sub-elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub-elements are not always well-shaped.

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“…Isogeometric shells have been successfully applied for large-deformation analysis [21], in conjunction with various nonlinear material models [22,23], and in contact and fluid-structure interaction problems [24][25][26][27]. 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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Higher‐order accurate integration of implicit geometries

Cited by 135 publications

References 51 publications

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Conformal higher‐order remeshing schemes for implicitly defined interface problems

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

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