Higher-order meshing of implicit geometries—Part I: Integration and interpolation in cut elements

Fries, Thomas‐Peter; Omerovic, Samir; Schöllhammer, Daniel; Steidl, Jakob Werner

doi:10.1016/j.cma.2016.10.019

Cited by 83 publications

(92 citation statements)

References 59 publications

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“…Clearly,

{normalΓ}_{q}^{h}

is defined parametrically through the map even if the original Γ was implicitly given, eg, by the zero isosurface of a level‐set function. See other works for the automatic generation of higher‐order meshes on zero isosurfaces. The discrete unit normal vector is

n_{Γ}^{h} = \frac{\partial_{r} x \times \partial_{s} x}{‖\partial_{r} x \times \partial_{s} x‖}

and is not smooth across element edges due to the C 0 ‐continuity of the surface mesh.…”

Section: Governing Equationsmentioning

confidence: 99%

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Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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Summary Stationary and instationary Stokes and Navier‐Stokes flows are considered on two‐dimensional manifolds, ie, on curved surfaces in three dimensions. The higher‐order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Streamline‐upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented, which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained, and higher‐order convergence rates are confirmed.

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“…Clearly,

{normalΓ}_{q}^{h}

n_{Γ}^{h} = \frac{\partial_{r} x \times \partial_{s} x}{‖\partial_{r} x \times \partial_{s} x‖}

and is not smooth across element edges due to the C 0 ‐continuity of the surface mesh.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Clearly, Γ h q is defined parametrically through the map (26) even if the original Γ was implicitly given, eg, by the zero isosurface of a level-set function. See other works [39][40][41] for the automatic generation of higher-order meshes on zero isosurfaces. The discrete unit normal vector is…”

Section: Surface Fem For Flows On Manifolds 331 Surface Meshesmentioning

confidence: 99%

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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“…Yet, it cannot be used in general to generate an analysis‐suitable FEM mesh, as it does not need to satisfy requirements on shape and regularity at cell edges and faces, eg, at the transition to the neighboring cells. This approach shares some similarities with the method presented by Fries et al The integration meshes for both considered approaches are displayed in Figure .…”

Section: Numerical Experimentsmentioning

confidence: 83%

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

Elhaddad

Zander

Bog

et al. 2018

Numer Methods Biomed Eng

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This work presents a numerical discretization technique for solving 3-dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high-order fictitious domain approach, is used for the numerical approximation of the solution without a boundary-conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub-domain and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchical hp-refinement scheme is used to locally refine the FCM meshes to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for nonsmooth problems. A series of numerical experiments with 2- and 3-dimensional benchmark problems is presented, showing that the proposed hp-refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra-implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description.

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“…Using higher‐order polynomials for the blending, isoparametric transformations are obtained. Blending techniques in the context of extended finite element methods are for instance investigated in previous works …”

Section: Discretizationmentioning

confidence: 99%

A fictitious domain method for the simulation of thermoelastic deformations in NC‐milling processes

Byfut

Schröder

2017

Numerical Meth Engineering

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Summary This paper presents a (higher‐order) finite element approach for the simulation of heat diffusion and thermoelastic deformations in NC‐milling processes. The inherent continuous material removal in the process of the simulation is taken into account via continuous removal‐dependent refinements of a paraxial hexahedron base‐mesh covering a given workpiece. These refinements rely on isotropic bisections of these hexahedrons along with subdivisions of the latter into tetrahedrons and pyramids in correspondence to a milling surface triangulation obtained from the application of the marching cubes algorithm. The resulting mesh is used for an element‐wise defined characteristic function for the milling‐dependent workpiece within that paraxial hexahedron base‐mesh. Using this characteristic function, a (higher‐order) fictitious domain method is used to compute the heat diffusion and thermoelastic deformations, where the corresponding ansatz spaces are defined for some hexahedron‐based refinement of the base‐mesh. Numerical experiments compared to real physical experiments exhibit the applicability of the proposed approach to predict deviations of the milled workpiece from its designed shape because of thermoelastic deformations in the process.

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Higher-order meshing of implicit geometries—Part I: Integration and interpolation in cut elements

Cited by 83 publications

References 59 publications

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

A fictitious domain method for the simulation of thermoelastic deformations in NC‐milling processes

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