The finite cell method for polygonal meshes: poly-FCM

Duczek, Sascha; Gabbert, Ulrich

doi:10.1007/s00466-016-1307-x

Cited by 21 publications

(18 citation statements)

References 121 publications

(235 reference statements)

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“…This concept leads to a non–boundary‐fitted finite element mesh, whose elements can be arbitrarily intersected by the domain boundary (see Figure ) and constitutes a significant simplification for meshing geometrically complex domains. It is independent of a specific type of finite element basis and has been successfully applied with integrated Legendre functions, splines, and polyhedral functions . We first define an embedding domain of simple geometry that can be meshed easily and subsequently remove all elements without support in the physical domain.…”

Section: The Voxel Finite Cell Methodsmentioning

confidence: 99%

“…It is independent of a specific type of finite element basis and has been successfully applied with integrated Legendre functions, 29,30 splines, 48,49 and polyhedral functions. 50 We first define an embedding domain of simple geometry that can be meshed easily and subsequently remove all elements without support in the physical domain.…”

Section: Discretization With Non-boundary-fitted Elementsmentioning

confidence: 99%

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Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

Nguyen

Stoter

Baum

et al. 2017

Numer Methods Biomed Eng

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The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

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Section: The Voxel Finite Cell Methodsmentioning

confidence: 99%

Section: Discretization With Non-boundary-fitted Elementsmentioning

confidence: 99%

Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

Nguyen

Stoter

Baum

et al. 2017

Numer Methods Biomed Eng

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“…The adaptive subdivision-based quadrature scheme can be easily generalized to different element types. 56,76,77 For tetrahedral elements applied in this work, the basic building block is the split of an intersected tetrahedron into 8 tetrahedral subcells as shown in Figure 9. From an algorithmic viewpoint, we implement recursive subdivision in a "bottom-up" fashion.…”

Section: Adaptive Quadrature Based On Recursive Subdivisionmentioning

confidence: 99%

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Nguyen

Stoter

Ruess

et al. 2017

Numerical Meth Engineering

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Summary We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

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“…To address the second challenge, the finite cell mesh needs to be locally refined. Alternative versions of the FCM make use of unstructured meshes and apply a local mesh refinement to resolve local solution features . However, these mesh‐based refinement approaches sacrifice the uniform grid structure of the FCM, which is particularly useful for image‐based geometries that are common in biomechanical applications.…”

Section: Introductionmentioning

confidence: 99%

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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The finite cell method for polygonal meshes: poly-FCM

Cited by 21 publications

References 121 publications

Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

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

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