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

Nguyen, Lam; Stoter, Stein K. F.; Ruess, Martin; Uribe, Manuel Alejandro Sánchez; Schillinger, Dominik

doi:10.1002/nme.5628

Cited by 22 publications

(61 citation statements)

References 79 publications

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“…This choice resulted in stable finite element computations in all simulations, while an influence of the stabilization term on the convergence behavior could not be observed in the numerical tests. For more information on accuracy, convergence, and stabilization, interested readers are referred to the computational study in the Appendix and further results reported in Nguyen et al 61 that focuses on the diffuse Nitsche method from a numerical analysis viewpoint. In particular, the latter provides a generalization of the eigenvalue-based estimation of the stabilization parameter and a comparison with consistent penalty-type methods derived, for example, in Li et al 27…”

Section: Dirichlet Boundary Conditionsmentioning

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 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: Dirichlet Boundary Conditionsmentioning

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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“…A phase field function ϕ is defined as an approximation to the step function to represent a diffuse boundary as illustrated in Figure 1. The volume integral over a function Q can be written as 16‐18

\int_{Ω} Q d normalΩ = \int_{{normalΩ}_{0}} italicQH d Ω_{0} \approx \int_{{normalΩ}_{0}} italicQϕ d Ω_{0} .

…”

Section: Methodsmentioning

confidence: 99%

“…In order to consider the boundary conditions on a surface Γ, the Dirac distribution δ Γ is approximated by the absolute gradient of the phase field function. A diffuse boundary integral of a function h over a boundary Γ is modeled as follows 16,18 :

\int_{Γ} h d normalΓ = \int_{{normalΩ}_{0}} h δ_{Γ} d normalΩ \approx \int_{{normalΩ}_{0}} h ∣ \nabla ϕ ∣ d normalΩ .

…”

Section: Methodsmentioning

confidence: 99%

Diffuse domain method for needle insertion simulations

Jerg

Austermühl

Roth

et al. 2020

Numer Methods Biomed Eng

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We present a new strategy for needle insertion simulations without the necessity of meshing. A diffuse domain approach on a regular grid is applied to overcome the need for an explicit representation of organ boundaries. A phase field function captures the transition of tissue parameters and boundary conditions are imposed implicitly. Uncertainties of a volume segmentation are translated in the width of the phase field, an approach that is novel and overcomes the problem of defining an accurate segmentation boundary. We perform a convergence analysis of the diffuse elastic equation for decreasing phase field width, compare our results to deformation fields received from conforming mesh simulations and analyze the diffuse linear elastic equation for different widths of material interfaces. Then, the approach is applied to computed tomography data of a patient with liver tumors. A three‐class U‐Net is used to automatically generate tissue probability maps serving as phase field functions for the transition of elastic parameters between different tissues. The needle tissue interaction forces are approximated by the absolute gradient of a phase field function, which eliminates the need for explicit boundary parameterization and collision detection at the needle‐tissue interface. The results show that the deformation field of the diffuse domain approach is comparable to the deformation of a conforming mesh simulation. Uncertainties of tissue boundaries are included in the model and the simulation can be directly performed on the automatically generated voxel‐based probability maps. Thus, it is possible to perform easily implementable patient‐specific elastomechanical simulations directly on voxel data.

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“…( 1) are evaluated the method's effectiveness crucially depends on the choice of β. From other works [34] it is also clear that the diffuse interface approach is very sensitive to the choice of . Hence a numerical study is now considered that allows to characterize the influence of β on the resulting error while considering in the diffuse interface approach.…”

Section: Annular Plate Benchmarks 311 Problem Definitionmentioning

confidence: 98%

Enforcing essential boundary conditions on domains defined by point clouds

Hartmann¹,

Kollmannsberger²

2021

Preprint

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This paper develops and investigates a new method for the application of Dirichlet boundary conditions for computational models defined by point clouds. Point cloud models often stem from laser or structured-light scanners which are used to scan existing mechanical structures for which CAD models either do not exist or from which the artifact under investigation deviates in shape or topology. Instead of reconstructing a CAD model from point clouds via surface reconstruction and a subsequent boundary conforming mesh generation, a direct analysis without pre-processing is possible using embedded domain finite element methods. These methods use non-boundary conforming meshes which calls for a weak enforcement of Dirichlet boundary conditions. For point cloud based models, Dirichlet boundary conditions are usually imposed using a diffuse interface approach. This leads to a significant computational overhead due to the necessary computation of domain integrals. Additionally, undesired side effects on the gradients of the solution arise which can only be controlled to some extent. This paper develops a new sharp interface approach for point cloud based models which avoids both issues. The computation of domain integrals is circumvented by an implicit approximation of corresponding Voronoi diagrams of higher order and the resulting sharp approximation avoids the side-effects of diffuse approaches. Benchmark examples from the graphics as well as the computational mechanics community are used to verify the algorithm. All algorithms are implemented in the FCMLab framework and provided at https://gitlab.lrz.de/cie_sam_public/fcmlab/. Further, we discuss challenges and limitations of point cloud based analysis w.r.t. application of Dirichlet boundary conditions.

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

Cited by 22 publications

References 79 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

Diffuse domain method for needle insertion simulations

Enforcing essential boundary conditions on domains defined by point clouds

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