Finite cell method

Parvizian, J.; Düster, Alexander; Rank, Ernst

doi:10.1007/s00466-007-0173-y

Cited by 482 publications

(278 citation statements)

References 15 publications

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“…To overcome this drawback, the quadtree-based SBFEM is coupled with the spectral cell method (SCM) to capture the modeled geometry more accurately. The SCM is a fictitious domain method that is usually defined on a regular structured finite element mesh [3,4]. Rather than modifying the shape of each element through an adequate mapping, the geometry is considered during the integration of the stiffness and mass matrices.…”

Section: A Coupled Sbfem/scm Approachmentioning

confidence: 99%

Automatic quadtree‐based modeling using a coupled SBFEM/SCM approach

Gravenkamp

Duczek²,

Birk

2017

Proc Appl Math and Mech

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We present an approach for the automatic image-based modeling in computational mechanics. The geometry is divided into subdomains by employing a quadtree-decomposition. The scaled boundary finite element method (SBFEM) can directly be applied to quadtree meshes without introducing hanging nodes, while the spectral cell method (SCM) allows for an accurate representation of curved boundaries. Hence, a combination of both methods can lead to highly accurate and reliable results in a fully automatic simulation environment. The performance of this coupled SBFEM/SCM approach is demonstrated by means of a modal analysis of a violin soundboard. A coupled SBFEM/SCM approachThe quadtree decomposition is often recognized as a highly efficient way to discretize arbitrary geometries, particularly in the context of automatic image-based simulations. A typical quadtree discretization consists of square-shaped cells only, while each square can successively be subdivided into four quadrants depending on the inhomogeneities or gradients that need to be resolved. Unfortunately, this strategy is not applicable to conventional finite element implementations in a straightforward fashion without introducing hanging nodes at the interfaces between elements of different size. It has been demonstrated recently that the scaled boundary finite element method (SBFEM) [1] can directly exploit quadtree meshes since in this method only the boundaries of each cell need to be discretized by an arbitrary number of line elements [2]. The SBFEM offers the additional advantage that the order of interpolation can be chosen individually for each subdomain, depending on its size and material parameters. On the other hand, the convergence of quadtree-based methods can be poor due to the introduced geometry error. Considering structures with complex curved boundaries, a staircase representation of the boundary is often insufficient. To overcome this drawback, the quadtree-based SBFEM is coupled with the spectral cell method (SCM) to capture the modeled geometry more accurately. The SCM is a fictitious domain method that is usually defined on a regular structured finite element mesh [3,4]. Rather than modifying the shape of each element through an adequate mapping, the geometry is considered during the integration of the stiffness and mass matrices. In the proposed approach, the SCM concept is only employed for cells that are intersected by the boundary of the physical domain (cut/broken cells), while all other cells are computed as SBFEM subdomains. This methodology results in a smoothed, more accurate and more realistic approximation of the actual geometry. Fig. 1 explains this concept: The generic quadtree mesh is suitable to create a rough approximation of the geometry. All cells that are intersected by the physical boundary (green line) are treated as SCM-cells (gray cells), i.e. the local distribution of material parameters is considered during the integration of the stiffness and mass matrices. For efficiency and accuracy, the integration is perfo...

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Section: A Coupled Sbfem/scm Approachmentioning

confidence: 99%

Automatic quadtree‐based modeling using a coupled SBFEM/SCM approach

Gravenkamp

Duczek²,

Birk

2017

Proc Appl Math and Mech

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“…Next, the weights λ i,0 , λ i,1 represent the actual intersection point with the interface on an edge and are computed correspondingly (lines [34][35][36][37][38][39][40][41][42]…”

Section: Matrix Assemblymentioning

confidence: 99%

“…end for 33: end procedure 34: procedure ComputeWeights(weights λ, level-set-function φ, hexahedron E, tetrahedron T ) 35:…”

Section: Matrix Assemblymentioning

confidence: 99%

“…Starting from classical FEM and "enriching" the FE spaces by additional basis functions to incorporate discontinuities has been exploited with the "Extended Finite Element Methods" (XFEM) [4]. The "Finite Cell Method" [35] is based on the idea of extending the PDE outside the actual object domain such that a domain-independent mesh for FEM can be used.…”

Section: Introductionmentioning

confidence: 99%

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Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

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Abstract. We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through e. g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method's efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

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“…The approach used in this work is in some ways related to other methods for dealing with complex boundaries, such as the Immersed Boundary Method (IBM), Immerse Finite Element Method (IFEM) [37], and the Finite cell method (FCM) [24]. IBM and IFEM are widely used in fluid mechanics.…”

Section: Introductionmentioning

confidence: 99%

Volumetric parametrization from a level set boundary representation with PHT-splines

Chan

Anitescu

Rabczuk

2017

Computer-Aided Design

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A challenge in isogeometric analysis is constructing analysis-suitable volumetric meshes which can accurately represent the geometry of a given physical domain. In this paper, we propose a method to derive a splinebased representation of a domain of interest from voxel-based data. We show an efficient way to obtain a boundary representation of the domain by a level-set function. Then, we use the geometric information from the boundary (the normal vectors and curvature) to construct a matching C 1 representation with hierarchical cubic splines. The approximation is done by a single template and linear transformations (scaling, translations and rotations) without the need for solving an optimization problem. We illustrate our method with several examples in two and three dimensions, and show good performance on some standard benchmark test problems.

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Finite cell method

Cited by 482 publications

References 15 publications

Automatic quadtree‐based modeling using a coupled SBFEM/SCM approach

Automatic quadtree‐based modeling using a coupled SBFEM/SCM approach

Composite finite elements for 3D image based computing

Volumetric parametrization from a level set boundary representation with PHT-splines

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