Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach

Gravenkamp, Hauke; Duczek, Sascha

doi:10.1007/s00466-017-1424-1

Cited by 35 publications

(20 citation statements)

References 97 publications

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“…When comparing with the conventional FEM on the other hand, the SBFEM/SCM still yields slightly smaller relative errors, while the main advantage lies in the fully automatic image-based modeling of complex geometries. Several different examples can be found in [5]. While the convergence is problem-dependent, the SBFEM/SCM consistently shows at least similar errors compared to conventional FEM analyses while in some cases the reduction in terms of degrees of freedom is more significant.…”

Section: Applicationmentioning

confidence: 93%

“…This methodology minimizes the required number of degrees of freedom and avoids the need for a two-dimensional quadrature in all cells that do not contribute to the approximation of the boundary. A detailed analysis of the coupled SBFEM/SCM approach is presented in a forthcoming article by the authors [5].…”

Section: A Coupled Sbfem/scm Approachmentioning

confidence: 99%

“…In this example, a modal analysis has been performed and the convergence of the 33rd eigenfrequency * Corresponding author: e-mail hauke.gravenkamp@uni-due.de, phone +49 201 183 6949 c 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim has been studied in comparison with a conventional finite element analysis as well as the pure SCM and the quadtree-based SBFEM, see Fig. 3 (for details the reader is referred to [5]). The results show that the coupled SBFEM/SCM approach provides a significant reduction of the number of degrees of freedom compared with the pure SCM or SBFEM.…”

Section: Applicationmentioning

confidence: 99%

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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: Applicationmentioning

confidence: 93%

Section: A Coupled Sbfem/scm Approachmentioning

confidence: 99%

Section: Applicationmentioning

confidence: 99%

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

Gravenkamp

Duczek²,

Birk

2017

Proc Appl Math and Mech

Self Cite

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“…These geometry errors are acceptable in many applications-particularly in image-based analyses, where the available initial geometry description is a pixel-based image. In other cases, it is desirable to improve the description of the boundaries by applying smoothing algorithms, cutting the cubes that are intersected by the boundary [38], or combining the octree decomposition with fictitious domain concepts [24]. Such approaches are the subject of other publications and will not be discussed here in detail.…”

Section: Octree Meshmentioning

confidence: 99%

Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions

2020

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The scaled boundary finite element method (SBFEM) has recently been employed as an efficient tool to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed, and each cubic cell is treated as an SBFE subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with a particular class of transition elements on the subdomains’ surfaces. Thus, a triangulation of these surfaces as executed in previous works is avoided, and consequently, the number of surface elements and degrees of freedom is reduced. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly.

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“…This method allows the construction and coupling of star-convex polygonal elements consisting of an arbitrary number of edges, where the type and order of interpolation along each edge can be chosen independently. The SBFEM has been applied successfully to quadtree/octree decompositions [61][62][63][64], as well as polytopal meshes [65,66]. For linear static problems, the SBFEM achieves high order convergence while only requiring nodes on the element edges.…”

Section: Introductionmentioning

confidence: 99%

High order transition elements: The xy-element concept—Part I: Statics

Duczek

Saputra

Gravenkamp

2020

Computer Methods in Applied Mechanics and Engineering

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Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element -xNy-element -which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h-and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.

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Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach

Cited by 35 publications

References 97 publications

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

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

Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions

High order transition elements: The xy-element concept—Part I: Statics

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