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

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Abstract

Quadtree-based domain decomposition algorithms offer an efficient option to create meshes for automatic image-based analyses. Without introducing hanging nodes the scaled boundary finite element method (SBFEM) can directly operate on such meshes by only discretizing the edges of each subdomain. However, the convergence of a numerical method that relies on a quadtree-based geometry approximation is often suboptimal due to the inaccurate representation of the boundary. To overcome this problem a combination of the SBFEM with the spectral cell method (SCM) is proposed. The basic idea is to treat each uncut quadtree cell as an SBFEM polygon, while all cut quadtree cells are computed employing the SCM. This methodology not only reduces the required number of degrees of freedom but also avoids a two-dimensional quadrature in all uncut quadtree cells. Numerical examples including static, harmonic, modal and transient analyses of complex geometries are studied, highlighting the performance of this novel approach.

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Notes

  1. Cells intersected by the boundary of the physical domain. In the literature often also referred to as broken cells.

  2. Cut leaf cells are the smallest cells of a quadtree decomposition that have no children (leaf). They are typically intersected by the physical boundary of the domain (cut). These cells are responsible for the approximation of the geometry of the computational domain.

  3. The point \(\xi =0\) is commonly referred to as scaling center in the context of the SBFEM.

  4. Using the same set of shape functions for the interpolation in Eqs. (1) and (3) means that isoparametric elements are employed on the boundary which is a common but not necessary assumption. Furthermore, it is not required to use nodal shape functions, though Lagrange shape functions are the most common choice in the context of the SBFEM.

  5. \(\varepsilon ^*\): IEEE 754 machine precision = \(2^{-53} \approx 1.16\times 10^{-16}\).

  6. In a spacetree decomposition scheme each cell that has no children is referred to as leaf cell. In contrast, the cell representing the whole domain is commonly denoted as root cell.

  7. Here, the Gauss–Lobatto–Legendre integration rule is applied, and thus for the regular elements, the nodes coincide with the displayed quadrature points.

  8. This issue could be partly alleviated by an anisotropic FCM approach based on integrated Legendre polynomials as shape functions. Here, the polynomial degrees for the edge modes and bubble modes can be chosen independently such that homogeneous domains are approximated with a lower polynomial degree [20].

  9. Leaf cells of the quadtree decomposition.

Abbreviations

BEM:

Boundary element method

BVP:

Boundary value problem

CAD:

Computer-aided design

CT:

Computed tomography

DOF:

Degrees of freedom

EOM:

Equation of motion

FCM:

Finite cell method

FDC:

Fictitious domain concept

FEM:

Finite element method

FE:

Finite element

GLL:

Gauss–Lobatto–Legendre (integration points)

IGA:

Isogeometric analysis

NURBS:

Non-uniform rational B-splines

ODE:

Ordinary differential equation

PDE:

Partial differential equation

SBFEM:

Scaled boundary finite element method

SBFE:

Scaled boundary finite element

SCM:

Spectral cell method

SD:

Subdomain

SEM:

Spectral element method

XFEM:

Extended finite element method

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Acknowledgements

Dr. S. Duczek would like to thank the German Research Foundation (DFG) for its financial support under Grant DU 1613/1-1.

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Author notes

  1. Hauke Gravenkamp and Sascha Duczek contributed equally to this project and should be considered co-first authors.

Authors and Affiliations

  1. University of Duisburg-Essen, Universitaetsstrasse 15, 45141, Essen, Germany

    Hauke Gravenkamp

  2. Otto von Guericke University Magdeburg, Universitaetsplatz 2, 39106, Magdeburg, Germany

    Sascha Duczek

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Correspondence to Hauke Gravenkamp.

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Gravenkamp, H., Duczek, S. Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach. Comput Mech 60, 559–584 (2017). https://doi.org/10.1007/s00466-017-1424-1

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  • DOI: https://doi.org/10.1007/s00466-017-1424-1

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