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

Abstract

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...