Automatic 3D modeling by combining SBFEM and transfinite element shape functions

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient means 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 SBFEM 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 certain transition elements on the subdomains' surfaces. Thus, we avoid the triangulation of surfaces employed in previous works and consequently reduce the number of surface elements and degrees of freedom. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly

A Combination of the Scaled Boundary Finite Element Method with the Mortar Method

Connecting different domains is one possibility to increase the performance of a numerical solution method. The Mortar Method is one of the well-established methods for this task. In this contribution, we focus on the solution of the elastodynamic wave equation by means of the scaled boundary finite element method and demonstrate that it is straightforward to connect different polygonal meshes by employing the Mortar Method in two dimensions. Examples show the stability for higher-order shape functions when performing h-refinement or p-refinement

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

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.Comment: 51 pages, 44 figures, 4 table

Simulation of Guided Waves in Cylinders Subject to Arbitrary Boundary Conditions Using the Scaled Boundary Finite Element Method

The scaled boundary finite element method (SBFEM) excels as a tool for numerical analysis at particular problem setups where the analytical solution in the scaling direction can be exploited to improve computational efficiency by reducing the number of required degrees of freedom (DOF). This is especially the case for simulating axisymmetric waveguides in the high-frequency range, allowing a significant decrease of computational costs (both memory and CPU time). Then, only the radial direction in a cylindrical coordinate system is discretized and the axial direction is solved analytically. A full threedimensional formulation is possible via the Fourier transform to include asymmetries. This contribution presents such an axisymmetric formulation, which is extended to allow the definition of circumferential as well as arbitrarily shaped dynamic boundary conditions (BCs). Furthermore, the required number of DOF depends on the frequency content. Hierarchical shape functions allow to dynamically adapt the DOF, further increasing efficiency. It will be shown that the results are in good agreement with standard finite element procedures, while greatly reducing computational time

Computation of leaky waves in layered structures coupled to unbounded media by exploiting multiparameter eigenvalue problems

We present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media. This problem is of relevance, e.g., for the simulation of guided ultrasound in embedded plate structures or seismic waves in soil layers over an elastic half-space. We employ a semi-analytical formulation to describe the layers, thus discretizing the thickness direction by means of finite elements. For a free layer, this technique leads to a well-known quadratic eigenvalue problem for the mode shapes and corresponding horizontal wavenumbers. Incorporating the coupling conditions to account for the adjacent half-spaces gives rise to additional terms that are nonlinear in the wavenumber. We show that the resulting nonlinear eigenvalue problem can be cast in the form of a multiparameter eigenvalue problem whose solutions represent the wave numbers in the plate and in the half-spaces. The multiparameter eigenvalue problem is solved numerically using recently developed algorithms. Matlab implementations of the proposed methods are publicly available

A Massively Parallel Explicit Solver for Elasto-Dynamic Problems Exploiting Octree Meshes

Typical areas of application of explicit dynamics are impact, crash test, and most importantly, wave propagation simulations. Due to the numerically highly demanding nature of these problems, efficient automatic mesh generators and transient solvers are required. To this end, a parallel explicit solver exploiting the advantages of balanced octree meshes is introduced. To avoid the hanging nodes problem encountered in standard finite element analysis (FEA), the scaled boundary finite element method (SBFEM) is deployed as a spatial discretization scheme. Consequently, arbitrarily shaped star-convex polyhedral elements are straightforwardly generated. Considering the scaling and transformation of octree cells, the stiffness and mass matrices of a limited number of unique cell patterns are pre-computed. A recently proposed mass lumping technique is extended to 3D yielding a well-conditioned diagonal mass matrix. This enables us to leverage the advantages of explicit time integrator, i.e., it is possible to efficiently compute the nodal displacements without the need for solving a system of linear equations. We implement the proposed scheme together with a central difference method (CDM) in a distributed computing environment. The performance of our parallel explicit solver is evaluated by means of several numerical benchmark examples, including complex geometries and various practical applications. A significant speedup is observed for these examples with up to one billion of degrees of freedom and running on up to 16,384 computing cores