17 research outputs found
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
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
On the numerical convergence and performance of different spatial discretization techniques for transient elastodynamic wave propagation problems
Prismatic semi-analytical elements for the simulation of linear elastic problems in structures with piecewise uniform cross section
Transient thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method
To model fracture in functionally graded materials (FGMs), the scaled boundary finite element method (SBFEM) is extended to examine the effects of fully coupled transient thermoelasticity. Previously developed SBFEM supplementary shape functions are utilized to model thermal stresses. The spatial variation of thermal and mechanical properties of FGMs are approximated by polynomial functions facilitating the semi-analytical evaluation of coefficient matrices. The dynamic stress intensity factors (SIFs) are also evaluated semi-analytically from their definitions without the need for additional post-processing. Scaled boundary polygon elements are employed to facilitate the meshing of complex crack geometries. Both isotropic and orthotropic materials with different material gradation functions are considered. To study the transient effects of thermoelasticity on fracture parameters, several numerical examples with different crack configurations and boundary conditions are considered. The current approach is validated by comparing the results of dynamic SIFs with available reference solutions. © 2023 Elsevier Lt