On mass lumping and explicit dynamics in the scaled boundary finite element method

We present, for the first time, the application of explicit time-stepping schemes within the context of the scaled boundary finite element method (SBFEM). To this end, we discuss in detail how and under which circumstances diagonal mass matrices can be obtained. In addition, we propose an approach to improving the interpolation accuracy of quadratic scaled boundary-shape functions. In essence, our results show that mass lumping can be achieved without loss of accuracy if linear or quadratic node-based shape functions are deployed for the interpolation on the subdomain boundaries. This finding also applies to non-convex elements, which allow coarse discretizations — and, hence, comparably large time steps — in problems involving singularities. We discuss the application to polygonal and severely distorted meshes. Numerical examples include problems from the fields of acoustics, elasticity, and thermal diffusion

Shape optimization of acoustic devices using the Scaled Boundary Finite Element Method

In this study, the Scaled Boundary Finite Element Method (SBFEM) was used to perform analyses and evaluate the objective function in shape optimization of devices relying on acoustic wave propagation. Similar to the Boundary Element Method (BEM), the SBFEM requires only the discretization of the boundary of the computational domain. However, unlike BEM, there is no need for a fundamental solution; thus, the SBFEM provides a flexibility similar to that of the Finite Element Method (FEM). The dimension reduction is achieved by representing the solution analytically inside the domain and numerically on the boundary. Consequently, the SBFEM provides a flexible platform for shape optimization and alleviates the re-meshing difficulties encountered in FEM. It was shown that domain boundaries can be optimized with a minimum number of design variables, while the existing accurate transparent boundary conditions effectively eliminate the artificial numerical reflections for a wide range of frequencies

Three-dimensional image-based numerical homogenisation using octree meshes

The determination of effective material properties of composites based on a three-dimensional representative volume element (RVE) is considered in this paper. The material variation in the RVE is defined based on the colour intensity in each voxel of an image which can be obtained from imaging techniques such as X-ray computed tomography (XCT) scans. The RVE is converted into a numerical model using hierarchical meshing based on octree decompositions. Each octree cell in the mesh is modelled as a scaled boundary polyhedral element, which only requires a surface discretisation on the polyhedron's boundary. The problem of hanging (incompatible) nodes – typically encountered when using the finite element method in conjunction with octree meshes – is circumvented by employing special transition elements. Two different types of boundary conditions (BCs) are used to obtain the homogenised material properties of various samples. The numerical results confirm that periodic BCs provide a better agreement with previously published results. The reason is attributed to the fact that the model based on the periodic BCs is not over-constrained as is the case for uniform displacement BCs

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

Thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method

The scaled boundary finite element method is extended to model fracture in functionally graded materials (FGM) under coupled thermo-mechanical loads. The governing equations of coupled thermo-mechanical equilibrium are discretized using scaled boundary shape functions enriched with the thermal load terms. The material gradient is modeled as a series of power functions, and the stiffness matrix is calculated semi-analytically. Stress intensity factors and T−stress are directly calculated from their definition without any need for additional post-processing techniques. Arbitrary-sided polygon elements are employed for flexible mesh generation. Several numerical examples for isotropic and orthotropic FGMs are presented to validate the proposed technique