Description:
In this paper, an adaptive refinement strategy based on the scaled boundary finite element method on quadtree meshes for linear elasticity problems is discussed. Within this framework, the elements with hanging nodes are treated as polygonal elements and thus does not require special treatment. The adaptive refinement is supplemented with a novel error indicator. The local error is estimated directly from the solution of the scaled boundary governing equations. The salient feature is that it does not require any stress recovery techniques. The efficacy and the robustness of the proposed approach are demonstrated with a few numerical examples.
Description:
The quadtree is a hierarchical-type data structure where each parent is recursively divided into four children. This structure makes it particularly efficient for adaptive mesh refinement in regions with localised gradients. Compared with unstructured triangles, mesh generation is more efficient using quadtree decompositions. The finite number of patterns in the quadtree decomposition makes it efficient for data storage and retrieval. Motivated by these advantages, a crack propagation modelling approach using a quadtree-based scaled boundary finite element method (SBFEM) is developed. Starting from the formulation of an arbitrary n-sided polygon element, each quadrant in the quadtree mesh is treated as a polygon within the framework of the SBFEM. Special techniques to treat the hanging nodes are not necessary. Moreover, the SBFEM enables accurate calculation of the stress intensity factors directly from its solutions without local mesh refinement or asymptotic enrichment functions. When a crack propagates, it is only necessary to split each quadrant cut by the crack into two. These quadrants are polygons that can be directly modelled by the SBFEM. Changes to the mesh are minimal. The efficiency of this approach is demonstrated using numerical benchmarks.
Description:
This manuscript presents the development of novel high-order complete shape functions over star-convex polygons based on the scaled boundary finite element method. The boundary of a polygon is discretised using one-dimensional high order shape functions. Within the domain, the shape functions are analytically formulated from the equilibrium conditions of a polygon. These standard scaled boundary shape functions are augmented by introducing additional bubble functions, which renders them high-order complete up to the order of the line elements on the polygon boundary. The bubble functions are also semi-analytical and preserve the displacement compatibility between adjacent polygons. They are derived from the scaled boundary formulation by incorporating body force modes. Higher-order interpolations can be conveniently formulated by simultaneously increasing the order of the shape functions on the polygon boundary and the order of the body force mode. The resulting stiffness-matrices and mass-matrices are integrated numerically along the boundary using standard integration rules and analytically along the radial coordinate within the domain. The bubble functions improve the convergence rate of the scaled boundary finite element method in modal analyses and for problems with non-zero body forces. Numerical examples demonstrate the accuracy and convergence of the developed approach. Copyright (c) 2016 John Wiley & Sons, Ltd.
Description:
Three different displacement based finite element formulations over arbitrary polygons are studied in this paper. The formulations considered are the conventional polygonal finite element method (FEM) with Laplace interpolants, the cell-based smoothed polygonal FEM with simple averaging technique and the scaled boundary polygon formulation. For the purpose of numerical integration, we employ the sub-triangulation for polygonal FEM and classical Gaussian quadrature for the smoothed FEM and the scaled boundary polygon formulation. The accuracy and the convergence properties of these formulations are studied with a few benchmark problems in the context of linear elasticity and the linear elastic fracture mechanics. The extension of scaled boundary polygon to higher order polygons is also discussed.
Description:
This manuscript presents an extension of the recently-developed hybrid polygon-quadtree-based scaled boundary finite element method to model crack propagation in concrete. This hybrid approach combines the use of quadtree cells with arbitrary sided polygons for domain discretization. The scaled boundary finite element formulation does not distinguish between quadtree cells and arbitrary sided polygons in the mesh. A single formulation is applicable to all types of cells and polygons in the mesh. This eliminates the need to develop transitional elements to bridge the cells belonging to different levels in the quadtree hierarchy. Further to this, the use of arbitrary sided polygons facilitate the accurate discretization of curved boundaries that may result during crack propagation. The fracture process zone that is characteristic in concrete fracture is modelled using zero-thickness interface elements that are coupled to the scaled boundary finite element method using a shadow domain procedure. The scaled boundary finite element method can accurately model the asymptotic stress field in the vicinity of the crack tip with cohesive tractions. This leads to the accurate computation of the stress intensity factors, which is used to determine the condition for crack propagation and the resulting direction. Crack growth can be efficiently resolved using an efficient remeshing algorithm that employs a combination of quadtree decomposition functions and simple Booleans operations. The flexibility of the scaled boundary finite element method to be formulated on arbitrary sided polygons also result in a flexible remeshing algorithm for modelling crack propagation. The developed method is validated using three laboratory experiments of notched concrete beams subjected to different loading conditions.
Description:
A recently developed scaled boundary finite element formulation that can model the response of functionally graded materials is further developed to model crack propagation in two-dimensions. This formulation can accurately model the stress singularity at the crack tip in heterogeneous materials. The asymptotic behaviour at the crack tip is analytically represented in the scaled boundary shape functions of a cracked polygon. This enables accurate stress intensity factors to be computed directly from their definitions. Neither local mesh refinement nor asymptotic enrichment functions are required. This novel formulation can be implemented on polygons with an arbitrary number of sides. When modelling crack propagation, the remeshing process is more flexible and leads to only minimal changes to the global mesh structure. Six numerical examples involving crack propagation in functionally graded materials are modelled to demonstrate the salient features of the developed method.
Description:
In this paper, we discuss two alternate techniques to treat hanging nodes in a quadtree mesh. Both the techniques share similarities, in that, they require only boundary information. Moreover, they do not require an explicit form of the shape functions, unlike the conventional approaches, for example, as in the work of Gupta (Int J Numer Methods Eng 12:35, 1978) or Tabarraei and Sukumar (Finite Elem Anal Des 41:686, 2005). Hence, no special numerical integration technique is required. One of the techniques relies on the strain projection procedure, whilst the other is based on the scaled boundary finite element method. Numerical examples are presented to demonstrate the accuracy and the convergence properties of the two techniques.
Description:
In this paper, we discuss two alternate techniques to treat hanging nodes in a quadtree mesh. Both the techniques share similarities, in that, they require only boundary information. Moreover, they do not require an explicit form of the shape functions, unlike the conventional approaches, for example, as in the work of Gupta (Int J Numer Methods Eng 12:35, 1978) or Tabarraei and Sukumar (Finite Elem Anal Des 41:686, 2005). Hence, no special numerical integration technique is required. One of the techniques relies on the strain projection procedure, whilst the other is based on the scaled boundary finite element method. Numerical examples are presented to demonstrate the accuracy and the convergence properties of the two techniques.
Description:
The stress intensity factors and the electrical displacement intensity factor for functionally graded piezoelectric materials (FGPMs) are influenced by: (a) the spatial variation of the mechanical property and (b) the electrical and mechanical boundary conditions. In this work, a semi-analytical technique is proposed to study the fracture parameters of FGPMs subjected to far field traction and electrical boundary conditions. A scaled boundary finite element formulation for the analysis of functionally graded piezoelectric materials is developed. The formulation is linearly complete for uncracked polygons and can capture crack tip singularity for cracked polygons. These salient features enable the computation of the fracture parameters directly from their definition. Numerical examples involving cracks in FGPMs show the accuracy and efficiency of the proposed technique.