Adaptive analysis using scaled boundary finite element method in 3D
Introduction
Numerical methods have been widely used in engineering due to their feasibility and reliability in handling problems with complex geometries and boundary conditions. The finite element method (FEM) is one of the most popular numerical method, in which a problem domain is spatially discretized into small subdomains with simple shapes, called elements. The discretization process is called mesh generation. The results obtained from the numerical methods are usually not exact as the actual continuous variables are approximated by a set of discrete functions, which inevitably introduces discretization error. However, the approximate solution will monotonically converge to the exact value when certain refinement techniques are adopted, such as refinement, refinement and refinement [1], [2]. In refinement, the element size is reduced to better approximate the geometry and the distribution of the field variables, while refinement relies on increasing the order of the finite elements. In refinement, the nodes of the elements are relocated to increase mesh density in some locations, and the topology of the original mesh is preserved [3]. The most commonly used method in engineering practice is refinement due to its flexibility and compatibility with existing software packages. However, uniform refinement of the whole mesh usually results in a waste of computational resources. Therefore, it is preferable to localize the mesh refinement only in the region where it can improve the accuracy most effectively.
Adaptive analysis is a technique to perform numerical analysis and refine the mesh locally based on a posterior error estimation, which has attracted significant attention in many engineering fields since its inception [4], [5], [6], [7]. Reviews of the development of adaptive analysis can be found in [2], [8], [9], [10]. The error estimation in discontinuous Galerkin’s method is studied in [11] and singularly perturbed problems have been investigated in [12]. Posteriori error estimates in mixed finite element discretizations of convection–diffusion–reaction equations are reported in [13], [14], while adaptive method for partial differential equations of diffusion types have been studied in [15]. There are two important techniques which are necessary in an adaptive analysis: (i) a posterior error estimator and (ii) a mesh updating technique.
An error estimator aims to select the elements which need to be refined based on the numerical result obtained from the previous analysis. There are mainly two classes of error estimators [9]: (i) the residual based method and (ii) the recovery based method. The first method is based on evaluating the residuals of the approximate solution, which involves a direct computation of the interior element residuals and the jumps at the element boundaries [16], [17], [18]. Though the estimation obtained from this method does not accurately bound the actual error, it is inexpensive in terms of computational cost and is sufficient to guide the adaptive mesh refinement, therefore it is often referred to as an “error indicator”. More accurate information about the error can be obtained by solving additional auxiliary problems [19]. In a recovery based method, a smoothed stress field is recovered to better represent the real stress distribution [20]. A commonly used approach to calculate nodal stress values is averaging the results obtained from the elements surrounding each node, and the stress inside each element can be interpolated using the element shape function [21], [22]. The difference between the raw stress field and the recovered stress field is used as an error estimator, which is usually measured in the energy norm [23]. The optimal convergence rate can be obtained by a super convergent patch recovery technique which was reported by Zienkiewicz and Zhu [24], [25]. There are also different methods to select the elements to be refined based on the estimated error, which are also known as marking algorithms [26]. The first method defines an “absolute threshold” [27], in which all the elements with an estimated error greater than this threshold will be refined. Obviously, if the threshold is small, the first several refinement steps will be almost identical to a uniform refinement. The second commonly used method is based on “relative threshold”, in which a fixed percentage of elements with the highest estimated error are selected. Alternatively, the method proposed by Dörfler [28] selects the elements contributing up to a fixed percentage of the total estimated error, which has been applied in [29]. Some comparisons between these methods can be found in [3], [27]. More recently, error estimation methods involving singularities have been developed by Bordas and Duflot [30] for extended finite element method, in which a smooth strain field is recovered using extended moving least square derivative recovery. An extended global recovery method is later developed in [31] to investigate 2D and 3D fracture mechanics problems, while the application in adaptive mesh refinement has been reported in [32].
Another important aspect of an adaptive analysis is the mesh updating technique, which includes refinement and coarsening of the mesh [33]. In this work, only mesh refinement is implemented as coarsening is usually less frequently required, especially when the initial mesh size is reasonably coarse. In the FEM, usually the whole mesh needs to be regenerated in each step even if only a small portion of the elements need to be refined, which is time consuming. Furthermore, in nonlinear problems, a mapping algorithm is required to transfer the data if the whole mesh is regenerated [8]. This disadvantage is caused by the restriction on element shapes in the FEM, i.e. only triangular, quadrilateral, tetrahedral and hexahedral elements can be directly used in the FEM, which means the local refinement in some region will cause incompatibility between elements, or the “hanging node” problem [8]. A large amount of research has been dedicated to overcome this difficulty in order to avoid regenerating mesh for the whole model. Quadtree/octree based algorithms have attracted extensive attention as they are highly efficient and produce fast mesh size transitions, therefore they are especially suitable for the local refinement of quadrilateral/hexahedral meshes [34], [35], [36]. A novel transition element was developed by Wu et al. [37], Lo et al. [38] which can connect hexahedral elements with different sizes. In [3], [39], tetrahedral elements were introduced in the size transition zone of hexahedral mesh, and different patterns for tetrahedralization were enumerated. A polytree algorithm was proposed to refine unstructured polygonal meshes by Spring et al. [40], which divides a polygon with nodes into quadrilaterals. Another polytree based polygonal FEM was proposed by Nguyen-Xuan et al. [41], [42], Chau et al. [43], which is able to improve the quality of the refined polygons even if the initial polygonal element is distorted.
Adaptive analysis has been extensively applied to other numerical methods as well, such as meshfree particle method [44], [45], [46], isogeometric analysis (IGA) [26], [27], [47], [48], boundary element method [49] and discrete element method [50]. These methods aim to integrate the geometric model from computer aided design (CAD) to the numerical model in the analysis. Recently, isogeometric boundary element method (IGABEM) [51] is developed to overcome the difficulties in mesh generation in traditional FEM. With the advantage of tight integration of design and analysis, IGABEM has been successfully applied to structural shape optimization [52], [53], [54], crack propagation [55], [56] and others [57], [58]. Furthermore, to bypass the constraint on spline spaces in IGABEM, Geometry-independent field approximation (GIFT) is developed by Xu et al. [59], in which the spline spaces used for the geometry and the field variables can be chosen independently. This advantage provides additional flexibility in adaptive analysis [60], which has been investigated by Atroshchenko et al. [61].
In this paper, an adaptive analysis method based on the scaled boundary finite element method (SBFEM) is presented. The SBFEM is a numerical method first proposed by Song and Wolf [62], which was initially developed for dynamic analysis in unbounded domains and later extended to a broad range of applications [63], [64], [65], [66]. The basic idea is to divide the problem domain into subdomains satisfying the so-called scaling requirement (Section 2.1). A semi-analytical approach is adopted to construct an approximate solution in a subdomain. To this end, only the boundary of the subdomain is discretized, and the solution in the radial direction must be obtained analytically. This feature of the SBFEM enables the polytopal (polygonal [67]/polyhedral [68]) elements with an arbitrary number of nodes, edges and faces to be used in the analysis. Due to the versatility in element shapes, the difficulty in mesh generation can be greatly reduced. In recent years, researchers have endeavoured to apply this method to solve image based analysis [69], [70], acoustics [71], [72], contact [73], [74], domain decomposition [75] and many other problems [76], [77], [78], [79], [80], [81], [82]. The SBFEM has been combined with adaptive analysis in many cases, e.g. in [83] a stress recovery and error estimation technique inside each subdomain is developed. A recovered stress field is constructed by smoothing the stress in the circumferential direction but not radial direction. The smoothing is based on individual strain modes in each subdomain, therefore the error is integrated over the volume analytically. A novel error indicator without stress recovery is reported by Song et al. [84], in which quadtree algorithm is used for mesh refinement. In [85] an adaptive algorithm is combined with the phase field method to model quasi-static brittle fracture problems.
In this work, an adaptive analysis technique is developed which can be applied to general polyhedral meshes in 3D, including hexahedral and tetrahedral meshes. A residual based error indicator using the discontinuity of the stress field on element boundaries is adopted, which is effective to guide the mesh refinement with low computational cost. A local mesh refinement technique based on the polytree algorithm is developed making use of the flexibility of element shapes in the SBFEM, therefore it is not necessary to regenerate mesh for the whole model during the successive iterations. The polyhedral elements in the mesh size transition zone can be formulated by discretizing their boundaries only, therefore no tetrahedralization is required. The main focus of this work is on the local mesh refinement using polytree algorithm and the formulation of polyhedral transition element based on the SBFEM, therefore a relatively simple error indicator is sufficient in guiding mesh refinement. More sophisticated techniques of error estimation will be investigated and compared in the forthcoming publications.
The remainder of this paper is organized as follows: In Section 2, a brief description of polyhedral elements constructed using the SBFEM is introduced. The error estimator and mesh refinement algorithm are presented in Section 3. In Section 4, the proposed method is verified using five numerical examples. The results are compared to available reference solutions and the convergence behaviors are examined. Remarks and summaries are stated in Section 5.
Section snippets
The scaled boundary finite element method
In this section, the construction of polyhedral elements using the SBFEM is briefly discussed. For the sake of brevity, only the key equations that are necessary for the implementation are presented. Readers interested in a detailed derivation and additional explanations are referred to the monograph by Song [86].
Adaptive mesh refinement
In this section, an error indicator is implemented using the discontinuity of stress field between adjacent elements together with a mesh refinement technique based on polytree algorithm. As a result, this technique can be applied to perform local mesh refinement after each iteration of analysis using the SBFEM. The complete adaptive analysis procedure is presented at the end of this section in Fig. 11.
Numerical examples
In this section, five numerical examples are presented to verify the proposed method. In the first example, a spherical hole in an infinite domain subject to a remote stress is considered, the analytical solution of which is available from elasticity theory. Both uniform and adaptive refinements are performed to compare the convergence rates. In the second example, a cube with a missing corner is modeled. Two different marking algorithms based on absolute and relative thresholds are considered.
Conclusions
In this paper, an adaptive refinement technique based on the SBFEM was proposed. An explicit residual based error indicator was developed together with a polytree based local mesh refinement algorithm. This method can be applied to general polyhedral meshes, including hexahedral and tetrahedral meshes, which elegantly avoided the need to regenerate mesh for the whole model. No special element was introduced in the mesh size transition zone. The polyhedral elements can be used in analysis with
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The work presented in this paper is partially supported by the Australian Research Council through Grant Number DP180101538.
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