Robust modelling of implicit interfaces by the scaled boundary finite element method
- Authors: Dsouza, Shaima , Pramod, A. L. N. , Ooi, Ean Tat , Song, Chongming , Natarajan, Sundararajan
- Date: 2021
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 124, no. (2021), p. 266-286
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- Description: In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O(h2) in L2 norm and O(h) in the H1 semi-norm. Further the condition number increases with the mesh size similar to the FEM. © 2021 Elsevier Ltd
Treatment of multiple input uncertainties using the scaled boundary finite element method
- Authors: Dsouza, Shaima , Varghese, Tittu , Ooi, Ean Tat , Natarajan, Sundararajan , Bordas, Stephane
- Date: 2021
- Type: Text , Journal article
- Relation: Applied Mathematical Modelling Vol. 99, no. (2021), p. 538-554
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- Description: This paper presents a non-intrusive scaled boundary finite element method to consider multiple input uncertainties, viz., material and geometry. The types of geometric uncertainties considered include the shape and size of inclusions. The inclusions are implicitly defined, and a robust framework is presented to treat the interfaces, which does not require explicit generation of a conforming mesh or special enrichment techniques. A polynomial chaos expansion is used to represent the input and the output uncertainties. The efficiency and the accuracy of the proposed framework are elucidated in detail with a few problems by comparing the results with the conventional Monte Carlo method. A sensitivity analysis based on Sobol’ indices using the developed framework is presented to identify the critical input parameter that has a higher influence on the output response. © 2021 Elsevier Inc.
A novel error indicator and an adaptive refinement technique using the scaled boundary finite element method
- Authors: Song, Chongmin , Ooi, Ean Tat , Pramod, Aladurthi , Natarajan, Sundararajan
- Date: 2018
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 94, no. (2018), p. 10-24
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- 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.
A quadtree-polygon-based scaled boundary finite element method for crack propagation modeling in functionally graded materials
- Authors: Chen, Xiaojun , Luo, Tao , Ooi, Ean Tat , Ooi, Ean Hin , Song, Chongmin
- Date: 2018
- Type: Text , Journal article
- Relation: Theoretical and Applied Fracture Mechanics Vol. 94, no. (2018), p. 120-133
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- Description: This paper presents a method to improve the computational efficiency of the scaled boundary finite element formulation for functionally graded materials. Both isotropic and orthotropic functionally graded materials are considered. This is achieved using a combination of quadtree and polygon meshes. This hybrid meshing approach is particularly suitable to be used with the SBFEM for functionally graded materials because of the significant amount of calculations required to compute the stiffness matrices of the polygons/cells in the mesh. When a quadtree structure is adopted, most of the variables required for the numerical simulation can be pre-computed and stored in the memory, retrieved and scaled as required during the computations, leading to an efficient method for crack propagation modeling. The scaled boundary finite element formulation enables accurate computation of the stress intensity factors directly from the stress solutions without any special post-processing techniques or local mesh refinement in the vicinity of the crack tip. Numerical benchmarks demonstrate the efficiency of the proposed method as opposed to using a purely polygon-mesh based approach. © 2018 Elsevier Ltd
Hydraulic fracture at the dam-foundation interface using the scaled boundary finite element method coupled with the cohesive crack model
- Authors: Zhong, Hong , Li, Hongjun , Ooi, Ean Tat , Song, Chongmin
- Date: 2018
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 88, no. (2018), p. 41-53
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- Description: The scaled boundary finite element method coupled with the cohesive crack model is extended to investigate the hydraulic fracture at the dam-foundation interface. The concrete and rock bulk are modeled by the scaled boundary polygons. Cohesive interface elements model the fracture process zone between the crack faces. The cohesive tractions are modeled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. Accurate displacement field, stress field and stress intensity factors can be obtained without asymptotic enrichment or local mesh refinement. The proposed procedure is verified by the hydraulic fracture of a rectangular embankment on rigid foundation and applied to the modeling of hydraulic fracture on the dam-foundation interface of a benchmark dam. Different distributions of water pressure inside the crack are investigated. It is found that the water pressure inside the crack decreases the peak overflow to less than 20% of the case without water in the crack. Considering the water lag or not is significant to the response, while different distribution of pressure following the water lag region in the fracture process zone has negligible influence.
A scaled boundary finite element formulation over arbitrary faceted star convex polyhedra
- Authors: Natarajan, Sundararajan , Ooi, Ean Tat , Saputra, Albert , Song, Chongmin
- Date: 2017
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 80, no. (2017), p. 218-229
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- Description: In this paper, a displacement based finite element framework for general three-dimensional convex polyhedra is presented. The method is based on a semi-analytical framework, the scaled boundary finite element method. The method relies on the definition of a scaling center from which the entire boundary is visible. The salient feature of the method is that the discretizations are restricted to the surfaces of the polyhedron, thus reducing the dimensionality of the problem by one. Hence, an explicit form of the shape functions inside the polyhedron is not required. Conforming shape functions defined over arbitrary polygon, such as the Wachpress interpolants are used over each surface of the polyhedron. Analytical integration is employed within the polyhedron. The proposed method passes patch test to machine precision. The convergence and the accuracy properties of the method is discussed by solving few benchmark problems in linear elasticity. © 2017 Elsevier Ltd
Finite element computations over quadtree meshes : Strain smoothing and semi-analytical formulation
- Authors: Natarajan, Sundararajan , Ooi, Ean Tat , Song, Chongmin
- Date: 2013
- Type: Text , Journal article
- Relation: International Journal of Advances in Engineering Sciences and Applied Mathematics Vol. 7, no. 3 (2013), p. 124-133
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- 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.