Convergence and accuracy of displacement based finite element formulations over arbitrary polygons: Laplace interpolants, strain smoothing and scaled boundary polygon formulation
- Authors: Natarajan, Sundararajan , Ooi, Ean Tat , Chiong, Irene , Song, Chongmin
- Date: 2014
- Type: Text , Journal article
- Relation: Finite Elements in Analysis and Design Vol. 85, no. (August 2014 2014), p. 101-122
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- 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.
SBFEM for fracture analysis of piezoelectric composites under thermal load
- Authors: Li, Chao , Ooi, Ean Tat , Song, Chongmin , Natarajan, Sundararajan
- Date: 2015
- Type: Text , Journal article
- Relation: International Journal of Solids and Structures Vol. 52, no. 1 (2015), p. 114-129
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- Description: This paper extends a semi-analytical technique, the so-called scaled boundary finite element method (SBFEM), to analyze fracture behaviors of piezoelectric materials and piezoelectric composites under thermal loading. In this method, only the boundary is discretized leading to a reduction of the spatial dimension by one. The temperature field in the domain is obtained using the SBFEM and expressed as a series of power functions of the radial coordinate. The resulting stress and electric displacement distribution along the radial direction is represented analytically. This permits the generalized stress and electric displacement intensity factors to be directly evaluated from the solution by following standard stress recovery procedures in the finite element method (FEM). Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique. © 2014 Elsevier Ltd.
Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method
- Authors: Hirshikesh , Pramod, Aladurthi , Annabattula, Ratna , Ooi, Ean Tat , Song, Chongmin , Natarajan, Sundararajan
- Date: 2019
- Type: Text , Journal article
- Relation: Computer Methods in Applied Mechanics and Engineering Vol. 355, no. (2019), p. 284-307
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- Description: In this work, we propose an adaptive phase field method (PFM) to simulate quasi-static brittle fracture problems. The phase field equations are solved using the scaled boundary finite element method (SBFEM). The adaptive refinement strategy is based on an error indicator evaluated directly from the solutions of the SBFEM without any need for stress recovery techniques. Quadtree meshes are adapted to perform mesh refinement. The polygons with hanging nodes in the quadtree decomposition are treated as n−sided polygons within the framework of the SBFEM and do not require any special treatment in contrast to the conventional finite element method. Several benchmark problems are used to demonstrate the robustness and the efficacy of the proposed technique. The adaptive refinement strategy reduces the mesh burden when adopting the PFM to model fracture. Numerical results show an improvement in the computational efficiency in terms of the number of elements required in the standard PFM without compromising the accuracy of the solution.
Virtual and smoothed finite elements : A connection and its application to polygonal/polyhedral finite element methods
- Authors: Natarajan, Sundararajan , Bordas, Stéphane , Ooi, Ean Tat
- Date: 2015
- Type: Text , Journal article
- Relation: International Journal for Numerical Methods in Engineering Vol. 104, no. 13 (2015), p. 1173-1199
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- Description: We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O(dofs)1.1 in case of the conventional polygonal FEM, while it scales as O(dofs)0.7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. © 2015 John Wiley & Sons, Ltd.
Numerical estimation of stress intensity factors in cracked functionally graded piezoelectric materials - a scaled boundary finite element approach
- Authors: Pramod, A. , Ooi, Ean Tat , Song, Chongmin , Natarajan, Sundararajan
- Date: 2018
- Type: Text , Journal article
- Relation: Composite Structures Vol. 206, no. (2018), p. 301-312
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- 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.
Scaled boundary finite element method for compressible and nearly incompressible elasticity over arbitrary polytopes
- Authors: Aladurthi, Lakshmi , Natarajan, Sundararajan , Ooi, Ean Tat , Song, Chongmin
- Date: 2019
- Type: Text , Journal article
- Relation: International Journal for Numerical Methods in Engineering Vol. 119, no. 13 (2019), p. 1379-1394
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- Description: In this paper, a purely displacement-based formulation is presented within the framework of the scaled boundary finite element method to model compressible and nearly incompressible materials. A selective reduced integration technique combined with an analytical treatment in the nearly incompressible limit is employed to alleviate volumetric locking. The stiffness matrix is computed by solving the scaled boundary finite element equation. The salient feature of the proposed technique is that it neither requires a stabilization parameter nor adds additional degrees of freedom to handle volumetric locking. The efficiency and the robustness of the proposed approach is demonstrated by solving various numerical examples in two and three dimensions.
Construction of high-order complete scaled boundary shape functions over arbitrary polygons with bubble functions
- Authors: Ooi, Ean Tat , Song, Chongmin , Natarajan, Sundararajan
- Date: 2016
- Type: Text , Journal article
- Relation: International Journal for Numerical Methods in Engineering Vol. 108, no. 9 (2016), p. 1086-1120
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- 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.
Extension of the scaled boundary finite element method to treat implicitly defined interfaces without enrichment
- Authors: Natarajan, Sundararajan , Dharmadhikari, Prasad , Annabattula, Ratna , Zhang, Junqi , Ooi, Ean , Song, Chongmin
- Date: 2020
- Type: Text , Journal article
- Relation: Computers and Structures Vol. 229, no. (2020), p.
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- Description: In this paper, the scaled boundary finite element method (SBFEM) is extended to solve the second order elliptic equation with discontinuous coefficients and to treat weak discontinuities. The salient feature of the proposed technique is that: (a) it requires only the boundary to be discretized and (b) does not require the interface to be discretized. The internal boundaries are represented implicitly by the level set method and the zero level sets are used to identify the different regions. In the regions containing the interface, edges along the boundary are assigned different material properties based on their location with respect to the zero level set. A detailed discussion is provided on the implementation aspects, followed by a few example problems in both two and three dimensions to show the robustness, accuracy and effectiveness of the proposed approach in modelling materials with interfaces. The proposed technique can easily be integrated to any existing finite element code. © 2019 Elsevier Ltd
An efficient forward propagation of multiple random fields using a stochastic Galerkin scaled boundary finite element method
- Authors: Mathew, Tittu , Pramod, A. L. N. , Ooi, Ean Tat , Natarajan, Sundararajan
- Date: 2020
- Type: Text , Journal article
- Relation: Computer Methods in Applied Mechanics and Engineering Vol. 367, no. (2020), p.
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- Description: This paper serves to extend the existing literature on the Stochastic Galerkin Scaled Boundary Finite Element Method (SGSBFEM) in two ways. The first part of this work deals with the formulation of multiple non-correlated Gaussian random fields using the conventional Karhunen–Loéve expansion technique and its forward propagation through the Spectral Stochastic Scaled Boundary Finite Element setting using the polynomial surface fit method in terms of the scaled boundary coordinates. The advantages in adopting such a forward propagation technique in capturing the statistical moments of Quantities of Interest (QoI) across the domain, are highlighted using carefully chosen linear elastic problems having large to least correlated random fields as inputs. The second contribution is the extension of the proposed forward Uncertainty Quantification (UQ) to take into account multiple independent random fields, followed by Polynomial Chaos Expansion (PCE) based sensitivity analysis. Both the computational efficiency and the accuracy of the proposed framework under different input random field correlation settings are elaborated upon by comparing their results against that obtained using the current existing SGSBFEM in the literature. Moreover, the stochastic results are validated for all the numerical examples using the Monte Carlo method. © 2020 Elsevier B.V.
Application of adaptive phase-field scaled boundary finite element method for functionally graded materials
- Authors: Pramod, Aladurthi , Hirshikesh , Natarajan, Sundararajan , Ooi, Ean Tat
- Date: 2021
- Type: Text , Journal article
- Relation: International Journal of Computational Methods Vol. 18, no. 3 (2021), p.
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- Description: In this paper, an adaptive phase-field scaled boundary finite element method for fracture in functionally graded material (FGM) is presented. The model accounts for spatial variation in the material and fracture properties. The quadtree decomposition is adopted for refinement, and the refinement is based on an error indicator evaluated directly from the solutions of the scaled boundary finite element method. This combination makes it a suitable choice to study fracture using the phase field method, as it reduces the mesh burden. A few standard benchmark numerical examples are solved to demonstrate the improvement in computational efficiency in terms of the number of degrees of freedom. © 2021 World Scientific Publishing Company.