Showing papers by "Sundararajan Natarajan published in 2020"

Isogeometric analysis of thin Reissner-Mindlin shells: locking phenomena and B-bar method

Abstract: We propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders as the projection bases. The introduced formulation is much cheaper computationally than the classical $$\bar{B}$$ method. We show the numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulation alleviates locking and yields good accuracy even for slenderness ratios of $$10^5$$ , and has the ability to capture deformations of thin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to locking with irregular meshes.

Development of User Element Routine (UEL) for Cell-Based Smoothed Finite Element Method (CSFEM) in Abaqus

Abstract: In this paper, we discuss the implementation of a cell-based smoothed finite element method (CSFEM) within the commercial finite element software Abaqus. The salient feature of the CSFEM is that it...

A combined virtual element method and the scaled boundary finite element method for linear elastic fracture mechanics

Abstract: In this paper, we propose a framework that combines the recently introduced virtual element method (VEM) and the scaled boundary finite element method (SBFEM) to evaluate the fracture parameters. The domain is discretized with arbitrary polygons and the element that contains the crack tip is treated within the framework of the SBFEM. This facilitates a semi-analytical treatment of the crack tip singularity allowing the fracture parameters are estimated directly from the definition. The VEM is employed for the rest of the domain. The salient feature of the VEM is that the terms in the stiffness matrix are computed without requiring higher order quadrature schemes. As both the methods satisfy partition of unity and the compatibility condition, the matrices are assembled as in the conventional FEM. The accuracy of the proposed formulation is demonstrated with two standard benchmark examples. The proposed VEM-SBFEM framework yields accurate results.

Adaptive phase field modelling of crack propagation in orthotropic functionally graded materials

Abstract: In this work, we extend the recently proposed adaptive phase field method to model fracture in orthotropic functionally graded materials (FGMs). A recovery type error indicator combined with quadtree decomposition is employed for adaptive mesh refinement. The proposed approach is capable of capturing the fracture process with a localized mesh refinement that provides notable gains in computational efficiency. The implementation is validated against experimental data and other numerical experiments on orthotropic materials with different material orientations. The results reveal an increase in the stiffness and the maximum force with increasing material orientation angle. The study is then extended to the analysis of orthotropic FGMs. It is observed that, if the gradation in fracture properties is neglected, the material gradient plays a secondary role, with the fracture behaviour being dominated by the orthotropy of the material. However, when the toughness increases along the crack propagation path, a substantial gain in fracture resistance is observed.

A locally refined adaptive isogeometric analysis for steady-state heat conduction problems

Abstract: This paper presents an isogeometric analysis with adaptivity using locally refined B-splines (LR B-splines) for steady-state heat conduction simulations in solids. Within this framework, the LR B-splines, which have an efficient and simple local refinement algorithm, are used to represent the geometry, and are also employed for spatial discretization, thus providing a seamless interaction between the CAD models and the numerical analysis. A Zienkiewicz-Zhu a posteriori error estimator in terms of the temperature gradient recovery is used to identify the regions for local mesh refinement. The accuracy and convergence properties of the proposed framework are demonstrated through several two-dimensional isotropic examples. Numerical results indicate good performance of the present method as the adaptive refinement technique yields more accurate solution compared to the uniform global refinement with an improved convergence rate.

Polytopal composite finite elements for modeling concrete fracture based on nonlocal damage models

Abstract: The paper presents an assumed strain formulation over polygonal meshes to accurately evaluate the strain fields in nonlocal damage models. An assume strained technique based on the Hu-Washizu variational principle is employed to generate a new strain approximation instead of direct derivation from the basis functions and the displacement fields. The underlying idea embedded in arbitrary finite polygons is named as Polytopal composite finite elements (PCFEM). The PCFEM is accordingly applied within the framework of the nonlocal model of continuum damage mechanics to enhance the description of damage behaviours in which highly localized deformations must be captured accurately. This application is helpful to reduce the mesh-sensitivity and elaborate the process-zone of damage models. Several numerical examples are designed for various cases of fracture to discuss and validate the computational capability of the present method through comparison with published numerical results and experimental data from the literature.

Size effects in nanostructured Li-ion battery cathode particles

Abstract: Abstract Cathode materials for Li-ion batteries exhibit volume expansions on the order of 10% upon maximum lithium insertion. As a result internal stresses are produced and after continuous electrochemical cycling damage accumulates, which contributes to their failure. Battery developers resort to using smaller particle sizes in order to limit damage and some models have been developed to capture the effect of particle size on damage. In this paper, we present a gradient elasticity framework,which couples the mechanical equilibrium equations with Li-ion diffusion and allows the Young’s modulus to be a function of Li-ion concentration. As the constitutive equation involves higher order gradient terms, the conventional finite element method is not suitable, while, the two-way coupling necessitates the need for higher order shape functions. In this study, we employ B-spline functions with the framework of the iso-geometric analysis for the spatial discretization. The effect of the internal characteristic length on the concentration evolution and the hydrostatic stresses is studied. It is observed that the stress amplitude is significantly affected by the internal length, however, using either a constant Young’s modulus or a concentration dependent one yields similar results.

Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes

Abstract: We consider the discretization of the nonlocal elliptic problem of Kirchhoff type using the virtual element method (VEM) over polygonal meshes. The nonlocal diffusion coefficient is approximated by using the L 2 projection operator, which is directly computable from the degrees of freedom. However, the presence of the nonlocal term reduces the sparsity of the Jacobian matrix, which would increase the computational burden. To avoid this and to retain the sparsity of the Jacobian, the nonlinear system is replaced by an equivalent system inspired by the work on the FEM by Gudi (2012). The numerical results show that the proposed methodology yields optimal convergence rate in the L 2 norm. Theoretical estimates are derived for H 1 norms and are verified by solving a numerical example.

Virtual element methods for nonlocal parabolic problems on general type of meshes

Abstract: In this paper, we consider the discretization of a parabolic nonlocal problem within the framework of the virtual element method. Using the fixed point argument, we prove that the fully discrete scheme has a unique solution. The presence of the nonlocal term makes the problem nonlinear, and the resulting nonlinear equations are solved using the Newton method. The computational cost of the Jacobian of the nonlinear scheme increases in the presence of nonlocal coefficient. To reduce the computational burden in computing the Jacobian, which otherwise is inevitable in the usual approach, in this paper, we propose an equivalent formulation. A priori error estimates in the L2 and the H1 norms are derived. Furthermore, we employ a linearized scheme without compromising the rate of convergence in the respective norms. Finally, the theoretical convergence results are verified through numerical experiments over polygonal meshes.

Roll-over shape of a prosthetic foot: a finite element evaluation and experimental validation

Abstract: Prosthetic feet have generally been designed experimentally by adopting a trial-and-error technique. The objective of this research is to introduce a novel numerical approach for the a priori evaluation of the roll-over shape (ROS) of a prosthetic foot for application in its systematic design and development. The ROS was achieved numerically by employing a non-linear finite element model incorporating the augmented Lagrangian and multi-point constraint contact formulations, a hyperelastic material model and a higher-order strain definition. The Ottobock Solid Ankle Cushion Heel (SACH) foot was chosen to experimentally validate the numerical model. The geometry of the foot was evaluated from optical scans, and the material properties were obtained from uniaxial tensile, shear and volumetric compression tests. A new setup was designed for an improved experimental determination of the ROS, with the inclusion of an extended moment arm and variable loading. Error analysis of the radius of curvature of the ROS between the numerical and experimental results showed the percentage error to be 7.52%, thereby establishing the validity of the model. A numerical design model of this kind can be utilised to vary the input design parameters to arrive at a prosthetic foot with specified performance characteristics effectively and economically.

A stochastic Galerkin cell-based smoothed finite element method (SGCS-FEM)

Abstract: In this paper, the cell-based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young’s Modulus and...

A Non-Intrusive Stochastic Isogeometric Analysis of Functionally Graded Plates with Material Uncertainty

Abstract: A non-intrusive approach coupled with non-uniform rational B-splines based isogeometric finite element method is proposed here. The developed methodology was employed to study the stochastic static bending and free vibration characteristics of functionally graded material plates with inhered material randomness. A first order shear deformation theory with an artificial shear correction factor was used for spatial discretization. The output randomness is represented by polynomial chaos expansion. The robustness and accuracy of the framework were demonstrated by comparing the results with Monte Carlo simulations. A systematic parametric study was carried out to bring out the sensitivity of the input randomness on the stochastic output response using Sobol’ indices. Functionally graded plates made up of Aluminium (Al) and Zirconium Oxide (ZrO2) were considered in all the numerical examples.

Design of a Biomimetic SACH Foot: An Experimentally Verified Finite Element Approach

Abstract: The Solid Ankle Cushioned Heel (SACH) foot is a commonly prescribed prosthetic foot for the rehabilitation of lower limb amputees. From the viewpoint of its biomechanical performance, the foot is known to cause drop-off effect and asymmetry in amputee gait. Therefore, the objective of this work is to improvise the effective foot length ratio (EFLR) and the progression of the centre of pressure (CoP) of the SACH foot by providing a novel design approach that utilizes finite element analysis. Boundary conditions employed for evaluating the roll-over characteristics of prosthetic feet were numerically incorporated in this work. The non-linear mechanical behavior of the foot was included with the incorporation of large deformation, a hyperelastic material model and the Augmented Lagrangian contact formulation. Outcomes from the simulations were experimentally verified using an inverted pendulum-like apparatus, thereby substantiating the numerical approach. The design process of the SACH foot involved the modification of the elastic modulus of its components for enhancing the parameters of interest. Results obtained presented a 5.07% increase in the EFLR and a 9.29% increase in the anteroposterior progression of the CoP, which may improve amputee stability. The design solution presented may support the large user base of the SACH foot towards achieving enhanced gait characteristics during ambulation. Moreover, this work successfully demonstrates a novel design procedure for a prosthetic foot through an effective numerical implementation.

Continuum modelling of stress diffusion interactions in an elastoplastic medium in the presence of geometric discontinuity.

Abstract: Chemo-mechanical coupled systems have been a subject of interest for many decades now. Previous attempts to solve such models have mainly focused on elastic materials without taking into account the plastic deformation beyond yield, thus causing inaccuracies in failure calculations. This paper aims to study the effect of stress-diffusion interactions in an elastoplastic material using a coupled chemo-mechanical system. The induced stress is dependent on the local concentration in a one way coupled system, and vice versa in a two way coupled system. The time-dependent transient coupled system is solved using a finite element formulation in an open-source finite element solver FEniCS. This paper attempts to computationally study the interaction of deformation and diffusion and its effect on the localization of plastic strain. We investigate the role of geometric discontinuities in scenarios involving diffusing species, namely, a plate with a notch/hole/void and particle with a void/hole/core. We also study the effect of stress concentrations and plastic yielding on the diffusion-deformation. The developed code can be from this https URL

Linear smoothed finite element method for quasi-incompressible hyperelastic media

Abstract: This work presents a linear smoothing scheme over high-order triangular elements within the framework of the cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme is that it unlike the classical SFEM, it does not require the subdivision of the finite element cells into smoothing sub-cell. The other features of the classical SFEM are retained, such as: it does not require an explicit form of the derivatives of the basis functions, all the computations are done in the physical space, and the results are less sensitive to mesh distortion. A series of benchmark tests are done to demonstrate the validity and the stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using quadratic triangular element and the exact solutions.

Shape Optimization of Beams with Scaled Boundary Finite Element Method and B-Splines

Abstract: Shape optimization is a method to find the optimal shape of a structure by minimizing the objective function, i.e., volume and/or compliance under the design and limit constraints. The boundary of the structure is discretized using B-splines which allow local control and these control points are taken as design variables for the optimization formulation. The scaled boundary finite element method (SBFEM) is a semi-analytical approach, which has the combined advantage of both finite element method and the boundary element method. The SBFEM discretizes the boundary of the element, which reduces the computational domain size by one. Both the B-splines and the SBFEM discretizes the boundary of the structure, so the control points of the B-spline and the nodes of the SBFEM discretization can coincide to form the design variables. In this paper, we design a cantilever beam by using the shape optimization with the B-splines and the SBFEM. The cantilever beam is designed for minimum volume under a UDL and a point load with displacement constraint. The results obtained by the FEM and the SBFEM are compared.

Stress diffusion interactions in an elastoplastic medium in the presence of geometric discontinuity

Abstract: This paper aims to study the effect of stress-diffusion interactions and its effect on the localization of the plastic strain in an elastoplastic material using a fully coupled chemo-mechanical sys...

An adaptive polytree approach to the scaled boundary boundary finite element method

Abstract: In this paper, a h-adaptive methodology based on the polytopal meshes is proposed for capturing high stress gradients at the materials corners and the stress singularities at the vicinity of a crack tip. The adaptive refinement is based on the error indicator directly computed from the displacement solutions of the scaled boundary finite element method. Based on the error indicator, a polygon of n-sides which has an error exceeding a specified tolerance is subdivided recursively into $$(n+1)$$ child polygons. The salient features of the proposed framework are: (a) circumvents a need for post-processing techniques for error estimation; (b) elements with hanging nodes are treated as polygons without a need for special treatment and (c) stress gradients and stress singularities are accurately captured due to the semi-analytical formulation. The robustness and the convergence properties of the proposed framework is demonstrated with three benchmark examples.