Showing papers by "Sundararajan Natarajan published in 2017"

Linear smoothed polygonal and polyhedral finite elements

Abstract: The strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes. The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations. The accuracy is also improved, and all the methods tested pass the patch test to machine precision.

A dual scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

Abstract: A novel technique to formulate arbritrary faceted polyhedral elements in three-dimensions is presented The formulation is applicable for arbitrary faceted polyhedra, provided that a scaling requirement is satisfied and the polyhedron facets are planar A triangulation process can be applied to non-planar facets to generate an admissible geometry The formulation adopts two separate scaled boundary coordinate systems with respect to: (i) a scaling centre located within a polyhedron and; (ii) a scaling centre on a polyhedron’s facets The polyhedron geometry is scaled with respect to both the scaling centres Polygonal shape functions are derived using the scaled boundary finite element method on the polyhedron facets The stiffness matrix of a polyhedron is obtained semi-analytically Numerical integration is required only for the line elements that discretise the polyhedron boundaries The new formulation passes the patch test Application of the new formulation in computational solid mechanics is demonstrated using a few numerical benchmarks

Crack propagation modelling in concrete using the scaled boundary finite element method with hybrid polygon–quadtree meshes

Abstract: 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.

Nonlinear Dynamic Thermal Buckling of Sandwich Spherical and Conical Shells with CNT Reinforced Facesheets

Abstract: Owing to their superior mechanical and thermal properties, carbon nanotube (CNT) reinforced composite materials have wide range of applications in various technical areas such as aerospace, automobile, chemical, structural and energy. In this paper, the nonlinear axisymmetric dynamic behavior of sandwich spherical and conical shells made up of CNT reinforced facesheets is studied. The shell is subjected to thermal loads and discretized with three-noded axisymmetric curved shell element based on field consistency approach. The in-plane and the rotary inertia effects are included within the transverse shear deformation theory in the element formulation. The present model is validated with the available analytical solutions from the literature. A detailed parametric study is carried out to showcase the effects of the shell geometry, the volume fraction of the CNT, the core-to-face sheet thickness and the environment temperature on the dynamic buckling thermal load of spherical caps.

A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities

Abstract: In this paper, we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique. This is accomplished by combining the Gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub-triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion.

Linear smoothed extended finite element method

Abstract: The extended finite element method was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak, and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and the convergence of the numerical solution. The smoothed extended finite element method (XFEM), for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in the literature that the strain smoothing is inaccurate when non‐polynomial functions are in the basis. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure which provides better approximation to higher‐order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics are solved and the results are compared with existing approaches. We observe that the stress intensity factors computed through the proposed linear smoothed XFEM is more accurate than that obtained through smoothed XFEM.

Linear smoothed extended finite element method

Abstract: The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. Moreover, in the case of open surfaces and singularities, special, usually non-polynomial functions must also be integrated. A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and the convergence of the numerical solution. The smoothed extended finite element method (SmXFEM) [1], for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in [1, 2] that the strain smoothing is inaccurate when non-polynomial functions are in the basis. This is due to the constant smoothing function used over the smoothing domains which destroys the effect of the singularity. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure [3] which provides better approximation to higher order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics (LEFM) are solved to compare the standard XFEM, the constant-smoothed XFEM (Sm-XFEM) and the linear-smoothed XFEM (LSm-XFEM). We observe that the convergence rates of all three methods are the same. The stress intensity factors (SIFs) computed through the proposed LSm-XFEM are however more accurate than that obtained through Sm-XFEM. To conclude, compared to the conventional XFEM, the same order of accuracy is achieved at a relatively low computational effort.

Static and free vibration analysis of cross-ply laminated plates using the Reissner-mixed variational theorem and the cell based smoothed finite element method

Abstract: Static bending and free vibration of cross-ply laminated plates with simply supported boundary conditions are studied using layerwise description for field variables. The layerwise approach accounts for the through-the-thickness deformations. The equations of motion and the boundary conditions are obtained by the Carrera's unified formulation. The stiffness matrix is computed by using the Reissner mixed variational theorem (RMVT), in which the transverse stresses are also treated as independent variables apart from the displacements. To this end, a mixed form of Hooke's law is defined. A cell-based smoothed finite element method is employed to compute the terms in the stiffness matrix. The influence of various parameters on the static bending and free vibration are numerically studied.

A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials

Abstract: This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials.

An Abaqus UEL implementation of the smoothed finite element method

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 does not require an explicit form of the derivative of the shape functions and there is no isoparametric mapping. This implementation is accomplished by employing the user element subroutine (UEL) feature of the software. The details on the input data format together with the proposed user element subroutine, which forms the core of the finite element analysis are given. A few benchmark problems from linear elastostatics in both two and three dimensions are solved to validate the proposed implementation. The developed UELs and the associated input files can be downloaded from Github repository link: this https URL\_in\_Abaqus.

A linear smoothed higher-order CS-FEM for the analysis of notched laminated composites

Abstract: Higher-order elements with highly accurate solutions are attractive for stress analysis and stress concentration problems. However, the distorted eight-node serendipity quadrilateral element is known to yield inaccurate results and sub-optimal convergence rate. In this paper, we present a higher-order CS-FEM to alleviate the effect of distorted mesh and guarantee the quality of solutions by employing a linear smoothing technique over eight-node quadratic serendipity elements. The modified strain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion of the weak form. The proposed method eliminates the need for isoparametric mapping and numerical studies demonstrate that the proposed method is insensitive to mesh distortion. The improved accuracy and superior convergence rates are numerically demonstrated with a few benchmark problems. The analysis of the stress concentration around cutouts also proves that the present method has good performance for the laminated composites.

Reconstrucción de tensiones para el método de elementos finitos con mallas poligonales

Abstract: El metodo de los elementos finitos es una de las herramientas numericas mas utilizadas para el diseno en ingenieria. En los ultimos anos se han desarrollado nuevas aproximaciones numericas para extender el uso del metodo de elementos finitos a mallas poligonales. Dichas aproximaciones mejoran la precision de la solucion y aumentan la flexibilidad en el mallado. Sin embargo, como toda aproximacion, es necesario cuantificar el valor del error inducido para poder validar los resultados obtenidos. En este trabajo se presenta el uso de una tecnica de estimacion del error de discretizacion para mallas de elementos finitos poligonales. La tecnica esta basada en la reconstruccion de la solucion en tensiones mediante un procedimiento de minimos cuadrados ponderados que considera la influencia de las ecuaciones de equilibrio. Se ha utilizado un problema con solucion exacta para evaluar la efectividad del estimador, obteniendo buenos resultados a nivel local y global.

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

Abstract: We propose a local type of B-bar formulation, addressing locking in degenerated Reissner-Mindlin plate and shell formulations 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 order as the projection bases. The present 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 $1 \times 10^5$, 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 and mesh distortion.

Erratum to: Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

Abstract: © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ERRATUM Hirshikesh et al. Asia Pac. J. Comput. Engin. (2017) 4:4 DOI 10.1186/s40540-017-0022-1