Quadrilateral Elements Research Articles

Quadrilateral scaled boundary spectral shell elements for static and dynamic analyses of functionally graded plates and shells

This paper presents new shell elements for static and dynamic analyses of functionally graded plates and shells with the material properties varying through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. In the formulation, a shell element is treated as a three-dimensional linear elastic body and its middle surface is represented with a quadrilateral spectral element. Along the thickness direction, the shell geometry is described by scaling the middle surface while the displacements are approximated by interpolating the displacements on the top, middle and bottom surfaces using quadratic Lagrange shape functions. The assumed natural strain method is applied to eliminate transverse shear locking, membrane locking and curvature thickness locking. The developed shell elements are featured by requiring only the shell mid-surface to be discretized, containing no rotational degree of freedom, adopting the full three-dimensional constitutive law and involving only the in-plane numerical integration. The validity and performance of the formulation are demonstrated through benchmark examples covering plates, cylindrical shells and spherical shells with varying span-to-thickness ratio and material gradation index. Static solutions of two functionally graded shell problems under gravity loads are also provided.

On the Use of Quadrilateral Meshes for Enhanced Analysis of Waveguide Devices with Manhattan-Type Geometry Cross-Sections

This work addresses the suitability of using structured meshes composed of quadrilateral finite elements, instead of the classical unstructured meshes made of triangular elements. These meshes are used in the modal analysis of waveguides with Manhattan-like cross-sections. For this problem, solved with the two-dimensional Finite Element Method, there are two main quality metrics: eigenvalue and eigenvector accuracy. The eigenvalue accuracy is first considered, showing how the proposed structured meshes are, given comparable densities, better, especially when dealing with waveguides presenting pairs of modes with the same cutoff frequency. The second metric is measured through a practical problem, which commonly appears in microwave engineering: discontinuity analysis. In this problem, for which the Mode-Matching technique is used, eigenvectors are needed to compute the coupling between the modes in the discontinuities, directly influencing the quality of the transmission and reflection parameters. In this case, it is found that the proposed analysis performs better given low-density meshes and mode counts, thus proving that quadrilateral-element structured meshes are more resilient than their triangular counterparts to higher-order eigenvectors.

Polygonal composite elements for stress-constrained topology optimization of nearly incompressible materials

We in this paper propose an efficient and vigorous approach based on flexible polygonal meshes for solving stress-constrained topology optimization problems involving both compressible and nearly incompressible materials. The core idea is to employ a polygonal composite finite element technique to deal with volumetric locking phenomena in nearly incompressible material limit. Then, to effectively solve topology optimization problems with local stress constraints, an augmented Lagrangian technique that allows treating problems with a large number of constraints is exploited. By employing the augmented Lagrangian method, the solution of the original optimization problem with a lot of constraints could be obtained by solving a series of unconstrained optimization ones and updating the design variables based on the method of moving asymptotes. The significant contribution of this work is to promote a unified formulation for stress-constrained topology optimization possessing the following salient features: (a) being valid for both compressible and nearly incompressible materials; (b) being valid for triangular, quadrilateral and polygonal elements; (c) sole displacement field involved; and (d) no constraint aggregation technique required. Through several numerical examples, for the first time, distinguishing and intriguing features of the optimized topology for nearly incompressible materials with stress constraints are exhibited in the present work.

Performance and accuracy of hybridized flux reconstruction schemes

We present a unified framework for the hybridization of flux reconstruction (FR) schemes. Specifically, we discuss the hybridized flux reconstruction (HFR) and interior-embedded flux reconstruction (EFR) methods and analyze their performance and accuracy for a range of correction functions on quadrilateral elements. HFR uses discontinuous traces on the faces of the elements, and EFR considers globally continuous trace functions. We perform two-dimensional von Neumann analysis to characterize the numerical error of EFR methods in relation to HFR and FR. HFR and FR are equivalent for linear advection, but EFR is only equivalent to FR one-dimensional advection directions. It is observed that EFR methods generally introduce additional numerical error for lower-order simulations, but behave closer to FR methods for higher orders. In addition, the behavior for different correction functions follows the same trend as the conventional FR formulation. Verification with linear and nonlinear numerical examples showed the expected order of accuracy of p+1 was obtained for all formulations, where p is the polynomial degree of the solution. With appropriate choices of correction functions, EFR methods can be more accurate than other HFR and FR methods. Furthermore, it is observed that the choice of correction function has an impact on the performance of the implicit solver by a factor of up to 2 for FR methods for an airfoil-vortex interaction problem, but had a smaller influence on the hybridized forms. For this problem, we observed speedups factors of up to 4.3 and 6.7 for HFR and EFR, respectively, when compared with FR. Hence, hybridized methods are a viable approach to reducing the computational cost of implicit FR solvers.

Utjecaji i dinamičko ponašanje interakcije tla i okvirne konstrukcije

The effect and behaviour of soil - framed-structure interaction (SSI) under dynamic load conditions are presented in the paper. All civil engineering structures involve various types of structural elements that are in direct contact with soil. Hence, a new thin-layer interface element, based on the concept of the finite element method, is formulated. The formulation is elaborated using a combination of degrees of freedom of the top and bottom sides of the interface elements. The compatibility conditions of displacements between beam elements and quadrilateral soil elements are applied. Thus, a numerical program integrating the thin-layer interface element is developed for this purpose. The obtained results show that interaction between the soil and a framed structure has a considerable influence on the structural dynamic response of system components. Additionally, a parametric study has been elaborated to quantify the significance of interface behaviour on the soil and on the framed structure under dynamic load.

Finite Element Modelling for Bending and Vibration Analysis of Composite and Sandwich Spherical Shells

Finite element modelling is used to illustrate the bending and vibration response of composite and sandwich spherical shells. ABAQUS is used to model the shell as a three-dimensional deformable solid part. Composites are modelled using mixed modelling technique. The quadrilateral continuum shell element, which has eight nodes, is used for finite element meshing. For a three-layered composite spherical shell with simply supported boundary conditions, the accuracy of the employed element is first evaluated. After that, three potential end conditions are evaluated for further analysis: cantilever (CFFF), two opposing sides clamped (CFCF), and all sides clamped (CCCC). Bending response is produced by applying two types of pressure loads to the exterior surfaces of composite and sandwich spherical shells: uniformly distributed load (UDL) and sinusoidal load (SINO). For various span to thickness ratios, free vibration response and forced vibration response (Time History) are provided under step pressure (UDL and SINO) loading. The element type being used is found to be quite precise and sturdy.

WITHDRAWN: A quadrilateral flat-shell element for the analysis of composite and sandwich cylindrical, globular and conical shell panels

This article has been withdrawn at the request of the author(s) and/or editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at https://www.elsevier.com/about/our-business/policies/article-withdrawal .

A stabilized cell‐based smoothed finite element method against severe mesh distortion in non‐Newtonian fluid–structure interaction

AbstractFailure is an inevitability for the finite element method (FEM) to be performed on seriously degenerated four‐node quadrilateral (Q4) elements whose Jacobians become negative. This article proposes a stabilized selective integration formulation of the cell‐based smoothed FEM (CSFEM) for modeling non‐Newtonian fluid–structure interaction (NNFSI) where that acute mesh distortion arises. The Carreau–Yasuda and Oldroyd‐B fluids, respectively, interact with a geometrically nonlinear solid in NNFSI. The CSFEM is applied to discretize both the fluid and solid media in space. As the fluid mesh accounts for a substantial part of the discretized NNFSI system, seriously distorted Q4 elements are specified exclusively for the fluid field. In this case, each convex Q4 element is subdivided into four smoothing cells (SCs) whereas its concave counterpart is regarded as one single SC. In the meantime, the solid elements are treated as usual. To stabilize the one‐SC integral in smoothed Galerkin weak form, an hourglass control is subsequently introduced into the fluid momentum equations as well as the viscoelastic constitutive equation. The transient non‐Newtonian fluid equations are advanced with dual time steps of the characteristic‐based split scheme in time to further enhance the numerical stability and accuracy. After discussing other aspects of computational NNFSI, two benchmark problems are presented to demonstrate the effectiveness and performance of the developed methodology under the harsh environment.

Fracture analysis of single edge notched specimen using phase field approach

For the damage tolerant design of cracked structural components, two important parameters namely, residual strength and remaining life are very important. To evaluate these, a reliable fracture parameter such as stress intensity factor (SIF) is to be determined. Phase field (PF) based finite element formulation for a brittle fracture to determine crack propagation trajectory is presented in this paper. PF methodologies have been formulated by coupling partial differential equations (PDEs) after minimizing the total potential energy, which includes elastic strain energy and fracture energy, with respect to the displacement field and PF variable. Solutions of these PDEs for PF variable resulted in identifying crack path geometry, while PDEs displacement field obtained near the crack field is used to determine SIF. In the present study, extending PF formulation to fracture analysis of a structural component is first of its kind. To simulate PF phenomena, a four noded 2-D bilinear quadrilateral element with three degrees of freedom (DOF) at each node (two translations and one for phase field) has been developed to analyze single edge notched (SEN) specimen under tensile and shear loading. By combining the displacement results of PF method with displacement extrapolation technique, SIF has been determined for various crack lengths representing crack propagation in SEN specimens. It is then compared with the analytical SIF solution available in Handbook, and also with the contour integral method SIF results. It is found that the SIF obtained by PF method is in good agreement (±10%) with the other solutions.

On the choice of finite element for applications in geodynamics

Abstract. Geodynamical simulations over the past decades have widely been built on quadrilateral and hexahedral finite elements. For the discretization of the key Stokes equation describing slow, viscous flow, most codes use either the unstable Q1×P0 element, a stabilized version of the equal-order Q1×Q1 element, or more recently the stable Taylor–Hood element with continuous (Q2×Q1) or discontinuous (Q2×P-1) pressure. However, it is not clear which of these choices is actually the best at accurately simulating “typical” geodynamic situations. Herein, we provide a systematic comparison of all of these elements for the first time. We use a series of benchmarks that illuminate different aspects of the features we consider typical of mantle convection and geodynamical simulations. We will show in particular that the stabilized Q1×Q1 element has great difficulty producing accurate solutions for buoyancy-driven flows – the dominant forcing for mantle convection flow – and that the Q1×P0 element is too unstable and inaccurate in practice. As a consequence, we believe that the Q2×Q1 and Q2×P-1 elements provide the most robust and reliable choice for geodynamical simulations, despite the greater complexity in their implementation and the substantially higher computational cost when solving linear systems.

Nodal integral methods in general 2D curvilinear coordinates - applied to convection–diffusion equation in domains discretized using quadrilateral elements

Nodal integral method (NIM) is derived in general, 2D, curvilinear coordinates and applied to solve the convection–diffusion equation in domains discretized using quadrilateral elements. The quadrilateral elements in the Cartesian system are transformed into rectangular elements in curvilinear coordinates using bi–linear Lagrangian interpolation functions. An approximation for the transverse-integration operator is developed in curvilinear coordinates. The convection–diffusion equation and the continuity conditions at the common edge between two adjacent nodes are derived in curvilinear coordinates. A new approximation of the cross–derivative terms resulting from transforming the governing equation to curvilinear coordinates is developed using the discrete unknowns of NIM (line–averaged variables). Three numerical test problems are solved to assess the accuracy and efficiency of the newly developed scheme. The results show that the scheme developed in this paper is second–order accurate in space and time, which is the same order of accuracy for traditional NIM for regularly-shaped elements. The results show that the accuracy of NIM for quadrilateral elements is maintained even for coarse mesh and highly distorted elements.

Vibration analysis of two-dimensional micromorphic structures using quadrilateral and triangular elements

PurposeThe paper aims to presents a numerical analysis of free vibration of micromorphic structures subjected to various boundary conditions.Design/methodology/approachTo accomplish this objective, first, a two-dimensional (2D) micromorphic formulation is presented and the matrix representation of this formulation is given. Then, two size-dependent quadrilateral and triangular elements are developed within the commercial finite element software ABAQUS. User element subroutine (UEL) is used to implement the micromorphic elements. These non-classical elements are capable of capturing the micro-structure effects by considering the micro-motion of materials. The effects of the side length-to-length scale parameter ratio and boundary conditions on the vibration behavior of 2D micro-structures are discussed in detail. The reliability of the present finite element method (FEM) is confirmed by the convergence studies and the obtained results are validated with the results available in the literature. Also, the results of micromorphic theory (MMT) are compared with those of micropolar and classical elasticity theories.FindingsThe study found that the size effect becomes very significant when the side length of micro-structures is close to the length scale parameter.Originality/value The study is to analyze the free vibrations of 2D micro-structures based on MMT; to develop a 2D formulation for micromorphic continua within ABAQUS; to propose quadrilateral and triangular micromorphic elements using UEL and to investigate size effects on the vibrational behavior of micro-structures with various geometries.

Thermo-mechanical analysis of laminated composites shells exposed to fire

This paper describes the research performed within the scope of H2020 project FIBRESHIP in the development and validation of a thermo-mechanical model to assess the fire performance of composite structures. A one-dimensional thermal model with pyrolysis is used to obtain the temperature profile across the thickness, and later, introduced in the thermo-mechanical model with a quadrilateral shell element approach. The composite constitutive model employed is the so-called Serial/Parallel Rule of Mixtures (SPROM) which has been modified to introduce the effect of the thermal deformation. A set of experimental tests are then used to validate the correctness of the numerical method proposed. The experimental data used to validate the thermal model is the classic Henderson experimental test. The thermo-mechanical coupling is validated against an original vertical furnace test of an FRP ship’s bulkhead, following on the 2010 FTP Code standards. These validations demonstrate the correctness and accuracy of the proposed decoupled thermo-mechanical formulation.

Free vibration response of cylindrical and spherical functionally graded material shell panels using a quadrilateral finite element

A quadrilateral element is developed for analysing the free vibration responses of moderately thick spherical and cylindrical FGM shell panels. The element has seven degrees of freedom per node namely, two transverse shear strain components, two bending rotations, two in-plane, and one transverse deflections. The transformation matrix is developed by using two separate co-ordinate systems namely: Cartesian and Surface, to convert the actions and displacements from global to local directions and vice versa. The local degrees of freedom related to displacements are transformed to a Cartesian coordinates system (x, y, z). The degrees of freedoms which are related to rotations are transformed to surface-coordinate system (ξ1, ξ2, ξ3), in which ξ3 is taken normal to the shell surface at every node of the element. The material property variation through the thickness is considered accordingly Mori-Tanaka scheme. The results for cylindrical and spherical shell panels having different geometrical configurations, volume fraction configurations and different boundary conditions are compared with the available 2D analytical and 2D finite elements results available in the literature. For some cases, the present results of non-dimentionalised frequencies are compared with those obtained using ANSYS (APDL). It is observed that the performance of the numerical evaluation with the present element (DKTOTS-FG) is quite satisfactory for free vibration response of functionally graded material spherical and cylindrical shells in this study.

Study on the Vector Quadrilateral Isoparametric Shell Element with Shear Deformation

Based on the linear superposition of the quadrilateral isoparametric membrane element and quadrilateral Mindlin plate element, basic vector formulas for the quadrilateral isoparametric shell element with the shear deformation is derived. Two different coordinate modes for updating the particle displacement and the element node internal force are put forward. The uniform numerical solution for an irregular quadrilateral element is realized by the isoparametric integral scheme along the element plane. Then, the bilinear elastoplastic constitutive of material is introduced into the vector finite shell element, and the nonlinear effect of the material is realized based on the integral method along the thickness. On this basis, the nonlinear calculation and analysis program of the quadrilateral isoparametric shell element is developed. Numerical example results show that the static and dynamic analysis, large deformation and large rotation analysis, and buckling analysis can all be well performed for the shell structures by the developed program, which verifies the validity of theoretical derivation and computer program.

Static and free vibration analysis of sandwich shell panels using quadrilateral flat shell finite element

A quadrilateral flat-shell element with nine DOF (Degrees of Freedom) per node is developed for the static (deflections and stresses) and free vibration analysis (natural frequencies and accompanying mode shapes) of sandwich cylindrical and globular shell panels. The element is based on zigzag theory and is augmented using the discrete Kirchhoff quadrilateral bending plate element having seven DOF per node. It is vital to convert the rotations and translations degrees of freedom from local to global direction for the plate bending element to transform into a flat shell element. DKZigTS is developed by adding two fictitious degrees of freedom and has nine local and global DOF per node. The transformation matrix of size 36 × 36 is used for DKZigTS. It does not face the problem of the ill-conditioned stiffness matrix. The developed element is judged for its performance for simply supported and clamped shell panels. The DKZigTS element’s result are compared with results gotten using the 3 D finite element (FE) results of ANSYS (Mechanical APDL). To establish the confidence the comparison is also done with those 2 D analytical, 2 D numerical and 3 D elasticity results available in the literature. The numerical evaluation with DKZigTS element exhibits a very satisfactory performance for the static and free vibration response of moderately thick and thick sandwich spherical and cylindrical shell panels.

A quadrilateral flat-shell element for the static and dynamic analysis of composite and sandwich cylindrical, spherical and conical shell panels

Abstract A quadrilateral flat-shell element is developed for analysing the deflections, stresses and natural frequencies along with their allied mode shapes of cylindrical, spherical, and conical shell panels made up of layered composite and sandwich material. The developed element (DKZigTS1) is based on zigzag theory and has seven local as well as global DOF per node. The concept of obtaining transformation matrix is used for transforming actions and reactions from local to global direction to convert the plate bending into a flat-shell element. The two separate coordinate systems are used to transform rotational and translational degrees-of-freedom (DOF), from local to global direction. The local translational DOF are transformed to global Cartesian coordinates (x, y, z) and the local rotational DOF are transformed to the surface coordinate system (ξ 1, ξ 2, ξ 3), in which ξ 3 is perpendicular to the surface. The DKZigTS1 element gives fairly accurate results that align with the 2D analytical and the 3D elasticity solutions, reported in the literature for moderately thick and thick shell panel. The present results are also in good agreement with the 3D finite element solutions for shallow and deep shell panels having various material properties, boundary restrained environments, and geometrical shapes considered in this study.

Computational analysis of a sandwich beam with FGM face sheets under flexural loading

Functionally graded materials (FGM) are nonhomogeneous materials at the microscopic level, which have a gradation in their structure and composition with the properties of the material varying along one or more directions. The applications of these materials are in the fields: Aerospace, Automotive, Biomedical-Medical, Optoelectronics-electronics, Machinery, Energy, Marine, etc. In the present work the problem of a sandwich beam with metal core and FGM face sheets and a sandwich beam with ceramic core and FGM face sheets under flexural loading is studied numerically via the Finite Element Method. The load is a distributed load on the upper face sheet of the beam. The elements that will be used are 8-node two-dimensional plane quadrilateral isoparametric elements. The elements used for the FGM layers, are graded finite elements where as for the core are homogeneous elements. The advantage of this study is the utilization of 8-node two-dimensional plane quadrilateral graded isoparametric elements in the FGM face sheets, for the study of a sandwich beam. The effect of the volume fraction index on the stress fields is examined.

Electromagnetic eigen analysis: Performance comparison of four node and four edge quadrilateral elements with the effect of distortion

In edge finite elements, electric fields are obtained across edges of each element instead of nodal output as obtained in nodal finite element. In the nodal framework, the potential formulation provides required tangential continuity and normal discontinuity for electromagnetic analysis whereas this continuity requirement is directly attained in the edge framework. In this work, we have used the conversion technique to transfrom four node quadrilateral element to four edge quadrilateral element. Then, we have compared the performance of these elements by performing the eigenvalue analysis for two standard examples. Finally, we also analyzed the effect of distortion in those elements.

Modal analysis of rotating Pre-twisted functionally graded sandwich blade

Modal analysis of rotating turbine blades is an inevitable step during the design process to avoid resonance conditions and related catastrophic failures. The need for such analysis arises because of vibration of supporting machine components like rotor and other dynamic forces involved. Mimicking actual structural and environmental conditions for the analysis to achieve close-form solution is very challenging, therefore approximations of real-time situations are adopted to analysis further. Here, functionally graded sandwich pre-twisted cantilever shell structure with metal core and functionally graded (metal/ceramic) face sheets is considered as thermal barrier coated turbine blades. The present work focuses on the modal analysis of rotating pre-twisted blade model under elevated temperature. Material gradation has been done using power-law distribution and Voigt’s model. The kinematics of the present model is based on the first-order shear deformation theory. The Hamilton principle is adopted to obtain the motion equations. Subsequently, the blade model is discretized and solved using 2D finite element approach via quadrilateral elements. The validation of the proposed work has been confirmed by comparing with the earlier published work. In addition, frequency responses are computed for different composite blade parameters such as twist angle, volume fractions, rotational velocity. The occurrence mode shape exchange phenomena with these parametric variations have been observed and discussed in detail.