Quadratic Elements Research Articles

Modelling delaminations using adaptive cohesive segments with rotations in dynamic explicit analysis

Multiple delamination planes can form when a composite structure is subjected to out-of-plane stresses during static over-loading or impact loading. Numerical modelling of such events is often prohibitively expensive because large numbers of cracks can co-exist and interact, and fracture models usually affect the time step size in explicit solutions. Here a new method called Adaptive Mesh Segmentation is proposed, which introduces segmentation ‘on-the-fly’ in meshes of quadratic finite elements with six degrees of freedom per node, without any intervention from the user and without any reductions in time step size for solution stability. A novel cohesive formulation with rotational degrees of freedom is introduced which increases the resolution of the numerical cohesive zone and allows the use of relatively coarse meshes. Once a critical stress criterion is met, new degrees of freedom are added at element boundaries to model strong discontinuities. A new moment–damage relationship is proposed to link the discontinuity in rotational degrees of freedom with the cohesive zone law which is translational by definition. A method for initiating cohesive tractions and moments with minimal disturbances to the surrounding stress field is also presented. Finally, the model is applied in the analysis of composite delamination benchmarks using relatively coarse meshes and modest model sizes. Considerable improvements in accuracy are observed when compared to conventional methods.

Static analysis of defective sandwich beam by Chebyshev quadrature element method

This paper proposes a quadrature element method based on Chebyshev polynomials for the static analysis of the defective sandwich beam with general boundary conditions. Gauss-Lobatto nodes are utilized. The discrete governing equation is obtained by the high-order sandwich beam theory and the principle of minimum potential energy. A series of numerical examples of the defective sandwich beam with different boundary conditions, geometric parameters, and material properties subjected to various loadings are carried out. The results are compared with the existing methods and the finite element method to verify the proposed method, demonstrating that the present approach can converge exponentially and yield very accurate displacements and stresses. The influence of the defect on the stiffness of the sandwich beam is analyzed. The current method can provide a theoretical basis for the design of the multifunctional beam structure.

Superconvergence recovery of cubic edge elements for Maxwell’s equations

In this article, a new recovery method is designed and analyzed for curl conforming elements on cuboid mesh. The proposed recovery method fully exploits the potential of symmetry to obtain the superconvergence of recovered edge finite element solution in discrete ℓ2 norm. The idea is to identify a symmetry sub-domain such that a polynomial of the same degree is recovered by local L2 projection, based on which the recovered values follow. Combined with extrapolation and the least square fitting for the linear edge elements and quadratic edge elements, respectively, we obtain global superconvergence for recovered quantities in L2 norm. Numerical experiments are provided to validate our theoretical findings.

TSDT theory for free vibration of functionally graded plates with various material properties

In this work, an implicit algorithm is used for analyzing the free dynamic behavior of Functionally Graded Material (FGM) plates. The Third order Shear Deformation Theory (TSDT) is used to develop the proposed model. In this contribution, the formulation is written without any homogenization technique as the rule of mixture. The Hamilton principle is used to establish the resulting equations of motion. For spatial discretization based on Finite Element Method (FEM), a quadratic element with four and eight nodes is adopted using seven degrees of freedom per node. An implicit algorithm is used for solving the obtained problem. To study the accuracy and the performance of the proposed approach, we present comparisons with literature and laminate composite modeling results for vibration natural frequencies. Otherwise, we examine the influence of the exponent of the volume fraction which reacts the plates "P-FGM" and "S-FGM". In addition, we study the influence of the thickness on "E-FGM" plates.

Convolution quadrature time-domain boundary element method for anisotropic pure SH wave with viscosity

Convolution quadrature time-domain boundary element method for anisotropic pure SH wave with viscosity

Quadratic Finite Volume Element Schemes over Triangular Meshes for a Nonlinear Time-Fractional Rayleigh-Stokes Problem

In this article, we study a 2D nonlinear time-fractional Rayleigh-Stokes problem, which has an anomalous sub-diffusion term, on triangular meshes by quadratic finite volume element schemes. Time-fractional derivative, defined... | Find, read and cite all the research you need on Tech Science Press

Nonlinear first ply failure study of laminated composite skew plates

The present work reports the first ply failure investigation of laminated composite clamped skew plates subjected to uniformly distributed transverse static load considering geometric linear and nonlinear strains. An eight-noded isoparametric curved quadratic plate element based on Sanders’ and Von Karman's nonlinearity is employed to develop the finite element code used in the present research. The first ply failure loads are evaluated based on various well-acknowledged failure theories like maximum stress, maximum strain, Hoffman, Tsai – Hill, Tsai – Wu, Hashin and Puck failure criteria. Effect of variation of skew angles on first ply failure loads of laminated composite skew plates for various stacking pattern is reported and studied in depth. The failure criterion corresponding to which the failure load value is minimum, for each skew angle, is only reported in this paper. The effects of skew angle on the first ply failure load are summarised in the form of charts and working equations. The engineers and researchers may use the charts and equations directly to predict the first ply failure load provided the geometry of the plate is known. Besides first ply failure loads the failed plies as well as the failure modes are also reported in the present paper.

LOAD PREDICTION ON METAL FORMING PROCESS (FORGING) USING FINITE ELEMENT METHOD.

Metal forming process is a widely used manufacturing process especially in high volume metal production system. In this paper, the main objective is using Bubnov-Galerkin finite element model to derive the pressure field set up at various cross-sections of a metal blank during a forging process, and the four Lagrange quadratic elements were assembled to represent the various metal blank. The governing equation adopted for this paper is a one-dimensional differential equation describing the pressures exerted on the forging process. During the analysis, the various metal blanks are divided into a finite number of elements and the weighted integral form for each element were formed after applying the Bubnov-Galerkin weighted residual method. A matrix form under certain boundary conditions from the weighted residual method were used to obtain the pressure distribution across the cross-section of the various metal blanks. Finite element results are obtained for a value of a circular disc diameter, thickness, coefficient of friction, principal stress, length, and radius of a circular material. Finite element method and the Exact solution approach are used to achieve and compare both results. Furthermore, the combination of both methods shows that there are potentials for using this approach towards the optimization of metal forming in manufacturing processes and some engineering practices. Keywords: Forging; LaGrange Interpolation Function; Bubnov-Galerkin Weighted Residual Method; Finite Element Method.

Stress recovery for the particle-in-cell finite element method

The particle-in-cell finite element (PIC-FE) method has been widely used in geodynamic numerical modelling due to its efficiency in dealing with large deformations without the requirement of remeshing. However, material deformation within a Eulerian mesh frame will mix particles of contrasting strength properties (e.g., viscosity in Stokes problems) in a single element requiring some form of averaging to project particle properties to integration points. The numerical solutions are thus dependent on the way how the particle properties are projected to the integration points. An intra-element property discontinuity may introduce severe stress oscillations along the interfaces. In this study, we assess three preprocessing methods to smooth the viscosity contrast within one element. For simplified models with analytical solutions, the accuracy and convergence rate in L2 norm are systematically studied with ensembles. It is found that using higher-order quadrature elements does not improve the convergence rate for either the velocity or stress solution, both close to one. Additionally, the convergence rate of the maximum stress error, which exists adjacent to the mixed-material elements, is much less than one for all cases studied here. Comparing each component of the stress tensor, we find that the stress tensor component with the highest strain rate gradient across the material interface produces the maximum stress error. Such errors can be reduced by averaging the particle properties to the Gaussian quadrature point with an inverse-distance-weighted harmonic mean.

Effects of extensible modelling on composite riser mechanical responses

The change from steel risers to composites comes with uncertainties that led to large safety factors. One area of uncertainty is the predicted response and stresses derived from commercial packages that are based on formulations that assume in-extensible riser. However, composite pipes exhibit a lower axial stiffness and therefore the velocity of the axial waves will change with a corresponding change in dynamic response. To determine the effect of this assumption, this paper assesses the effect of extensibility on the time-domain response. It is found that the in-extensible model predicts 3 times the number of high frequency tension cycles in the 20 kN tension range. To determine the impact of this change on the stress, the accuracy of available composite pipe models is benchmarked using shell, continuum-shell and solid elements. The quadratic and continuum-shell elements provide a maximum percentage difference of 4% compared to solid elements but the continuum-shell is selected as it has a lower computational cost. The response from the extensible and in-extensible models are input into the pipe model, they provide similar Tsai-Wu failure factors, alleviating concerns when modelling the strength. However, the change in dynamics remains a concern for other applications such as machine-learning or digital-twins.

Impact of power law fluid and magnetic field on double diffusive mixed convection in staggered porous cavity considering Dufour and Soret effects

This study deals the influence of power law fluid and inclined magnetic field on a porous medium in staggered cavity. The effect of Soret and Dufour parameters are also given attention. A two dimensional system of partial differential equations has been discretized by employing Galerkin finite element method. A finite element method involving the cubic polynomials (ℙ3) has been implemented to compute for velocity, temperature and concentration fields while the pressure is approximated by quadratic (ℙ2) finite element space of functions. The system of discretized equations is simplified using the adaptive Newton's method. Simulations are performed for various ranges of pertinent parameters such as power law index (between 0.6 and 1.8), Hartmann number (between 0 and 100), Lewis number (between 1 and 10), Dufour/Soret numbers (between −3 and 3), Darcy number (between 10−5 and 10−2), magnetic field inclination (between 0∘ and 90∘) and buoyancy ratio (between 0 and 10). It is inferred that the retarding effect of the rising Lorentz force is pronounced on fluid flow. The convective heat transfer is enhanced with the change of Dufour number from negative to positive. The suppression on convection at the inclination angle γ = 90∘ is much more than γ = 0∘. The increment in Dufour and Soret numbers has an improving influence on both the average Nusselt and Sherwood numbers, and thus the convective heat and mass transfer at buoyancy ratio N = 10.

Spectral extended finite element method for band structure calculations in phononic crystals

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt a rational Bézier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples are presented to demonstrate the advantages of p-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver O(10−8) accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

Enhanced nodal gradient finite elements with new numerical integration schemes for 2D and 3D geometrically nonlinear analysis

The consecutive-interpolation procedure (CIP) has been recently proposed as an enhanced technique for traditional finite element method (FEM) with various desirable properties such as continuous nodal gradients and higher accuracy without increasing the total number of degrees of freedom (DOFs). It is common knowledge that linear finite elements, e.g., four-node quadrilateral (Q4) or eight-node hexahedral (HH8) elements, are not highly suitable for geometrically nonlinear analysis. The elements with quadratic interpolation functions have to be used instead. In this paper, the CIP-enhanced four-node quadrilateral element (CQ4), and the CIP-enhanced eight-node hexahedral element (CHH8), are for the first time extended to investigate geometrically nonlinear problems of two- (2D) and three-dimensional (3D) structures. To further enhance the efficiency of the present approaches, novel numerical integration schemes based on the concept of mid-point rules, namely element mid-points (EM) and element mid-edges (EE) are integrated into the present CQ4 element. For CHH8, the 3D-version of EM (namely 3D-EM) and the element mid-faces (EF) scheme are investigated. The accuracy and computational efficiency of the two novel quadrature schemes in both regular and irregular (distorted) meshes are analyzed. Numerical results indicate that the new integration approaches perform more efficiently than the well-known Gaussian quadrature while gaining equivalent accuracy. The performance of the CIP-enhanced elements, which is examined through numerical experiments, is found to be equivalent to that of quadratic Lagrangian finite element counterparts, while having the same discretization with that by the linear finite elements. In addition, we also apply the present CQ4 and CHH8 elements associated with different numerical integration techniques to nearly incompressible materials.

A fast multipole BEM with higher-order elements for 3-D composite materials

The present work is to develop the FMBEM with quadratic elements to the analysis of 3-D linear elastic structures containing subdomains. To the numerical integration, an adaptive method with subdivision of elements is applied. The BEM formulation for perfectly bonded subdomains, in which interface traction forces are eliminated from the system of equations is implemented. The method is applied to problems of solids with spherical inclusions: a single inclusion, two interacting inclusions, many regularly distributed inclusions and randomly distributed inclusions. Different values of volume fraction of inclusions, up to 0.5 for the cubic arrangement of inclusions, and contrast in elastic properties between the matrix and inclusions, up to 1000 for the ratio of Young’s moduli of the constituents, are considered. Stresses or effective elastic constants are computed. The results are compared to analytical models and a good agreement is observed. The proposed method shows a perspective of modelling 3-D composites efficiently.

Linear smoothed finite element method for quasi-incompressible hyperelastic media

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.

An energy–momentum conserving scheme for geometrically exact shells with drilling DOFs

An energy–momentum conserving temporal integration scheme is presented for a recently proposed geometrically exact shell formulation with drilling degrees of freedom. The scheme is based on a novel idea of defining mixed discrete derivatives for holonomic constraint functions with displacements and rotations. By defining general discrete derivative expressions with unknown terms, the mixed discrete derivatives with second-order accuracy are constructed according to deformation modes to satisfy directionality and orthogonality properties simultaneously, thus preserving conservation laws of total energy and momenta. The analysis of shell structures is conducted using the weak form quadrature elements to ensure exact incorporation of constraints and conservation of total energy after discretization, as well as circumvent shear and membrane locking phenomena. Benchmark numerical examples are presented to demonstrate the validity of the present scheme.

SEVEN-POINT FINITE ELEMENT OF INTEGRAL TYPE APPLIED TO CONFORMING - NONCONFORMING METHOD

The paper deals with integral variants of a seven-point triangular finiteelement. This is an enriched quadratic element constructed by adding a non-negativebasis bulb-function which vanishes on the element boundary. The polynomial spacewhich seven-point element uses is P 2 + span{x 2 y + xy 2 }. The main goal here is to determine two-sided bounds of eigenvalues for a second-order elliptic operator. The method consists of finite element solving of the problemby means of seven-point conforming element and then constructing an nonconforminginterpolant of the approximate conforming eigendunctions. Thus, solving only oncethe eigenvalue problem, we get upper and lower bounds for the exact eigenvalues.Furthermore, the fact that the nonconforming interpolants uses the nodal values of theconforming approximate eigenfunctions gives an obvious computational advantage.Finally, numerical experiments confirming the efficiency of the proposed algorithmare also provided.

Convergence study on ultrasonic guided wave propagation modes in an axisymmetric cylindrical waveguide

Guided wave propagation and its dispersion phenomenon of infinite solid elastic rods are encountered in several applications including, mechanical and civil engineering fields. In this paper, the elastic stress wave propagation in the axisymmetric circular cross-section of a high strength steel wire with cylindrical waveguide is investigated using a semi-analytical finite element (SAFE) method. The error analyses are carried out on fundamental modes, namely, flexural longitudinal and torsional modes. The theoretical framework for finite element (FE) discretization is established for the cylindrical waveguide. A three-node triangular linear element is used for solving SAFE dispersion solutions such as the wavenumber-frequency curve, phase velocity, and group velocity curves. The convergence and accuracy of the method are analyzed by comparing it with the calculation results of the transcendental Pochhammer frequency equation, and the meshing criterion is proposed. The use of higher-order (quadratic) elements are proposed for lower computational burden and effective method for solving eigenvalue problems. The effect of using 6-node triangular quadratic elements in the SAFE method for improving frequency accuracy is discussed in a detailed manner by proposing the meshing criterion. The calculation accuracy of a quadratic element semi-analytical discretization exceeds that of a linear element discretization with four times its radial circumference. The statistical histograms are demonstrated to prove the results of the proposed semi-analytical discretization method which can be used for solving cylindrical waveguides.

Geometrical and Material Nonlinear Finite Analysis of Fiber Reinforced Concrete Slabs

Geometrical nonlinearity arises from the continuous change of shape under increasing loads on thin or long span members, and the materials nonlinearity due to the nonlinear behavior of the materials composing the structural element. In this study the finite element method is used to analyze fibrous reinforced concrete slabs, material and geometric nonlinearities were considered. Concrete is represented by degenerate quadratic thick shell element. Concrete behavior in compression is modeled by elastic perfectly plastic or strain hardening plasticity. A tensile strength criterion is used to follow crack initiation and propagation. A parabolic stress degradation function, which is based on the fracture energy concept, is used to model the tension stiffening effect of steel fibers. The reinforcement is considered as smeared within the concrete layers, and assumed as either an elastic perfectly plastic material or as an elastic-plastic material with linear strain hardening. Von Karman assumptions which is based on the total Lagrangian approach is used for the geometric nonlinearity. The numerical results using for both normal and lightweight fiber reinforced concrete slabs were predicted satisfactorily. The average predicted failure load of the investigated cases to the experimental failure load was 0.899 with a standard deviation of 0.092.

Impact of fins and inclined magnetic field in double lid-driven cavity with Cu–water nanofluid

This study deals the influence of fins and inclined magnetic field with nanofluid in single lid-driven and double lid-driven cavities. The effect of fins length and distance between them is also given attention. A two dimensional system of partial differential equations has been discretized by employing Galerkin finite element method. A finite element method involving the cubic polynomials (P3) has been implemented to compute for velocity and temperature fields while the pressure is approximated by quadratic (P2) finite element space of functions. The system of discretized equations is simplified using the adaptive Newton’s method. The implemented finite element code is validated with an experimental study. Simulations are performed for various ranges of pertinent parameters such as distance among fins (between 0.2 and 0.5), Hartmann number (between 0 and 100), length of fins (between 0.25 and 0.50), Richardson numbers (between 0.01 and 1), number of fins (between 2 and 5) and magnetic field inclination (between 0° and 90°). Two cases of fins such as adiabatic and isothermal are also discussed in view of average Nusselt number. It is inferred that the fluid flow and heat transfer are significantly affected in the presence of fins comparing to the absence of fins. The smallest convective heat transfer is achieved when the distance between adiabatic fins is 0.3. Fluid flows faster at the angle γ=90∘ than γ=0∘ in both cases. Also, isotherms form larger red hot zones inside the cavity at γ=90∘. Isothermal fins have much more weakening effect on convective heat transfer than the adiabatic fins.