Quadrilateral Finite Element Research Articles

Analysis of crack problems in functionally graded materials under thermomechanical loading using graded finite elements

This paper represents a comprehensive study of the fracture behaviour of linear elastic isotropic functionally graded materials (FGMs) subjected to mechanical and thermal loading. For this purpose, an improved graded 9-node quadrilateral finite element which includes the variation of elastic (Young's modulus and Poisson's ratio) and thermal properties (coefficient of thermal expansion and heat conductivity) at the element level is developed. The performance of the new element in conducting thermomechanical fracture analysis is evaluated by conducting a set of simulations to compare the fracture parameters in FGMs with sharp material and temperature gradients against those with mild material and temperature gradients. The reliability of the proposed element is verified by comparing with existing solutions. The thermomechanical coupling and its influence on the coupled modes of fracture which originates from the material gradient in FGMs are investigated. The influence of the steady-state temperature distribution, as well as the material gradient on the crack tip local fields and fracture parameters, such as stress intensity factors, mode mixity, and energy release rate, are studied. More importantly, it was found that compared with conventional homogeneous finite elements the current model is more reliable for modelling crack problems in FGMs.

Numerical approach of free and forced elastic vibrations using high-regularity Hermitian partition of unities

In this paper, shape functions with regularity up to $$C^{2}$$ were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the $$C^{0}$$ Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field.

Free Vibration of FGSW Plates Partially Supported by Pasternak Foundation Based on Refined Shear Deformation Theories

A refined third-order shear deformation theory (RTSDT), in which the transverse displacement is split into bending and shear parts, is employed to formulate a four-node quadrilateral finite element for free vibration analysis of functionally graded sandwich (FGSW) plates partially supported by a Pasternak foundation. An element based on the refined first-order shear deformation theory (RFSDT) which requires a shear correction factor is also derived for comparison purpose. The plates consist of a fully ceramic core and two functionally graded skin layers with material properties varying in the thickness direction by a power gradation law. The Mori–Tanaka scheme is employed to evaluate the effective moduli. The elements are derived using Lagrangian and Hermitian polynomials to interpolate the in-plane and transverse displacements, respectively. The numerical result reveals that the frequencies obtained by the RTSDT element are slightly higher than the ones using the RFSDT element. It is also shown that the foundation supporting area plays an important role on the vibration of the plates, and the effect of the material distribution on the frequencies is dependent on this parameter. A parametric study is carried out to highlight the effects of the material inhomogeneity, the foundation stiffness parameters, and the foundation supporting area on the frequencies and vibration modes. The influence of the layer thickness and aspect ratios on the frequencies is also examined and highlighted.

Mixed finite elements for nonlocal elastic multilayered composite plate refined theories

A novel mixed finite element formulation for the layerwise analysis of nonlocal multilayered composite plates is presented. The finite elements are formulated starting from the weak form of a set of governing equations for the laminate layers that were deduced via the Reissner Mixed Variational Theorem. The primary variables, namely displacements and out-of-plane stresses, are expressed at layer level as through-the-thickness expansions of suitable selected functions with coefficients approximated by the finite element scheme. The through-the-thickness expansion order is considered as a free parameter. This way, finite elements for different refined higher order plate theories can be systematically developed by assembling the layers contributions associated with the variable expansion terms. These contributions are called fundamental nuclei and their definition is formally unique whatever the considered expansion order. The obtained finite elements inherently ensure stresses and displacements continuity at the layer interfaces and they allow to associate different values of the nonlocal parameter to the laminate layers. Standard 9-node and 16-node isoparametric, quadrilateral finite elements have been implemented to verify the viability of the proposed formulation. The obtained results compare favourably with literature solutions and highlight the characteristics of the approach. Original results are proposed also to serve as benchmarking data.

Seismic control of multi degree of freedom structure outfitted with sloped bottom tuned liquid damper

Application of TLD in controlling the unwanted vibration of the structure is of practical importance. But the studies are limited to circular or rectangular shape TLD. However, it is not always feasible to use conventional shape TLD due to geometrical and space constraints of the building floor plan. Hence, a new shape of the tank is used in this study to verify its efficiency as a damping device. The present research is focused on controlling the vibration of a multidegree of freedom structure rigidly supporting a sloped bottom tuned liquid damper subjected to real earthquake ground excitation. Three actual earthquake ground motion classified in the category of low, intermediate, and high-frequency contents are used for the investigation. Numerical simulation of the TLD-structure coupled model is developed, which uses a finite element method to simulate sloshing in sloped bottom TLD and a spring-mass model to determine structural response. The confined liquid in TLD is discretized as a combination of three-node triangular and four-node quadrilateral finite elements. A parametric study is carried out on mass ratio and tuning ratio to study the effectiveness of TLD in controlling the structural vibration. It is observed that increase in mass ratio significantly control the top floor structural vibration. However, the tuning ratio plays an important role.

Elastoplastic analysis of plane structures using improved membrane finite element with rotational DOFs: Elastoplastic analysis of plane structures

In this work, the small-strain elastoplastic behavior of structures is analyzed using an improved nonlinear finite element formulation. In this framework, an eight-node quadrilateral finite element denoted PFR8 (Plane Fiber Rotation) that belongs to the set of elements with rotational degrees of freedom is developed. Its formulation stems from the plane adaptation of the Space Fiber Rotation (SFR) concept that considers virtual rotations of nodal fiber within the element. This approach results in an enhancement of the displacement vector approximation. Von-Mises yield criteria have been applied for yielding of the materials along with the associated flow rule. Newton-Raphson method has been used to solve the nonlinear equations. To assess the performance of the proposed element, benchmark problems are addressed and the results are compared with some analytical and numerical solutions from the literature.

C1 conforming quadrilateral finite elements with complete second-order derivatives on vertices and its application to Kirchhoff plates

The classical problem of the construction of C 1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method. In order to achieve the C 1 conformity on the interfaces of quadrilateral elements, complete second-order derivatives are used at the element vertices, and the information of geometrical mapping is also considered into the construction of shape functions. It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes. However, the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes. Therefore, very fast convergence can be achieved. A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements. Since the present elements are over conforming on element vertices, an approach for handling problems of material discontinuity is also proposed. Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.

Free Vibration with Mindlin Plate Finite Element Based on the Strain Approach

This paper presents the development of a quadrilateral plate bending finite element for the computation of natural frequencies of plates with arbitrary geometry. This element is based on the Reissner–Mindlin plate theory using assumed strains rather than displacements and contains only the three physical degrees of freedom at each of the four corner nodes. Tests of convergence are first established for square plates with both simply supported and clamped on their edges where it is shown that a good rate of convergence is obtained using few elements. Other tests are then applied to rectangular, circular, skew and stepped plates, in addition to a rectangular plate with a central cutout as well. The numerical results obtained show that the present element has successfully passed patch tests and its results of the natural frequencies and the associated modes of vibration are in excellent agreement when compared with analytical and other available numerical solutions.

Comparisons of Conventional, SRI, EAS and Hybrid-Stress Plane Q4 and Q9 Finite Element Models in Displacement and Energy Norms

Advanced finite element models have been being pursued for several decades. For plane models, the popularly employed formulations include the selectively reduced integration, enhanced assumed strain, hybrid/mixed methods, stabilization methods, among others. The accuracies of the advanced finite element models with respect to the standard element models are often demonstrated by some popular numerical tests in which displacement and stress components at a few selected points are benchmarked. The more comprehensive but laborious means of assessing the accuracy by computing the relative errors in displacement norm (Sobolev norm of order zero) and energy norm are less popularly employed. In this paper, these norms of the standard and some advanced 4-node and 9-node quadrilateral finite element models for a number of tests are computed and reported. In contrast to the common perception, the advanced models are not always more accurate than the standard ones. The discussions in this paper can be helpful for the understanding and usage of the numerical benchmark tests.

Correction to: An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

This article was published with an erroneous version of Eq. 6 and therefore has been corrected.

Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

Nonconforming quadrilateral finite element method (FEM) of nonlinear Kirchhoff‐type equation with damping is studied on anisotropic meshes. Based on the property of the nonlocal term of this equation, unconditional optimal error estimates of and ( , the spatial parameter, and , the time step) in the broken norm are deduced for the semidiscrete and a linearized fully discrete schemes without any restrictions of through a distinct approach compared with the methods used for other partial differential equations, respectively. Besides, the damping term appearing in the Kirchhoff‐type equation is solved with a novel technique, which is the major difficulty in the theoretical analysis. Finally, some numerical results are provided to verify the theoretical analysis.

Dynamic analysis of the elasto-plastic behaviour of buildings and structures in the SCAD ++ software package

The problem formulation of nonlinear equations of motion obtained by the finite element method applying to the dynamic analysis of buildings and structures is presented. The elasto-plastic behaviour of reinforced concrete structural elements is described using the plastic flow theory, moreover, for concrete, the Drucker-Prager yield criterion is used for bending elements, and the yield surface, which coincides in shape with the Geniev strength surface, for compressed-bending elements. Concrete degradation due to crack opening is simulated by the descending branch of the σ − ε diagram. The plastic flow theory with von Mises yield criterion describes the behaviour of reinforcement. Using the aforementioned constitutive models, a finite element library was developed, including plane shell quadrilateral and triangular finite elements based on the Mindlin-Reissner shear theory as well as a two-node spatial frame finite element based on the Timoshenko shear theory. The seismic analysis of the 3-D model of multi-storey building is considered as an example.

Hybrid-mixed shell quadrilateral that allows for large solution steps and is low-sensitive to mesh distortion

We compare three nearly optimal quadrilateral finite elements for geometrically exact inextensible-director shell model. Two of them are revisited and one is novel. The assumed natural strain (ANS) element of Ko et al. (Comput Struct 185:1–14, 2017) shows low sensitivity to mesh distortion and excellent convergence behavior for most types of shell problems. The Hu–Washizu element with ANS shear strains of Wagner and Gruttmann (Int J Numer Methods Eng, 64:635–666, 2005) allows for large solution steps and is computationally fast. However, both formulations have undesirable weak spots, which we clearly identify by a comprehensive set of numerical examples. We show that a straightforward combination of both formulations results in a novel element that synergizes the positive features and eliminates the weak spots of its predecessors.

A mixed finite element for the nonlinear analysis of in‐plane loaded masonry walls

SummaryA mixed membrane eight‐node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history‐dependent 2D stress‐strain constitutive law is used to model masonry material, the element derivation is based on a Hu‐Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral‐type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement‐based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.

Seismic behavior of partially filled liquid tank with sloped walls

Abstract The seismic characteristics of liquid sloshing in partially filled sloped bottom tank are investigated numerically. Six different ground motions classified on the basis of low, intermediate and high frequency contents are selected to determine the dynamic characteristics of liquid in sloped bottom tanks. The effect of different slope angle on the frequency contents of earthquakes are investigated. The liquid domain is discretized as the combination of three node triangular and four node quadrilateral finite elements. The individual contribution of impulsive and convective response components of hydrodynamic behavior in terms of base shear force and hydrodynamic pressure distribution along the walls of the containers are successfully measured. The efficacy of the developed numerical model is validated with the existing published results.

Quadrilateral overlapping elements and their use in the AMORE paradigm

We give the formulation of new 4-node overlapping quadrilateral finite elements and demonstrate their performance in several numerical examples. The elements are insensitive to mesh distortions and lead overall to accurate numerical solutions. We show how the quadrilateral elements are utilized in the AMORE scheme of “automatic meshing with overlapping and regular elements”. Since in this paradigm the finite elements interior to the domain of analysis are undistorted regular elements and the distortion-tolerant overlapping elements are used for discretizations near the boundaries, high solution accuracy is achieved. This accuracy is obtained with much less effort for meshing than in traditional finite element analysis and with a reasonable computational expense.

Analysis of two low-order equal-order finite element pairs for Stokes equations over quadrilaterals

Two quadrilateral low-order equal-order finite element schemes are analyzed for Stokes equations. Both of these schemes adopt the quadrilateral P1-nonconforming finite element to approximate the pressure over a coarser mesh. The velocity spaces are constructed over a finer mesh, where the standard Q1-conforming element space and the quadrilateral P1-nonconforming element space are selected, respectively. The stability assertion is given for each pair. Moreover, the superconvergence property of the pressure is obtained over uniform rectangular meshes. All the analyses above are verified by numerical tests.

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Finite element moment diagram of solvation of thermoplastic elastic deformation problems

The calculated finite element moment diagram (MSEC) for the solution of geometrically nonlinear problems of thermoplastic elasticity with regard to material damage for axisymmetric rotation bodies was obtained. The use of quadrilateral finite elements of arbitrary shape, taking into account the variability of the components of the metric tensor, provides high efficiency of the approach. This allows us to determine nonlinear deformations and their variations due to displacement, whereby the form of the obtained expressions for nonlinear deformations in shape coincides with the relations for linear deformations. This allows us to determine nonlinear deformations and their variations due to displacement, whereby the form of the obtained expressions for nonlinear deformations in shape coincides with the relations for linear deformations. This is realized by changing the value of the transformation tensor components accordingly, which enables us to effectively solve problems in a geometrically nonlinear formulation. The method of solving the system of nonlinear equations is the orientation of the step algorithm in combination with the iterative procedure of Newton-Kantorovich.

Functional Reconstitution of Reissner’s Mixed Variational Theorem for Finite Element Applications

The functional reconstitution technique is used for the first time to theoretically and a priori assess the numerical performance of Reissner’s Mixed Variational Theorem in the case of linear quadrilateral finite element, Generalized Unified Formulation, and composite materials. From the finite element discretization, the semi-complementary energy is reconstructed in integral form. Analytical expressions for the exact and spurious terms are obtained. Using the elasticity solution for a simply supported plate, it is numerically shown that these spurious contributions do not significantly alter the quality of the approximation of the total level of energy, with overall excellent computational performance. On the other hand, when the starting functional is the strain energy and the principle of virtual displacement is adopted, the finite-element-introduced additional terms and in particular the contributions associated to the transverse shear stresses are demonstrated to add a very large error, which is a priori proven to be eliminated with selective integration. The presented technique could be extended to analyze the performance of other finite elements (e.g., higher-order triangular ones) within Reissner’s Mixed Variational Theorem and Generalized Unified Formulation.