Four-node Isoparametric Element Research Articles

Improved chaos game optimization algorithm for optimal frequency prediction of variable stiffness curvilinear composite plate

The optimal fundamental frequency of the variable stiffness laminated composite (VSCL) plate are predicted using an improved chaos game optimization technique (ICGO). In this analysis, the design parameters related to the fibre orientations within the layers are considered the discrete integer, i.e. the values vary between [Formula: see text] and [Formula: see text]. The fibre orientations are changed linearly concerning the longitudinal geometry coordinate in each ply. The numerical solutions are computed by adopting a four-noded isoparametric element (five degrees of freedom per node). The proposed optimization algorithm’s effectiveness, reliability and accuracy have been verified by comparing its optimum results with those of other algorithms in the literature. Also, the effect of the different relevant structural parameters, i.e. the aspect ratios, the edge conditions and the number of layers, on the optimal design, are reported in detail.

An efficient curved beam element for thermo-mechanical nonlinear analysis of functionally graded porous beams

This research is aimed to develop an efficient and high-performance four-node iso-parametric beam element, which is composed of Functionally Graded Material (FGM). In addition, different patterns of material distribution will be considered through the height of element. On the other hand, beam’s imperfection, presented here with porosity, is taken into account by using the rule of mixture. In order to alleviate the shear locking, Mixed Interpolation of Tensorial Components (MITC) is utilized by using tying points. Strain interpolation at some tying points reduces the order of strain functions. Therefore, three Gauss points can be employed for numerical integration instead of four Gauss points. Furthermore, the geometrically nonlinear effects are incorporated by using Green-Lagrange strains. Since the material properties are considered to be thermal-independent, they remain constant during the analysis. Finally, some benchmark problems are solved to illustrate the correctness of formulation and accuracy of the proposed element. Several parameters, including porosity percentage, FGM patterns and corresponding power indices, are investigated in the other examples. It is observed that the proposed element is more accurate for linear and nonlinear analyses of the thin beam with large deformations and rotations, even by using fewer numbers of elements compared to other developed elements. On the other hand, both axial and transverse displacements decrease when the value of the exponent of sigmoid pattern increases. On the contrary, the exponent of power pattern has a different effect on the axial displacement.

Numerical model of time-dependent diffusion of chlorides in the concrete based on 2D four-node isoparametric element

The enhancement of 2D model for the diffusion of chloride ions into reinforced concrete considering the time-dependent diffusion coefficient is discussed in the article. The non-stationary Finite Element Model is based on the four-node isoparametric element. The algorithm is implemented in MatLab compatible environment and an important part of source code is presented. The results of Finite Element Analysis are compared with the results of 2D analytical diffusion model.

Stability analysis of underground space with discontinuous planes using a four-node iso-parametric element numerical manifold method with rigid body rotation

In a large-scale rock structure with discontinuous planes in a rock mass, the pattern of the discontinuous planes strongly influences the deformation and stress distribution behaviors along the structure. A jointed rock mass, however, is generally assumed to be partially and imperfectly connected rather than completely discontinuous. Earlier the authors presented an elastic-plastic analysis of a discontinuous model of a jointed rock mass using a four-node iso-parametric manifold method (Sasaki and Ohnishi, 2001). The authors adapted Drucker-Prager’s elastic-plastic yield criterion, the Jaumann co-rotational derivative to determine the objectivity of stresses for rotation, and a rock-bolting element.In discontinuous models such as the original numerical manifold method (NMM), however, rigid rotation is not introduced in the degrees of freedom in the manner seen in the conventional finite element method (FEM). This paper describes the co-rotational formulation theory (Crisfield and Moita, 1996) introduced for a four-node iso-parametric NMM, along with examples of a model of rockfall activity in underground space. The results compare very well with the values obtained without a rigid body rotation analysis, especially those obtained for block shapes with large rotations.

Parametric Resonance Characteristics of Woven Fiber Metal Laminated Plates

The present investigation deals with the assessment of parametric resonance behavior of new aircraft material, i.e., woven fiber metal laminated (FML) plates subjected to in-plane static and harmonic loading using finite element (FE) technique and Bolotin’s method. In this analysis, a four-node isoparametric element with five degrees of freedom per node is adopted. Based on the first-order Reissner–Mindlin theory, the parametric instability of FML plate subjected to in-plane harmonic loading is examined. A MATLAB code is developed for the parametric study on the dynamic stability of FML plates. The reliability of present formulation is checked by comparing numerical results obtained from present FE analysis with the published researches in the field. The influences of several factors, viz. static load factor, aspect ratio, length-to-thickness ratio, number of layers, ply orientation and boundary conditions on the dynamic instability regions are discussed. Significant variations of these factors on dynamic instability zones of FML plates are observed. The instability zones can be used as guidelines for the prediction of the dynamic behavior of FML plates.

Vibration Analysis of Woven Fiber Metal Laminated Plates — Experimental and Numerical Studies

The present study deals with numerical and experimental investigations on the vibration behavior of fiber-metal-laminated (FML) plates, a new aircraft material. A finite element (FE)-based formulation is established for the plate using the first-order Reissner–Mindlin theory, including both fibers and metals of different material properties in alternate layers. A four-node isoparametric quadratic element with five degrees of freedom per node is adopted in the analysis. Convergence studies and comparison with previous studies are made to validate the present FE formulation. A set of experiments was conducted to get natural frequencies of vibration for glass FML (GFML) plates using Bruel and Kjaer (B&K) Fast Fourier Transform (FFT) analyzer with PULSE platform. The effects of different parameters such as aspect ratio, side-to-thickness ratio, ply orientation, and boundary conditions on the dynamic behavior of the FMLs are studied. Good agreement is achieved between the numerical and experimental results. Both results indicate that increasing the aspect ratio can increase the natural frequency of the FML plate, while the increase in the side-to-thickness ratio decreases the natural frequency of vibration. The boundary conditions can significantly affect the natural frequency of the FML plates due to the restraint effect at the edges.

Four-Node Quadrilateral Element with Continuous Nodal Stress for Geometrical Nonlinear Analysis

In this paper, the performance of a hybrid ‘FE-Meshfree’ quadrilateral element with continuous nodal stress (Quad4-CNS) is investigated for geometrical nonlinear solid mechanic problems. By combining finite element method (FEM) and meshfree method, this Quad4-CNS synergizes the individual strengths of these two methods, which leads to higher accuracy, better convergence rate, as well as high tolerance to mesh distortion. Therefore, Quad4-CNS is attractive for geometrical nonlinear solid mechanic problems where excessive distorted meshes occur. For geometrical nonlinear analysis, numerical results show that the results of Quad4-CNS element are much better than those of four-node isoparametric quadrilateral element (Quad4), and are comparable to quadratic quadrilateral element (Quad8) and other hybrid ‘FE- Meshfree’ elements.

A partition-of-unity based three-node triangular element with continuous nodal stress using radial-polynomial basis functions

A partition-of-unity (PU) based “FE-Meshfree” three-node triangular element (Trig3-RPIM) was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element (Trig3) and radial-polynomial basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom (DOFs), the nodal stress field through Trig3-RPIM element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.

Application of the ‘FE-Meshfree’ QUAD4 with continuous nodal stress using radial-polynomial basis functions for vibration and geometric nonlinear analyses

A hybrid ‘FE-Meshfree’ four-node quadrilateral element with continuous nodal stress using radial-polynomial basis functions (Quad4-RPIMcns), was recently proposed for static analysis. The Quad4-RPIMcns element can be considered as a development of the previous partition-of-unity (PU) based ‘FE-Meshfree’ QUAD4 element (Quad4-RPIM) which uses FE shape functions to construct the PU and radial-polynomial basis functions to construct the local approximation (LA), so as to synergize the individual strengths of finite element and meshfree methods. As a result, high order global approximations in Quad4-RPIMcns element could be easily constructed without adding extra nodes and DOFs, thereby achieving high accuracy and convergence rate. In this paper, the element is further applied to conduct free vibration, forced vibration and geometric nonlinear analyses of two-dimensional solids. Several numerical test problems are solved and the performance of the element is compared with that of the three-node triangular element (Trig3) and four-node isoparametric quadrilateral element (Quad4). Numerical results show that Quad4-RPIMcns element has higher tolerance to mesh distortion and gives more accurate solution as compared to Trig3 and Quad4 elements.

A Modified Smoothed Finite Element Method for Static and Free Vibration Analysis of Solid Mechanics

The smoothed finite element method (S-FEM) proposed recently is more accurate and has higher convergence rate compared with standard four-node isoparametric finite element method (FEM). In this work, a modified S-FEM using four-node quadrilateral elements is proposed, which greatly reduces further the computation cost while maintaining the high accuracy and convergence rate. The key idea of the proposed modification is that the strain of the element is a weighted average value of the smoothed strains in the smoothing cells (SCs), which means that only one integration point is required to construct the stiffness matrix, similar to the single cell S-FEM. A stabilization item is proposed using the differences of the smoothed strains obtained in four SCs, which installs the stability of algorithm and increases the accuracy. To verify the efficiency, accuracy and stability of the present formulation, a number of numerical examples of static and free vibration problems, are studied in comparison with different existing numerical methods.

A four-node quadrilateral element fitted to numerical manifold method with continuous nodal stress for crack analysis

Formulations of a four-node quadrilateral element fitted to numerical manifold method (NMM) with continuous nodal stress called Quad4-CNS (NMM) element for two-dimensional (2D) crack analysis are presented. This Quad4-CNS (NMM) element can be considered as a development of the recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS). In contrast to the four-node iso-parametric quadrilateral element (Quad4), the Quad4-CNS element has higher order of global approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity (PU) based methods with high order global approximations. Due to the adoption of two cover systems, namely, the mathematical cover and physical cover, the NMM is capable of solving continuous and discontinuous problems in a unified way. The purpose of this paper is to synergize the advantages of both the Quad4-CNS element and the NMM to precisely model linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Quad4-CNS (NMM) element.

An approach for bending and transient dynamic analysis of integrated thermal protection system with temperature-dependent material properties

A C0 higher-order layerwise finite element model combined with homogenization techniques for bending and transient dynamic analysis of integrated thermal protection system (ITPS) panel is presented. The ITPS panel is homogenized as an equivalent sandwich plate, and a reduced temperature field for the entire panel is derived. An advanced analytical equivalent model based on energy approach and Castigliano’s theorem for the mechanical properties of the trapezoidal corrugated core is presented, which considers the effects of temperature-dependent material properties. The proposed layerwise theory which is developed for a sandwich plate, assumes linear, quadratic and cubic polynomial distributions of displacements for middle layer and linear polynomial distributions of displacements for top and bottom layers. And the effect of transverse normal strains is taken into account in the layerwise theory. A four-noded isoparametric element with 16 degrees of freedom per node is employed to model the equivalent sandwich plate. The accuracy of the proposed method is verified by comparison with the results by three-dimensional (3D) finite element analysis. It has been shown that the proposed method requires significantly less computational effort and agrees well with the finite element results.

Free and forced vibration analyses using the four-node quadrilateral element with continuous nodal stress

The recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS) is extended to free and forced vibration analyses of two-dimensional solids. The Quad4-CNS element can be regarded as a partition-of-unity (PU) based ‘FE-Meshfree’ element which inherits better accuracy, higher convergence rate, and high tolerance to mesh distortion from the meshfree methods, while preserving the Kronecker-delta property of the finite element method (FEM). Moreover, the Quad4-CNS element is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several free and forced vibration problems are solved and the performance of the element is compared with that of the four-node isoparametric quadrilateral element (Quad4) and eight-node isoparametric quadrilateral element (Quad8). The results show that, for regular meshes, the performance of the element is superior to that of Quad4 element, and comparable to that of Quad8 element. For distorted meshes, the present element has better mesh-distortion tolerance than Quad4 and Quad8 elements.

A Simple Finite Element with Five Degrees of Freedom Based on Reddy’s Third-Order Shear Deformation Theory

A simple four-node isoparametric finite element with five degrees of freedom, based on Reddy’s third-order shear deformation theory, is elaborated and used in a model for analizing the bending of laminated plates. The results obtained are compared with solutions given by the three-dimensional elasticity and other theories.

A partition-of-unity based ‘FE-Meshfree’ triangular element with radial-polynomial basis functions for static and free vibration analysis

A new ‘FE-Meshfree’ three-node triangular element (Trig3-RPIM) is developed based on the partition of unity (PU) concept. The Trig3-RPIM element employs the shape function of classical three-node triangular element (Trig3) to construct the PU and the radial-polynomial basis function which is free from the possible singularity of the moment matrix to construct the nodal approximation. The Trig3-RPIM element synergizes the individual strengths of finite element method and meshfree method. Moreover, it is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several linear, nonlinear and free vibration test problems are solved and the performance of the element is compared with those of the well-known three-node triangular element (Trig3) and four-node iso-parametric quadrilateral element (Quad4). The results show that, for regular meshes, the performance of the element is superior to those of Trig3 and Quad4 elements. For distorted meshes, the present element has better mesh-distortion tolerance than Trig3 and Quad4 elements.

Construct ‘FE-Meshfree’ Quad4 using mean value coordinates

The present work uses mean value coordinates to construct the shape functions of a hybrid ‘FE-Meshfree’ quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the ‘FE-Meshfree’ quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4.

A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions

In the present work, a novel hybrid FE-Meshless quadrilateral element with continuous nodal stress is developed using radial-polynomial basis functions, named as Quad4-RPIMcns. Quad4-RPIMcns can be regarded as the development of the previous FE-Meshless quadrilateral element with radial-polynomial basis functions (Quad4-RPIM) and quadrilateral element with continuous nodal stress (Quad4-CNS). Similar to Quad4-RPIM, radial-polynomial basis functions are used to construct nodal approximations of Quad4-RPIMcns in the context of partition of unity, which avoids the possible singularity problem of constructing nodal approximations. The derivative of Quad4-RPIMcns shape function is continuous at nodes. Therefore, nodal stress can be obtained without any extra operation. Quad4-RPIMcns possesses Kronecker-delta property which is a very important property to impose essential boundary conditions directly as in the FEM. The numerical tests in this paper demonstrate that Quad4-RPIMcns gives better accuracy and higher convergence rate as compared to four-node iso-parametric quadrilateral element (Quad4). Additionally, Quad4-RPIMcns seems to have higher tolerance to mesh distortion than Quad4.

Study on bending of hard core sandwich plates based on Reissner’s sandwich theory

In the traditional theories of sandwich plates the core is soft so the in-plane stress components and bending stiffness are neglected. This paper explores the bending problem of hard core sandwich plates based on the revised Reissner's hypothesis. The basic equations and boundary conditions expressed with three generalized displacements are established and simplified according to the principle of minimum potential energy. In addition this paper presents the finite element model for the bending problem of hard core sandwich plates. Based on this model a four-node quadrilateral isoparametric element is constructed, the element stiffness matrix and equivalent nodal load vector are derived. It is built the solid model for ANSYS. According to the numerical analysis the finite element solutions converge to the analytical solutions with sufficient accuracy and also consistent with solutions of ANSYS.

A radial interpolation finite element method for 2D homogeneous and multifluid coupling acoustic problems

In the acoustics Finite Element Method (FEM), the typical problems of the four-node isoparametric element are low accuracy and high sensitivity to the quality of mesh in the finite element mesh model. Therefore, a Radial Interpolation Finite Element Method (RIFEM), whose shape function is constructed by using the meshless radial interpolation technique and the partition-of-unity method, is proposed for the 2D homogeneous and multifluid coupling acoustic problems. In acoustic RIFEM, to maintain the integral accuracy of the sound pressure derivatives and apply the accurate boundary conditions easily, the acoustic system stiffness matrix and the vectors of the boundary integrals are constructed by using the bilinear shape function. On the contrary, to improve the interpolation accuracy of the approximated sound pressure function, the acoustic system mass matrix is constructed by the shape function of RIFEM using the four-node isoparametric element. Numerical examples on a 2D homogeneous acoustic tube, the 2D homogeneous acoustic cavity of a car and a 2D multifluid coupling tube verify that RIFEM achieves higher accuracy, when compared with the linear FEM and the smoothed FEM.

Interference of Two Asymmetric Closely Spaced Strip Footings Resting on Nonhomogeneous and Linearly Elastic Soil Bed

AbstractThis paper presents a study using the FEM on the effect of interference on the settlement of two closely spaced rough rigid strip footings resting on the surface of linearly elastic finite and infinite nonhomogeneous soil beds with the modulus of elasticity linearly varying with the depth. Both symmetric and asymmetric cases with respect to its geometry and applied loading have been considered in the analysis. The finite-element (FE) mesh was created using four-noded isoparametric elements. The results are presented in terms of the interaction factors, which are defined as the ratio of the settlement of the interfering footings with the corresponding value of the settlement of the isolated footing identifying the influence of the size of the footings, loading on the footings, and clear spacing between the footings. The settlement of the interfering footings is observed to be greater than the settlement of the isolated footing of the same width and loading. With an increase in the stiffness of the ...