QUAD4 Element Research Articles

Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions.

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

Development of a four-node quadrilateral element-based high order numerical manifold method without linear dependency

A high-order numerical manifold method (HONMM) is developed using four-node quadrilateral (QUAD4) element to enhance accuracy and computational efficiency in solid mechanical problems. In the current study, the high-order global approximations are built by increasing the order of local approximations without facing the linear dependence (LD) problem, which is one of the main issues in the partition of unity (PU)-based methods with high-order approximations. To remove the LD problem, a new and simple scheme is proposed. Four problems are utilized to evaluate the efficiency of the QUAD4-based HONMM (QHONMM). A cantilever beam example is analyzed to compare different orders of QHONMM. Also, a block example with different loads and boundary conditions is used to investigate the sensitivity of the QHONMM results. Moreover, to demonstrate the capabilities of the QHONMM in dynamic analysis, free and forced vibrations of a simple beam under distributed and moving loads are analyzed. The results showed that using QUAD4 elements, the proposed QHONMM performs better than the conventional NMM in terms of accuracy and computational costs.

FE‐Meshfree QUAD4 Element with Modified Radial Point Interpolation Function for Structural Dynamic Analysis

The partition‐of‐unity method based on FE‐Meshfree QUAD4 element synthesizes the respective advantages of meshfree and finite element methods by exploiting composite shape functions to obtain high‐order global approximations. This method yields high accuracy and convergence rate without necessitating extra nodes or DOFs. In this study, the FE‐Meshfree method is extended to the free and forced vibration analysis of two‐dimensional solids. A modified radial point interpolation function without any supporting tuning parameters is applied to construct the composite shape functions. The governing equations of elastodynamic problem are transformed into a standard weak formulation and then discretized into time‐dependent equations which are solved via Bathe time integration scheme to conduct the forced vibration analysis. Several numerical test problems are solved and compared against previously published numerical solutions. Results show that the proposed FE‐Meshfree QUAD4 element owns greater tolerance for mesh distortion and provides more accurate solutions.

Reverse engineering in finite element analysis of the behaviour of lignocellulosic materials subjected to cyclic stresses

In all engineering fields, continuous improvement of products and processes is being pursued. Reverse engineering is a fast and innovative method of converting a physical model into a virtual one based on its physical and mechanical characteristics. The paper aims to develop the geometric and numerical virtual model of the behavior of a lignocellulosic material subjected to cyclical stresses (tensile-compression) based on mechanical characteristics determined through experimental tests. The importance of this study consists of identifying the simulation method by comparing the experimental results with the numerical results (engineering objective) and developing the working methodology with effect in the education for sustainable development. Modeling with the finite element method (FEM) based on the physical model aimed to determine the results spectrum which are interpolating with the experimentally ones. Thus, the CAD model of the real sample was made by attributing its physical and elastic characteristics using the HyperMesh (preprocessing) and HyperWorks software (post-processing). The model was meshed into QUAD4 elements, after which it were simulated the cyclical stresses on tensile-compression in successive stages, keeping in memory of each stage the changes made at the structural level in the previous stage. Finally, the theoretical results were compared with the experimental ones. The obtained results provide information about distribution of stresses and strains on each layer of material according to the type of stress, traction or compression, for each loading cycle. The values obtained with FEA are close to the experimental ones. The minor differences are due to using an approximate decrease of the modulus of elasticity after each stress cycle. To obtained the real values of Young’s Modulus after each cycle would have been necessary a big consumption of samples. The reverse engineering method has the advantage of material economy and the ability to carry out a large number of numerical models to replicate the material’s behavior more accurately to cyclical loading and even improve its properties by designing a new material.

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.

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.

Numerical investigations of a partition-of-unity based “FE-Meshfree” QUAD4 element with radial-polynomial basis functions for acoustic problems

Recently, a mixed finite element-least square point interpolation method (FE-LSPIM) has been extended to deal with 2D acoustic problem by the authors. That element employed radial-polynomial basis functions for the local approximation (LA). This paper presents a FE-Meshfree QUAD4 element for analyzing the two dimensions (2D) and three dimensions (3D) acoustic problem by combining the excellent property of FE and radial-polynomial basis point interpolation shape functions by utilizing the partition of unity (PU) principles methods. In this work, the acoustic domain is discretized by quadrilateral mesh, and then the shape functions of quadrilateral element and the radial point interpolation are used for the LA. The radial-polynomial basis capacitates the proposed method to free from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions. The present method also offers an “appropriate-stiff” model for reducing the numerical dispersion error as compared to the previous FEM or FE-LSPIM with pure polynomial basis, especially for the high wave number problem. These findings have been validated through several numerical test problems.

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.

A hybrid ‘FE-Meshfree’ QUAD4 element with nonlocal features

A new element is developed by introducing a nonlocal-integrated displacement interpolation into the `FE-Meshfree' QUAD4 element recently published. Both nodal displacement and elemental strain are constructed using the nonlocal concept so that the elemental stiffness matrix and the resulting deformation fields are naturally nonlocal, leading to a significant difference when compared with nonlocal simulations using existing finite elements where local fields are output firstly in each incremental step and their nonlocal counterparts are obtained by averaging. The proposed element has the capability of predicting the nonlocal characteristics of localized deformations, with the intrinsic material length included into the nonlocal averaging. Several examples are employed to demonstrate the advantage of the proposed element in capturing highly localized deformations.

Study of Performance of US-FE-LSPIM QUAD4 Element on Mesh Distortion

This US-FE-LSPIM QUAD4 element is formed by using two different sets of shape functions for the trial and test functions, viz. sets of FE-LSPIM QUAD4 element shape functions and sets of classical isoparametric shape functions. For some test problems, the US-FE-LSPIM QUAD4 element has good accuracy. And by compared with FE-LSPIM QUAD4 element, the proposed element does not need to use Penalty method or Lagrange multiplier method to ensure fulfilment of exact essential boundary condition along the entire length of the edge. This paper further studies the performance of US-FE-LSPIM QUAD4 element on mesh distortion. Numerical test examples show that the element exhibits high precisions even under the mesh distortions. The US-FE-LSPIM QUAD4 element displays good tolerance to the mesh distortions

A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses

Recently, a partition-of-unity (PU) based finite element called ‘FE-Meshfree’ QUAD4 element has been published. That element employed polynomial basis functions for the local approximation (LA). In the present paper, a new FE-Meshfree QUAD4 element employing hybrid radial-polynomial basis functions for the LA is proposed. An advantage of radial-polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions. Another advantage is the improved accuracy of finite element solution. The new element has been applied to several linear and nonlinear test problems. The results demonstrate that the new element gives much better performance as compared to the previous FE-Meshfree QUAD4 element with pure polynomial basis. Even with a lower order basis, viz., with just three or four polynomial terms, the present element is capable of giving more accurate solution than the previous element which employs a six-term polynomial basis. The new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements. Moreover, the linear dependence problem, otherwise associated with many of the partition-of-unity (PU) based elements, is completely eliminated from the present element.

A new finite element procedure for fatigue life prediction of AL6061 plates under multiaxial loadings

An energy-based fatigue life prediction framework was previously developed by the authors for prediction of axial, bending and shear fatigue life at various stress ratios. The framework for the prediction of fatigue life via energy analysis was based on a new constitutive law, which states the following: the amount of energy required to fracture a material is constant. In the first part of this study, energy expressions that construct the constitutive law are equated in the form of total strain energy and the distortion energy dissipated in a fatigue cycle. The resulting equation is further evaluated to acquire the equivalent stress per cycle using energy based methodologies. The equivalent stress expressions are developed both for biaxial and multiaxial fatigue loads and are used to predict the number of cycles to failure based on previously developed prediction criterion. The equivalent stress expressions developed in this study are further used in a new finite element procedure to predict the fatigue life for two and three dimensional structures. In the second part of this study, a new Quadrilateral fatigue finite element is developed through integration of constitutive law into minimum potential energy formulation. This new QUAD-4 element is capable of simulating biaxial fatigue problems. The final output of this finite element analysis both using equivalent stress approach and using the new QUAD-4 fatigue element, is in the form of number of cycles to failure for each element on a scale in ascending or descending order. Therefore, the new finite element framework can provide the number of cycles to failure at each location in gas turbine engine structural components. In order to obtain experimental data for comparison, an Al6061-T6 plate is tested using a previously developed vibration based testing framework. The finite element analysis is performed for Al6061-T6 aluminum and the results are compared with experimental results.

A partition of unity‐based ‘FE–Meshfree’ QUAD4 element for geometric non‐linear analysis

AbstractThe recently published ‘FE–Meshfree’ QUAD4 element is extended to geometrical non‐linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh‐distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non‐linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well‐known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh‐distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.

‘FE-Meshfree’ QUAD4 element for free-vibration analysis

The FE-LSPIM QUAD4 element, which was recently introduced for the static analysis, uses new shape functions that combine the meshfree and finite element shape functions so as to synergize the individual strengths of meshfree and finite element methods. As a result, the element inherits the completeness properties of meshfree shape functions and the compatibility properties of finite element shape functions. In this paper, the element is extended to free vibration analysis of two-dimensional solids. Typical benchmark problems are solved to test the efficacy of the element. The results show that the element gives more accurate solution than the peer elements particularly under distorted mesh.

A “FE-meshfree” QUAD4 element based on partition of unity

A new 4-node quadrilateral element is developed by combining the shape functions of finite element and meshfree methods. The objective is to synergise the attractive features of finite element and meshfree shape functions using the concepts of partition of unity (PU) methods. The shape functions of the well known QUAD4 element are used for PU and the least-square point interpolation for local approximation. This enables the proposed element inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods. Numerical results for the bench mark problems show that the proposed element has a good tolerance to distorted meshes. The element also seems to be free from the linear dependence problem which is known to plague some of the PU-based elements.

O.366 The classification and management of late post-traumatic enophthalmos

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.

Thermal buckling response of shear flexible laminated anisotropic plates using a three-node isoparametric element

A recently developed three-node plate element with full integration scheme is extended to solve the problems of thermal buckling response of laminated composite square and skew plates. The plate element is based on the first-order shear deformation theory primarily suitable for moderately thick plates, but is also capable of handling thin plates without difficulty. The shear locking effects are nullified through use of shear correction terms in the transverse shear strain components. Thermal buckling temperatures including the critical one and mode shapes are numerically investigated. The present element shows excellent performance in the moderately thick to very thin plate regime. Results computed using the present element are generally in close agreement with the available analytical and higher order finite element methods solutions. The results computed using the present element are also compared with those due to QUAD4 element of NASTRAN (which supports only symmetric laminates) and SHELL99 element of ANSYS. Interesting results pertaining to the effects of shear deformation, lamination, skewness, etc., are discussed.

A weighted element patch based superconvergent stress extraction strategy

A superconvergent element patch based stress extraction strategy is proposed for general FE postprocessing and/or error estimation procedures in adaptive finite element analysis. Generalized weight functions are proposed for discrete least-square functionals, and a corrected conjoint polynomial fitting procedure is presented to ensure accurate stress extraction from the element domain once the primary level stress parameters have been evaluated. A numerical example is presented to fix the parameters of the weight functions. Several plane stress examples are solved using QUAD4 elements, and results are compared to those of the node patch based extraction method using the conventional conjoint polynomial method. It is shown that the proposed technique yields more accurate results with enhanced local convergence. Although the method may be applied to higher-order elements too, only QUAD4 elements are chosen for the present examples since the present secondary refitting strategy assumes superconvergent points only to be at element centroids.

A comparison of active, passive and hybrid damping in structural design

The QUAD4 element found in ASTROS is modified to include piezoelectric layers and a viscoelastic layer. These features are developed and combined to allow for active damping using piezoelectrics, passive damping using constrained layer damping (CLD) and hybrid damping in the form of active constrained layer damping (ACLD). These features are implemented in the QUAD4 element as additional layers in the composite layup definition. The development of these features are presented and the integrity of the implementation is validated by comparing with results found in the literature. A patch placement process is developed which allows the damping treatment to move on the structure and to distort its shape so as to improve the damping characteristics of the design. The structure and damping treatment are simultaneously designed for minimum weight with constraints on the frequency and damping ratio. Both beam and plate structures are used to illustrate this concept. A proportional-derivative controller is used for the active control of the beam and the linear quadratic regulator (LQR) is used to design the active controller for the plates.

Shear force predictions from RBF corrected QUAD4 elements

The four-node plate bending element (QUAD4) is the workhorse element in all production run FEA work. Its performance in most applications as measured by convergence of deflections under loads is limited by the simple bilinear functions used. Attempts have been made to enhance its performance by introducing what is called the residual bending flexibility (RBF) correction.”’ Since these elements have a provision to include shear deformation, the RBF correction takes advantage of this fact by modifying the shear energy term to compensate for the deficient bending energy term so that superior performance is obtained. Recently it was shown that this is strictly an extra-variational trick (or ‘crime’), using as an example the two node beam element, which is the one-dimensional equivalent of the QUAD4 element. It is not difficult to argue from this that the RBF correction, as it is frequently applied to QUAD4 elements, will also disturb some properties of its performance, and especially its accuracy in stress prediction. Our short exercise here is motivated from this point of view. Expected (predicted) results will be compared with observed (computed) results using QUAD4 elements with and without RBF correction to substantiate our argument.