Quadrilateral Area Coordinate Method Research Articles

Real-time temperature field reconstruction using a few measurement points and RPIM-AGQ6 interpolation

In thermal management systems, temperature field monitoring is vital for the structural safety of electronic equipment. However, there are practical challenges when it comes to installing temperature sensors on some monitoring points directly. To solve this problem, a novel temperature field reconstruction method is proposed. This strategy offers the advantage of using only a limited number of discrete values to predict temperature fields across an entire plane. Initially, the approach constructs a composite interpolation function, named RPIM-AGQ6, by coupling the mesh-free radial point interpolation method (RPIM) with the quadrilateral area coordinate method (AGQ6) The temperature field is discretized and represented as the product of the nodal temperatures and the RPIM-AGQ6. Next, ill-posed equations are established and solved using Bayesian Multivariate Linear Regression (BMLR) to obtain the temperature in the key points. Subsequently, a novel optimization strategy is proposed to achieve the cooperative optimization of the number of sensors and reconstruction accuracy. Finally, the simulated calculation and experimental tests are used to demonstrate the effectiveness and accuracy of the proposed approach. The results show that the proposed methodology can accurately reconstruct the plane temperature field of plate structure with one and two hot sources and the relative percentage error is within 2.6 %. Therefore, it can be utilized as an effective tool to real-time monitor structural surface temperature.

Simulating Indentation Experiment by Finite Element Method Using Mechanism-based Strain Gradient Plasticity and Quadrilateral Area Coordinate Method

Simulating Indentation Experiment by Finite Element Method Using Mechanism-based Strain Gradient Plasticity and Quadrilateral Area Coordinate Method

Koiter–Newton analysis of thick and thin laminated composite plates using a robust shell element

The Koiter–Newton method is a novel reduced-order modeling technique to efficiently trace the geometrically nonlinear equilibrium path of the structure in the presence of buckling. In this paper, the existing method is extended for nonlinear buckling analysis of thick and thin laminated composite plates. A 4-node quadrilateral element S4AT is developed as a geometric linear element in the co-rotational formulation of the Koiter–Newton method. The developed element S4AT is a combination of the membrane part and the plate-bending part. The quadrilateral area coordinate method is applied to make the element insensitive to the mesh distortion. The Reissner–Mindlin laminated composite plate theory is used to consider the transverse shear deformation effect, and the shear-locking phenomenon is eliminated using the Timoshenko laminated composite beam theory. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The flat shell element is constructed by combining a quadrilateral area coordinate method (QAC) based membrane element AGQ6-II, and a Timoshenko beam function (TBF) method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory (FSDT), for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical results show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.

Several Treatments on Nonconforming Element Failed in the Strict Patch Test

For nonconforming finite elements, it has been proved that the models whose convergence is controlled only by the weak form of patch tests will exhibit much better performance in complicated stress states than those which can pass the strict patch tests. However, just because the former cannot provide the exact solutions for the patch tests of constant stress states with a very coarse mesh (strict patch test), their usability is doubted by many researchers. In this paper, the non-conforming plane 4-node membrane element AGQ6-I, which was formulated by the quadrilateral area coordinate method and cannot pass the strict patch tests, was modified by three different techniques, including the special numerical integration scheme, the constant stress multiplier method, and the orthogonal condition of energy. Three resulting new elements, denoted by AGQ6M-I, AGQ6M-II, and AGQ6M, can pass the strict patch test. And among them, element AGQ6M is the best one. The original model AGQ6-I and the new model AGQ6M can be treated as the replacements of the well-known models Q6 and QM6, respectively.

The third form of the quadrilateral area coordinate method (QACM-III): Theory, application, and scheme of composite coordinate interpolation

Two quadrilateral area coordinate methods (hereinafter referred to as QACM-I and QACM-II) have already been proposed in 1999 and 2008, respectively. Many successful applications demonstrate that these new coordinate systems are efficient tools for developing robust quadrilateral finite element models. In this paper, another new quadrilateral area coordinate method (hereinafter referred to as QACM-III), containing two components T 1 and T 2, is systematically established. This new coordinate system (QACM-III) not only keeps the most important advantages of the QACM-I and QACM-II, but also possesses other distinguishing characters. Then, the QACM-III is applied to formulate two new 4-node membrane elements with internal parameters. The first new element, completely formulated by the QACM-III and denoted as QACIII6, exhibits much better performance than that of Wilson's Q6 element in benchmark problems, and has the simplest formulations when compared with the similar models formulated by the QACM-I and -II. The second element, denoted as CQAC6, is developed by a new scheme of composite coordinate interpolation, in which the QACM-I and -III are both used. It can strictly pass Iron's patch test and exhibits better accuracy than the Taylor's QM6 element. Finally, the new scheme of composite coordinate interpolation is employed again. Three different quadrilateral area coordinate methods, QACM-I, -II and -III, are used simultaneously to develop a new 8-node membrane element CQAC-Q8. On the one hand, element CQAC-Q8 is insensitive to mesh distortion, which the traditional isoparametric element Q8 cannot achieve; on the other hand, its construction procedure and formulations are quite simpler than the models formulated only by any single area coordinate method. The new coordinate method, the element models, and the scheme of composite coordinate interpolation presented in this paper provide new tools and ideas for high-performance finite element analysis.

On convergence of nonconforming convex quadrilateral finite elements AGQ6

The 4-node quadrilateral membrane elements AGQ6-I and II are two novel incompatible models formulated by the Quadrilateral Area Coordinate Method (QACM). In this paper, the sufficient conditions for their convergence are established. It is further shown theoretically that the convergence in the energy norm is linear, and the convergence in the L 2 norm is quadratic provided that certain geometric conditions are met requiring asymptotically parallelograms meshes to be used. The necessity of conditions is also discussed. The results of numerical examples completely confirm the theoretical findings.

Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM‐II)

AbstractA novel strategy for developing low‐order membrane elements with analytical element stiffness matrices is proposed. First, some complete low‐order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method (QACM‐II). Then, these solutions are taken as the trial functions for developing new membrane elements. Thus, the interpolation formulae for displacement fields naturally possess second‐order completeness in physical space (Cartesian coordinates). Finally, by introducing nodal conforming conditions, new 4‐node and 5‐node membrane elements with analytical element stiffness matrices are successfully constructed. The resulting models, denoted as QAC‐ATF4 and QAC‐ATF5, have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added. These two elements exhibit excellent performance in various bending problems with mesh distortion. It is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method, and the QACM‐II is a powerful tool for constructing high‐performance quadrilateral finite element models. Copyright © 2008 John Wiley & Sons, Ltd.

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.

A new quadrilateral area coordinate method (QACM‐II) for developing quadrilateral finite element models

AbstractThe quadrilateral area coordinate method proposed in 1999 (hereinafter referred to as QACM‐I) is a new and efficient tool for developing robust quadrilateral finite element models. However, such a coordinate system contains four components (L1, L2, L3, L4), which may make the element formulae and their construction procedure relatively complicated. In this paper, a new category of the quadrilateral area coordinate method (hereinafter referred to as QACM‐II), containing only two components Z1 and Z2, is systematically established. This new coordinate system (QACM‐II) not only has a simpler form but also retains the most important advantages of the previous system (QACM‐I). Hence, as an application, QACM‐II is used to formulate a new 4‐node membrane element with internal parameters. The whole process is similar to that of the famous Wilson's Q6 element. Numerical results show that the present element, denoted as QACII6, exhibits much better performance than that of Q6 in benchmark problems, especially for MacNeal's thin beam problem. This demonstrates that QACM‐II is a powerful tool for constructing high‐performance quadrilateral finite element models. Copyright © 2007 John Wiley & Sons, Ltd.

Application of the quadrilateral area coordinate method: a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate (QAC) method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF (5-DOF per node) quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-processing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Quadrilateral membrane element family formulated by the quadrilateral area coordinate method

The quadrilateral area coordinate (QAC) method is a new tool for developing quadrilateral finite element models. Compared with those traditional elements using isoparametric coordinates, models formulated by QAC method are less sensitive to mesh distortion. In this paper, a family of quadrilateral membrane elements with 4–8 nodes are successfully developed by employing the QAC method, the generalized conforming conditions and internal parameters. All the displacement fields of these elements possess second-order completeness in Cartesian coordinates, so they are more accurate and robust in various distorted meshes. Numerical results show that the presented elements exhibit excellent performance in all benchmark problems, such as MacNeal’s beam, thin curved beam problems, etc., which the traditional isoparametric elements can not easily achieve. The efficiency of the QAC method for developing simple, effective and reliable serendipity plane membrane elements is again demonstrated.

The analytical element stiffness matrix of a recent 4‐node membrane element formulated by the quadrilateral area co‐ordinate method

AbstractThe quadrilateral area co‐ordinate (QAC) method is a new tool for developing quadrilateral finite element models. Compared with those traditional elements using isoparametric co‐ordinates, models formulated by QAC method are less sensitive to mesh distortion. Furthermore, another advantage for QAC system is that the analytical expressions for element stiffness matrix can be obtained theoretically by basic formulae of QAC integrals, which are beneficial to finite element computation procedure. However, because of the relative complexity of the derivation process, no such explicit form of the stiffness matrix has been presented before. In this paper, the analytical element stiffness matrix of a recent 4‐node quadrilateral membrane element, AGQ6‐I, is given out for the first time. This element AGQ6‐I, which was constructed by QAC method and generalized conforming conditions, successfully overcomes various locking problems and exhibits much better performances than many isoparametric models. And its formulations have also been introduced in shell analysis by other researchers. Numerical examples show that obvious higher computation efficiency can be achieved by presented explicit formulations. So the analytical form of the element stiffness matrix of element AGQ6‐I possesses significance for its further applications. Copyright © 2006 John Wiley & Sons, Ltd.

A new approach to reduce membrane and transverse shear locking for one-point quadrature shell elements: linear formulation

In the last decade, one-point quadrature shell elements attracted many academic and industrial researchers because of their computational performance, especially if applied for explicit finite element simulations. Nowadays, one-point quadrature finite element technology is not only applied for explicit codes, but also for implicit finite element simulations, essentially because of their efficiency in speed and memory usage as well as accuracy. In this work, one-point quadrature shell elements are combined with the enhanced assumed strain (EAS) method to develop a finite element formulation for shell analysis that is, simultaneously, computationally efficient and more accurate. The EAS method is formulated to alleviate locking pathologies existing in the stabilization matrices of one-point quadrature shell elements. An enhanced membrane field is first constructed based on the quadrilateral area coordinate method, to improve element's accuracy under in-plane loads. The finite element matrices were projected following the work of Wilson et al. (Numerical and Computer Methods in Structural Mechanics, Fenven ST et al. (eds). Academic Press: New York, 1973; 43–57) for the incompatible modes approach, but the present implementation led to more accurate results for distorted meshes because of the area coordinate method for quadrilateral interpolation. The EAS method is also used to include two more displacement vectors in the subspace basis of the mixed interpolation of tensorial components (MITC) formulation, thus increasing the dimension of the null space for the transverse shear strains. These two enhancing vectors are shown to be fundamental for the Morley skew plate example in particular, and in improving the element's transverse shear locking behaviour in general. Copyright © 2005 John Wiley & Sons, Ltd.