- 1. A consecutive-interpolation quadrilateral element (CQ4): Formulation and applications Tinh Quoc Bui a,n , Dam Quang Vo b , Chuanzeng Zhang a , Du Dinh Nguyen c a Department of Civil Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen, Germany b Piping Department, Petrovietnam Engineering Company, Ho Chi Minh, Vietnam c Department of Civil Engineering, Lac Hong University, Dong Nai Province, Vietnam a r t i c l e i n f o Article history: Received 3 April 2013 Received in revised form 13 February 2014 Accepted 19 February 2014 Available online 12 March 2014 Keywords: FEM Consecutive-interpolation ﬁnite element Stress analysis Numerical methods Quadrilateral element a b s t r a c t An efﬁcient, smooth and accurate quadrilateral element with four-node based on the consecutive- interpolation procedure (CIP) is formulated. The CIP is developed recently by Zheng et al. (Acta Mech Sin 26 (2010) 265–278) for triangular element with three-node. In this setting the approximation functions handle both nodal values and averaged nodal gradients as interpolation conditions. Two stages of the interpolation are required; the primary stage is carried out using the same procedure of the standard ﬁnite element method (FEM), and the interpolation is further reproduced in the secondary step according to both nodal values and average nodal gradients derived from the previous interpolation. The new consecutive-interpolation quadrilateral element with four-node (CQ4) deserves many desirable characteristics of an efﬁcient numerical method, which involves continuous nodal gradients, continuous nodal stresses without smoothing operation, higher-order polynomial basis, without increasing the degree of freedom of the system, straightforward to implement in an existing FEM computer code, etc. Four benchmark and two practical examples are considered for the stress analysis of elastic structures in two-dimension to show the accuracy and the efﬁciency of the new element. Detailed comparison and some other aspects including the convergence rate, volumetric locking, computational efﬁciency, insensitivity to the mesh, etc. are investigated. Numerical results substantially indicate that the consecutive-interpolation ﬁnite element method (CFEM) with notable features pertains to high accuracy, convergence rate, and efﬁciency as compared with the standard FEM. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Design procedures of improving and enhancing the perfor- mance of engineering structures through stress analysis are often time-consuming and expensive. Nowadays, simulation technolo- gies using advanced numerical methods in engineering and science are popular and have been emerged rapidly. The motiva- tions are to accurately model practical problems as exact as the techniques can. The ﬁnite element method (FEM) [1–3] and the boundary element method (BEM) [4] have become very powerful and versatile numerical methods, which are the most common and extensively used methods in a broad range of engineering applications. Owing to the simplicity, the three-node triangular and four-node quadrilateral ﬁnite elements are often introduced and applied to solve engineering problems in two-dimensions (2D). Because of the linear approximations, the spatial derivatives of the ﬁeld variables are constant within each element [5]. Such constant-strain ﬁnite elements are easily formulated and imple- mented but their performance in practical applications is often unsatisfactory and, frequently low accuracy is obtained due to their low-order trial functions [5,6]. Moreover, the gradients on element-edges in both constant elements and mapped elements are discontinuous, and demanding smoothing operation in post- processing step is rigorous [7]. Other relevant issues involving volumetric locking and sensitivity to mesh, etc. for such elements can be found in Refs. [1–3,5–8] for instance. A number of advanced numerical methods have been devel- oped in order for improving the accuracy and efﬁciency of the conventional FEM methods. For instance, Hansbo proposed a non- conforming rotated Q1 tetrahedral element for linear elastic [9] and elastodynamic problems [10]. By containing the bilinear terms, the Q1 element performs substantially better than the standard constant-strain one in bending and allows for under- integration in nearly incompressible situations. Papanicolo- pulos and Zervos [11] presented a means for creating a class of triangular C1 ﬁnite element particularly suitable for model- ing problems where the underlying partial differential equation is of fourth-order (e.g., beam and plate bending, deformation of Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design http://dx.doi.org/10.1016/j.ﬁnel.2014.02.004 0168-874X & 2014 Elsevier B.V. All rights reserved. n Corresponding author. Tel.: þ49 2717402836; fax: þ49 2717404074. E-mail address: tinh.buiquoc@gmail.com (T.Q. Bui). Finite Elements in Analysis and Design 84 (2014) 14–31
- 2. strain-gradient-dependent materials). Liu and his co-workers [12,13] introduced the smoothed ﬁnite element method based on the smoothing strain technique. An extension of the SFEM to stationary dynamic crack analysis of 2D elastic solids is studied by the author [14]. In recent years Hughes et al. [15] introduced the isogeometric analysis (IGA) using higher-order basis functions (e.g., Non-Uniform Rational B-Splines) for constructing an exact geometrical model. The IGA soon after has been further extended to study many engineering problems. In the contrary to the mesh- based methods, the so-called meshfree or meshless methods, e.g., see Refs. [16–18], have introduced as alternative numerical approaches that do not require a predeﬁned mesh, whereof the trial functions can be constructed through the scattered nodes without the reliance of the elements. The meshfree methods on one side involve several desirable features of an attractive numer- ical scheme in modeling various engineering problems, but one of their primary disadvantages lies in the expensive cost of using the more complex shape functions, which substantially induces a higher cost of the computation as compared with that of the mesh-based approaches, e.g., the FEM. As well-known in the standard FEM frameworks that the gradients on the element-edges using the constant-strain ele- ments or mapped elements are discontinuous, and a smoothing operation is rigorously demanded in the post-processing. One possibility among others that can completely overcome such discontinuities without any smoothing operation is the novel consecutive-interpolation procedure (CIP) proposed by Zheng et al. [7]. They have already proved it for elasticity problems using their own CIP-based triangular element (CT3). Basically, the essence of the CIP technique is to enhance the trial functions by taking the continuous nodal gradients and a higher-order poly- nomial basis, by which several desirable characteristics when utilizing the consecutive-interpolation ﬁnite element method (CFEM) can be reached: (a) high accuracy in the ﬁeld variables; (b) high convergence rate; (c) nodal stresses can be produced continuously without the aid of any smoothing operation; (d) insensitivity to mesh distortions; (e) volumetric locking is avoided in incompressible materials; (f) the total number of the degrees of freedom (DOFs) of the system does not change which implies that the total number of the DOFs discretized by the CFEM is the same as that by the FEM; (g) shape functions with higher- order polynomials possess the Kronecker-delta function property; and so on. The CFEM approximation functions are constructed through two stages of the interpolation. Apart from the primary interpolation, the same procedure of the classical FEM, the inter- polation is further reproduced using the nodal values and aver- aged nodal gradients derived from the previous interpolation. The main objective of the present work is to precisely for- mulate a novel quadrilateral element with four-node based on the CIP approach (termed as CQ4), and then apply it to stress analysis of 2D elastic structures. The developed CQ4 element generally inherits all the superior characteristics and desirable properties of the CT3 element as pointed out above. Most importantly, the CFEM can be implemented straightforwardly from any existing computer FEM code. To show the accuracy and the efﬁciency of the proposed CQ4 element, four benchmark examples and two practical appli- cations with complex geometries are considered. The computed numerical results are then compared with analytical solutions, the CT3, the standard quadrilateral (Q4) and triangular (T3) solutions, as well as the FEM solutions using ANSYS. The convergence rate and the computational efﬁciency are investigated in detail. Addi- tionally, the volumetric locking phenomenon occurring in incom- pressible materials is also addressed and it shows that the CQ4 can treat such problem without any modiﬁcation. The paper is formed into ﬁve sections. After the introduction, the CIP technique is brieﬂy presented in Section 2. The formulation of the CQ4 element is detailed in Section 3. Numerical examples are presented and discussed in Section 4. Some conclusions from the study are drawn in Section 5. 2. A brief on the consecutive-interpolation technique Consider a 2D elastic body in the domain Ω bounded by Γ ¼ Γu þΓt and Γu Γt ¼ ∅ that can be described by the equili- brium equations as [1–3]. sij;j þbi ¼ 0 in Ω ð1Þ where bi denotes the components of the body force vector and sij is the stress tensor. The balance equations, Eq. (1), satisfy the following boundary conditions: sijnj ¼ ti on Γt and ui ¼ ui on Γu ð2Þ with ui representing the prescribed boundary displacements on Γu, ti being the traction components on Γt while ni being the unit outward normal vector. The variational weak-form for this static elastic problem can be expressed as [1–3] Z Ω δ∇sðuÞijDijkl∇sðuÞkl dΩ Z Ω δuibi dΩ Z Γt δuiti dΓ ¼ 0 ð3Þ where Dijkl is the elasticity tensor and ∇sðuÞij denotes the sym- metric part of the displacement gradients i.e., ∇sðuÞij ¼ ðui;j þuj;iÞ=2. In the FEM we approximate solutions to Eq. (3) by dividing Ω and the boundary Γ into small elements, and the interpolation is then determined by approximating the displacement ﬁeld in each element. The element stiffness matrix is derived and it is then assembled into the global stiffness matrix [1–3]. Generally, this step is accomplished almost identically for the FEM and the CFEM. Now, we start describing the CIP procedure in a general and brief way. In the subsequent sections the formulation of the novel CQ4 element with four-node is detailed. For the sake of brevity, the following presentation will be focused on the displacement component u1 ¼ u only. The function uðxÞ with x ¼ fx; ygΤ in 2D in the FEM can be approximated by uðxÞ ¼ ∑ n i ¼ 1 NiðxÞdi ¼ NðxÞd ð4Þ where n is the number of nodes, d is the nodal displacement vector, while NðxÞ is the vector of the shape functions, and Ni are the shape functions of node i. By assigning the approximation value at node i with u½iŠ ¼ uðxiÞ, and the vector of the shape functions at node i with N½iŠ ¼ NðxiÞ, the averaged nodal derivatives u½iŠ ;x (similar for u½iŠ ;y) can then be determined by [7] u½iŠ ;x ¼ N ½iŠ ;x d ð5Þ where N ½iŠ ;x are the averaged derivative of N½iŠ , and calculated by N ½iŠ ;x ¼ ∑ e ASi ðwe UN½iŠ½eŠ ;x Þ ð6Þ with N½iŠ½eŠ ;x being the derivative of N½iŠ computed in element e. In Eq. (6), Si are a set of elements containing all the elements connected to node i, while we is a weight-function dependent on the element-type and it will be detailed in the subsequent section for the quadrilateral element. The shortcoming in the discontinuous stresses and strains caused by the discontinuity of the nodal gradients is well known in the standard FEM [7]. In the present CIP approach, both the nodal values u½iŠ and the averaged nodal derivatives u½iŠ ;x are taken into the interpolations, which can substantially overcome such drawback of the discontinuities in the stress and strain ﬁelds. As a consequence the approximation functions in Eq. (4) can now be T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 15
- 3. rewritten by means of the CIP scheme as follows: ~ uðxÞ ¼ ∑ n i ¼ 1 ðϕiu½iŠ þϕixu½iŠ ;x þϕiyu½iŠ ;y Þ ¼ ∑ n i ¼ 1 ðϕiN½iŠ þϕixN ½iŠ ;x þϕiyN ½iŠ ;yÞd ð7Þ or ~ uðxÞ ¼ ~ NðxÞd ð8Þ where the shape functions are given by ~ N ¼ ∑ n i ¼ 1 ðϕiN½iŠ þϕixN ½iŠ ;x þϕiyN ½iŠ ;yÞ ð9Þ and ϕi, ϕix and ϕiy are the ﬁeld functions dependent on the element-type, which will be detailed in the subsequent section for the quadrilateral element. Here, one important point to be stated is that the unknowns still contain only nodal displacements irrespective of the extension of the approximation functions including the nodal values and the averaged nodal derivatives. It means that no additional DOFs in the system are required for the CIP approach, an advantageous feature of the development of the CFEM. Another essential issue is also worth noting that the general solution procedure of the CFEM is very similar to that of the standard displacement-based FEM. So, any other steps in realization of the FEM scheme regardless of the CIP procedure could be applied the same in the implementation of the CFEM approach. 3. Formulation of consecutive-interpolation quadrilateral elements (CQ4) In this section we shall formulate the quadrilateral element with four-node based on the CIP procedure. The four-node quad- rilateral element in the global coordinate system can be mapped to the four-node rectangular element in the natural one as depicted in Fig. 1. The geometry of the quadrilateral element can be described explicitly by x ¼ Liðr; sÞxi þLjðr; sÞxj þLkðr; sÞxk þLmðr; sÞxm; y ¼ Liðr; sÞyi þLjðr; sÞyj þLkðr; sÞyk þLmðr; sÞym; ð10Þ where Liðr; sÞ; Ljðr; sÞ; Lkðr; sÞ; and Lmðr; sÞ are considered as geo- metric interpolation functions as Liðr; sÞ ¼ 1 4 ð1 rÞð1 sÞ; Ljðr; sÞ ¼ 1 4 ð1þrÞð1 sÞ; Lkðr; sÞ ¼ 1 4 ð1þrÞð1þsÞ; Lmðr; sÞ ¼ 1 4 ð1 rÞð1þsÞ; ð11Þ and the partial derivatives can also be computed through the inverse Jacobian matrix J 1 by ∂ ∂x ∂ ∂y ( ) ¼ J 1 ∂ ∂r ∂ ∂s ( ) ð12Þ with J ¼ ∂x ∂r ∂y ∂r ∂x ∂s ∂y ∂s " # ¼ ∂Li ∂r ∂Lj ∂r ∂Lk ∂r ∂Lm ∂r ∂Li ∂s ∂Lj ∂s ∂Lk ∂s ∂Lm ∂s 2 4 3 5 xi yi xj yj xk yk xm ym 8 > > > > < > > > > : 9 > > > > = > > > > ; ð13Þ From Eqs. (10) and (11), the J-matrix can be rewritten as J ¼ 1 4 ð1 sÞ ð1 sÞ ð1þsÞ ð1þsÞ ð1 rÞ ð1þrÞ ð1þrÞ ð1 rÞ " # xi yi xj yj xk yk xm ym 8 > > > > < > > > > : 9 > > > > = > > > > ; ð14Þ 3.1. The ﬁrst stage of the interpolation: standard The approximation function at the point of interest x in a quadrilateral element in the framework of the conventional FEM can be expressed as uðxÞ ¼ Niu½iŠ þNju½jŠ þNku½kŠ þNmu½mŠ ð15Þ where Ni ¼ Li; Nj ¼ Lj; Nk ¼ Lk; Nm ¼ Lm ð16Þ Fig. 2 illustrates the application of the CIP method to the quadrilateral element described particularly in an irregular ﬁnite element mesh, in which the element sets Si; Sj; Sk and Sm contain all the neighboring elements of node i; j; k and m, respectively. It indicates that the supporting nodes for the point of interest x include all the nodes in the element sets Si; Sj; Sk and Sm. In any cases using the CFEM method, the support domain as the one shown in Fig. 2 is larger than that of the standard FEM. The weight-functions used for evaluating the averaged nodal deriva- tives of the ﬁeld variables of the quadrilateral element e are calculated similarly to that of the CT3 element, and they are given by we ¼ Δe ∑~ e A Si Δ~ e ; with eASi ð17Þ where Δe is the area of the element e. By deﬁning ds ¼ fd1; d2; …; dns gΤ as the displacement vector of the supporting nodes with ns being the number of the supporting Fig. 1. A quadrilateral element in (a) global coordinate and (b) natural coordinate systems. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 16
- 4. nodes connected to the point of interest x, the interpolation of the standard FEM for any point in Si; Sj; Sk; and Sm can be determined by uðxÞ ¼ ∑ ns l ¼ 1 NlðxÞdl ð18Þ The averaged derivatives of the ﬁeld variables on node i can then be computed as follows: u½iŠ ;x ¼ ∑ e A Si weu½iŠ½eŠ ;x ¼ ∑ ns l ¼ 1 ∑ e A Si weN½iŠ½eŠ l;x # dl u½iŠ ;y ¼ ∑ e A Si weu½iŠ½eŠ ;y ¼ ∑ ns l ¼ 1 ∑ e A Si weN½iŠ½eŠ l;y # dl; ð19Þ or in compact form u½iŠ ;x ¼ ∑ ns l ¼ 1 N ½iŠ l;xdl; u½iŠ ;y ¼ ∑ ns l ¼ 1 N ½iŠ l;ydl ð20Þ where N ½iŠ l;x ¼ ∑ eA Si weN½iŠ½eŠ l;x ; N ½iŠ l;y ¼ ∑ e ASi weN½iŠ½eŠ l;y ð21Þ 3.2. The second stage of the interpolation: consecutive As stated in the previous section, the consecutive stage is accomplished by taking the nodal values and averaged nodal derivatives into the interpolation scheme. The approximation function of the CQ4 element can be expressed as ~ uðxÞ ¼ ϕiu½iŠ þϕixu½iŠ ;x þϕiyu½iŠ ;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} related to nodei þϕju½jŠ þϕjxu½jŠ ;x þϕjyu½jŠ ;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node j þϕku½kŠ þϕkxu½kŠ ;x þϕkyu½kŠ ;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node k þϕmu½mŠ þϕmxu½mŠ ;x þϕmyu½mŠ ;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node m ð22Þ In Eq. (22), the ϕi; ϕix; and ϕiy functions shown below must satisfy the following conditions (see Appendix A for proving this condition) ϕiðxlÞ ¼ δil; ϕi;xðxlÞ ¼ 0; ϕi;yðxlÞ ¼ 0; ϕixðxlÞ ¼ 0; ϕix;xðxlÞ ¼ δil; ϕix;yðxlÞ ¼ 0; ϕiyðxlÞ ¼ 0; ϕiy;xðxlÞ ¼ 0; ϕiy;yðxlÞ ¼ δil; ð23Þ where l can be any one of the indices i; j; k; and m, and δij ¼ 1 if i ¼ j 0 if iaj ( ð24Þ We note that the above conditions have to be applied in a similar manner to other functions, i.e., ϕj; ϕjx; ϕjy, ϕk; ϕkx; ϕky and ϕm; ϕmx; ϕmy. Finally, the ϕi; ϕix and ϕiy functions can be computed for the CQ4 element as ϕi ¼ Li þL2 i Lj þL2 i Lk þL2 i Lm LiL2 j LiL2 k LiL2 m; ϕix ¼ ðxi xjÞðL2 i Lj þpLiLjLk þpLiLjLmÞ ðxi xkÞðL2 i Lk þpLiLkLm þpLiLkLjÞ ðxi xmÞðL2 i Lm þpLiLmLj þpLiLmLkÞ; ϕiy ¼ ðyi yjÞðL2 i Lj þpLiLjLk þpLiLjLmÞ ðyi ykÞðL2 i Lk þpLiLkLm þpLiLkLjÞ ðyi ymÞðL2 i Lm þpLiLmLj þpLiLmLkÞ ð25Þ In Eq. (25), p ¼ 1=2 and the ϕj; ϕjx; ϕjy; ϕk; ϕkx; ϕky and ϕm; ϕmx; ϕmy functions can be also computed in the same manner by a cyclic permutation of indices i; j; k and m. By substituting u½iŠ ;x ; u½iŠ ;y and ϕ into Eq. (22), we ﬁnally arrive at ~ uðxÞ ¼ ∑ ns l ¼ 1 ~ NlðxÞdl ¼ ~ NðxÞd ð26Þ with the consecutive-interpolation shape functions ~ Nl ¼ ϕiN½iŠ l þϕixN ½iŠ l;x þϕiyN ½iŠ l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} related to node i þϕjN½jŠ l þϕjxN ½jŠ l;x þϕjyN ½jŠ l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node j þϕkN½kŠ l þϕkxN ½kŠ l;x þϕkyN ½kŠ l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node k þϕmN½mŠ l þϕmxN ½mŠ l;x þϕmyN ½mŠ l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node m ð27Þ Once again, similar to the CT3 element the unknowns in the system using the CQ4 element contain only the nodal displace- ments, and no additional number of the DOFs is required for the Fig. 2. Schematic sketch of the consecutive-interpolation quadrilateral element (CQ4) in 2D. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 17
- 5. CQ4 element. This issue shows a signiﬁcant difference from the well-known conforming plate element, where the nodal deriva- tives are considered as additional DOFs [2,7]. Remark: In general, the proposed CFEM interpolation is com- parable to the Hermite interpolation as they use the nodal values and nodal gradients. However, it may be different in some details because the CFEM uses the “averaged nodal gradients” derived from ﬁnite element interpolation at each node instead. 3.3. Desirable properties of the shape functions Figs. 3a and 3b show a comparison of the 1D shape functions and their ﬁrst-order derivatives between the FEM and CFEM methods. It is observed in the ﬁgures that the curves of the CFEM shape functions and their derivatives are rather smooth and continuous as compared with those based on the FEM. The 2D shape functions derived from the Q4 and CQ4 elements are also visualized in Fig. 3c and d. Nevertheless, some important and desirable properties of the CFEM shape functions may be sum- marized as follows: (a) The approximation functions or the shape functions do not include any rational terms, and as a result they are advanta- geous for an accurate integration of the stiffness matrix. (b) The shape functions have high-order continuities, i.e., C1 inside the elements, C1 on nodes and C0 on the element edges. It is also noted that some nodes, for instance the nodes located on the essential boundary or on the interface of bi-materials, are required for the CQ4 to recover to a C0 continuity. Hence a slight modiﬁcation may be made on those nodes [7]. The CFEM will be degenerated into the FEM if all the nodes in the problem domain are supposed to be C0 . (c) Since the consecutive-interpolation passing through nodal values, the shape functions are hence said to be satisﬁed the Kronecker-delta function property. (d) The consecutive-interpolation is said to be linear consistency as it can exactly reproduce any functions in the basis functions i.e., ∑ns i ¼ 1 ~ NiðxÞ ¼ 1; ∑ns i ¼ 1 ~ NiðxÞxi ¼ x; ∑ns i ¼ 1 ~ NiðxÞyi ¼ y. 3.4. Stiffness matrix and numerical integration Before describing the stiffness matrix implementation and the numerical integration of the proposed CQ4 element, it must be stated here that the approximation function of the displacements using the FEM in Eq. (4) and the CFEM in Eq. (26) has a similar form. The only difference between them is the way of the construction of their shape functions. Therefore, the discrete equations derived from the weak-form in Eq. (3) shall be the same for both the FEM and the CFEM. As a result the element stiffness matrix Ke can be ﬁnally expressed as [1–3]. Ke ¼ Z Ωe BT e DBe dΩ ð28Þ In Eq. (28), the domain Ωe is different between the FEM and the CFEM, which fully depends on the supporting nodes and neigh- boring elements determined by the CFEM procedure. The matrix of the derivative of the shape functions Be is also different between the FEM and the CFEM. Obviously, the difference can be depicted explicitly, for instance, let us consider a quadrilateral element with Fig. 3. Comparison of the shape functions (a) and their ﬁrst-order derivatives (b) in 1D classical and consecutive FEM. Visualization of the shape functions: Q4 (c) and CQ4 (d) elements. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 18
- 6. four-node presented in this study, the matrix Be obtained by the FEM method can be expressed as BFEM e ¼ ∂Ni ∂x ∂Nj ∂x ∂Nk ∂x ∂Nm ∂x 0 0 0 0 0 0 0 0 ∂Ni ∂y ∂Nj ∂y ∂Nk ∂y ∂Nm ∂y ∂Ni ∂y ∂Nj ∂y ∂Nk ∂y ∂Nm ∂y ∂Ni ∂x ∂Nj ∂x ∂Nk ∂x ∂Nm ∂x 2 6 6 6 4 3 7 7 7 5 ð38Þ ð29Þ and the same but by the CFEM BCFEM e ¼ ∂ ~ N1 ∂x ∂ ~ N2 ∂x ⋯ ∂ ~ Nl ∂x ⋯ ∂ ~ Nns ∂x 0 0 ⋯ 0 ⋯ 0 0 0 ⋯ 0 ⋯ 0 ∂ ~ N1 ∂y ∂ ~ N2 ∂y ⋯ ∂ ~ Nl ∂y ⋯ ∂ ~ Nns ∂y ∂ ~ N1 ∂y ∂ ~ N2 ∂y ⋯ ∂ ~ Nl ∂y ⋯ ∂ ~ Nns ∂y ∂ ~ N1 ∂x ∂ ~ N2 ∂x ⋯ ∂ ~ Nl ∂x ⋯ ∂ ~ Nns ∂x 2 6 6 6 6 4 3 7 7 7 7 5 ð32nsÞ ð30Þ In Eq. (30), ns is the number of the supporting nodes, 1olons and ns 44. It is because that the displacements in the CFEM are not only interpolated from the nodal displacements of the con- sidered element, but also interpolated from the supporting nodes determined by other neighboring elements, as sketched in Fig. 2. The size of the matrix BFEM e is of ð3 8Þ and that is much smaller than the size of the matrix BCFEM e , ð3 2nsÞ, which results in an increased bandwidth of the stiffness matrix of the CFEM. In the above equations the derivative of the CFEM shape functions is calculated by ∂ ~ Nl ∂z ¼ ∂ϕi ∂z N i ½ Š l þ ∂ϕix ∂z N i ½ Š l;x þ ∂ϕiy ∂z N i ½ Š l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} related to node i þ ∂ϕj ∂z N j ½ Š l þ ∂ϕjx ∂z N j ½ Š l;x þ ∂ϕjy ∂z N j ½ Š l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node j þ ∂ϕk ∂z N k ½ Š l þ ∂ϕkx ∂z N k ½ Š l;x þ ∂ϕky ∂z N k ½ Š l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node k þ ∂ϕm ∂z N m ½ Š l þ ∂ϕmx ∂z N m ½ Š l;x þ ∂ϕmy ∂z N m ½ Š l;y |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} node m ð31Þ with z ¼ x; y. In order to perform the numerical integration of the stiffness matrix of the CFEM method, we adopt a set of 3 3 Gaussian quadrature points for all the implementations throughout the study. It is because, in general, the implementation of the numerical integration in the CFEM is realized similar to that in the conventional FEM. None of any special techniques is required for the numerical integrations of the CQ4 element. Any quadrature rules used for the FEM can be applied the same for the CFEM, but it may obey the relation of the number of Gaussian points ngp with respect to the polynomial of order p as indicated in Ref. [2], i.e., pr2ngp 1. Nonetheless, the variation and the inﬂuence of the Gauss quadrature rules on the accuracy of the solutions shall be studied numerically in the numerical example part. This is accomplished by considering the ﬁrst six two-dimensional Gaus- sian quadrature rules from 1 1 to 6 6 respectively, and the two error indicators in the displacement and energy norms, as well as the maximum and minimum values of the von Mises stresses will be investigated accordingly. 4. Numerical examples In this section four benchmark numerical examples are exam- ined to show the accuracy, the efﬁciency and the convergence of the present CQ4 element. In addition, other two examples are considered as practical applications to illustrate the applicability of the proposed CQ4 element in dealing with complex geometries. The materials used for all the four benchmark examples are assumed to be linear elastic with Young's modulus E ¼ 1000 and Poisson's ratio ν ¼ 0:3, whereas they are speciﬁed thereafter for the two practical examples. The units used in the examples can be any consistent unit based on the international standard unit system, if not speciﬁed otherwise. To accomplish the convergence study, two error indicators with respect to the displacement and energy norms are deﬁned as follows: [19,20] ed ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ∑n i ¼ 1ðuexact i unumer i Þ2 ∑n i ¼ 1ðuexact i Þ2 v u u t ; ð32Þ ee ¼ 1 A ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 Z Ω ðεexact εnumerÞT Dðεexact εnumerÞdΩ s ; ð33Þ where the superscript “exact” represents the analytical solutions while “numer” stands for the numerical solutions, and A is the area of the problem domain. 4.1. Standard patch test The satisfaction of the standard patch test in general requires the displacement ﬁelds of all the interior nodes of the patch that follow “exactly” the same linear functions of the imposed dis- placements on its boundaries [20]. As a consequence a sufﬁcient condition for the convergence of a numerical method is to precisely pass such standard patch test. A 1 1 unit square domain of a patch test discretized by 4 4 regular and irregular quadrilateral elements with four-node as shown in Fig. 4 is studied using the CFEM. The displacements are prescribed on all boundaries by the following linear functions: ux ¼ x; Fig. 4. Domain discretization of a unit square patch test using the CQ4 elements: regular (a) and irregular (b) meshes. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 19
- 7. uy ¼ y ð34Þ The relative errors in the displacement norm ed are found to be 2:0761e 016 and 2:4855e 016 for the regular and irregular meshes, respectively. As a result, these error results indicate that the present CQ4 element passes the standard patch test successfully. 4.2. Cantilever beam A cantilever beam with length L ¼ 8 and height D ¼ 2 subjected to a parabolic traction on the right end as depicted in Fig. 5 is considered. The cantilever is assumed to have a unit thickness and the corresponding analytical solutions of the displacements and stresses based on the plane-stress condition are given by [21] ux ¼ Py 6EI ð6L 3xÞxþð2þνÞ y2 D2 4 ! # ; uy ¼ P 6EI 3νy2 ðL xÞþð4þ5νÞ D2 x 4 þð3L xÞx2 # ; ð35Þ sx ¼ P L x ð Þy I ; sy ¼ 0; τxy ¼ P 2I D2 4 y2 # ð36Þ where I ¼ D3 =12 is the moment of inertia of the cantilever beam. For the plane-strain problem, E and ν in Eqs. (35) and (36) are replaced by E=ð1 ν2 Þ and ν=ð1 νÞ, respectively, and P ¼ 2. To serve the validation purpose, four elements listed in the following are considered: (a) the CIP-based quadrilateral element with four-node: CQ4 (b) the CIP-based triangular element: CT3 [7] (c) the standard quadrilateral element with four-node: Q4 (d) the standard triangular element with three-node: T3 The convergence rate of all the aforementioned elements is analyzed using both the regular and distorted meshes. However, Fig. 5. Geometry of a cantilever beam subjected to a parabolic traction on the right end. Fig. 6. Three discretized regular (a) and distorted (b) meshes using quadrilateral elements for a cantilever beam. Fig. 7. Comparison of the convergence rates for a cantilever beam using the regular meshes obtained by the CQ4, CT3, Q4 and T3 elements: relative errors in displacement norm (a) and energy norm (b). T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 20
- 8. only the regular and irregular meshes of 9 5, 16 10 and 30 20 quadrilateral elements with four-node are depicted in Fig. 6, and the meshes discretized by using the triangular elements of the CT3 and T3 elements are omitted for the brevity. Fig. 7 illustrates the convergence rates in both the energy and displacement norms using the regular meshes with respect to the nodal ratio h (e.g., h ¼ 1=9, 1/16, 1/30, 1/60) obtained by the CQ4, Q4, CT3, and T3 elements. It reveals in the ﬁgures that the proposed CQ4 element outperforms the T3 and Q4 elements as well as the CT3 once the mesh is further reﬁned. In terms of the accuracy, the CFEM provides much better results than the FEM and the CQ4 element performs the best. Further study of the convergence rate is conducted using the distorted meshes, and their obtained con- vergence rates in both the displacement and energy norms are then sketched in Fig. 8. Similar convergence rates for both the regular and distorted meshes are gained and, not surprisingly, the regular mesh desirably yields higher accuracy than the irregular one, as usual. However, the obtained results indicate that the irregular mesh has only a little effect on the accuracy. To investigate the accuracy of the developed CQ4 element, the beam is discretized using both the regular and irregular meshes of 16 10 elements. The numerical results of the displacement in y- direction along the neutral line and the shear stress along the middle line are obtained using the CQ4 (regular and irregular meshes), CT3, Q4 and T3 elements, and plotted in Figs. 9 and 10, respectively. The ﬁgures exhibit that the displacement and stress components obtained by the CQ4 and CT3 are all in good agree- ment with the analytical ones, whereas less accuracy can be found for the Q4 and T3 elements. For a better representation, the normal and shear stress distributions (e.g., sx, τxy) obtained by the CQ4, Q4, T3 and SQ4 elements are depicted in Fig. 11. Obviously, the stresses achieved by the Q4 and T3 elements are discontinuous and non-smooth whilst the developed CQ4 element works well, i.e., the stresses are continuous and smooth. It should be recalled that the developed CFEM is very smooth though no post-processing is performed. In Fig. 11, the SQ4 element is the standard Q4 but its ﬁnal results (see Fig. 11d and h) are smoothed out by further applying a smoothing stress recovery technique in the post-processing step. These SQ4 results are motivated since it may be interesting to see how smooth on the stresses obtained by the present CFEM method and the one using the stress recovery technique as usually done in practice. By accomplishing that, we merely adopt one of the simplest stress recovery techniques that have been found most useful in practice, the averaged nodal stresses [22]. The realization of this so-called unweighted averaging is carried out by assigning the same weight to all elements that meet at a node. It is obvious that the SQ4 results are as smooth as the CQ4 ones. To check the ability of the present CFEM in treating the volume locking phenomenon in the incompressible materials (i.e., Poisson's ratio tends toward 0.5), the same problem but the beam Fig. 8. Comparison of the convergence rates for a cantilever beam using the regular and distorted meshes obtained by the present CQ4 element: relative errors in displacement norm (a) and energy norm (b). Fig. 9. Comparison of the deﬂections along the neutral line of a cantilever beam obtained by the analytical, regular and irregular CQ4, CT3, Q4 and T3 solutions. Fig. 10. Comparison of the shear stress distributions along the line (x¼L/2) of a cantilever beam obtained by the analytical, regular and irregular CQ4, CT3, Q4 and T3 solutions. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 21
- 9. is now set to be under plane-strain condition and Poison's ratio is taken as ν ¼ 0:49 to simulate a nearly incompressible material. Only the deﬂection along the neural line obtained by the four aforementioned methods is presented in Fig. 12 together with the analytical solution. Note that results of the pressure and its convergence, or issues related to pressure modes, inf–sup satisfac- tion, etc. are not studied and do not cover in this manuscript because of simplicity. However, they are scheduled as our future research works. We see in the picture that the CFEM performs well as compared with the exact deﬂection, and it shows higher Fig. 11. Comparison of the stress distributions in the cantilever beam obtained by the CQ4 (a, e); Q4 (b, f); T3 (c, g) and SQ4 (d, h) elements: normal stress (a)–(d) and shear stress (e)–(h). Fig. 12. Comparison of the deﬂections along the neutral line of a cantilever beam obtained by the analytical, regular and irregular CQ4, CT3, Q4 and T3 solutions with ν ¼ 0:49 for nearly incompressible materials. Fig. 13. Comparison of the computational time for a cantilever beam subjected to a parabolic traction at the free-end obtained by different approaches. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 22
- 10. accuracy than the FEM. More importantly, based on these pre- liminary results it may be stated that the nearly incompressible materials can be treated by the CFEM with high accuracy, and the volumetric locking is alleviated. Another important issue in relevance to the computational efﬁciency of the developed CQ4 element is also studied. The computational time needed for the T3, Q4, CT3 and CQ4 elements tested on three different meshes of 9 5, 16 10 and 30 20 elements, is investigated. Here, only the time required for the computation of the global stiffness matrix is measured and estimated. It is because that the most difference among the aforementioned methods is induced by the implementation of the stiffness matrix, which substantially pertains to the difference of establishing the shape functions and their derivatives. The comparison is performed on the same PC of Intel(R) Pentium(R) Dual-Core 2.6 GHz, 2.GB RAM. 5 calculations are carried out for each mesh and the averaged computational time is then reported. Fig. 13 shows the required CPU time of different approaches using the same direct solver. Because of an extra task of the consecutive- interpolation implementation of the shape functions and their derivatives, the CQ4 and CT3 elements obviously require more time than the standard Q4 and T3 ones. On the other hand, the computational efﬁciency in terms of the relative errors in both the displacement and energy norms against the computational time (in seconds) is compared and depicted in Fig. 14 under a log–log plot. It is evident that with respect to the computational efﬁciency (computational time for the same accuracy) the CFEM is more efﬁcient as clearly shown in the relative error results in compar- ison with the FEM. From the practical point of view, it should be noted that to achieve acceptable solutions by using the standard FEM, demanding post-processing is often required and it may be a time-consuming task. In fact, it is difﬁcult and/or even impossible to estimate how much time such task would take. Furthermore, as mentioned in Section 3.4, we now study numeri- cally the effect of the Gaussian quadrature rules on the numerical results of the beam. This investigation is carried out by considering the ﬁrst six two-dimensional Gaussian quadrature rules from 1 1 to 6 6, and the two error indicators in the displacement (ed) and energy (ee) norms as well as the maximum and minimum values (smax,smin) of the von Mises stresses are estimated and explored. Two regular meshes of 16 8 and 32 16 discretized for the beam are considered. The corresponding obtained results are tabulated in Table 1. We observe in the table that as usual high error and less accuracy can be found for the quadrature rule of 1 1, and the results are similar for other quadrature rules. It implies that higher number of the Gaussian quadrature rules does not inﬂuence too much on the accuracy of the results, but it may increase the computational time. As expected, the errors obtained by the ﬁner mesh are found to be smaller than that by the coarser one. As a result we decide to use a set of 3 3 Gaussian quadrature rule for all the implementations throughout the study. 4.3. An inﬁnite plate with a central circular hole Next benchmark example considers an inﬁnite plate with a central circular hole of radius a and subjected to a unidirectional tensile loading Tx ¼ 1 as depicted in Fig. 15a. Only one quarter of Fig. 14. Comparison of the computational efﬁciency in terms of displacement (a) and energy (b) error norms for a cantilever beam subjected to a parabolic traction at the free-end obtained by different approaches. Table 1 Variation and the effect of the Gaussian quadrature rules on the numerical results of the cantilever beam. Mesh 1 1 2 2 3 3 4 4 5 5 6 6 16 8 smax 24.0883 23.2061 23.1883 23.1863 23.1863 23.1863 smin 1.0232 0.8472 0.8562 0.8563 0.8563 0.8563 ed 0.0169 0.0021 0.0019 0.0019 0.0019 0.0019 ee 0.0188 0.0446 0.0434 0.0442 0.0442 0.0442 32 16 smax 23.818 23.5251 23.5174 23.5165 23.5165 23.5165 smin 0.5771 0.4586 0.4543 0.4543 0.4543 0.4543 ed 0.0042 0.00027 0.00025 0.00026 0.00026 0.00026 ee 0.0051 0.0161 0.0155 0.0158 0.0158 0.0158 T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 23
- 11. the plate (b ¼ 5; a ¼ 1) shown in Fig. 15b is modeled due to the two-fold symmetry. The analytical solutions of the displacement and stress ﬁelds of the inﬁnite plate are given by [21] ur ¼ Tx 4μ r ðκ 1Þ 2 þ cos ð2θÞ þ a2 r ½1þð1þκÞ cos ð2θÞŠ a4 r3 cos ð2θÞ ; uθ ¼ Tx 4μ ð1 κÞ a2 r r a4 r3 sin ð2θÞ; ð37Þ sx ¼ Tx 1 a2 r2 3 2 cos ð2θÞþ cos ð4θÞ þ 3a4 2r4 cos ð4θÞ ; Fig. 15. Geometry of an inﬁnite plate with a central circular hole (a) and its quarter model (b). Fig. 16. Comparison of the convergence rates for an inﬁnite plate with a circular hole obtained by the CQ4, CT3, Q4 and T3 elements: relative errors in displacement norm (a) and energy norm (b). Fig. 17. Comparison of the stress distributions along the left boundary (a) and the bottom boundary (b) of the quarter plate with a circular hole subjected to a unidirectional tension. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 24
- 12. sy ¼ Tx a2 r2 1 2 cos ð2θÞ cos ð4θÞ þ 3a4 2r4 cos ð4θÞ ; τxy ¼ Tx a2 r2 1 2 sin ð2θÞþ sin ð4θÞ 3a4 2r4 sin ð4θÞ ; ð38Þ where μ ¼ E=ð2ð1þνÞÞ and κ is deﬁned in terms of Poisson's ratio by κ ¼ 3 4ν for the plane-strain case and by κ ¼ ð3 νÞ=ð1þνÞ for the plane-stress problem. In Eqs. (37) and (38), ðr; θÞ are the polar coordinates and θ is measured counter-clockwise from the positive x-axis. The problem is ﬁrst studied under a plane-strain assump- tion, the traction boundary conditions are imposed on the right and upper edges with the analytical stresses obtained using Eq. (38), and the following essential boundary conditions are imposed: uxðx ¼ 0Þ ¼ 0; ð1ryr5Þ; Fig. 18. Comparison of the normal stress (e.g., sx) distributions in an inﬁnite plate with a central circular hole obtained by the CQ4 (a), CT3 (b), Q4 (c) and T3 (d) elements. Fig. 19. Comparison of the displacement solutions of an inﬁnite plate with a central circular hole for plane-strain condition with ν ¼ 0:4999 obtained by the CQ4, Q4 and analytical methods. Displacement distribution along the bottom boundary (a) and the left boundary (b) of the quarter plate. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 25
- 13. uyðy ¼ 0Þ ¼ 0; ð1rxr5Þ ð39Þ Similar to the previous beam example, the convergence study is presented. Fig. 16 essentially shows the convergence rate in both the energy and displacement norms using a regular mesh with respect to the nodal ratio h (e.g., h ¼ 1=7, 1/13, 1/25) for different approaches. The obtained results of the convergence rate are not surprising, and as expected the developed CQ4 outperforms the CT3, T3 and Q4 elements when a mesh reﬁnement is made. The CFEM provides much better results than the FEM as clearly seen in their relative errors in both the displacement and energy norms, and here again the present CQ4 is the winner. In terms of the accuracy, Fig. 17 further provides us a compar- ison of the stress distributions along the left and bottom Fig. 20. Geometry of internally pressurized hollow cylinder (a) and its quarter model. Fig. 21. Regular (a) and irregular (b) meshes of 12 12 quadrilateral elements. Fig. 22. Comparison of the convergence rates of an internally pressurized hollow cylinder hole obtained by the CQ4, CT3, Q4 and T3 elements: relative errors in displacement norm (a) and energy norm (b). T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 26
- 14. boundaries of the quarter plate obtained by the CQ4 element using a regular mesh of 13 13 elements, and by the analytical solu- tions. The CFEM matches well with the exact solutions. Addition- ally, Fig. 18 shows the normal stress distributions of an inﬁnite plate obtained, respectively, by the CQ4, CT3, Q4 and T3 elements. It is again observed that the stresses obtained by the CFEM are continuous and smooth whereas the standard FEM does not guarantee such smoothness and continuity. Similarly, the volumetric locking phenomenon is again ana- lyzed numerically using a regular mesh of 13 13 elements with Poisson's ratio taken to be ν ¼ 0:4999. Fig. 19 shows a comparison of the displacement distributions along the bottom (Fig. 19a) and left (Fig. 19b) boundaries of the quarter plate obtained by the present CQ4, the Q4 and the analytical approaches. It shows that the CFEM can still achieve better results with a Poisson's ratio ν ¼ 0:4999 while large errors are found for the standard Q4 element using the same mesh. 4.4. Hollow cylinder under internal pressure Another benchmark example considers a hollow cylinder sub- jected to an internal pressure as depicted in Fig. 20 to further show the accuracy of the CFEM. The cylinder is designed with an inner radius of a, an outer radius of b and a unit thickness. A uniform pressure of p is applied to the inner surface at r ¼ a, whilst traction-free boundary condition is assigned at the outer surface r ¼ b. Only one-quarter of the cylinder is modeled due to the geometrical symmetry of the structure. The analytical solutions of the displacement and stress ﬁelds of this internally pressurized hollow cylinder are available and given by [21]. urðrÞ ¼ a2 pr Eðb 2 a2Þ 1 νþ b 2 r2 ð1þνÞ # ; uθ ¼ 0; ð40Þ srðrÞ ¼ a2 p ðb 2 a2Þ 1 b 2 r2 # ; sθðrÞ ¼ a2 p ðb 2 a2Þ 1þ b 2 r2 # ; τrθ ¼ 0 ð41Þ In the numerical investigation, the following parameters are used: a ¼ 1, b ¼ 5, p ¼ 1, and the plane-stress condition is assumed. We ﬁrst explore the convergence rate of the different methods by considering three regular and three irregular meshes of 6 6, 12 12, 22 22 and 32 32 elements. Only the case of the regular and distorted meshes with 12 12 elements is shown in Fig. 21. Fig. 22 shows a comparison of the convergence rates in both the energy and displacement norms with respect to the nodal ratio h (e.g., h ¼ 1=7, 1/13, 1/23, 1/33) obtained by the CQ4, CT3, Q4 and T3 elements. Again and similar to the previous examples, the numerical results clearly conﬁrm the high accuracy and conver- gence rate of the present CFEM. For the accuracy study, the problem is discretized by using the regular and irregular meshes with 12 12 elements, and the obtained numerical solutions using several aforementioned meth- ods are plotted in Figs. 23 and 24 for the displacement and stress ﬁelds, respectively. The results obtained by the CFEM agree well with the analytical solutions, but the standard FEM yields less accuracy. The radial stress component sr derived by the four approaches is also depicted in Fig. 25 and one clearly observes in the ﬁgures that the CFEM provides much smoother stresses than the FEM using the same mesh. 4.5. Practical example: corner bracket The major objective of the subsequent numerical examples is to further demonstrate the applicability of the CFEM method to the problems of complex geometry. The ﬁrst practical example considers a corner angle bracket with its geometry as depicted in Fig. 26a. The corner bracket is popularly used in many engineering applications. The bracket in this study is made of steel with a Young's modulus of 206.84 GPa, Poisson's ratio of 0.27, and the plane-stress condition is assumed [23]. The upper left-hand pin hole is constrained around its entire circumference, and a tapered pressure load is applied to the bottom of the lower right-hand hole. The corner bracket is discretized with irregular meshes (see Fig. 26b) using the present CQ4, the Q4 elements, and ANSYS with the PLANE42 element. The results com- puted by the ANSYS are used here as a reference solution, and the bracket is discretized with different reﬁnements using the standard 4- node element starting from approximately 700 elements up to almost 16,000 elements. A maximum value of the von Mises stress of 19.705 MPa is obtained for a ﬁne mesh. On the other hand, the Fig. 23. Displacement distribution along the boundary line x¼0 of an internally pressurized hollow cylinder. Fig. 24. Stress distributions along the boundary line x¼0 of an internally pressurized hollow cylinder. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 27
- 15. bracket is also discretized by an irregular mesh of 1282 elements (see Fig. 26b) using quadrilateral elements. The maximum von Mises stresses are obtained, respectively, as 19.8817 MPa by the CQ4 and 20.2707 MPa by the Q4 elements. As compared with the ANSYS-based von Mises value, one can see that the calculated CQ4 solution is closer than that of the Q4 element for the same mesh. Fig. 27 depicts the distribution of the von Mises stresses in the bracket by the CQ4 (a), Q4 (b) and ANSYS1 using PLANE42 with course (c) and ﬁne (d) meshes. A very good agreement in the stress distributions obtained by different approaches is found. However, the CQ4 element again provides much smoother stresses than the Q4 element. 4.6. Practical example: rotor of a micro-motor Finally, a common micro-actuator in the form of a side-driving electrostatic micro-motor used in MEMs devices is analyzed [3]. Such micro-motor is usually made from polysilicon using litho- graphic techniques. The material parameters of the polysilicon are Young's modulus of 169 GPa and Poisson's ratio of 0.262. Isotropic material properties are employed to simplify the problem. A real model of the micro-rotor can be found in Ref. [3], and due to its Fig. 25. Comparison of the radial stress (e.g., sr) of an internally pressurized hollow cylinder obtained by the CQ4 (a), CT3 (b), Q4 (c) and T3 (d) elements. Fig. 26. Geometry of a corner angle bracket (a) and its ﬁnite element mesh (b). 1 Note that we use the ANSYS's command such as (PLESOL, S, EQV) to plot the stresses. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 28
- 16. Fig. 27. Comparison of the von Mises stress distributions in a corner bracket obtained by the CQ4 (a), Q4 (b), ANSYS with coarse (c) and ﬁne (d) meshes. Fig. 28. Geometry of a quarter micro-motor rotor (a) and ﬁnite element mesh with 96 quadrilateral elements. Fig. 29. Comparison of the von Mises stresses in a micro-motor rotor obtained by the CQ4 (a) and Q4 (b) elements. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 29
- 17. symmetry only one quarter with the corresponding boundary conditions as depicted in Fig. 28a is taken. Typically, a ﬁnite element mesh of the domain discretized by 96 quadrilateral elements with four-node is depicted in Fig. 28b. Similar to the bracket example, reference solutions are also derived from the FEM using ANSYS. Thus, different reﬁnements from a coarse mesh to a very ﬁne one are performed, and the maximum von Mises stress at the corners of the rotor obtained by ANSYS using a very ﬁne mesh is 18.832 MPa. The solutions computed by the CQ4 and Q4 elements, respectively, are 18.837 MPa and 22.599 MPa, and as a result it again reveals the high accuracy of the CFEM. Fig. 29 similarly shows the von Mises stress distributions in the rotor obtained by the CQ4 (a) and Q4 (b) elements, respectively, and smoother stresses are achieved by using the CFEM rather than the FEM for the same mesh. 5. Conclusions An efﬁcient and accurate quadrilateral element with four-node based on the CIP is formulated. The developed CQ4 element is applied to the stress analysis of 2D elastic structures. The advan- tages of the CFEM are due to the fact that the stresses and strains at nodes are continuous, smooth and it requires no smoothing operation in the post-processing step. From the numerical results of the four benchmark examples, it conﬁrms the high accuracy of the CFEM that pertains to high convergence rate as compared with the standard FEM (Q4, T3) and even the CT3. Highly accurate results in the practical examples further conﬁrm the applicability of the present CFEM in the stress analysis of 2D elastic structures with complex conﬁguration. The CIP algorithm is general and has no limitations on its extension to other elements such as tetra- hedral and brick elements in 3D. The extension of the method is also attractive to the incompressible media, in which some relevant issues pertaining to the pressure modes, convergence, the inf–sup satisfaction, etc. should be investigated and addressed in detail. From the present research work, the following conclu- sions can be drawn: The shape functions of the present CQ4 element are higher- order polynomials and possess the Kronecker-delta function property, which allows a straightforward imposition of the essential boundary conditions. As compared with the traditional FEM (Q4, T3) and even the CT3 element, the CQ4 substantially pertains to higher accuracy, higher efﬁciency and higher convergence rate. Specially, nodal stresses are generated continuously without any smoothing operation; mesh distortions are insensitive; based on the preliminary results it shows that the volumetric locking is alleviated in incompressible materials; and without the increase of the DOFs in the system. The present CFEM method is straightforwardly implemented in any existing computer FEM code. No post-processing is required in the CFEM. Acknowledgment The support of the German Academic Exchange Service (DAAD, Project-ID: 54368781) is gratefully acknowledged. Appendix A As stated above that the ϕi; ϕix; and ϕiy functions in Eq. (25) must satisfy the conditions in Eq. (23), hence we shall present it in this Appendix. Let us consider a quadrilateral element with four-node as sketched in Fig. 1. The geometric interpolation functions are given in Eq. (11). For convenience in representation, the derivatives of Eq. (11) are ∂ ∂r ∂ ∂s ( ) Li Lj Lk Lm h i ¼ 1 4 ð1 sÞ ð1 sÞ ð1þsÞ ð1þsÞ ð1 rÞ ð1þrÞ ð1þrÞ ð1 rÞ # ðA1Þ The Jacobian matrix and its inverse are explicitly expressed as J ¼ 1 4 ð1 sÞ ð1 sÞ ð1þsÞ ð1þsÞ ð1 rÞ ð1þrÞ ð1þrÞ ð1 rÞ # xi yi xj yj xk yk xm ym 8 : 9 = ; ðA2Þ J 1 ¼ 1 4detðJÞ J1 J2 J3 J4 # with J1 ¼ yið1 rÞ yjð1þrÞþykð1þrÞþymð1 rÞ; J2 ¼ yið1 sÞ yjð1 sÞ ykð1þsÞþymð1þsÞ; J3 ¼ xið1 rÞþxjð1þrÞ xkð1þrÞ xmð1 rÞ; J4 ¼ xið1 sÞþxjð1 sÞþxkð1þsÞ xmð1þsÞ ðA3Þ First, the derivatives of the geometric interpolation functions can be expressed as ∂ϕi ∂Li ¼ 1þ2LiLj þ2LiLk þ2LiLm L2 j L2 k L2 m; ∂ϕi ∂Lj ¼ L2 i 2LiLj; ∂ϕi ∂Lk ¼ L2 i 2LiLk; ∂ϕi ∂Lm ¼ L2 i 2LiLm ∂ϕix ∂Li ¼ ðxi xjÞð2LiLj þpLjLk þpLjLmÞ ðxi xkÞð2LiLk þpLkLm þpLkLjÞ ðxi xmÞð2LiLm þpLmLj þpLmLkÞ; ∂ϕix ∂Lj ¼ ðxi xjÞðL2 i þpLiLk þpLiLmÞ ðxi xkÞðpLiLkÞ ðxi xmÞðpLiLmÞ; ∂ϕix ∂Lk ¼ ðxi xjÞðpLiLjÞ ðxi xkÞðL2 i þpLiLm þpLiLjÞ ðxi xmÞðpLiLmÞ; ∂ϕix ∂Lm ¼ ðxi xjÞðpLiLjÞ ðxi xkÞðpLiLkÞ ðxi xmÞðL2 i þpLiLj þpLiLkÞ; ∂ϕiy ∂Li ¼ ðyi yjÞð2LiLj þpLjLk þpLjLmÞ ðyi ykÞð2LiLk þpLkLm þpLkLjÞ ðyi ymÞð2LiLm þpLmLj þpLmLkÞ; ∂ϕiy ∂Lj ¼ ðyi yjÞðL2 i þpLiLk þpLiLmÞ ðyi ykÞðpLiLkÞ ðyi ymÞðpLiLmÞ; ∂ϕiy ∂Lk ¼ ðyi yjÞðpLiLjÞ ðyi ykÞðL2 i þpLiLm þpLiLjÞ ðyi ymÞðpLiLmÞ; ∂ϕiy ∂Lm ¼ ðyi yjÞðpLiLjÞ ðyi ykÞðpLiLkÞ ðyi ymÞðL2 i þpLiLj þpLiLkÞ ðA4Þ Now, we prove the condition: ϕiðxlÞ ¼ δil. When l i, then r ¼ 1; s ¼ 1 and Li ¼ 1; Lj ¼ Lk ¼ Lm ¼ 0, substituting them into Eqs. (11) and (25), we obtain: ϕiðxiÞ ¼ 1. Similarly, when l j; l k; or l m, we respectively obtain ϕiðxjÞ ¼ 0; ϕiðxkÞ ¼ 0 and ϕi xm ð Þ ¼ 0. Next, we prove the conditions: ϕi;xðxlÞ ¼ 0, ϕi;yðxlÞ ¼ 0. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 30
- 18. When l i, then r ¼ 1; s ¼ 1 and Li ¼ 1; Lj ¼ Lk ¼ Lm ¼ 0. Substituting them into Eqs. (A1) and (A3), we have ∂ ∂r ∂ ∂s ( ) Li Lj Lk Lm h i ¼ 1 4 2 2 0 0 2 0 0 2 ; J 1 ¼ 1 4detðJÞ 2yi þ2ym 2yi 2yj 2xi 2xm 2xi þ2xj # ; ∂ ∂x ∂ ∂y ( ) Li Lj Lk Lm h i ¼ J 1 ∂ ∂r ∂ ∂s ( ) Li Lj Lk Lm h i ¼ 1 4detðJÞ 2yi þ2ym 2yi 2yj 2xi 2xm 2xi þ2xj # 1 4 2 2 0 0 2 0 0 2 ¼ 1 4detðJÞ yj ym ym yi 0 yi yj xm xj xi xm 0 xj xi # ðA5Þ and into the ﬁrst sub-equations of Eq. (A4) ∂ϕ ∂Li ¼ 1; ∂ϕ ∂Lj ¼ 1; ∂ϕ ∂Lk ¼ 1 and ∂ϕ ∂Lm ¼ 1 ðA6Þ Then, we ﬁnally obtain ϕi;xðxlÞ ¼ 0, ϕi;yðxlÞ ¼ 0 as ϕi;xðxiÞ ¼ ∂ϕi ∂x ¼ ∂ϕi ∂Li ∂ϕi ∂Lj ∂ϕi ∂Lk ∂ϕi ∂Lm h i ∂Li ∂x ∂Lj ∂x ∂Lk ∂x ∂Lm ∂x 8 : 9 = ; ¼ 1 1 1 1 1 4detðJÞ yj ym ym yi 0 yi yj 2 6 6 6 6 4 3 7 7 7 7 5 ¼ 1 4detðJÞ ½ðyj ymÞþðym yiÞþðyi yjÞŠ ¼ 0 ϕi;y xi ð Þ ¼ ∂ϕi ∂y ¼ ∂ϕi ∂Li ∂ϕi ∂Lj ∂ϕi ∂Lk ∂ϕi ∂Lm h i ∂Li ∂y ∂Lj ∂y ∂Lk ∂y ∂Lm ∂y 8 : 9 = ; ¼ 1 1 1 1 1 4detðJÞ xm xj xi xm 0 xj xi 2 6 6 6 6 4 3 7 7 7 7 5 ¼ 1 4detðJÞ ½ðxm xjÞþðxi xmÞþðxj xiÞŠ ¼ 0 ðA7Þ Similarly, we straightforwardly prove the same for l j; l k; or l m, as well as other conditions. References [1] Hughes TJR, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, 1987. [2] J. Fish, T. Belytschko, A First Course in Finite Elements, John Wiley Sons Ltd, England, 2007. [3] G.R. Liu, S.S. Quek, The Finite Element Method: A Practical Course, Butter- worth-Heinemann, Elsevier Science, Burlington, MA, 2003. [4] G. Beer, I. Smith, C. Duenser, The Boundary Element Method with Program- ming – For Engineers and Scientists, Springer-Verlag, Wien, Germany, 2008. [5] C.R. Dohrmann, M.W. Heinstein, J. Jung, S.W. Key, W.R. Witkowski, Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, Int. J. Numer. Methods Eng. 47 (2000) 1549–1568. [6] C.R. Dohrmann, S.W. Key, M.W. Heinstein, J. Jung, A least-square approach for uniform strain triangular and tetrahedral ﬁnite elements, Int. J. Numer. Methods Eng. 42 (1998) 1181–1197. [7] C. Zheng, S.C. Wu, X.H. Tang, J.H. Zhang, A novel twice-interpolation ﬁnite element method for solid mechanics problems, Acta Mech. Sin. 26 (2010) 265–278. [8] J. Bonet, A.J. Burton, A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications, Commun. Numer. Methods Eng. 14 (1998) 437–449. [9] P. Hansbo, A nonconforming rotated Q1 approximation on tetrahedral, Comput. Methods Appl. Mech. Eng. 200 (2011) 1311–1316. [10] P. Hansbo, Nonconforming rotated Q1 tetrahedral element with explicit time stepping for elastodynamics, Int. J. Numer. Methods Eng. 91 (2012) 1105–1114. [11] S.A. Papanicolopulos, A. Zervos, A method for creating a class of triangular C1 ﬁnite element, Int. J. Numer. Methods Eng. 89 (2012) 1437–1450. [12] G.R. Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, Int. J. Comput. Methods 05 (2008) 136–199. [13] G.R. Liu, N. Nourbakhshnia, Y.W. Zhang, A novel singular ES-FEM method for simulating singular stress ﬁeld near the crack tips for linear fracture problems, Eng. Fract. Mech. 78 (2011) 863–876. [14] P. Liu, Q.T. Bui, Yu TT Zhang Ch, G.R. Liu, M.V. Golub, The singular edge-based smoothed ﬁnite element method for stationary dynamic crack problems in 2D elastic solids, Comput. Methods Appl. Mech. Eng. 233–236 (2012) 68–80. [15] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, ﬁnite elements, NURBS, exact geometry and mesh reﬁnement, Comput. Methods Appl. Mech. Eng. 194 (2004) 4135–4195. [16] S.N. Atluri, T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22 (1998) 117–127. [17] Q.T. Bui, N.M. Nguyen, Ch. Zhang, An efﬁcient meshfree method for vibration analysis of laminated composite plates, Comput. Mech. 48 (2011) 175–193. [18] Q.T. Bui, N.M. Nguyen, Zhang Ch, D.A.K. Pham, An efﬁcient meshfree method for analysis of two-dimensional piezoelectric structures, Smart Mater. Struct. 20 (2011) 065016. [19] G.R. Liu, G.Y. Zhang, Y.Y. Wang, Z.H. Zhong, G.T. Li, X. Han, A nodal integration technique for meshfree radial point interpolation method (NI-RPIM), Int. J. Solids Struct. 44 (2007) 3840–3860. [20] G.R. Liu, G.Y. Zhang, A novel scheme of strain-constructed point interpolation method for static and dynamic mechanics problems, Int. J. Appl. Mech. 1 (2009) 233–258. [21] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970. [22] C.A. Felippa, Introduction to Finite Element Methods. University of Colorado, Boulder. Available from: 〈http://www.colorado.edu/engineering/CAS/courses. d/IFEM.d/Home.html〉 (accessed 15.11.13). [23] ASNYS User's Manual, Release 12.1, ANSYS, Inc, Technology Drive Connosburg, PA, 2009. T.Q. Bui et al. / Finite Elements in Analysis and Design 84 (2014) 14–31 31