DSpace at VNU: FREE AND FORCED VIBRATION ANALYSIS USING THE n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE ELEMENT METHOD (nCS-FEM)

2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 International Journal of Computational Methods Vol 10, No (2013) 1340008 (19 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876213400082 FREE AND FORCED VIBRATION ANALYSIS USING THE n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE ELEMENT METHOD (nCS-FEM) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only T NGUYEN-THOI∗,†, ,∗∗ , P PHUNG-VAN† , T RABCZUKĐ , H NGUYEN-XUAN, and C LE-VANả Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, VNU-HCM 227 Nguyen Van Cu, Dist Hochiminh City, Vietnam †Division of Computational Mechanics Ton Duc Thang University 98 Ngo Tat To St., War 19 Binh Thanh Dist., Hochiminh City, Vietnam §Institute of Structural Mechanics Bauhaus-University Weimar Marienstrasse 15, 99423, Weimar ¶Department of Civil Engineering, International University VNU-HCM, Vietnam thoitrung76@gmail.com Received 14 March 2011 Accepted 10 June 2011 Published 18 January 2013 A n-sided polygonal cell-based smoothed finite element method (nCS-FEM) was recently proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of twodimensional (2D) dynamic problems A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM Keywords: Numerical methods; finite element method (FEM); cell-based smoothed finite element method (CS-FEM); polygonal element; n-sided polygonal cell-based smoothed finite element method (nCS-FEM) Introduction In the front of development of novel numerical methods, by incorporating the strain smoothing technique of meshfree methods [Chen et al (2001)] into the FEM, Liu ∗∗ Corresponding author 1340008-1 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only T Nguyen-Thoi et al and Nguyen-Thoi et al (2010) have formulated a series of smoothed FEM (S-FEM) models named as cell-based S-FEM (CS-FEM) [Bordas et al (2009); Cui et al (2008); Dai and Liu (2007); Dai et al (2007); Liu et al (2007); Liu et al (2007); Liu et al (2009); Nguyen-Thanh et al (2008); Nguyen-Thoi et al (2007); Nguyen-Xuan et al (2008a, 2008b); Nguyen-Xuan et al (2008); Nguyen-Xuan and Nguyen-Thoi (2009); Nguyen-Van et al (2008)], node-based S-FEM (NS-FEM) [Nguyen-Xuan et al (2010); Liu et al (2009); Liu et al (2010); Nguyen-Thoi et al (2009); NguyenThoi et al (2009); Nguyen-Thoi et al (2010)], edge-based S-FEM (ES-FEM) [Liu et al (2009); Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009); Nguyen-Xuan et al (2009); Nguyen-Xuan et al (2009); Tran et al (2010)], face-based (FS-FEM) [Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009)], and alpha-FEM [Liu et al (2008)] that use linear interpolations In these S-FEM models, the finite element mesh is used similarly as in the standard FEM However, these S-FEM models evaluate the weak form based on smoothing domains created from the entities of the element mesh such as cells/elements, or nodes, or edges or faces These smoothing domains can be located inside the elements (CS-FEM) or cover parts of adjacent elements (NS-FEM, ES-FEM, and FS-FEM) These smoothing domains are linear independent and hence ensure stability and convergence of the S-FEM models They cover parts of adjacent elements, and therefore the number of supporting nodes in smoothing domains is larger than that in elements This leads to the bandwidth of stiffness matrix in the S-FEM models to increase and the computational cost is hence higher than those of the FEM However, also due to contributing of more supporting nodes in the smoothing domains, the S-FEM models often produce the solution that is much more accurate than that of the FEM Therefore in general, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM models are more efficient than the counterpart FEM models [Liu et al (2009); Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009)] It is clear that these S-FEM models have the features of both models [Rabczuk et al (2006)]: meshfree and FEM The element mesh is still used but the smoothed gradients bring the information beyond the concept of only one element in the FEM: they bring in the information from the neighboring elements In these S-FEM models, only the CS-FEM applies the strain smoothing technique for elements seperately, while the others S-FEM models apply this technique for two or more adjacent elements As a result, the others S-FEM models are closer to meshfree methods [Liu and Nguyen Thoi (2010); Liu et al (2010); Rabczuk et al (2004); Rabczuk and Belytschko (2005)], while the CS-FEM model is closer to the standard FEM and simpler than other S-FEM models The CS-FEM is first developed for the quadrilateral elements [Liu et al (2007); Liu et al (2007); Liu et al (2009); Nguyen-Xuan et al (2008a)] to analyze elastic solid mechanics problems [Liu et al (2007)], and free and forced vibration analysis [Dai and Liu (2007)] The CS-FEM is then further developed for n-sided polygonal elements to analyze the 1340008-2 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Free and Forced Vibration Analysis Using the nCS-FEM elastic solid mechanics problems [Dai et al (2007)] In Dai et al [2007], the stability condition of the nCS-FEM is examined and some criteria are provided to avoid the presence of spurious zero-energy modes An approach to constructing nCS-FEM shape functions are also suggested with emphasis on a novel and simple averaging method A selective integration scheme is recommended to overcome volumetric locking for nearly incompressible materials In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of two-dimensional (2D) dynamic problems A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM The paper is outlined as follows In Sec 2, the briefing on the nCS-FEM is presented including the proposal of lump mass matrix Some numerical examples are presented and examined in Sec and some concluding remarks are made in the Sec Brief of Dynamic Analysis of the nCS-FEM 2.1 Brief of the finite element method (FEM ) The discrete equations of the FEM are generated from the Galerkin weak form and the integration is performed on the basis of element as follows Ω (∇s δu)T D(∇s u)d uT (b ă u cu)d ˙ − Γt δuT ¯ tdΓ = 0, (1) where b is the vector of external body forces, D is a symmetric positive definite (SPD) matrix of material constants, ¯ t is the prescribed traction vector on the natural boundary Γt , u is trial functions, δu is test functions, ρ is the mass density, c is the damping coefficient and ∇s u is the symmetric gradient of the displacement field [Bathe et al (1996); Liu and Quek (2003); Hughes (1987); Zienkiewicz and Taylor (2000)] The FEM uses the following trial and test functions uh (x) = Nn NI (x)dI ; δuh (x) = I=1 Nn NI (x)δdI , (2) I=1 where Nn is the total number of nodes of the problem domain, dI is the nodal displacement vector and NI (x) is the shape function matrix of Ith node By substituting the approximations, uh and δuh , into the weak form and invoking the arbitrariness of virtual nodal displacements, Eq (1) yields the standard discretized system of algebraic equation: ¨ + Cd˙ + KFEM d = f , Md 1340008-3 (3) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 T Nguyen-Thoi et al where M is the mass matrix; C is the damping matrix; KFEM is the stiffness matrix and f is the element force vector that are assembled with entries of MIJ = Ω CIJ = Ω KFEM = IJ Ω Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only fI = Ω NTI ρNJ dΩ (4) NTI cNJ dΩ, (5) BTI DBJ dΩ, (6) NTI (x)bdΩ + Γt NTI (x)¯ tdΓ (7) with the strain gradient matrix defined as BI (x) = ∇s NI (x) (8) 2.2 Smoothed stiffness matrix in the nCS-FEM In the nCS-FEM [Dai et al (2007)], the problem domain is also discretized into Ne Ωe and Ne polygonal elements of arbitrary number of sides such that Ω = e=1 Ωi ∩ Ωj = However, we will replace the stiffness matrix KFEM in Eq (3) by ˜ by using strain smoothing technique [Chen et al the smoothed stiffness matrix K (2001)] based on the triangular smoothing cells created inside the polygonal elements In such a cell-based integration process, each n-sided polygonal element Ωe (k) (k) n is divided into n triangular smoothing cells Ωe such that Ωe = k=1 Ωe and (i) (j) Ωe ∩ Ωe = , i = j, by connecting n nodes of the element to the central point of the element as shown in Fig O : field nodes nSC =6 : added node to form the smoothing cells Fig Division of a six-sided polygonal element into six sub-triangles by connecting n field nodes with the central point O 1340008-4 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM Applying the cell-based smoothing operation, the compatible strains ε = ∇s u used in Eq (1) is used to create a smoothed strain on the triangular smoothing cell (k) Ωe : ˜(k) ε e = (k) Ωe ε(x)Φ(k) (x)dΩ = ∇s u(x)Φ(k) (x)dΩ, (k) Ωe (9) where Φ(k) (x) is a given smoothing function that satisfies at least unity property (k) Φ(k) (x)dΩ = (10) Ωe Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Using the following constant smoothing function (k) 1/Ae Φ(k) (x) = (k) x ∈ Ωe (11) (k) x∈ / Ωe (k) (k) where Ae is the area of the triangular smoothing cell Ωe , and applying a divergence theorem, one can obtain the smoothed strain ˜(k) ε e = (k) Ae (k) Γe n(k) e (x)u(x)dΓ, (k) (k) (12) (k) where Γe is the boundary of the smoothing cell Ωe , and ne (x) is the outward (k) normal vector matrix on the boundary Γe and has the following form for 2D problems  (k)  nex   (k)  n(k) (13) ney  e (x) =   (k) (k) ney nex In the nCS-FEM, the trial function uh (x) is the same as in Eq (2) of the FEM and therefore the force vector f in the nCS-FEM is calculated in the same way as in the FEM Substituting Eq (2) into Eq (12), the smoothed strain on the smoothing cell (k) Ωe can be written in the following matrix form of n nodal displacements of n-sided polygonal element Ωe ˜(k) ε e = n i=1 (k) ˜ where B i (k) Ωe , ˜ (k) di , B i (14) is termed as the smoothed strain gradient matrix on the smoothing cell ˜ (k) B i  (k) ˜b ix  =  ˜b(k) iy   ˜b(k)  iy  ˜b(k) ix 1340008-5 (15) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 T Nguyen-Thoi et al and it is calculated numerically using ˜b(k) = ih (k) (k) Ae (k) nh (x)Ni (x)dΓ, (k) Γe (h = x, y) (16) (k) When a linear compatible displacement field along the boundary Γe is used, one (k) Gaussian point is sufficient for line integration along each segment Γej of boundary (k) Γe (k) of Ωe , the above equation can be further simplified to its algebraic form ˜b(k) = ih M (k) (k) Ae j=1 (k) nhj Ni (xGP j )lj , (h = x, y), (17) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only (k) is the midpoint where M is the total number of the boundary segments of Γe , xGP j (k) (Gaussian point) of the boundary segment of Γej , whose length and outward unit (k) (k) normal are denoted as lj and nhj , respectively Equation (17) implies that only shape function values at some particular points (k) along segments of boundary Γej are needed and no derivatives of the shape function are required This gives tremendous freedom in shape function construction In this paper, the simple averaging method [Dai et al (2007); Liu and Nguyen Thoi (2010)] for constructing nCS-FEM shape functions is used ˜ of the system is then assembled by a similar The smoothed stiffness matrix K process as in the FEM N ˜ e, ˜ = Ae K K (18) e=1 ˜ e is the stiffness matrix of the element Ωe and where A is assembled operator and K is calculated by n n k=1 k=1 ˜ (k) ˜e = A K K e = A n ˜ TI DB ˜ TI DB ˜ J dΩ = A B ˜ J A(k) B e (k) k=1 Ωe (19) 2.3 Lump mass matrix in the nCS-FEM In dynamic analysis using the nCS-FEM, we can use the usual consistent mass matrix defined in Eq (4) to compute However, this computational process will be rather difficult and cumbersome due to the sub-division of polygonal elements into smoothing cells, and also due to the sub-division of the shape function of polygonal elements into linear piecewise shape functions on smoothing cells [Liu and Nguyen Thoi (2010)] In order to avoid such difficulty and to increase the computational efficiency, in this paper, we propose the well-known lumped mass matrix for the n-sided polygonal elements Ωe such as Nn Nn p=1 p=1 M = A Mp = A ρtAp Ip , (20) where Mp is the lump mass matrix of pth node; Ip is the identity matrix of size 2×2; Nn is the total number of nodes of the problem domain; Ap is the area surrounding 1340008-6 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Α Fig Area Ap of field node p in a mesh of n-sided polygonal elements the pth node and is created by connecting sequentially the mid-edge-point to the central points of the surrounding n-sided polygonal elements of the node pth as shown in Fig 2; ρ and t are the mass density and the thickness of the element, respectively Note that the diagonal form of lumped mass matrix gives the superiority in terms of computational efficiency over the consistent mass matrix in solving transient dynamics problems [Liu and Quek (2003)] 2.4 Shape functions of the nCS-FEM The general shape functions of n-sided polygonal elements in the nCS-FEM was presented by Liu and Nguyen-Thoi [2010] However, in actual computation of the nCS-FEM, it is not necessary to use such shape functions to compute the smoothed ˜ Instead, as shown in sub-section 2.2, we only need to evaluate stiffness matrix K the shape function values at Gauss points along boundary segments of triangu˜ This computational process is very simple and lar smoothing cells to compute K performed in three steps as follows: • Step 1: For each n-sided polygonal element, write explicitly the available shape function values at the field nodes • Step 2: Evaluate the shape function values at the central point of the n-sided polygonal element by averaging the shape function values of n field nodes at Step • Step 3: Evaluate the shape function values at Gauss points along boundary segments of triangular smoothing cells by linear interpolation from the available shape function values of n field nodes and central point 1340008-7 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only T Nguyen-Thoi et al Fig Positions of Gauss points at mid-segment-points on segments of six triangular smoothing cells in an six-sided polygonal element Figure and Table presents explicitly the shape function values at different points of a six-sided polygonal element divided into six triangular smoothing cells The number of support field nodes for this six-sided element is (from #1 to #6) We have a total of 12 segments (1–2, 2–3, 3–4, 4–5, 5–6, 6–1, 1–O, 2–O, 3–O, 4– O, 5–O, 6–O) Each segment needs only one Gauss point (due again to the linear interpolation) Therefore, there are a total of 12 Gauss points (from g1, to g12) to be used for all the smoothing cells, and the shape function values at all these 12 Gauss points can be tabulated in Table by simple inspection It should be mentioned that the purpose of introducing of central points such as point O in Fig is to facilitate the evaluation of the values of shape functions at some discrete points inside and on the segments of the interested element There is no extra degrees of freedom are associated with these added points In other words, these points carry no additional independent field variable Therefore, the total degrees of freedom (DOFs) of a nCS-FEM model will be exactly the same as the standard FEM using the same set of nodes 2.5 Dynamic analyses of the nCS-FEM By replacing the stiffness matrix K of the FEM in Eq (3) by the smoothed stiffness ˜ in Eq (18) of the nCS-FEM, the dynamic discretized system of equations matrix K in the nCS-FEM is expressed as a set of differential equations with respect to time ă + Cd + Kd = f Md 1340008-8 (21) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Table Values of shape functions at different points within an n-sided polygonal element (Fig 3) Point Node Node Node Node Node Node Description O g1 1.0 0 0 1/6 7/12 1.0 0 0 1/6 1/12 0 1.0 0 1/6 1/12 0 1.0 0 1/6 1/12 0 0 1.0 1/6 1/12 0 0 1.0 1/6 1/12 Field node Field node Field node Field node Field node Field node Centroid point (k) Gauss point (mid-segment point of Γej ) g2 1/2 1/2 0 0 Gauss point (mid-segment point of Γej ) g3 1/12 7/12 1/12 1/12 1/12 1/12 Gauss point (mid-segment point of Γej ) g4 1/2 1/2 0 Gauss point (mid-segment point of Γej ) g5 1/12 1/12 7/12 1/12 1/12 1/12 Gauss point (mid-segment point of Γej ) g6 0 1/2 1/2 0 Gauss point (mid-segment point of Γej ) g7 1/12 1/12 1/12 7/12 1/12 1/12 Gauss point (mid-segment point of Γej ) g8 0 1/2 1/2 Gauss point (mid-segment point of Γej ) g9 1/12 1/12 1/12 1/12 7/12 1/12 Gauss point (mid-segment point of Γej ) g10 0 0 1/2 1/2 Gauss point (mid-segment point of Γej ) g11 1/12 1/12 1/12 1/12 1/12 7/12 Gauss point (mid-segment point of Γej ) g12 1/2 0 0 1/2 Gauss point (mid-segment point of Γej ) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) For simplicity, the Rayleigh damping is used, and the damping matrix C is ˜ assumed to be a linear combination of M and K, ˜ C = αM + β K, (22) where α and β are the Rayleigh damping coefficients Many existing standard schemes can be used to solve the second-order time dependent problems, such as the Newmark method, Crank-Nicholson method, etc [Smith and Griffiths (1998)] In this paper, the Newmark method is used When the ă0 ), we aim to find a new state (d1 , d˙ , current state at t = t0 is known as (d0 , d , d ă d1 ) at t1 = t0 + θ∆t where 0.5 ≤ θ ≤ 1, using the following formulations: α+ + θ∆t α+ ˜ d1 = θ∆tf1 + (1 − θ)∆tf0 M + (β + θ∆t)K θ∆t ˜ Md0 + Md˙ + [β − (1 − θ)∆t] Kd θ 1−θ ˙ (d1 − d0 ) − d0 , d = t ă0 ă1 = (d˙ − d˙ ) − − θ d d θ∆t θ 1340008-9 (23) (24) (25) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 T Nguyen-Thoi et al Without the damping and forcing terms, Eq (21) is reduced to a homogenous dierential equation: ă + Kd ˜ = Md (26) A general solution of such a homogenous equation can be written as d = D exp(iωt), (27) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only where t indicates time, D is the amplitude of the sinusoidal displacements and ω is the angular frequency On its substitution into Eq (26), the natural frequency ω can be found by solving the following eigenvalue equation ˜ − ω M]D = [K (28) Numerical Examples In this section some examples will be analyzed to demonstrate the effectiveness and accuracy of the nCS-FEM The results of the nCS-FEM will be compared with analytic solutions and with those of FEM using triangular elements (FEMT3), FEM using quadrilateral elements (FEM-Q4) and FEM using 8-node elements (FEM-Q8) 3.1 Free vibration analysis of a rectangular cantilever beam In this example, a rectangular cantilever beam is studied The parameters used are length L = 100 mm, height H = 10 mm, thickness t = 1.0 mm, Young’s modulus E = 2.1 × 104 kgf/mm2 , Poisson’s ratio ν = 0.3, and mass density ρ = 8.0 × 10−10 kgf s2 /mm4 A plane stress problem is considered This problem has also been investigated in Dai and Liu [2007] Using the Euler–Bernoulli beam theory we obtain the fundamental frequency f1 = 0.08276 × 104 Hz that can serve as a reference Three types of meshes of triangular, quadrilateral and polygonal elements are used for comparison purpose as shown in Fig Because the exact solution is not available, numerical results using the FEM-Q4 with a very fine mesh (100 × 10) for the same problem are computed and used as the reference solutions Table lists the first nine natural frequencies of the beam, and the first nine free vibration modes using nCS-FEM are plotted in Fig It is observed that the nCS-FEM does not have any spurious nonzero energy and all the modes obtained corresponds to physical modes In addition, the convergence of the first natural frequency of the beam using FEM-T3, FEM-Q4, and nCS-FEM is shown in Fig It is shown that the nCS-FEM converges to the reference solution much faster than FEM-T3 and FEM-T4 Figure plots the first nine natural frequencies of the beam using FEM-T3, FEM-Q4, and nCS-FEM compared to those of reference solution Again, it is shown that the results of nCS-FEM are closer to the reference solution 1340008-10 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM (a) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only (b) (c) Fig (Color online) Three types of meshes of the cantilever beam; (a) A triangular mesh; (b) A quadrilateral mesh; (c) An n-sided polygonal mesh Table First nine natural frequencies (×104 Hz) of a cantilever beam FEM-T3 FEM-Q4 nCS-FEM Reference FEM-Q4 (100 × 10) Mesh: 30 × Nodes: 124 Elements: 180 triangles or 90 quadrilateral elements or 90 polygonal elements 0.102 0.609 1.586 2.852 4.046 4.315 5.903 6.747 0.089 0.534 1.410 2.568 3.932 4.040 5.442 6.729 0.089 0.532 1.394 2.513 3.801 3.841 5.213 6.382 0.082 0.494 1.302 2.366 3.609 3.844 4.967 6.396 Mesh : 50 × 306 Nodes 500 T3 elements 250 Q4 elements 250 polygonal elements 0.092 0.552 1.347 1.446 2.614 3.970 4.039 5.450 6.718 0.087 0.522 1.346 1.375 2.495 3.803 4.035 5.238 6.710 0.084 0.507 1.282 1.332 2.416 3.677 3.843 5.056 6.392 0.082 0.494 1.282 1.302 2.366 3.609 3.844 4.967 6.396 Mesh: 90 × Nodes: 910 Elements: 1,620 triangles or 810 quadrilateral elements or 810 polygonal elements 0.088 0.527 1.346 1.384 2.505 3.807 4.034 5.228 6.705 0.086 0.518 1.346 1.360 2.465 3.750 4.032 5.154 6.635 0.082 0.497 1.282 1.309 2.377 3.624 3.844 4.988 6.395 0.082 0.494 1.282 1.302 2.366 3.609 3.844 4.967 6.396 Model 1340008-11 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 T Nguyen-Thoi et al Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only a Model b Model c Model d Model e Model f Model g Model Fig (Color online) First nine modes of the cantilever beam by nCS-FEM 1340008-12 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM h Model Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only i Model Fig (Continued ) Fig (Color online) Convergence of the first natural frequency of the beam using FEM-T3, FEM-Q4 and nCS-FEM 3.2 Transient vibration analysis of a cantilever beam A benchmark problem of a cantilever beam is investigated using the nCS-FEM with the Newmark method for time stepping The beam is subjected to a tip harmonic loading f (t) = cos ωf t in y-direction Plane strain problem is considered with nondimensional parameters as L = 4.0, H = 1.0, t = 1.0, E = 1.0, v = 0.3, ρ = 1.0, α = 0.005, β = 0.272, ωf = 0.05, θ = 0.5 The domain of the beam is represented with 10 × polygonal elements Two FEM models of FEM-T3 (10 × triangular elments) and FEM-Q4 (10 × quadrilateral elements) are also used in the analysis for comparison purposes The time step for time integration is set at ∆t = 1.57 From the dynamic responses shown in Fig 8, it is seen that the amplitude of the 1340008-13 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only T Nguyen-Thoi et al Fig (Color online) Comparison of the first nine natural frequencies of the beam using FEM-T3, FEM-Q4 and nCS-FEM Fig (Color online) Transient responses for the cantilever beam subjected to a harmonic loading 1340008-14 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM P t R θ O Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Fig A spherical shell subjected to a concentrated loading at its apex Fig 10 (Color online) Domain discretization of half of the spherical shell using n-sided polygonal elements Fig 11 (Color online) Transient responses for the spherical shell subjected to a harmonic loading nCS-FEM-T3 is closer to that of the FEM-Q4 as compared to the FEM-T3 This shows that the nCS-FEM-T3 using n-sided polygonal elements can be applied to transient vibration analysis to deliver results of good accuracy 3.3 Transient vibration analysis of a spherical shell As shown in Fig 9, a spherical shell is studied that subjected to a concentrated timedependent loading at its apex Due to the symmetry, only half of the spherical shell 1340008-15 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only T Nguyen-Thoi et al Fig 12 (Color online) Transient responses obtained using the nCS-FEM for the spherical shell subjected to a Heaviside step loading is modeled with 70 × polygonal elements as shown in Fig 10 Nondimensional numerical parameters are used: R = 12, t = 0.1, φ = 10.9◦, θ = 0.5, E = 1.0, v = 0.3, and ρ = 1.0 The time-dependence of the loading is first specified in the harmonic form of f (t) = cos ωf t and its dynamic responses are plotted in Fig 11 for the case of ωf = 0.02, ωf = 0.05 and time step ∆t = Without damping, it is seen from Fig 11 that the deflection at apex approaches in an oscillatory fashion a constant value with the increase in time Next, a Heaviside step load f (t) = is added at apex since t = With an inclusion of damping (α = 0.005, β = 0.272), the response is damped out with time as expected in Fig 12 Conclusions A n-sided polygonal cell-based smoothed finite element method (nCS-FEM) was recently proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of 2D dynamic problems A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM 1340008-16 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340008 Free and Forced Vibration Analysis Using the nCS-FEM Acknowledgments This work was supported by Vietnam National Foundation for Science & Technology Development (NAFOSTED), Ministry of Science & Technology, under the basic research program (Project No.: 107.02.2010.01) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only References Bathe, K J [1996] Finite Element Procedures (MIT Press/Prentice Hall, Cambridge, MA, Englewood Cliffs, NJ) Bordas, S., Rabczuk, T., Nguyen-Xuan, H., Nguyen-Vinh, P., 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and forced vibration analysis [Dai and Liu (2007)] The CS-FEM is then further developed for n-sided polygonal elements to analyze the 1340008-2 2nd Reading