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Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 Contents lists available at ScienceDirect Comput Methods Appl Mech Engrg journal homepage: www.elsevier.com/locate/cma A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis Tinh Quoc Bui a,b,⇑, Minh Ngoc Nguyen c, Chuanzeng Zhang a a Chair of Structural Mechanics, Department of Civil Engineering, University of Siegen, Paul-Bontz-Strasse 9-11, D-57076, Siegen, Germany Department of Computational Mechanics, Faculty of Mathematics and Computer Science, University of Natural Science-National University of Ho Chi Minh City, Viet Nam c Institute of Computational Engineering, Department of Civil Engineering, Ruhr University Bochum, Germany b a r t i c l e i n f o Article history: Received 25 June 2010 Received in revised form 16 December 2010 Accepted 21 December 2010 Available online 25 December 2010 Keywords: Dynamic analysis Vibration Meshfree method Moving Kriging interpolation a b s t r a c t In this paper, a meshfree method based on the moving Kriging interpolation is further developed for free and forced vibration analyses of two-dimensional solids The shape function and its derivatives are essentially established through the moving Kriging interpolation technique Following this technique, by possessing the Kronecker delta property the method evidently makes it in a simple form and efficient in imposing the essential boundary conditions The governing elastodynamic equations are transformed into a standard weak formulation It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard implicit Newmark time integration scheme Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in details As a consequence, it is found that the method is very efficient and accurate for dynamic analysis compared with those of other conventional methods Ó 2010 Elsevier B.V All rights reserved Introduction The analysis of structural dynamics problems is of great importance in the field of structural mechanics and computational mechanics Generally, the dynamic analysis needs more efforts in modeling because of acting of many different conditions of complicated external loadings than the static one To find an exact solution to the class of dynamic problems usually is a hard way and in principle it could be reachable only with a simple loading condition and geometrical configuration Due to many requirements of engineering applications in reality, such a task of finding a solution analytically is generally difficult and often impossible Therefore, numerical computational methods emerge as an alternative way in finding an approximate solution The finite element method (FEM), e.g see [1,2], formed into that issue and becomes the most popular numerical tool for dealing with these problems The necessity of such numerical computational methods is nowadays unavoidable In the past two decades, the so-called meshfree or meshless methods, e.g see [3–5], have emerged alternatively, where a set of scattered ‘‘nodes’’ in the domain is used instead of a set of ‘‘ele⇑ Corresponding author at: Chair of Structural Mechanics, Department of Civil Engineering, University of Siegen, Paul-Bonatz Strasse 9-11, D-57076, Siegen, Germany Tel.: +49 2717402836; fax: +49 2717404074 E-mail addresses: bui-quoc@bauwesen.uni-siegen.de, tinh.buiquoc@gmail.com (T.Q Bui) 0045-7825/$ - see front matter Ó 2010 Elsevier B.V All rights reserved doi:10.1016/j.cma.2010.12.017 ments’’ or ‘‘mesh’’ as in the FEM No meshing is generally required in meshfree methods Note that the meshing here means different from the concept of background cells which are usually needed for performing the domain integrations There is another concept of ‘‘truly’’ meshfree or meshless methods, in which no meshing at all including the background cells for the domain integrations is required, e.g see [5–7] In particular, the last author has developed the meshless local Petrov-Galekin (MLPG) method for analysis of static, dynamic and crack problems of nonhomogeneous, orthotropic, functionally graded materials as well as Reissner–Mindlin and laminated plates [8–13] Recently, Belytschko et al [14] proposed and promoted by Moes et al [15] an effective method by substantially adding an enrichment function into the traditional finite element approximation function; the extended finite element method (X-FEM), which aims at modeling of the discontinuity The present work belongs to the meshfree scheme, and a novel meshfree method based on a combination of the classical elementfree Galerkin (EFG) method [3] and the moving Kriging (MK) interpolation is further developed for analysis of structural dynamics problems Previously, the present method has been developed by the first author for static analysis [16] and recently [17] for free vibration analysis of Kirchhoff plates The MK interpolation-based meshfree method was first introduced by Gu [18] and its application to solid and structural mechanics problems is still young and more potential Gu [18] successfully demonstrated its applicability for solving a simple problem of steady-state heat conduction Dai et al [19] reported a comparison between the radial point 1355 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 interpolation method (RPIM) and the Kriging interpolations for elasticity Lam et al [20] introduced an alternative approach, a Local Kriging (LoKriging) method to two-dimensional solid mechanics problems, where a local weak-form of the governing partial differential equations was appplied Li et al [21] further developed the LoKriging method for structural dynamics analysis Furthermore, Tongsuk et al [22,23] and Sayakoummane et al [24] recently illustrated the applicability of the method to investigations of solid mechanics problems and shell structures, respectively Imposing essential boundary conditions is a key issue in meshfree methods because of the lack of the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as in the FEM Thus, many special techniques have been proposed to avoid such difficulty by various ways e.g Lagrange multipliers [3], penalty method [4], or coupling with the FEM [24–26], etc Due to the possession of the Kronecker delta property, the present method is hence capable of getting rid of such drawback of enforcing the essential boundary conditions Note here that a majority of meshfree methods has been developed by displacement-based approaches and, in contrast, Duflot et al [27] and the first author have also implemented an equilibrium-based meshfree method for elastostatic problems where a stress-based approach is taken into consideration, see [28,29] With respect to the linear structural dynamics analysis in twodimension, a variety of studies has been reported so far Gu et al [30] successfully used the meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids, while in a similar manner Hua Li et al [21] developed the LoKriging, Dai and Liu [31] proposed the smoothed finite element method (SFEM), Gu and Liu [32] presented a meshfree weak-strong form (MWS) approach, and recently, Liu et al [33] and Nguyen-Thanh et al [34] implemented the edge-based smoothed finite element method (ES-FEM) and an alternative alpha finite element method (AaFEM), respectively As mentioned above, the proposed method has a significant advantage in the treatment of the boundary conditions, which is easier than the classical EFG This present work essentially makes use of that good feature to structural dynamics analysis At the standing point of view and to the best knowledge of the authors, such a task has not yet been carried out while this work is being reported The paper is organized as follows The moving Kriging shape function is introduced in the second section The governing equations and their discretization of elastodynamic problems will then be presented in Section In Section 4, numerical examples for free and forced vibration analyses are investigated and discussed in details Finally, some conclusions from this study are given in Section Moving Kriging shape function Essentially, the MK interpolation technique is similar to the MLS approximation In order to approximate the distribution function u(xi) within a sub-domain Xx # X, this function can be interpolated based on all nodal values xi(i [1, nc]) within the sub-domain, with n being the total number of the nodes in Xx The MK interpolation uh(x), "x Xx is frequently defined as follows [16,18,22,23] uh ðxÞ ẳ ẵpT xịA ỵ rT xịBuxị 1ị or in a shorter form of h u xị ẳ n X /I xịuI ẳ Uxịu 2ị uxn ị T and /I(x) is the MK shape m X pj xịAjI ỵ j n X r k ðxÞBkI ð3Þ k The matrixes A and B are determined by A ẳ PT R1 Pị1 PT R1 4ị B ẳ R I PAị ð5Þ where, I is an unit matrix and the vector p(x) is the polynomial with m basis functions pxị ẳ f p1 ðxÞ p2 ðxÞ Á Á Á pm ðxÞ gT ð6Þ The matrix P has a size n  m and represents the collected values of the polynomial basis functions (6) as p1 ðx1 Þ p2 ðx1 Þ Á Á Á pm ðx1 Þ 7 P¼6 p1 ðx2 Þ p2 ðx2 Þ Á Á Á pm ðx2 Þ ð7Þ p1 ðxn Þ p2 ðxn Þ Á Á Á pm ðxn Þ and r(x) in Eq (1) is rxị ẳ f Rx1 ; xị Rx2 ; xÞ Á Á Á Rðxn ; xÞ gT ð8Þ where R(xi,xj) is the correlation function between any pair of the n nodes xi and xj, and it is belong to the covariance of the field value u(x): R(xi,xj) = cov[u(xi)u(xj)] and R(xi, x) = cov[u(xi)u(x)] The correlation matrix R[R(xi, xj)]nÂn is explicitly given by Rðx2 ; x1 ị RẵRxi ; xj ị ẳ 6 Rðx1 ; x2 Þ Á Á Á Rðx1 ; xn Þ Á Á Á Rðx2 ; xn Þ 7 Rðxn ; x1 Þ Rðxn ; x2 Þ Á Á Á ð9Þ Many different correlation functions can be used for R but the Gaussian function with a correlation parameter h is often and widely used to best fit the model hr2ij Rxi ; xj ị ẳ e 10ị where rij = kxi À xjk, and h > is a correlation parameter As studied in the previous work by the author [16], the correlation parameter has a significant effect on the solution In this work, the quadratic bassis functions p T xị ẳ ẵ x y x2 y2 xy are used for all numerical computations Furthermore, the MK shape function in one-dimension and its first-order derivatives used in the dynamic analysis are presented in Fig One of the most important features in meshfree methods is the concept of the influence domain where an influence domain radius is defined to determine the number of scattered nodes within an interpolated domain of interest In fact, no exact rules can be derived appropriately to all types of nodal distributions The accuracy of the method depends on the number of nodes inside the support domain of the interest point Therefore, the size of the support domain should be chosen by analysts somehow to ensure the convergence of the considered problems It might also be found in the same manner as in [4,5] Often, the following formula is employed to compute the size of the support domain dm ¼ adc I where u ẳ ẵ ux1 ị ux2 ị function and dened by Uxị ẳ /I xị ẳ 11ị where dc is a characteristic length regarding the nodal spacing close to the point of interest, while a stands for a scaling factor Other features related to the method can be found in [16,18,22,23] for more details 1356 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 with u0 and v0 being the initial displacements and velocities at the initial time t0, respectively, and nj standing for the unit outward normal to the boundary C = Cu [ Ct By using the principle of virtual work, the variational formulation of the initial-boundary value problems of Eq (12) involving the inertial and damping forces can be written as [1,2,31] 0.8 Z φ(x) 0.6 deT rdX À Z X € À cud _ XÀ duT ½b À qu Z du T tdC ¼ In the meshfree method, the approximation (2) is utilized to calculate the displacements uh(x) for a typical point x The discretized form of Eq (14) using the meshfree procedure based on the approximation (2) can be written as 0.4 0.2 ỵ Cu_ ỵ Ku ¼ f Mu 0 0.2 0.4 0.6 0.8 x b φ’(x) ð15Þ where u is known as the vector of the general nodal displacements, M,C,K and f stand for the matrixes of mass, damping and stiffness and force vector, respectively They are defined as follows Z MIJ ¼ UTI qUJ dX X Z UTI cUJ dX CIJ ¼ X Z BTI DBJ dX KIJ ẳ Z X Z UTI bI dX ỵ UTI tI dC fI ẳ 16ị 17ị 18ị 19ị Ct X where c in Eq (17) is the damping coefficient, U is the MK shape function defined in Eq (3), the elastic matrix D and the displacement gradient matrix B in Eq (18) are given, respectively, by −2 m E D¼ 4m 5ðplane stressÞ À m2 0 ð1 À mÞ=2 −4 −6 −8 ð14Þ Ct X 0.2 0.4 0.6 0.8 Fig The MK shape function (a) and its first-order derivative (b) /I;x BI ¼ /I;y ð20Þ /I;y ð21Þ /I;x 3.2 Free vibration analysis Meshfree elastodynamic formulation For the free vibration analysis, the damping and the external forces are not taken into account in the system Then, Eq (15) can be reduced to a system of homogeneous equations as [1] 3.1 Discrete governing equations Let us consider a deformable body occupying a planar linear elastic domain X in a two-dimensional configuration bounded by C subjected to the body force bi acting on the domain The strong form of the initial-boundary value problems for small displacement elastodynamics with damping can be written in the form qui ỵ cu_ i ẳ rij;j ỵ bi in X 12ị where q stands for the mass density, c is the damping coefficient, üi and u_ i are accelerations and velocities, and rij specifies the stress tensor corresponding to the displacement field ui, respectively The corresponding boundary conditions are given as i ui ¼ u on the essential boundary Cu t i ¼ rij nj ¼ t i on the natural boundary Ct ð13aÞ ð13bÞ ỵ Ku ẳ Mu 22ị A general solution of such a homogeneous equation system can be written as expixtị uẳu 23ị is the eigenvector where i is the imaginary unit, t indicates time, u and x is natural frequency or eigenfrequency Substitution of Eq (23) into Eq (22) leads to the following eigenvalue equation for the natural frequency x ẳ0 K x2 Mịu 24ị The natural frequencies and their corresponding mode shapes of a structure are often referred to as the dynamic characteristics of the structure i and ti are the prescribed displacements and tractions, with u respectively, and the initial conditions are defined by 3.3 Forced vibration analysis ux; t ị ẳ u0 xị; _ ux; t ị ẳ v ðxÞ; For the forced vibration analysis, the approximation function Eq (2) is a function of both space and time For the displacements, x2X ð13cÞ x2X ð13dÞ 1357 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 velocities and accelerations at time t + Dt, the dynamic equilibrium equations or equations of motion presented in Eq (15) are also considered at time t + Dt as follows 1000 There are many different methods available to solve the secondorder time dependent problems such as Houbolt, Wilson, Newmark, Crank–Nicholson, etc [2,31] In this study, the Newmark time integration scheme is adopted to solve the equations of motion expressed in Eq (25) at time step t + Dt The Newmark scheme can be given in the form [1,2] tỵDt ¼ u 1 €t À1 u utỵDt ut ị u_ t bDt b Dt 2b tỵDt Dt u_ tỵDt ẳ u_ t ỵ ẵ1 cịut ỵ cu 26bị Numerical results In order to demonstrate the efficiency and the applicability of the present method to analysis of structural dynamics problems, some typical numerical examples are considered for free and forced vibrations and their dynamic responses are reported correspondingly 4.1 Free vibration analysis 4.1.1 Cantilever beam A cantilever beam as shown in Fig is first considered as a benchmark example To so, the non-dimensional parameters in the computation have the length L = 48 and height D = 12 The beam is assumed to have a unit thickness so that plane stress condition is valid Young’s modulus E = 3.0  107, Poisson’s ratio m = 0.3, and mass density q = 1.0 [16] are used As confirmed by static analysis in the previous work [16], two important parameters involving the correlation coefficient h and the scaling factor a related to the interpolation function expressed in Eqs (10) and (11), respectively; have certain influences on the numerical solutions Thus, they are of importance to the present method and may also have effects on the dynamic analysis in the present work This implies that the choice of these two parameters must be in carefulness, and the choice might be different from the static analysis Fig shows the computed results of the natural frequency of the beam compared to those of available reference y x L=48 Fig The geometry of the cantilever beam 800 600 400 ð26aÞ By substituting both Eqs (26a) and (26b) into Eq (25) one can obtain the dynamic responses at time t + Dt Since the Newmark time integration scheme is an implicit method, the initial conditions of € Þ are thus assumed to be known and the state at t ¼ t ðu0 ; u_ ; u € Þ is needed to be determined the new state at t ¼ t ỵ Dtu1 ; u_ ; u correspondingly In addition, the choice of c = 0.5 and b = 0.25, unconditionally guarantees the stability of the Newmark scheme with c P 0.5 and b P 0.25(c + 0.5)2 D=12 LoKriging [21] FEM [21] θ=0.004 θ=0.2 θ=1 θ=5 θ=30 =1000 25ị Natural frequency tỵDt ỵ Cu_ tỵDt ỵ KutỵDt ẳ f tỵDt Mu 1200 200 Mode No 10 Fig Natural frequency versus the correlation parameter h for the cantilever beam (a = 2.8) solutions, where the correlation parameter is varied in an interval of 0.004 h 1000 whilst a = 2.8 is fixed A regular set of 189 scattered nodes is taken in this example and its distribution will be seen later A comparison of the obtained results of the present method to that of the LoKriging [21] and the FEM (4850 DOFs) [21] is given in Table below It is found that a good agreement can be reached if 0.004 h < is chosen, it fails with h < 0.004, and other h values are though possible but the error increases and a bad result is unavoidable Additionally, the corresponding percentage error is then estimated and presented in Fig Note that the FEM (4850 DOFs) derived from [21] is used as a reference solution for the verification purpose Similarly, the influence of the scaling factor a to the selected quantity of scattered nodes within the influence domain is also studied in the same manner The results are plotted in Fig The correlation coefficient h = 0.2 is kept unchanged in the computation Definitely, a smaller error is obtained with a scaling factor 2.4 a 3.0 Table listing the first ten frequencies shows a comparison of natural frequencies among LoKriging [21], FEM (4850 DOFs) [21] and the present method, in which two scattered nodes of coarse and fine node distributions with 55 and 189 are considered for the cantilever beam associated with the chosen parameters of h = 0.2 and a = 3.0 An excellent agreement with other solutions can be found Furthermore, the first twenty eigenmodes of the cantilever beam are also provided in Fig The applicability of the method to irregularly scattered nodes is also given in Table 2, which shows a very good result compared to that of the FEM [21] To analyze the influences of the density of nodal distributions and the convergence of the natural frequencies versus the nodal densities, five regular nodal distributions with  5; 11  5; 15  9; 21  and 21  16 are additionally applied to the beam problem and four of them are illustrated in Fig The corresponding results of the non-dimensional frequencies are calculated individually for each set of scattered nodes and presented in Table in comparison with those obtained by LoKriging [21] and the FEM [21] It shows a very good convergence of the frequencies to the reference solutions even with a coarse set of 55 nodes 4.1.2 A shear wall with four openings The next numerical example dealing with a shear wall with four openings as shown in Fig is considered This example has been solved using several different computational methods such as BEM [35], MLPG [30], SFEM [31], ES-FEM [33], AaFEM [34], etc 1358 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 Table Comparison of the natural frequencies for different node distributions for the cantilever beam Mode FEM [21](4850 DOFa) 10 27.72 140.86 179.71 323.89 523.43 536.57 730.04 881.28 899.69 1000.22 a 55 Nodes 189 Nodes MWS [32] LoKriging [21] Present method MWS [32] (regular) LoKriging [21] Present method 26.7693 141.3830 179.7013 327.0243 527.3999 539.0598 730.1131 886.5635 896.9009 1004.7952 28.16 142.94 178.90 329.08 529.77 535.58 733.34 882.40 902.75 1001.55 27.952 143.943 179.874 334.562 537.394 548.201 776.301 884.231 929.177 1046.214 27.8370 141.1300 179.9077 323.8497 522.3307 537.1464 727.2628 881.5703 896.1059 997.7824 27.76 140.46 178.81 323.83 523.96 534.12 731.11 877.89 899.46 999.39 27.781 142.525 179.781 331.385 538.608 542.063 763.999 888.505 921.521 1028.855 DOF-Degree of freedom 1st (27.7403Hz) 10 −10 Error (%) 10 −10 −10 θ=0.004 θ=0.08 θ=0.2 θ=1.0 θ=5.0 θ=10.0 θ=30.0 θ=1000 −15 −20 −25 Mode No 10 α=2.4 α=2.8 α=3.0 α=3.2 α=4.0 α=6.0 α=10.0 50 7th (747.897Hz) 50 8th (887.0532Hz) 10 −10 50 50 50 10th (1010.478Hz) 11th (1085.3417Hz) 12th (1181.1089Hz) 10 −10 10 −10 50 50 50 50 13th (1251.4518Hz) 14th (1261.0004Hz) 15th (1315.9708Hz) 16th (1350.3932Hz) 14 10 4th (326.8239Hz) 10 −10 10 −10 10 −10 10 −10 10 −10 10 −10 10 −10 50 50 50 50 17th (1428.2654Hz) 18th (1452.5852Hz) 19th (1469.6634Hz) 20th (1519.1815Hz) 10 −10 12 10 −10 10 −10 10 −10 3rd (179.8042Hz) 50 6th (538.1866Hz) 50 9th (904.1602Hz) Fig Influence of correlation parameter h on the natural frequency (a = 2.8) Error (%) 10 −10 50 5th (532.2126Hz) −5 −30 2nd (141.2899Hz) 10 −10 50 10 −10 50 10 −10 50 50 Fig The first twenty eigenmodes of the cantilever beam by the present method Table A comparison of natural frequencies of the cantilever beam for both regular and irregular node distributions (h = 0.2, a = 2.8) Mode −2 Mode No 10 Fig Influence of scaling factor a on the natural frequency of the beam (h = 0.2) The geometrical parameters of the shear wall can be found in Fig and other relevant material parameters are taken exactly the same as in [30,31] with E = 10000, m = 0.2, t = 1.0 and q = 1.0 in a plane stress state The first eight natural frequencies are given in Table 4 10 FEM [21] (4850 DOFs) 27.72 140.86 179.71 323.89 523.43 536.57 730.04 881.28 899.69 1000.22 Present method 200 Nodes 325 Nodes Irregular Regular Irregular Regular 27.788 142.050 179.755 328.248 533.935 537.464 748.069 884.706 913.835 1018.345 27.752 139.152 178.403 322.164 515.272 535.144 728.552 881.552 922.056 995.822 27.711 138.977 178.957 319.114 530.419 529.225 729.687 879.388 907.737 1009.411 27.720 139.515 179.300 318.097 518.802 530.712 742.449 889.150 913.706 1080.830 Two sets of 261 and 556 scattered nodes are used, as well as h = 0.2 and a = 3.0 are specified in the computation As a consequence, it is found that the present solutions are in a good agreement with the 1359 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 a 20 1.8 18 3.0 16 −2 −4 14 1.8 −6 b 10 20 30 40 50 12 3.0 10 1.8 −2 −4 3.0 −6 10 20 30 40 50 1.8 c 21 3.0 0 −2 3.0 3.0 4.8 10 12 −4 Fig A shear wall with four openings with 556 scattered nodes −6 d 10 20 30 40 50 −2 −4 −6 10 20 30 40 50 Fig Various regular nodal distributions:  (a), 11  (b), 21  9(c) and 21  16 (d) chosen The beam is considered to be in plane stress condition with parameters E =  107, m = 0.3, mass density q = 1.0, and the thickness t = 1.0, respectively A regular set of 189 scattered nodes is used for all implementations of the forced vibration analysis Three main kinds of dynamic loadings depicted in Fig 11 as harmonic loading, Heaviside step loading, and transient loading with a finite decreasing time are analyzed associated with a traction at the free end of the beam by P = 1000  g(t), where g(t) is the time-dependent function The implicit Newmark time integration scheme is applied The vertical displacement or deflection at point A as depicted in Fig 10 is computed, and the detailed results obtained by the present method are then compared either to those of the ANSYS FEM software or other available solutions ones obtained by BEM, FEM and MLPG Additionally, the first twelve eigenmodes are also presented in Fig for the shear wall 4.2.1 Harmonic loading The loading in this case is shown in Fig 10(a) with the loading function g(t) given by 4.2 Forced vibration analysis gtị ẳ sin xf t Regarding the analysis of the forced vibration, a benchmark cantilever beam in two-dimensional setting shown in Fig 10 is in which xf is the forced frequency of the traction loading P, and xf = 27 is used in the computation in this example Fig 12 shows ð27Þ Table Convergence of the natural frequencies with various nodal densities of the cantilever beam Mode 5Â5 11  15  21  21  16 LoKriging [21] FEM [21] (4850DOF) 10 31.384 149.236 162.053 314.614 326.115 365.320 551.751 1112.647 1209.457 1396.263 27.952 143.943 179.874 334.562 537.394 548.201 776.301 894.231 929.177 1046.214 27.648 142.474 179.653 331.419 538.284 545.519 765.665 889.478 925.343 1032.822 27.781 142.525 179.781 331.385 538.608 542.063 763.999 888.505 921.521 1028.855 27.725 141.770 179.433 324.355 524.267 537.306 738.306 884.180 899.173 1002.132 27.76 140.46 178.81 323.83 523.96 534.12 731.11 877.89 899.46 999.39 27.72 140.86 179.71 323.89 523.43 536.57 730.04 881.28 899.69 1000.22 1360 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 Table Comparison of the first eight natural frequencies of a shear wall with four openings among different methods Mode MLPG[30] FEM[30] Brebbia et al.[35] 2.069 7.154 7.742 12.163 15.587 18.731 20.573 23.081 2.073 7.096 7.625 11.938 15.341 18.345 19.876 22.210 2.079 7.181 7.644 11.833 15.947 18.644 20.268 22.765 1st (2.1176rad/s) Present method 261 Nodes 556 Nodes 2.180 7.385 7.631 13.008 16.088 18.475 20.059 22.438 2.117 7.199 7.616 12.317 15.633 18.014 19.841 22.241 2nd (7.1992rad/s) 3rd (7.6161rad/s) 20 10 10 10 −10 10 20 −10 4th (12.3174rad/s) 20 10 10 10 20 20 10 20 10 20 −10 10 20 −10 11th (25.3297rad/s) 20 10 10 10 10 20 10 20 −10 10 20 10 20 12th (26.3196rad/s) 20 −10 20 9th (23.2189rad/s) 20 10 10 −10 10th (23.553rad/s) 6th (18.0144rad/s) 8th (22.2409rad/s) 10 −10 −10 10 −10 7th (19.8407rad/s) 10 10 5th (15.6329rad/s) 20 −10 0 −10 10 20 Fig The first twelve eigenmodes for the shear wall with four openings by the present method Fig 10 A cantilever beam subjected to a tip uniform traction the computed results of the vertical displacement uy of point A by different time-steps Obviously, the results obtained including for large time-steps are very stable compared with that of the FEM Additionally, Fig 13 illustrates the influence of the correlation parameter h on the displacement uy of point A, and it shows that an acceptable result can be yielded even though the h value takes up to 500 Various different time-steps are then applied for this loading case and the results of the computed displacement uy at point A are presented in Fig 14 It can be found that all considered time-steps could give stable results and they have a good agreement with the one obtained by the FEM except for Dt =  10À2s This implies that the accuracy of the present method will be decreased if the time-step is taken too large, which is known as the numerical damping effect Similar to the free vibration analysis, the effect of the densities of the nodal distributions on the dynamic response is also investigated here numerically and presented in Fig 15 for five different nodal distributions, which are the same as used in the free vibration analysis of the cantilever beam Compared with the FEM solution, it shows that a large error may occur when a very coarse set of  nodes is taken whereas all other nodal densities can yield good agreements even with a coarse set of 11  nodes Many time-steps are further studied to check the stability of the method involving damping effect by comparing the displacement uy obtained at point A versus the forced frequency of the dynamic loading xf The numerical results till to 20s are plotted in Fig 16 In this figure, the computed responses with xf = 18 on the left and with xf = 27 on the right are given A damping coefficient c = 0.4 is selected for the two cases and the time-step is taken as Dt =  10À3s In fact, this example has also been investigated by Li et al in [21], and the same conclusion is found here by comparing Fig 16(a) and (b) That is, the amplitude in the case of xf = 18 is smaller than about six times of that with xf = 27, because xf = 27 is close to the first natural frequency of the beam and a resonance is hence occurred The present results are very stable compared with the results in [21,30] Without damping, i.e c = 0, the present method can also give stable dynamic response with many timesteps as shown in Fig 17 When using a time integration technique to elastodynamic analysis, dissipation error representing the amplitude decay is known as a critical issue in measuring of the accuracy of the method As concluded in [1,37,38], the standard Newmark method with c = 1/2 leads to no numerical dissipation, whereas with values of c > 1/2 it gives rise to numerical dissipation In the present study, the valuec = 1/2 in conjunction with b = 1/4 is chosen to eliminate the numerical dissipation for the case of constant average acceleration By using these two values i.e.c = 1/2, b = 1/4, the method is always stable To verify the increase of the numerical dissipation for values of c > 1/2, the harmonic loading condition of the cantilever beam for several specific values of c > 1/2 such as 0.5; 0.8; 1.5; 2.0 and 10.0 are considered, respectively The corresponding results are given in Fig 18 for a typical peak in comparison with the one obtained by the FEM (ANSYS) It is evident that the amplitudes decay gradually when the c parameter is increasing, especially a very large error is observed with c = 10.0 The dispersion error is related to the nodal distributions (or mesh density in FEM) and the time-step used in the time integration technique Generally speaking, a straightforward way of reducing the dispersion error is to use a finer mesh in FEM and smaller time-step size [38] This issue is investigated numerically and the corresponding results are presented in Figs 14 and 15 for various time-step sizes and for various nodal distributions, respectively Consequently, it can be concluded that a sufficiently small time-step and a sufficiently small nodal density can yield very good agreement between the results obtained by the present method and the FEM, whereas a large dispersion error can be found for a large time-step as well as a coarse nodal distribution It is worth noting that such a refinement in time-step or nodal distribution always results in more computational effort and thus a more efficient technique for reducing the dispersion errors is desirable More details on dissipation and dispersion errors arising in structural dynamics by using numerical time-integration techniques can be found in [1,37–44] 1361 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 Fig 11 Schematic diagram of dynamic loadings: (a) harmonic loading, (b) Heaviside step loading and (c) transient loading with a finite decreasing time 0.2 0.025 −3 FEM(ANSYS) (Δt=1x10 ) 0.15 Δt=1x10−3 0.02 Δt=1x10−4 0.1 −2 Δt=1x10 Displacement uy Displacement u y 0.015 0.01 0.005 0.05 −0.05 −3 FEM(ANSYS) (Δt=1x10 ) −0.1 −3 Δt=1x10 −0.005 Δt=1x10−4 −0.15 −2 Δt=1x10 −0.01 Δt=5x10−2 −0.2 −0.015 0.05 0.1 Time t 0.15 0.2 Fig 12 Displacement uy at point A using Newmark time integration scheme (c = 0.5, b = 0.25) and the scaling and correlation parameters a = 2.8 and h = 0.2 for time-harmonic loading 1.5 0.2 FEM(ANSYS) θ = 0.1 θ=1 θ=5 θ = 10 θ = 50 θ = 100 θ = 500 θ = 1000 0.015 0.01 0.005 0.15 0.1 Displacement uy 0.02 y Time t Fig 14 Displacement uy at point A with various time-steps using Newmark method (c = 0.5, b = 0.25) and a = 2.8 and h = 0.2 for time-harmonic loading 0.025 Displacement u 0.5 0.05 −0.05 −0.005 −0.1 −0.01 −0.15 −0.015 FEM (ANSYS) 5x5 11x5 15x9 21x9 21x16 −0.2 0.05 0.1 Time t 0.15 0.2 Fig 13 Influence of the correlation parameter h on the displacement uy at point A using Newmark method (c = 0.5, b = 0.25) and a = 2.8 for time-harmonic loading 4.2.2 Heaviside step loadings In this section, three different types of dynamic Heaviside step loadings are examined The scaling factor and the correlation parameter are taken as a = 2.8 and h = 0.2 in all the computations 0.5 Time t 1.5 Fig 15 Displacement uy at point A with various nodal densities using Newmark method (c = 0.5, b = 0.25), Dt =  10À3 and a = 2.8 and h = 0.2 for time-harmonic loading 4.2.2.1 Heaviside step loading with an infinite duration When the loading function is specified as gtị ẳ Htị 28ị 1362 a T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 0.025 FEM (ANSYS) γ =0.5 γ =0.8 γ =1.0 γ =1.5 γ =2.0 γ =10.0 0.1 −3 Δt=5x10 , ω = 18 0.02 0.08 0.015 Displacement u Displacement u y y 0.01 0.005 −0.005 0.06 0.04 0.02 −0.01 −0.015 −0.02 −0.02 −0.025 b 10 Time t 15 20 0.85 0.9 0.95 Time t Fig 18 Dissipation verification at point A without damping (c = 0) for c = 0.5; 0.8; 1.0; 1.5; 2.0 and 10.0 (b = 0.25) using Newmark method for time-harmonic loading with D t =  10À3 0.25 Δt=5x10−3, ω = 27 0.2 0.8 0.15 0.05 −0.002 −0.004 −0.05 −0.006 Displacement u y Displacement u y 0.1 −0.1 −0.15 −0.2 −0.25 −0.01 −0.012 10 Time t 15 20 −0.014 Fig 16 Response at point A for different loading frequency using Newmark method (c = 0.5,b = 0.25), Dt =  10À3 and c = 0.4 for time-harmonic loading.x = 18 (a) and x = 27(b) −3 Δt=4x10 , c=0 −0.016 −0.018 0.5 Time t 1.5 Fig 19 Response at point A under Heaviside step loading with an infinite duration and without damping 0.3 0.2 Displacement uy −0.008 0.1 −0.1 −0.2 −0.3 10 Time t 15 20 Fig 17 Response at point A without damping (c = 0) using Newmark method (c = 0.5, b = 0.25) and Dt =  10À3 for time-harmonic loading as depicted in Fig 11(b), the dynamic loading is often referred to as impact loading and this type of dynamic analysis is often defined as dynamic relaxation [21,32] The dynamic relaxation implies that the loading will keep unchanged once a constant loading is suddenly applied to the structure The static analytical solution at point A is uy,exact = À0.0089 [36] The results with and without damping specified by c = and c = 0.4, respectively, are plotted in Figs 19 and 20 It is evidently found that the response will become a steady state harmonic vibration with the static deformation of the beam as the equilibrium position if the damping is neglected, and the response converges to the static deformation once a damping is introduced Both obtained results are obviously very stable and have an excellent agreement with those computed by different methods such as MWS [32] and LoKriging [21] In addition, the results for a damping with c = 0.4 are listed in Table for several time-steps at t % 50s Fig 20 conforms that the response converges to the static solution uy = À0.00888811 The computed percentage errors compared to the exact solution for both the LoKriging [21] and the present methods are also estimated in Fig 21 This result implies that the present method gives a remarkable convergence with a smaller error about 0.1% compared to that of about 0.6% obtained by the LoKriging [21] As a consequence, it is hence demonstrated that the present method works very well and accurate for the forced vibration analysis 1363 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 0.9 Δt=4x10 ,c=0.4 −0.002 0.8 −0.004 0.7 −0.006 0.6 y −0.008 Error (%) Displacement u Present LoKriging −3 −0.01 −0.012 u = −0.008888112425049 y 0.5 0.4 0.3 −0.014 0.2 −0.016 0.1 −0.018 10 15 20 Time t 25 30 35 40 Fig 20 Response at point A under Heaviside step loading with an infinite duration and with damping 47 47.5 Time (s) Displacement uy LoKriging [21] Present method 11750 11875 12000 12125 12250 12375 12500 0.470000E+02 0.475000E+02 0.480000E+02 0.485000E+02 0.490000E+02 0.495000E+02 0.500000E+02 À0.00883283 À0.00883255 À0.00883264 À0.00882592 À0.00883220 À0.00884123 À0.00884174 À0.008888400762486 À0.008889026243629 À0.008890959168876 À0.008885829057817 À0.008888068307071 À0.008889391269139 À0.008888348924774 48.5 Time t 49 49.5 50 Fig 21 A comparison of the percentage errors between the LoKriging [21] and the present methods with damping Table Computed results at several time-steps (about t % 50s) under dynamic Heaviside step loading with an infinite duration Number of time-steps 48 g(t) 1.0 t=0.5s t(s) Fig 22 The dynamic Heaviside step loading with a finite duration Exact: uy = À0.0089 [36] 4.2.2.2 Heaviside step loading with a finite duration The dynamic Heaviside rectangular step loading with a finite duration has a similar form as the dynamic impact loading considered previously but suddenly vanished at time t = 0.5 s as depicted in Fig 22 This can be considered as a special case of the dynamic relaxation, and the loading function is given by The results computed for both c = and c = 0.4 are compared to that of the FEM (ANSYS) and given in Figs 23 and 24, respectively Several different time-steps are chosen for c = 0, and the results show again that the accuracy of the present method decreases for large time-steps, while in other cases the results fit well with the FEM’s solution Alternatively, these results with and without damping can be compared with the results of Gu et al [30] and a good agreement could be found It is worth noting that after 0.5 s the response oscillates around zero as an equilibrium position because of the vanishing loading on the system 4.2.2.3 Heaviside step loading with a finite rise time Furthermore, we consider the dynamic Heaviside step loading with a finite rise time as depicted in Fig 25 In this case, the loading function is defined by t t gtị ẳ Htị Hðt À t Þ t0 t0 0.005 y 29ị 0.01 Displacement u gtị ẳ Htị Ht 0:5Þ 0.015 −0.005 −0.01 −3 FEM (ANSYS)(Δt=5x10 ) Δt=1x10−4 −0.015 Δt=1x10−3 Δt=5x10−2 −0.02 0.5 Time t 1.5 Fig 23 Transient displacement uy at point A without damping under Heaviside step loading with a finite duration and 27 that the displacement response always converges to the static deformation given by the static analytical solution ð30Þ The corresponding numerical results are presented in Figs 26 and 27, respectively As can be seen from the results shown in Figs 20 4.2.3 Transient loading with a finite decreasing time For the transient loading with a finite decreasing time as depicted in Fig 11c, the loading function is determined by 1364 T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 0.015 −0.001 −3 Δt=5x10 ,c=0.4 0.01 −3 Δt=1x10 ,c=0.4 −0.002 g(t) 0.005 y Displacement u Displacement u y −0.003 −0.005 1.0 −0.004 −0.005 −0.006 t t(s) −0.007 −0.01 −0.008 −0.015 −0.009 −0.02 10 Time t 15 −0.01 20 Fig 24 Transient displacement uy at point A with damping for many time-steps under Heaviside step loading with a finite duration 10 Time t 15 20 Fig 27 Response at point A of the beam under Heaviside step loading with a finite rise time and with damping 0.01 g(t) 0.005 1.0 t0 Displacement u y t(s) Fig 25 The dynamic Heaviside step loading with a finite rise time −0.01 −0.015 Δt=4x10−3,c=0 −3 Δt=1x10 ,c=0 −0.001 −0.002 −0.02 g(t) −0.003 1.0 0.5 1.5 Time t 2.5 3.5 Fig 28 Transient displacement uy at point A without damping under transient loading with a finite decrease time This result can be compared to that of the MSW method in [32] y Displacement u −0.005 −0.004 −0.005 t t(s) −0.006 0.01 −0.007 −0.008 0.005 −0.009 Time t Fig 26 Response at point A of the beam under Heaviside step loading with a finite rise time and without damping y 0 Displacement u 0.01 0.005 0.01 gtị ẳ tịẵHtị Ht 1ị 31ị 0.015 The corresponding results with and without damping are provided in Figs 28–30 by using different time-steps, respectively The different time-steps are selected in such a way that the corresponding numerical results can be compared with others Because the input values of the problem are set up the same as that of [21,32], thus the responses obtained in Figs 28–30 can be directly compared with the results presented in [21,32] Here again, a very good agree- −3 Δt=5x10 ,c=0 −0.02 10 15 Time t Fig 29 Transient response uy at point A without damping under transient loading with finite decreasing time This result can be compared with that of the LoKrging method in [21] T.Q Bui et al / Comput Methods Appl Mech Engrg 200 (2011) 1354–1366 References 0.01 0.005 Displacement u y −0.005 −0.01 −0.015 −0.02 1365 −3 Δt=5x10 ,c=0.5 10 15 Time t Fig 30 Transient displacement uy at point A with damping under transient loading with a finite decreasing time This result can be compared to that of both the MSW [32] and LoKriging [21] methods ment can be found Furthermore, it can be observed in Figs 28 and 29 that the amplitude decreases as the time increases from to 1.0 s The response oscillates in a steady state way after 1.0 s because damping affect is ignored in the system On the other hand, a damping in the system results an amplitude decrease to zero as the time increases, as illustrated in Fig 30 Very stable results for the forced vibration analysis are obviously achieved by the present meshfree method Conclusions Free and forced vibration analyses of two-dimensional solid mechanics problems are presented in this paper Numerical examples are investigated in details for many different dynamic loading cases using the Newmark time integration scheme It demonstrates a successful application of the present moving Kriging interpolation-based element-free Galerkin method for structural dynamics problems By making use of the good feature of the possession of the Kronecker delta property, the method is effective in enforcing the essential boundary conditions, and the procedure is straightforward as in the FEM As a result, the combination between the novel MK interpolation technique and the standard EFG method is an attractive method Numerical results presented in this paper show an excellent agreement with other reference solutions The present moving Kriging meshfree method is efficient, accurate and stable for solving the structural dynamics problems Therefore, the proposed meshfree method has a good potential and is promising to be extended to non-linear problems, crack problems and so forth, especially for numerical modeling of multifield coupled problems in smart materials such as piezoelectric, magnetoelectroelastic media, etc Furthermore, studies on the effects of the size of the domain of influence on the numerical solutions of higher modes in free vibration analysis by using an adaptive procedure as proposed in [45] are useful in the future research Also, a detailed investigation of dispersion and 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