DSpace at VNU: AN APPLICATION OF THE ES-FEM IN SOLID DOMAIN FOR DYNAMIC ANALYSIS OF 2D FLUID-SOLID INTERACTION PROBLEMS

DSpace at VNU: AN APPLICATION OF THE ES-FEM IN SOLID DOMAIN FOR DYNAMIC ANALYSIS OF 2D FLUID-SOLID INTERACTION PROBLEMS...

2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 International Journal of Computational Methods Vol 10, No (2013) 1340003 (26 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876213400033 AN APPLICATION OF THE ES-FEM IN SOLID DOMAIN FOR DYNAMIC ANALYSIS OF 2D FLUID–SOLID INTERACTION PROBLEMS 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, District Hochiminh City, Vietnam †Division of Computational Mechanics Ton Duc Thang University, 98 Ngo Tat To Street Ward 19, Binh Thanh District Hochiminh City, Vietnam §Institute of Structural Mechanics, Bauhaus-University Weimar Marienstrasse 15, 99423, Weimar ¶Department of Civil Engineering, International University VNU-HCM, Vietnam ngttrung@hcmus.edu.vn thoitrung76@gmail.com Received 16 June 2011 Accepted 29 August 2011 Published 18 January 2013 An edge-based smoothed finite element method (ES-FEM-T3) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the solid mechanics analyses In this paper, the ES-FEM-T3 is further extended to the dynamic analysis of 2D fluid–solid interaction problems based on the pressure-displacement formulation In the present coupled method, both solid and fluid domain is discretized by triangular elements In the fluid domain, the standard FEM is used, while in the solid domain, we use the ES-FEM-T3 in which the gradient smoothing technique based on the smoothing domains associated with the edges of triangles is used to smooth the gradient of displacement This gradient smoothing technique can provide proper softening effect, and thus improve significantly the solution of coupled system Some numerical examples have been presented to illustrate the effectiveness of the proposed coupled method compared with some existing methods for 2D fluid–solid interaction problems Keywords: Numerical methods; edge-based smoothed finite element method (ES-FEM); finite element method (FEM); fluid–solid interaction problems; gradient smoothed; dynamic analysis ∗∗Corresponding author 1340003-1 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 Introduction The need of computing the dynamic behavior of two-dimensional (2D) fluid–solid system arises in many important engineering problems The dam-reservoir interaction during earthquakes and fluid storage containers subjected to dynamic loads are examples of this class of problems However, predicting the response of fluid–solid coupled systems is generally a difficult task In most practical problems, it is not possible to obtain closed form analytical solutions for the coupled systems As a result, much effort has been performed in order to develop the different numerical methods for these coupled systems Numerical analysis of fluid–solid interaction problems involves the modeling of fluid domain, solid domain, and the interaction between these two domains The finite element method (FEM), the boundary element method (BEM) and the meshfree methods are currently the most preferred tools for the simulation of the fluid–solid interaction problems [Zienkiewicz and Bettess (1978); Wilson and Khalvati (1983); Chen and Taylor (1990); Brunner et al (2009); Everstine and Henderson (1990); He et al (2010); He et al (2010); Bathe et al (1995); Wang and Bathe (1997); Rabczuk et al (2006); Rabczuk et al (2010); Wall and Rabczuk (2008); Rabczuk et al (2007)] The numerical solution of the fluid–solid interaction problems can be performed using only FEM or a coupled BEM/FEM with a displacement–displacement formulation [Zienkiewicz and Bettess (1978); Wilson and Khalvati (1983); Chen and Taylor (1990)] or a pressure-displacement formulation [Brunner et al (2009); Everstine and Henderson (1990); He et al (2010); He et al (2010)], or a combination of these [Bathe et al (1995); Wang and Bathe (1997)] In numerical computation using the FEM for 2D solid mechanics problems, the 3-node linear triangular element (FEM-T3) are preferred by many engineers due to its simplicity and efficiency of adaptive mesh refinements However, the FEM-T3 element possesses “overly stiff” property which causes the following certain drawbacks: (1) They overestimate excessively the stiffness of the problem which leads to poor accuracy in solutions; (2) They are subjected to locking in the problems with bending domination and incompressible materials In order to overcome these disadvantages, some new finite elements were proposed Allman [1984, 1988] introduced rotational degrees of freedom at the element nodes to achieve an improvement for the overly stiff behavior Elements with rotational degrees of freedom were also considered by Bergan and Felippa [1985] and Cook [1991] Piltner and Taylor [2000] combined the rotational degrees of freedom and enhanced strain modes to give a triangular element which can achieve a higher convergence in energy and deal with the nearly incompressible plane strain problems However, using more degrees of freedom at the nodes limits the practical application of those methods Dohrmann et al [1998] presented a weighted least-squares approach in which a linear displacement field is fit to an element’s nodal displacements The method is claimed to be computationally efficient and avoids the volumetric locking problems However, 1340003-2 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only An Application of the ES-FEM in Solid Domain for Dynamic Analysis more nodes are required on the element boundary to define the linear displacement field Recently, in order to “soften” the system using FEM-T3, Liu and NguyenThoi [2010] incorporated the gradient smoothing technique of meshfree methods [Chen et al (2001)] into the FEM to formulate a series of smoothed FEM (S-FEM) models named as cell-based S-FEM (CS-FEM) [Liu et al (2007); Liu et al (2007); Liu et al (2009); Liu et al (2010); Dai et al (2007); Nguyen-Thoi et al (2007); Nguyen-Xuan and Nguyen-Thoi (2009)], node-based S-FEM (NS-FEM) [Liu et al (2009); Nguyen-Thoi et al (2009a); Nguyen-Thoi et al (2010)], edge-based S-FEM (ES-FEM) [Liu et al (2007)], 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 FEM models 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 Among these S-FEM models, the ES-FEM-T3 [Liu et al (2009)] using triangular elements shows some following excellent properties for the 2D solid mechanics analyses: (1) The numerical results are often found super-convergent and very accurate; (2) The method is stable and works well for dynamic analysis; (3) The implementation of the method is straightforward and no penalty parameter is used The ES-FEM-T3 has been developed for n-sided polygonal elements [Nguyen-Thoi et al (2009b)], visco-elastoplastic analyses [Nguyen-Thoi et al (2009)], 2D piezoelectric [Nguyen-Xuan et al (2009)], plate [Nguyen-Xuan et al (2009)] and primal-dual shakedown analyses [Tran-Thanh et al (2010)] The idea of the ES-FEM-T3 is also quite straightforward to extend for the 3D problems using tetrahedral elements to give a so-called the face-based smoothed finite element method (FS-FEM) [NguyenThoi et al (2009); Nguyen-Thoi et al (2009)] This paper hence attempts to extend the ES-FEM-T3 to the dynamic analyses of 2D fluid–solid interaction problems based on the pressure-displacement formulation In this coupled method, both solid and fluid domain is discretized by triangular elements In the fluid domain, the standard FEM-T3 is used, while in the solid domain, the gradient smoothing technique based on the smoothing domains associated with the edges of triangles is used to smooth the gradient of displacement This gradient smoothing technique can provide proper softening effect, which will effectively relieve the overly stiff behavior of the standard FEM model and thus improve significantly the solution of coupled system Some numerical examples have been presented to illustrate the effectiveness of the proposed coupled method compared with some existing methods for 2D fluid–solid interaction problems Governing Equations for Fluid–Solid Interaction Problems The fluid–solid interaction problem is schematically sketched in Fig It consists of a fluid domain, Ωf , and a solid domain, Ωs The interaction boundary between the 1340003-3 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 fluid domain and the solid domain is denoted, ∂Ωsf ; two remaining fluid boundaries are given by prescribed pressure, p = p¯ on ∂Ωp , and a prescribed normal pressure ¯ on ∂Ωz ; the remaining solid boundaries are given by prescribed gradient nf ∇p = w ¯ on ∂Ωu , and prescribed force vector, ns σ s = ¯ ts on ∂Ωt displacement, us = u For the fluid–solid system, the solid is described by the differential equation of motion for a continuum body assuming small deformations and the fluid is described by the wave equation in which the fluid is inviscid, irrotational, and only undergoes small translations Coupling conditions at the interaction boundary between the solid and fluid domains ensure the continuity in displacement and pressure between the domains Hence, the governing equations and boundary conditions were described in general as [Carlsson (1992)]:  ∂ p ∂q   in Ωf − c20 ∇2 p = c20    ∂t ∂t   p = p¯ on ∂Ωp , (1) Fluid:   n ∇p = w ¯ on ∂Ω  f z     + and the initial conditions   ∇Ts σ s + bs = ρs ∂ us in Ωs    ∂t2   us = u ¯ on ∂Ωu Solid: , (2)   ¯ n σ = t on ∂Ω  s s s t     and the initial conditions Coupling: us |n = uf |n on ∂Ωsf , (3a) σ s |n = −p on ∂Ωsf (3b) Fig (Color online) A model of the fluid–solid interaction problems 1340003-4 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis where for the fluid, p(t) is dynamic pressure; qf (t) is the added fluid mass per unit volume; c0 is the speed of sound; ∇ = [∂/∂x ∂/∂y]T and ∇2 = ∇ · ∇ = ∂ /∂x2 + ∂ /∂y ; nf = [nf x nf y ] is the boundary normal vector pointing outward from the fluid domain; and for the solid, σ s = [σx σy σxy ]T is the stress; us = [usx usy ]T is the displacement; bs = [bsx bsy ]T is the body force; ρs is the density of the material; ns is the boundary normal matrix pointing outward from the solid domain written as Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only ns = nsx nsy nsy nsx (4) and ∇s is 2D differential operator written as   ∂   ∂x    ∂    ∇s =  ∂y       ∂ ∂  ∂y ∂x (5) In the solid, the displacement us and strains εs are related by the kinematic relation εs = ∇s us (6) and the stresses σ s and strains εs are related by the Hook’s law σ s = Ds εs , (7) where Ds (3 × 3) is a symmetric positive definite (SPD) matrix of material constants A Coupled FEM/ES-FEM for the Fluid–Solid Interaction Problems 3.1 Brief on the FEM for fluid domain The weak form of the differential equation is derived by multiplying the first term in Eq (1) with a weight function, vf ∈ H01 , and integrating over the fluid domain, Ωf , vf Ωf ∂2p ∂qf − c20 ∇2 p − c20 dΩ = ∂ 2t ∂t (8) Using Green–Gauss theorem on the second term in Eq (8), the weak form can be written vf Ωf ∂2p dΩ + c20 ∂ 2t Ωf (∇vf )T ∇pdΩ = c20 ∂Ωf 1340003-5 vf nf ∇pdΓ + c20 vf Ωf ∂qf dΩ ∂t (9) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 T Nguyen-Thoi et al Due to ∂Ωf = ∂Ωsf ∪ ∂Ωp ∪ ∂Ωz , Eq (9) is rewritten vf Ωf ∂2p dΩ + c20 ∂2t Ωf (∇vf )T ∇pdΩ = c20 ∂Ωsf + c20 vf nf ∇pdΓ + c20 ∂Ωz vf nf ∇pdΓ+c20 ∂Ωp vf nf ∇pdΓ vf Ωf ∂qf dΩ ∂t (10) Using the relation nf ∇p = w ¯ on the boundary ∂Ωz in Eq (1), and vf = on ∂Ωp (due to vf ∈ H01 ), we get Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only vf Ωf ∂2p dΩ + c20 ∂2t = c20 ∂Ωsf (∇vf )T ∇pdΩ Ωf vf nf ∇pdΓ + c20 ∂Ωz vf wdΓ ¯ + c20 vf Ωf ∂qf dΩ ∂t (11) Supposing the fluid domain Ωf is discretized into Nfnod nodes and Nfel triangular elements, then the pressure field p ∈ H and test weight function vf ∈ H01 can be approximated by p = Nf p; vf = Nf cf , (12) where p = [ p1 p2 · · · pNfnod ]T contains the approximate pressure values at nodes; cf = [cf cf · · · cf Nfnod ]T contains the chosen test values at nodes; and Nf = [Nf Nf · · · Nf Nfnod ] contains the finite element shape functions at nodes for the fluid domain First, by choosing cf = [1 · · · 0]T , we get vf = Nf Similarly, by choosing Nfnod linear independent vectors cf such that [vf vf · · · vf Nfnod ] = [Nf Nf · · · Nf Nfnod ] = Nf , we obtain the finite element formulation for the fluid domain from Eq (11) as Ωf NTf Nf dă p + c20 = c20 sf (Nf )T Nf dΩp Ωf NTf nf ∇pdΓ + c20 ∂Ωz NTf wdΓ ¯ + c20 Ωf NTf ∂qf dΩ, ∂t (13) and the governing system of equations for the fluid domain can be written Mf p ă + Kf p = f q +f s , (14) where Mf = Ωf fs = c20 NTf Nf dΩ; ∂Ωsf Kf = c20 NTf nf ∇pdΓ; (∇Nf )T ∇Nf dΩ, Ωf fq = c20 ∂Ωz NTf wdΓ ¯ + c20 Ωf NTf ∂qf dΩ ∂t (15) in which fs represents the force caused by the solid domain at the interface between the fluid and solid domains, and fq represents the force in the fluid domain 1340003-6 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis 3.2 Brief on the FEM for solid domain The weak form of the differential equation is derived by multiplying the first term in Eq (2) with a weight function, vs ∈ H10 , and integrating over the solid domain, Ωs , Ωs vsT ∇Ts σ s + bs − ρs ∂ us dΩ = ∂t2 (16) Using Green–Gauss theorem, the first term in Eq (16) becomes Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only Ωs vsT ∇Ts σ s dΩ = ∂Ωs vsT ns σ s dΓ − (∇s vs )T σ s dΩ (17) Ωs Due to ∂Ωs = ∂Ωsf ∪ ∂Ωu ∪ ∂Ωt , Eq (17) is rewritten Ωs vsT ∇Ts σ s dΩ = ∂Ωsf vsT ns σ s dΓ + + ∂Ωt ∂Ωu vsT ns σ s dΓ − vsT ns σ s dΓ (∇s vs )T σ s dΩ (18) Ωs ts on the boundary ∂Ωt , and vs = on ∂Ωu (due to Using the relation ns σ s = ¯ vs ∈ H10 ), Eq (18) becomes Ωs vsT ∇Ts σ s dΩ = ∂Ωsf vsT ns σ s dΓ + ∂Ωt ts dΓ − vsT ¯ (∇s vs )T σ s dΩ (19) Ωs Substituting Eq (19) into Eq (16), the weak form of the solid domain becomes ∂Ωsf vsT ns σ s dΓ + + Ωs ∂Ωt vsT bs dΩ − vsT ¯ ts dΓ − Ωs vsT ρs (∇s vs )T σ s dΩ Ωs ∂ us dΩ = ∂t2 (20) or Ωs vsT ρs ∂ us dΩ + ∂t2 = ∂Ωsf (∇s vs )T σ s dΩ Ωs vsT ns σ s dΓ + ∂Ωt ts dΓ + vsT ¯ Ωs vsT bs dΩ (21) Substituting Eqs (6) and (7) into the second term in Eq (21), the weak form of the solid domain becomes Ωs vsT ρs ∂ us dΩ + ∂t2 = ∂Ωsf (∇s vs )T Ds ∇s us dΩ Ωs vsT ns σ s dΓ + ∂Ωt 1340003-7 ts dΓ + vsT ¯ Ωs vsT bs dΩ (22) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 T Nguyen-Thoi et al Supposing the solid domain Ωs is discretized into Nsnod nodes and Nsel triangular elements, then the displacement field u ∈ H1 and test weight function vs ∈ H10 can be approximated by vs = Ns cs , Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only us = Ns ds ; (23) where vector ds contains the approximate displacement values at nodes; cs contains the chosen test values at nodes; and Ns contains the finite element shape functions for the solid domain Similarly as in the fluid domain, by choosing Nsnod linear independent vectors cs such that [vs1 vs2 · · · vsNsnod ] = [Ns1 Ns2 · · · NsNsnod ] = Ns , we obtain the finite element formulation for the solid domain from Eq (22) as s ăs + NTs s Ns dd = ∂Ωsf (∇s Ns )T Ds ∇s Ns dΩds Ωs NTs ns σ s dΓ + NTs ¯ ts dΓ + ∂Ωt Ωs NTs bs dΩ (24) and the governing system of equations for the solid domain can be written ăs +Ks ds = f f +f b , Ms d (25) where Ms = Ωs ff = NTs ρs Ns dΩ; ∂Ωsf NTs ns σ s dΓ; (∇s Ns )T Ds ∇s Ns dΩ, Ks = Ωs fb = ∂Ωt (26) ts dΓ + NTs ¯ Ωs NTs bs dΩ in which ff represents the force caused by the fluid domain at the interface between the fluid and solid domains, and fb represents the force in the solid domain 3.3 FEM for the fluid–solid interaction system At the interaction boundary between the solid and fluid domains, denoted ∂Ωsf , the fluid particles and the solid moves together in the normal direction of the boundary Introducing the normal vector n = [nx ny ] = [nf x nf y ] = [−nsx −nsy ], the continuous boundary condition in displacement can be written us |∂Ωsf = uf |∂Ωsf , or nus = nuf on ∂Ωsf (27) and the continuity in pressure is written σ s |n = ns σ s = −p nsx nsy 1340003-8 = nT p, (28) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis where uf = [uf x uf x ]T is the displacement of the fluid particles and p is the fluid pressure Using Eq (28), the force vector ff in Eq (26) can be expressed in the fluid pressure by ff = ∂Ωsf NTs ns σ s dΓ = ∂Ωsf NTs nT pdΓ = ∂Ωsf NTs nT Nf dΓpf (29) For the fluid partition, the coupling is introduced in the force term fs (in Eq (14)) Using the relation between pressure and acceleration in the fluid domain ∂ uf (t) (30) ∂t2 and the boundary condition in Eq (27) , the force acting on the fluid can be described in terms of structural acceleration, Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only ∇p = −ρ0 ∂ uf (t) ∂ us (t) ă s |sf | = n |sf = −ρ0 nNs d (31) ∂Ω sf ∂t2 ∂t2 and the boundary force term of the fluid domain, fs in Eq (14), can be expressed in structural acceleration n∇p|∂Ωsf = −ρ0 n fs = c20 ∂Ωsf NTf nf ∇pdΓ = c20 sf NTf npd = c20 sf ăs NTf nNs dΓd (32) The introduction of a spatial coupling matrix H= ∂Ωsf NTs nT Nf dΓ (33) allows the coupling forces to be written as ff = Hpf (34) ăs fs = c20 HT d (35) and The fluid–solid interaction problem can then be described by an unsymmetrical system of equations Ms ρ0 c20 HT Mf ¨s Ks d + p ¨f −H Kf ds f = b pf fq (36) 3.4 ES-FEM-T3 using triangular elements for the solid domain Similar to the FEM-T3, the ES-FEM-T3 also uses a mesh of triangular elements The shape functions used in the ES-FEM-T3 are also identical to those in the FEM-T3, and hence the displacement field in the ES-FEM-T3 is also ensured to be continuous on the whole problem domain However, being different from the FEMT3 which computes the stiffness matrix K based on the elements, the ES-FEM-T3 uses the gradient smoothing technique [Chen et al (2001)] to compute the stiffness matrix based on the edges The stiffness matrix in the ES-FEM-T3 hence is called 1340003-9 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 T Nguyen-Thoi et al H boundary edge m (m) D Γ E A Ω inner edge k Γ (k) Ω C Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only (k) (m) G B : field node : centroid of triangles Fig (Color online) Triangular elements and the smoothing domains associated with edges in ES-FEM-T3 ˜ In this process, the finite element the smoothed stiffness matrix and symbolized K (k) mesh in the solid domain is divided into smoothing domains Ωs based on edges ed Ns (k) (i) (j) Ωs and Ωs ∩ Ωs = , i = j, in which Nsed is of elements such that Ωs = k=1 the total number of edges of the finite element mesh For triangular elements, the (k) smoothing domain Ωs associated with the edge k is created by connecting two endpoints of the edge to centroids of adjacent elements as shown in Fig Applying the edge-based smoothing operation, the compatible displacement gra˜ s us on dient ∇s us in Eq (22) is used to create a smoothed displacement gradient ∇ (k) the smoothing domain Ωs associated with edge k such as: ˜ s u(k) ∇ s (x) = (k) Ωs ∇s us (x)Φ(k) s (x)dΩ, (37) where Φ(k) s (x) is a given smoothing function that satisfies at least unity property (k) (k) Φs (x)dΩ =1 Using the following constant smoothing function Ω s (k) Φ(k) s (x) (k) where As = = (k) 1/As x ∈ Ωs x∈ / Ωs (k) , (38) (k) and applying a divergence theorem, one ˜ s u(k) that is constant over the can obtain the smoothed displacement gradient ∇ s (k) domain Ωs as follows (k) Ωs dΩ is the area of Ωs ˜ s u(k) (x) = ∇ s (k) As (k) Γs T (n(k) s (x)) us (x)dΓ, 1340003-10 (39) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 T Nguyen-Thoi et al Equation (44) implies that only shape function values at some particular points (k) along segments of boundary Γsi are needed and no derivatives of the shape function ˜ s of the system is then assembled by are required The smoothed stiffness matrix K a similar process as in the FEM Nsed ˜s = K k=1 ˜ (k) , K sIJ (45) (k) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only ˜ where K sIJ is the smoothed stiffness matrix associated with edge k and is calculated by ˜ (k) = K sIJ (k) Ωs ˜ sJ dΩ = B ˜ TsI DB ˜ sJ A(k) ˜ TsI DB B s (46) 3.5 A coupled FEM-T3/ES-FEM-T3 method for the fluid–solid interaction problems As shown in Sec 3.4, the only difference between the FEM-T3 and the ES-FEM-T3 in the solid domain is the way to compute the stiffness matrix In the FEM-T3, the stiffness matrix Ks is computed based on the elements While in the ES-FEM-T3, ˜ s is computed based on the edge-based smoothing the smoothed stiffness matrix K domains through the gradient smoothing technique [Chen et al (2001)] Hence, based on the system of Eq (36) for the fluid–solid interaction problems using the FEM, the system of equations for the 2D fluid–solid interaction problems using the coupled FEM-T3/ES-FEM-T3 will be expressed in the following form Ms ρ0 c20 HT Mf ăs d p ăf + s K −H Kf ds fb = , pf fq (47) ˜ s in Eq (45) where Ks in Eq (36) is replaced by K Dynamic Analysis Because the ES-FEM-T3 is both spatially and temporally stable [Liu et al (2010); Liu et al (2009); Nguyen-Thoi et al (2009b)], it suits well for dynamic problems, such as free and forced vibrations analyses If the damping forces are also considered in the dynamic equilibrium equations, the system of Eq (47) for the fluid–solid interaction problems using the ES-FEM-T3 can be expressed as follows: + Cx + Mă Kx x = F, (48) where x= ds pf ; ˜ ˜ = Ks K −H ; ˜f K M= Ms ρ0 c20 HT Mf 1340003-12 ; F= fb fq (49) 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis and C is the damping matrix Using the Rayleigh damping, matrix C is assumed ˜ and M, to be a linear combination of K Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only ˜ C = αM + β K, (50) 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 ă0 ), we aim to nd a new state the current state at t = t0 is known as (x0 , x , x ă1 ) at t1 = t0 + θ∆t where 0.5 ≤ θ ≤ Note that the Newmark method is (x1 , x˙ , x a one step implicit method for solving the transient problem, and by choosing the parameter θ such that 0.5 ≤ θ ≤ 1, we ensure that the method is stable and the dissipation and dispersion errors can be ignored without affecting to the stability of solutions [Smith and Griffiths (1998)] The formulation of the method is expressed as follows: ˜ x1 = θ∆tF1 + (1 − θ)∆tF0 + α + M + (β + θ∆t)K Mx0 α+ θ∆t θ∆t ˜ 0, + Mx˙ + [β − (1 − θ)∆t]Kx (51) θ 1−θ (x1 − x0 ) − x˙ , (52) x˙ = θ∆t θ 1−θ (x˙ − x ) x ă0 (53) x ă1 = θ∆t θ Without the damping and forcing terms, Eq (48) is reduced to a homogenous differential equation: ˜ + Mă Kx x = (54) A general solution of such a homogenous equation can be written as x=x ¯ exp(iωt), (55) where t indicates time, x ¯ is the amplitude of the sinusoidal displacements and ω is the angular frequency On its substitution into Eq (54), the natural frequency ω can be found by solving the following eigenvalue equation ˜ − ω M]¯ x = [K (56) Numerical Examples In this section, two numerical examples are performed to show the advantageous properties of the proposed coupling FEM-T3/ES-FEM-T3 method for 2D fluid– solid interaction problems The numerical results of coupled FEM-T3/ES-FEM-T3 will be compared with those of the coupled FEM-T3/FEM-T3 using standard triangular elements for both fluid and solid domains, and of the coupled FEM-Q4/FEMQ4 using quadrilateral elements for both fluid and solid domains In addition, to illustrate the convergent property of the numerical methods, the reference solution 1340003-13 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 T Nguyen-Thoi et al by the coupled FEM-Q8/FEM-Q8 using 8-node elements with a very fine element mesh for both solid and fluid domain will be used Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only 5.1 2D deformable solid backed by a closed box filled with water The 2D deformable solid in this example has the dimension of 10 m × m The solid is fixed supported at two ends and is given by the following datas: density of the material ρs = 2,500 kg/m2 ; elastic modulus E = 2.1 × 109 N/m2 and poisson’s ratio υ = 0.3 A closed box filled with water is attached to the solid and has the dimension of 10 m × m The following data were used in the fluid, ρ = 1,000 kg/m2 and speed of air c = 1,500 m/s2 The model is shown in Fig and a discretization of the model using triangular elements for both solid domain and fluid domain is shown in Fig 5.1.1 Free vibration analysis The eigenmodes analysis for the fluid–solid system is first investigated The solution of FEM-Q8/FEM-Q8 with 729 degree of freedom (DOFs) for fluid and 1,290 DOFs for solid is used as reference solution Figure shows the convergence of the first coupled eigenmode by different coupled methods: FEM-T3/FEM-T3, FEMQ4/FEM-Q4 and FEM-T3/ES-FEM-T3 The results show that with the same of DOFs, the FEM-T3/ES-FEM-T3 is the best one Figure shows the shape and value of nine first coupled eigenmodes by the FEMT3/ES-FEM-T3 It is seen that the shapes of eigenmodes express suitably the real physical modes without having any of the spurious nonzero energy modes Figure compares nine values of coupled eigenmodes by different coupled methods Again, it is seen that the results of the FEM-T3/ES-FEM-T3 is closest to the reference solution In addition, in order to check the sensitivity of higher eigenmodes to the distorted meshes, an eigenmode analysis for a very distorted mesh as shown in Fig Fig (Color online) Model of the 2D deformable solid backed by a closed box filled with water 1340003-14 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis Interface between fluid and solid A(2,5) B(2,3) Solid domain Fluid domain Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only 0 10 11 Fig (Color online) A discretization of the 2D deformable solid backed by a closed box filled with water using triangular elements for both solid and fluid domains Fig (Color online) Convergence of the first coupled eigenmode by three different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and FEM-T3/ES-FEM-T3 is performed and the results are shown in Fig It is seen that the solutions of higher modes are not sensitive to the distorted meshes 5.1.2 Forced vibration analysis The forced frequency response analysis for the fluid–solid system by the FEMT3/ES-FEM-T3 is now investigated First, the force applied to the structure is a 1340003-15 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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) Shape of nine coupled eigenmodes of the fluid-solid system by FEM-T3/ESFEM-T3 Fig (Color online) Comparison of nine coupled eigenmodes of the fluid-solid system by three different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and FEM-T3/ES-FEM-T3 1340003-16 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only An Application of the ES-FEM in Solid Domain for Dynamic Analysis Fig (Color online) A very distorted mesh using triangular elements for both solid and fluid domains Fig (Color online) Comparison of nine coupled eigenmodes of the fluid-solid system discretized by a uniform mesh and a very distorted mesh by the FEM-T3/ES-FEM-T3 harmonic vertical point load F (x, ω) = δc (x)iωeiωt , where i is the imaginary unit and δc (x) is the Dirac function at the point x Considering the case t = 0, the point x is put at point A (2.0, 5.0) as shown in Fig 4, and ω/2π is changed from Hz to 17 Hz where the value of three first coupled eigenfrequencies appears as shown in Fig The displacement responses measured also at the loaded point A (2.0, 5.0) are shown in Fig 10 It is seen clearly that the peaks of the three first responses occur exactly at the values of three first eigenfrequencies as shown in Fig Similarly, if 1340003-17 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 10 (Color online) Forced frequency response by the FEM-T3/ES-FEM-T3 at the point A (2.0, 5.0) in the solid domain with the force applied also to the the point A (2.0, 5.0) Fig 11 (Color online) Forced frequency response by the FEM-T3/ES-FEM-T3 at the point A (2.0, 5.0) in the solid domain with the force applied to the the point B (2.0, 3.0) in the fluid domain 1340003-18 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only An Application of the ES-FEM in Solid Domain for Dynamic Analysis we change the position of loaded point x to the new coordinate in the fluid domain B (2.0, 3.0) and keep everything unchanged, we get the displacement responses measured at the point A (2.0, 5.0) as shown in Fig 11 Again, it is seen clearly that the peaks of the three first responses occur exactly at the values of three first eigenfrequencies These results imply that the forced frequency response analysis by the FEM-T3/ES-FEM-T3 can provide sufficient information about the eigenfrequencies of the fluid–solid interaction system, where the peaks of the response occur In addition, the forced frequency response analysis also shows clearly the difference of eigenfrequencies between the solid system without coupling with the fluid and the fluid–solid interaction system as shown in Table and Fig 12 It is seen that the eigenfrequencies of the solid system without coupling with the fluid are smaller than those of the fluid–solid interaction system It is hence necessary to Table Difference of the value of eight first coupled and uncoupled eigenmodes Structure without Coupling Structure with Coupling Mode Mode Mode Mode Mode Mode Mode Mode 3.1259 8.0027 14.4589 15.3279 21.9963 30.2898 30.5920 39.1541 5.6467 10.7650 15.2903 18.4016 25.6971 30.5001 34.9768 42.7242 Fig 12 (Color online) Compare the forced frequency response at the point A (2.0, 5.0) in the solid domain with the force applied also to the the point A (2.0, 5.0) by the FEM-T3/ES-FEM-T3 and by the ES-FEM-T3 for solid domain without coupling with the fluid domain 1340003-19 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 13 (Color online) Compare the forced frequency response at the point A (2.0, 5.0) in the solid domain with the force applied also to the the point A (2.0, 5.0) between the FEM-T3/ES-FEM-T3 and FEM-Q4/FEM-Q4 use the fluid–solid interaction system to model accurately the solid system coupling with the fluid Figure 13 compares the forced frequency response analysis between the FEMT3/ES-FEM-T3 and FEM-Q4/FEM-Q4 It is seen that the displacement responses of the FEM-T3/ES-FEM-T3 is closest to those of the reference solution by FEMQ8/FEM-Q8 using very fine mesh In addition, as depicted in Fig 13, at lower frequencies, both the coupled FEM-T3/ES-FEM-T3 and FEM-Q4/FEM-Q4 can provide quite good results However, with the increase in the frequency, the deviation between the coupled FEM-T3/ES-FEM-T3 and the reference is much smaller than the deviation between the coupled FEM-Q4/FEM-Q4 and the reference Figure 14 compares the transient response between the FEM-T3/ES-FEMT3 and FEM-Q4/FEM-Q4 Again, it is seen that the displacement responses of the FEM-T3/ES-FEM-T3 is closest to those of the reference solution by FEMQ8/FEM-Q8 using very fine mesh This numerical example hence shows that the coupled FEM-T3/ES-FEM-T3 provides more accurate results in eigenfrequencies, frequency responses and transient responses than those of many existing coupled FEM/FEM models for the fluid–solid interaction problems 5.2 2D deformable water dam — A free vibration analysis A deformable water dam, with dimensions as shown in Fig 15, is fixed at the dam’s foundation and is given by the following datas: density of the material ρs = 2,500 kg/m2 ; elastic modulus E = 2.1 × 109 N/m2 and poisson’s ratio υ = 0.3 The 1340003-20 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only An Application of the ES-FEM in Solid Domain for Dynamic Analysis Fig 14 (Color online) Compare the transient response at the point A (2.0, 5.0) in the solid domain with the force applied also to the the point A (2.0, 5.0) between the FEM-T3/ES-FEM-T3 and FEM-Q4/FEM-Q4 Fig 15 (Color online) Model of 2D deformable water dam water is attached to the dam and has the dimension of 10 m × m The following data were used in the fluid, ρ = 1,000 kg/m2 and speed of air c = 1,500 m/s2 The model is shown in Fig 15, and a discretization using triangular elements for both solid domain and fluid domain is shown in Fig 16 The eigenmodes analysis for the fluid–solid system is investigated The solution of FEM-Q8/FEM-Q8 with 697 DOFs for fluid and 1,570 DOFs for solid is used as reference solution Figure 17 shows the convergence of the first coupled 1340003-21 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 16 (Color online) A discretization using triangular elements for both water dam and fluid domain Fig 17 (Color online) Convergence of the first coupled eigenmode by three different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and FEM-T3/ES-FEM-T3 eigenmode by three different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEMQ4 and FEM-T3/ES-FEM-T3 The results show that with the same of DOFs, the FEM-T3/ES-FEM-T3 is the best one Figure 18 shows the shape and value of nine first coupled eigenmodes by the FEM-T3/ES-FEM-T3 It is seen that the shapes of eigenmodes express suitably the real physical modes without having any of the spurious nonzero energy modes In addition, Fig 19 compares nine values of coupled eigenmodes by different coupled methods Again, it is seen that the results of the FEM-T3/ES-FEM-T3 is closest to the reference solution 1340003-22 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only An Application of the ES-FEM in Solid Domain for Dynamic Analysis Fig 18 (Color online) Shape of nine coupled eigenmodes of the fluid–solid system by the FEMT3/ES-FEM-T3 Fig 19 (Color online) Comparison of nine coupled eigenmodes of the fluid–solid system by three different methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and FEM-T3/ES-FEM-T3 1340003-23 2nd Reading January 16, 2013 16:7 WSPC/0219-8762 196-IJCM 1340003 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 Conclusion The ES-FEM-T3 is further extended to the dynamic analysis of 2D fluid–solid interaction problems based on the pressure-displacement formulation In the present coupled method, both solid and fluid domain is discretized by triangular elements In the fluid domain, the standard FEM is used, while in the solid domain, we use the ES-FEM-T3 in which the gradient smoothing technique based on the smoothing domains associated with the edges of triangles is used to smooth the gradient of displacement This gradient smoothing technique can provide proper softening effect, which will effectively relieve the overly stiff behavior of the standard FEM model and thus improve significantly the solution of coupled system Some numerical examples have been presented to illustrate the effectiveness of the proposed coupled method compared with some existing methods for 2D fluid–solid interaction problems The numerical examples show that the coupled FEM-T3/ES-FEM-T3 provides more accurate results in eigenfrequencies, frequency responses and transient responses than those of many existing coupled FEM/FEM models 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) References Allman, D J [1984] “A compatible triangular element including vertex rotations 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[1978] “Fluid–structure dynamic interaction and wave forces: An introduction to numerical treatment,” Int J Numer Method Eng 13, 1–16 1340003-26 ... 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis and C is the damping matrix Using the Rayleigh damping, matrix C is assumed ˜ and M, to be a linear combination of K Int... dΩ in which ff represents the force caused by the fluid domain at the interface between the fluid and solid domains, and fb represents the force in the solid domain 3.3 FEM for the fluid? ?solid interaction. .. 1340003 An Application of the ES-FEM in Solid Domain for Dynamic Analysis 3.2 Brief on the FEM for solid domain The weak form of the differential equation is derived by multiplying the first term in