A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems
Applied Mathematics and Computation 232 (2014) 324–346 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems T Nguyen-Thoi a,b,⇑, P Phung-Van b, S Nguyen-Hoang a, Q Lieu-Xuan c a Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Viet Nam Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, VNU-HCMC, Viet Nam c Faculty of Civil Engineering, Nguyen Tat Thanh University, Hochiminh City, Viet Nam b a r t i c l e i n f o Keywords: Numerical methods Node-based smoothed finite element method (NS-FEM) n-Sided polygonal edge-based smoothed finite element method (nES-FEM) Fluid–solid interaction problems Gradient smoothing technique Dynamic analysis a b s t r a c t A node-based smoothed finite element method (NS-FEM-T3) using triangular elements and a n-sided polygonal edge-based smoothed FEM (nES-FEM) using polygonal elements is combined to give the smoothed coupled NS/nES-FEM for dynamic analysis of twodimensional (2D) fluid–solid interaction problems based on the pressure–displacement formulation In the present method, the NS-FEM-T3 is used for the fluid domain and the gradient of pressure is smoothed, while the nES-FEM is used for the solid domain and the gradient of displacement is smoothed This gradient smoothing technique can provide proper softening effect, which will effectively relieve the overly stiff behavior of the FEM model and thus improve significantly the solution of coupled system Some numerical examples have been presented to illustrate the effectiveness of the coupled NS/nES-FEM compared with some existing methods for 2D fluid–solid interaction problems Ó 2014 Elsevier Inc All rights reserved 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 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 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 [1– 15] 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 [2,10,11], or a pressure–displacement formulation [1,3–5], or a coupling ES-FEM/ BEM [14], or immersed smoothed finite element method [15], or a combination of these [12,13] Related to some new ideas, we can mention to Refs [16,17] using Nitsche’s ideas in some fluid–structure interaction problems, or a scheme using higher-order XFEM [18] for handling the interface of three-dimensional fluid–structure interaction In numerical computation using the conventional FEM for 2D solid mechanics problems, the use of triangular and quadrilateral elements is well-established However, there is significant bottleneck in generating quality meshes using polygonal elements for complex geometries The use of elements with an arbitrary number of sides will provide greater flexibility and ⇑ Corresponding author at: Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Viet Nam E-mail addresses: trungnt@tdt.edu.vn, thoitrung76@gmail.com, ngttrung@hcmus.edu.vn (T Nguyen-Thoi) 0096-3003/$ - see front matter Ó 2014 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.amc.2014.01.052 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 325 better accuracy to solve problems that arise in solid mechanics and biomechanics Since material microstructure in polycrystalline alloys and piezoelectrics, and bone can be described through polygonal sub-domains, the use of n-sided polygonal finite elements (nFEM) in such applications is a natural choice However so far, although there have been many researches [19–27] about the nFEM using polygonal elements, these nFEM still possesses some among four following main disadvantages: (1) the construction of shape functions is complicated; (2) the numerical integration on the polygonal elements is difficult; (3) they overestimate excessively the stiffness of the problem which leads to poor accuracy in solutions; (4) they are subjected to locking in the problems with bending domination and incompressible materials In the 2D fluid–solid interaction problems, the geometrical domain of fluid is usually simpler than that of the solid domain Hence, the three-node linear triangular elements (FEM-T3) are preferred due to its simplicity, robustness, and efficiency of adaptive mesh refinements However, the FEM-T3 element also 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 of both triangular element and n-sided polygonal element, Liu and Nguyen Thoi [28] incorporated the gradient smoothing technique of meshfree methods [29] into the FEM to formulate a series of the smoothed FEM models (S-FEM) named as cell-based S-FEM (CS-FEM) [30–38], node-based S-FEM (NS-FEM) [39–41], edge-based S-FEM (ES-FEM) [42–48], that use linear interpolations For polygonal elements, there are three smoothed FEM models (nS-FEM) including n-sided polygonal cell-based smoothed FEM (nCS-FEM) [49], n-sided polygonal node-based smoothed FEM (nNS-FEM) [39,41], and n-sided polygonal edge-based smoothed FEM (nES-FEM) [50] In the 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 These smoothing domains can be located inside the elements (CS-FEM and nCS-FEM) or cover parts of adjacent elements (NS-FEM, nNS-FEM, ES-FEM and nES-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 [41,42,50] It is clear that these S-FEM models have the features of both models: meshfree [51] 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 The essence of the S-FEM is that only shape function values at points on the boundaries of the smoothing cells are needed and only the compatibility of the nodal shape functions on the boundaries of these smoothing cells is required in the formulation This gives tremendous freedom to compute these shape function values for the S-FEM, and they can be easily obtained using the simple linear point interpolation method without mapping [52] In addition, Wachspress interpolants computed in the physical coordinate system are also well suited to compute the shape function values for the S-FEM [53], especially when elements are heavily distorted or n-sided polygonal elements are used Each of these smoothed FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems The S-FEM models have also been further investigated and applied to various problems such as plates and shells [54–61], piezoelectricity [62], fracture mechanics using enriched smoothed elements [63,64], and some other applications [65,66], etc For the 2D solid mechanics analyses, the NS-FEM-T3 [39,41] shows some important properties such as: (1) it possesses the upper bound property in strain energy; (2) it is immune naturally from the volumetric locking; (3) it achieves superaccurate and super-convergent properties of stress solutions; (4) it can use linear triangular and tetrahedral elements; (5) the stress at nodes can be computed directly from the displacement solution without using any post-processing These six properties are very promising to apply the NS-FEM-T3 effectively in the more complicated non-linear problems The NS-FEM-T3 was then extended to perform adaptive analysis [67], linear elastostatics and vibration problems [68], viscoelastoplastic analyses [69], and alpha finite element method [70] Among these nS-FEM models using n-sided polygonal elements, the nES-FEM [50] shows some following excellent properties for the 2D solid mechanics analyses: (1) the numerical results are often found super-convergent and much more accurate than those of nFEM using polygonal elements with the same sets of nodes; (2) there are no spurious non-zeros energy modes found and hence the method is also stable and works well for dynamic analysis; (3) the implementation of the method is straightforward and no penalty parameter is used, and the computational efficiency is better than nFEM using the same sets of nodes; (4) the smoothed stiffness matrix can be computed directly from the shape function values at Gauss points along boundary segments of smoothing domains, without constructing the shape functions explicitly as in the FEM This computational process hence becomes very simple This paper hence attempts to combine the NS-FEM-T3 and nES-FEM to give the coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems based on the pressure–displacement formulation In the present method, the NS-FEMT3 is used for the fluid domain and the gradient of pressure is smoothed, while the nES-FEM is used for the solid domain and the gradient of displacement is smoothed This gradient smoothing technique can provide proper softening effect, which will effectively relieve the overly stiff behavior of the FEM model and thus improve significantly the solution of coupled system 326 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Using the NS-FEM-T3 for the fluid domain helps combine the simple advantage of discretizing the domain by three-node triangular elements with the advantage of node-based smoothing technique [39,41] While using the nES-FEM for the solid domain helps combine the flexible and practical advantage of discretizing the domain by n-sided polygonal elements with the advantage of edge-based smoothing technique [50] Some numerical examples have been presented to illustrate the effectiveness of the coupled NS/nES-FEM compared with some existing methods for 2D fluid–solid interaction problems Note that the proposed coupled approach is different from two reference approaches, the coupling ES-FEM/BEM [14] and immersed smoothed finite element method [15], in two aspects: (1) the solid domain in the present approach is discretized by n-sided polygonal elements which are suitable for many material microstructures in polycrystalline alloys and biomechanics, while the three-node triangular elements are still used in two reference approaches [14,15]; (2) the present approach uses only the strain smoothed techniques in the analysis which is more simple than two reference approaches [14,15] using different complicated coupled techniques The coupling ES-FEM/BEM [14] uses the smoothed finite element method (ES-FEM) for the solid domain and the boundary element method (BEM) for the fluid domain While the immersed smoothed finite element method [15] combined the smoothed finite element method with the finite difference method Governing equations for fluid–solid interaction problems The fluid–solid interaction problem is schematically sketched in Fig It consists of an fluid domain, Xf , and a solid domain, Xs The boundary between the fluid domain and the solid domain is denoted, @ Xsf ; two remaining fluid boundaries are on @ Xz ; the remaining solid on @ Xp , and a prescribed normal pressure gradient nf rp ¼ w given by prescribed pressure, p ¼ p on @ Xu , and prescribed force vector, ns rs ¼ ts on @ Xt boundaries are given by prescribed displacement, us ¼ 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 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 [71]: Fluid : Solid : @ p > c20 r2 p ¼ c20 @q in Xf ; > @t > @t > < p¼ p on @ Xp ; > n r p ¼ w on @ Xz ; > f > > : and the initial conditions: 1ị > rTs rs ỵ bs ẳ qs @@tu2s in Xs ; > > > < us ¼ u on @ Xu ; > on @ Xt ; > ns rs ¼ ts > > : and the initial conditions: Coupling : us j n ¼ uf j n on @ Xsf ; rs jn ¼ p on @ Xsf ; ð2Þ ð3Þ where for the fluid, pðtÞ is dynamic pressure; qf ðtÞ is the added fluid mass per unit volume; c0 is the speed of sound; nfy is the boundary normal vector pointing outward from T T T the fluid domain; and for the solid, rs ẳ ẵ rx ry rxy is the stress; us ẳ ẵ usx usy is the displacement; bs ẳ ẵ bsx bsy is r ẳ ½ @=@x @=@y T and r2 ¼ r r ẳ @ =@x2 ỵ @ =@y2 ; nf ¼ ½ nfx ∂Ω p ∂Ω z Fluid Ωf ∂Ωsf ∂Ωu Solid Ωs ∂Ωt Fig A model of the fluid–solid interaction problems 327 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 the body force; qs is the density of the material; ns is the boundary normal matrix pointing outward from the solid domain written as ns ¼ nsx nsy nsy nsx ð4Þ and rs is 2D differential operator written as @ @x @ @y 5: rs ¼ 40 @ @y @ @x ð5Þ In the solid, the displacement us and strains es are related by the kinematic relation es ẳ rs us 6ị and the stresses rs and strains es are related by the Hook’s law rs ẳ Ds es ; 7ị where Ds 3Þ is a symmetric positive definite (SPD) matrix of material constants A coupled NS/nES-FEM for the fluid–solid interaction problems 3.1 Brief on the FEM for fluid domain [71] vf The weak form of the differential equation is derived by multiplying the first term in Eq (1) with a weight function, H10 , and integrating over the fluid domain, Xf , Z vf Xf @2p @2t c20 r2 p c20 ! @qf dX ẳ 0: @t 8ị Using Green–Gauss theorem on the second term in Eq (8), the weak form can be written Z vf Xf @2p @ t dV ỵ c20 Z Xf rv f ÞT rp dV ¼ c20 Z @ Xf v f nf rp dS ỵ c20 Z vf Xf @qf dV: @t 9ị Due to @ Xf ẳ @ Xsf [ @ Xp [ @ Xz , Eq (9) is rewritten Z Z Z Z Z Z @2p v f dV ỵ c20 rv f ịT rp dV ẳ c20 v f nf rp dS ỵ c20 v f nf rp dS ỵ c20 v f nf rp dS ỵ c20 @ t Xf Xf @ Xsf @ Xp @ Xz Xf on the boundary @ Xz in Eq (1), and Using the relation nf rp ¼ w Z Xf vf @2p @ t dV ỵ c20 Z Xf rv f ịT rp dV ẳ c20 Z @ Xsf v f nf rp dS ỵ c20 vf Z @ Xz ¼ on @ Xp (due to v f w dS ỵ c20 Z vf Xf vf vf @qf dV: @t ð10Þ H10 ), we get @qf dV : @t ð11Þ Supposing that the fluid domain Xf is discretized into N nf nodes and N ef triangular elements, then the pressure field p H1 and test weight function v f H10 can be approximated by p ẳ Nf p; vf ẳ Nf cf ; 12ị T T where p ẳ ẵ p1 p2 pNnf contains the approximate pressure values at nodes; cf ẳ ẵ cf cf cfNnf contains the chosen test values at nodes; and Nf ẳ ẵ N f N f N fNnf contains the finite element shape functions at nodes for the fluid domain First, by choosing cf ẳ ẵ T , we get v f ¼ N f Similarly, by choosing N nf linear independent vectors cf such that v ½ f v f v fNnf ẳ ẵ N f N f N fNnf ¼ Nf , we obtain the finite element formulation for the fluid domain from Eq (11) as Z Xf ỵ c20 NTf Nf dX p Z Xf ðrNf ÞT rNf dX p ẳ c20 Z @ Xsf NTf nf rpdC ỵ c20 Z @ Xz dS ỵ c20 NTf w Z Xf NTf @qf dX @t ð13Þ and the governing system of equations for the fluid domain can be written ỵ Kf p ẳ f q ỵ f s ; Mf p where ð14Þ 328 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Mf ¼ Z Xf NTf Nf dX ; Z f s ¼ c20 @ Xsf Kf ¼ c20 NTf nf rpdC Z rNf ịT rNf dX; Xf f q ẳ c20 Z C ỵ c20 NTf wd @ Xz Z Xf ð15Þ @qf dX: @t NTf 3.2 Brief on the FEM for solid domain [71] The weak form of the differential equation is derived by multiplying the first term in Eq (2) with a weight function, v s H10 , and integrating over the solid domain, Xs , Z Xs ! @ us dX ¼ 0: @t v Ts rTs rs ỵ bs qs 16ị Using GreenGauss theorem, the first term in Eq (16) becomes Z Xs v Ts rTs rs dX ¼ Z @ Xs v Ts ns rs dC Z ðrs v s ÞT rs dX: 17ị Xs Due to @ Xs ẳ @ Xsf [ @ Xu [ @ Xt , Eq (17) is rewritten Z Xs v Ts rTs rs dX ¼ Z v Ts ns rs dC ỵ @ Xsf Z @ Xu v Ts ns rs dC ỵ Z @ Xt Using the relation ns rs ¼ ts on the boundary @ Xt , and Z Xs v Ts rTs rs dX ẳ Z v Ts ns rs dC ỵ @ Xsf Z @ Xt v Ts ts dC Z v Ts ns rs dC v s ¼ on @ Xu Z ðrs v s ÞT rs dX: ð18Þ Xs (due to v s H10 ), Eq (18) becomes ðrs v s ÞT rs dX: ð19Þ Xs Substituting Eq (19) into Eq (16), the weak form of the solid domain becomes Z @ Xsf v Ts ns rs dC ỵ Z @ Xt v Ts ts dC Z rs v s ịT rs dX ỵ Xs Z Xs v Ts bs dX Z Xs v Ts qs @ us dX ¼ @t ð20Þ vTs bs dX: ð21Þ or Z Xs v Ts qs @ us dX ỵ @t Z rs v s ịT rs dX ẳ Xs Z @ Xsf v Ts ns rs dC ỵ Z @ Xt v Ts ts dC ỵ Z Xs Substituting Eqs (6) and (7) into the second term in Eq (21), the weak form of the solid domain becomes Z Xs v Ts qs @ us dX ỵ @t Z rs v s ịT Ds rs us dX ẳ Xs Z @ Xsf v Ts ns rs dC ỵ Z @ Xt vTs ts dC ỵ Z Xs v Ts bs dX: ð22Þ Supposing that the solid domain Xs is discretized into N ns nodes and N es triangular elements, then the displacement field u H1 and test weight function v s H10 can be approximated by u s ¼ N s ds ; v s ¼ Ns cs ; ð23Þ 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 N ns linear independent vectors cs such that ½ v s1 v s2 v sNns ¼ ½ Ns1 Ns2 NsNns ¼ Ns , we obtain the finite element formulation for the solid domain from Eq (22) as Z Xs s ỵ NTs qs Ns dXd Z ðrs Ns ÞT Ds rs Ns dXds ¼ Xs Z @ Xsf NTs ns rs dC ỵ Z @ Xt NTs ts dC ỵ Z NTs bs dX Xs ð24Þ and the governing system of equations for the solid domain can be written € s þ Ks ds ¼ f þ f ; Ms d f b 25ị where Ms ẳ Z Xs NTs qs Ns dX; Ks ẳ Z Xs rs Ns ịT Ds rs Ns dX; f f ¼ Z @ Xsf NTs ns rs dC; fb ¼ Z @ Xt NTs ts dC ỵ Z Xs NTs bs dX: 26ị T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 329 3.3 FEM for the coupled fluid–solid system [71] At the boundary between the solid and fluid domains, denoted @ Xsf , the fluid particles and the solid moves together in the normal direction of the boundary Introducing the normal vector n ẳ ẵ nx ny ẳ ẵ nfx nfy ẳ ẵ nsx nsy , the continuous boundary condition in displacement can be written us j@ Xsf ¼ uf j@ Xsf or nus ¼ nuf on @ Xsf ð27Þ and the continuity in pressure is written rs jn ¼ ns rs ¼ p nsx nsy ¼ nT p; 28ị T where uf ẳ ẵ ufx ufx is the displacement of the fluid particles and p is the fluid pressure Using Eq (28), the force vector f f in Eq (26) can be expressed in the fluid pressure by ff ¼ Z @ Xsf NTs ns rs dC ¼ Z @ Xsf NTs nT pdC ¼ Z @ Xsf NTs nT Nf dC pf : ð29Þ For the fluid partition, the coupling is introduced in the force term f s (in Eq (14)) Using the relation between pressure and acceleration in the fluid domain rp ¼ q0 @ uf ðtÞ @t ð30Þ and the boundary condition in Eq (27), the force acting on the fluid can be described in terms of structural acceleration nrpj@Xsf ¼ q0 n @ uf ðtÞ @ us tị sj j@ Xsf ẳ q0 n j@Xsf ẳ q0 nNs d @ Xsf @t @t ð31Þ and the boundary force term of the fluid domain, f s in Eq (14), can be expressed in structural acceleration f s ¼ c20 Z NTf nf rpdC ¼ c20 @ Xsf Z @ Xsf NTf nrp dC ¼ q0 c20 Z @ Xsf €: NTf nNs dC d s ð32Þ The introduction of a spatial coupling matrix H¼ Z @ Xsf NTs nT Nf dC ð33Þ allows the coupling forces to be written as f f ẳ Hpf 34ị s f s ẳ c20 q0 HT d 35ị and The fluid–solid interaction problem can then be described by an unsymmetrical system of equations Ms q T c0 H " # Ks ds ỵ Mf f p H Kf " ds pf # ¼ fb fq ð36Þ : 3.4 NS-FEM-T3 for the fluid domain in the coupled fluid–solid system Similar to the FEM-T3, the NS-FEM-T3 also uses a mesh of triangular elements The shape functions used in NS-FEM-T3 are also identical to those in the FEM-T3, and hence the displacement field in the NS-FEM-T3 is also ensured to be continuous on the whole problem domain However, being different from the FEM-T3 which computes the stiffness matrix Kf based on the elements, the NS-FEM-T3 uses the gradient smoothing technique [29] to compute the stiffness matrix based on the ~ f In this pronodes The stiffness matrix in the NS-FEM-T3 hence is called the smoothed stiffness matrix and symbolized K ðkÞ cess, the finite element mesh in the fluid domain is further divided into smoothing domains Xf based on nodes of elements N nod f such that Xf ẳ S kẳ1 iị jị Xkị and Xf \ Xf ẳ ;, i j, in which N nod is the total number of nodes of the finite element mesh For f f ðkÞ triangular elements, the smoothing domain Xf associated with the node k is created by connecting sequentially the mid-edge-points to the centroids of the surrounding triangular elements of the node k as shown in Fig 330 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Applying the node-based smoothing operation, the pressure gradient rp in Eq (9) is used to create a smoothed pressure ~ pðkÞ on the smoothing domain XðkÞ associated with node k such as: gradient r f ~ pkị ẳ r Z kị Xf rp UfðkÞ ðxÞdX; ð37Þ ðkÞ where Uf ðxÞ is a given smoothing function that satisfies at least unity property stant smoothing function kị f xị U ẳ < 1=Akị R kị Xf Ufkị xịdX ẳ Using the following con- ðkÞ x Xf ; f ð38Þ ðkÞ :0 x R Xf ; R kị kị where Af ẳ Xkị dX is the area of Xf and applying a divergence theorem, one can obtain the smoothed pressure gradient f ~ pðkÞ that are constants over the smoothing domain XðkÞ as follows r f ~ pkị ẳ r kị where Cf ðkÞ Af Z T ðkÞ CðkÞ f pðxÞðnf ðxÞÞ dC; ð39Þ h ðkÞ as shown in Fig 2, and nf xị ẳ nkị fx kị is the boundary of the domain Xf ðkÞ the boundary Cf ðkÞ nfy i is the outward normal vector on In the NS-FEM-T3, the trial pressure function pðxÞ is the same as in Eq (12) of the FEM and therefore the force vectors f s and f q in the NS-FEM-T3 are calculated in the same way as in the FEM ~ pðkÞ on XðkÞ associated with node k can be Substituting pðxÞ in Eq (12) into Eq (39), the smoothed pressure gradient r f written in the following matrix form of nodal displacements ~ pkị ẳ r X ~ fI xk ịp ; B I ð40Þ ðkÞ I2N f ðkÞ ~ fI ðxk Þ is termed as the smoothed where N f is the total number of nodes of elements containing the common node k, and B ðkÞ pressure gradient matrix on Xf , ~ fI xk ị ẳ B " ~ ðx Þ b fIx k ~ b ðx Þ fIy k # ð41Þ and its components are calculated numerically using ~ x ị ẳ b fIx k kị Af Z NfI xịnfx xịdC; ~ x ị ẳ b fIy k ðkÞ Af Z N fI ðxÞnfy ðxÞdC: ðkÞ CfðkÞ CðkÞ f ð42Þ ðkÞ node k (k) Ω (k) Γ : field node : centroid of triangle ðkÞ Fig Triangular elements and smoothing domains Xf : mid-edge point (shaded area) associated with the nodes in the NS-FEM-T3 331 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Using the linear shape function of triangles as in Eq (12) of the FEM-T3, the pressure field in the NS-FEM-T3 is linear comðkÞ patible along the boundary Cf Hence, one Gaussian point is sufficient for line integration along each segment of boundary ðkÞ ðkÞ Cfi of Cf , the above equation can be further simplified to its algebraic form M X kị kị ~ x ị ẳ b NfI ðxGP fIx k i Þnfxi lfi ; kị Af iẳ1 43ị M X kị kị ~ x Þ ¼ b NfI ðxGP fIy k i Þnfyi lfi ; kị Af iẳ1 kị where M is the total number of the boundary segments of Cfi , xGP is the midpoint (Gaussian point) of the boundary segment i ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ of Cfi , whose length and outward unit normal vector are denoted as lfi and nfi ẳ ẵ nfxi nfyi , respectively ~ The smoothed stiffness matrix Kf of the system is then assembled by a similar process as in the FEM nod ~f ¼ K Nf X ~ kị ; K fIJ 44ị kẳ1 ~ ðkÞ is the smoothed stiffness matrix associated with node k and is calculated by where K fIJ ~ ðkÞ ¼ c2 K fIJ Z ðkÞ ~T ~ ~T B ~ B fI fJ dX ¼ c BfI BfJ Af : ðkÞ Xf ð45Þ 3.5 nES-FEM for the solid domain [50] in the coupled fluid–solid system In the nES-FEM, the domain discretization is still based on polygonal elements of arbitrary number of sides However, being different from the FEM which computes the stiffness matrix Ks based on the elements, the nES-FEM uses the gradient smoothing technique [29] to compute the stiffness matrix based on the edges The stiffness matrix in the nES-FEM hence is ~ s In this process, the solid domain Xs is divided into smoothing called the smoothed stiffness matrix and symbolized K domains associated with edges such that Xs ¼ N Sed k¼1 jị Xskị and Xiị s \ Xs ẳ ;, ij, in which N ed is the total number of edges of the solid domain For n-sided polygonal elements, the smoothing domain XðkÞ s associated with the edge k is created by connecting two endpoints of the edge to the two central points of the two adjacent elements as shown in Fig Applying the edge-based smoothing operation, the compatible displacement gradient rs us in Eq (22) is used to create a ~ s us on the smoothing domain XðkÞ associated with edge k such as: smoothed displacement gradient r s ~ s ukị xị ẳ r s Z kị Xs rs us ðxÞUðkÞ s ðxÞdX; ð46Þ boundary edge m (AB) A B Ω O (m) I (ABI ) (m) Γ (AB, BI, IA) D edge k (CD) (k) Ω C (CKDO) K Γ (k) (CK,KD,DO,OC) : field node : central point of elements (I, O, K) Fig Domain discretization and the smoothing domains (shaded areas) associated with edges of n-sided polygonal elements in the nES-FEM 332 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 where UðkÞ s ðxÞ is a given smoothing function that satisfies at least unity property stant smoothing function ðkÞ s ðxÞ U ẳ ( 1=Akị s x Xskị ; x R Xskị ; R kị Xs Uskị xịdX ẳ Using the following con- 47ị R where Askị ẳ XðkÞ dX is the area of the smoothing domain XsðkÞ and applying a divergence theorem, one can obtain the s ~ s usðkÞ that is constant over the domain XðkÞ as follows smoothed displacement gradient r s ~ s ukị xị ẳ r s Akị s Z Ckị s T ðnsðkÞ ðxÞÞ us ðxÞdC; ð48Þ ðkÞ where CsðkÞ is the boundary of the domain XsðkÞ as shown in Fig 3, and ns ðxÞ is the outward normal matrix on the boundary CsðkÞ and has the form " ðkÞ # kị nsy nsx nkị xị ẳ 49ị s kị ðkÞ nsy nsx In the nES-FEM, the trial displacement function us ðxÞ is the same as in Eq (23) of the FEM and therefore the force vectors f f and f b in the nES-FEM are calculated in the same way as in the FEM ~ s uðkÞ Substituting us ðxÞ in Eq (23) into Eq (48), the smoothed displacement gradient r on the smoothing domain XsðkÞ s associated with edge k can be written in the following matrix form of nodal displacements ~ s ukị ẳ r s X ~ sI ðxk ÞdI B ð50Þ ðkÞ I2N s ðkÞ ðkÞ where N ðkÞ s is the total number of nodes of elements containing the common edge k (N s ¼ for boundary edges and N s ¼ ~ for inner edges as shown in Fig 3), and BsI ðxk Þ is termed as the smoothed displacement gradient matrix on the smoothing domain XðkÞ s , ~ sI xk ị ẳ B ~sIx ðx Þ b k ~sIy ðx Þ b k ~sIx ðx Þ ~sIy ðx Þ b b k k ð51Þ and its nonzero components are calculated numerically using Z kị ~sIx x ị ẳ b N sI ðxÞnsx ðxÞdC; k AsðkÞ CsðkÞ Z kị ~sIy x ị ẳ N sI xịnsy xịdC: b k ðkÞ CðkÞ As s ð52Þ When a linear compatible displacement field along the boundary CsðkÞ is used, one Gaussian point is sufficient for line inteðkÞ gration along each segment of boundary Csi of CðkÞ s , the above equation can be further simplified to its algebraic form M X kị kị ~ x ị ẳ b NsI xGP sIx k i ịnsix lsi ; Askị iẳ1 M X kị kị ~sIy x ị ẳ b NsI xGP k i ịnsiy lsi ; Askị iẳ1 53ị kị where M is the total number of the boundary segments Csi , xGP is the midpoint (Gaussian point) of the boundary segment i C ðkÞ si , ðkÞ ðkÞ ðkÞ whose length and outward unit normal are denoted as lsi and nsi ẳ ẵ nsxi kị nsyi , respectively kị Eq (53) implies that only shape function values at some particular points along segments of boundary Csi 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 [28,49] for constructing nES-FEM shape functions is used, and is briefted in Section 4.2 ~ s of the system is then assembled by a similar process as in the FEM The smoothed stiffness matrix K ed ~s ¼ K Ns X ~ ðkÞ K sIJ ð54Þ k¼1 ~ ðkÞ is the smoothed stiffness matrix associated with edge k and is calculated by where K sIJ ~ ðkÞ ¼ K sIJ Z XðkÞ s ~ T DB ~ sJ dX ẳ B ~ T DB ~ sJ Akị : B sI sI s ð55Þ 333 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 3.6 A coupled NS/nES-FEM for the fluid–solid interaction problems As shown in Sections 3.4 and 3.5, the difference between the FEM and the smoothed FEM is the way to compute the stiffness matrix In the FEM, the stiffness matrices Kf and Ks is computed based on the elements While in the NS-FEM-T3, the ~ f is computed based on the node-based smoothing domains, and K ~ s is computed based on smoothed stiffness matrices K the edge-based smoothing domains 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 NS/nES-FEM will be expressed in the following form " € # " ~ Ks ds ỵ Mf f p Ms q0 c20 HT H ~f K #" ds pf # ¼ fb fq ; ð56Þ ~ s in Eq (54) and K ~ f in Eq (44) where Ks and Kf in Eq (36) are replaced, respectively, by K Note that the present dynamic approach requires the concident nodes at the interactive boundary of the fluid and solid domains, while for the static analysis, the mesh density of fluid domain and solid domain can be performed independently It is because in the present dynamic analysis, whole the coupled fluid–solid system is analyzed at the same time and the continuity of the displacement fields and traction fields should be automatically satisfied in the formulation through the concident nodes on the interactive boundary While in the static analysis, the boundary force of the solid domain affected to the fluid domain f s in Eq (32) is small and can be ignored In this case, the system of fluid–solid problem can be separated into two discrete problems in fluid domain and solid domain, and the mesh density of fluid domain and solid domain can be performed independently And the fluid domain can be analyzed in advance to find the values of pressure at nodes of fluid domain And the boundary force of the fluid domain affected to the solid domain f f can be determined by approximating a polynomial function of interactive forces at the interactive boundary and then transformed into the forces at nodes on the solid domain as shown in Fig Finally, the static analysis of the solid domain can be performed independently to find the values of the displacement field at nodes of solid domain Dynamic analysis for the fluid–solid interaction problems Because nES-FEM is spatially and temporally stable [28,42,50], the coupled NS/nES-FEM 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 (56) for the fluid–solid interaction problems using the coupled NS/nES-FEM can be expressed as follows: ~ ỵ Cx_ ỵ Mx ẳ F; Kx 57ị where x¼ " ds pf # ; ~¼ K " ~s K # H ; ~f K M¼ Ms T c0 H q Mf Fẳ ; fb fq 58ị ~ and M, and C is the damping matrix Using the Rayleigh damping, matrix C is assumed to be a linear combination of K ~ C ẳ aM ỵ bK; 59ị where a and b are the Rayleigh damping coefficients node p Αp : field node : mid-edge point : central point of n-sided polygonal elements Fig Area Ap of field node p in a mesh of n-sided polygonal elements 334 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Many existing standard schemes can be used to solve the second-order time dependent problems, such as the Newmark method, Crank–Nicholson method, etc [72] In this paper, the Newmark method is used When the current state at t ¼ t is € Þ, we aim to find a new state ðx1 ; x_ ; x €1 Þ at t1 ẳ t0 ỵ hDt where 0:5 h 1, using the following known as ðx0 ; x_ ; x formulations: aỵ ~ x1 ẳ hDtF1 ỵ hịDtF0 ỵ a ỵ Mx0 ỵ Mx_ ỵ ẵb hịDt Kx ~ 0; M ỵ b þ hDtÞK h Dt h Dt h ð60Þ x_ ¼ 1h x_ ; ðx1 x0 Þ hDt h 61ị ẳ x 1h : x ðx_ x_ Þ hDt h ð62Þ Without the damping and forcing terms, Eq (57) is reduced to a homogenous differential equation: ~ ỵ Mx ẳ 0: Kx 63ị A general solution of such a homogenous equation can be written as expixtị; xẳx ð64Þ is the amplitude of the sinusoidal displacements and x is the angular frequency On its substitution where t indicates time, x into Eq (63), the natural frequency x can be found by solving the following eigenvalue equation ~ x2 Mx ẳ 0: ẵK 65ị 4.1 Lump mass matrix in the nES-FEM In dynamic analysis using the nES-FEM, we can use the usual consistent mass matrix defined in Eq (26) to compute However, this computational process will be rather difficult and cumbersome due to the sub-division of polygonal elements into A n(k) s g2 n(k) s (k) n(k) s g3 g4 B : field node g1 Ωs n(k) s (k) Γs : central point of n-sided polygonal element : Gauss point Fig Gauss points of the smoothing domains associated with edges for n-sided polygonal elements in the nES-FEM Fig A scheme for determining the forces at nodes on the solid domain by approximating a polynomial function of interactive forces at the interactive boundary 335 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Table Shape function values at different sites on the smoothing domain boundary associated with the edge 1–6 in Fig Site Node Node Node Node Node Node Node Node Node Description A B g1 1.0 0 0 0 0 1/6 1/5 7/12 1.0 0 0 0 1/6 1/12 0 1.0 0 0 0 1/6 1/12 0 1.0 0 0 1/6 1/12 0 0 1.0 0 0 1/6 1/12 0 0 1.0 0 1/6 1/5 1/12 0 0 0 1.0 0 1/5 0 0 0 0 1.0 0 1/5 0 0 0 0 1.0 1/5 Field node Field node Field node Field node Field node Field node Field node Field node Field node Centroid of element Centroid of element g2 1/12 1/12 1/12 1/12 1/12 7/12 0 Mid-segment point of Csi g3 1/10 0 0 6/10 1/10 1/10 1/10 Mid-segment point of Csi g4 6/10 0 0 1/10 1/10 1/10 1/10 Mid-segment point of Csi ðkÞ Mid-segment point of Csi ðkÞ ðkÞ ðkÞ smoothing cells, and also due to the sub-division of the shape function of polygonal elements into linear piecewise shape functions on smoothing cells [28] In order to avoid such difficulty and to incearse the computational efficiency, in this paper, we propose the well-known lumped mass matrix for the n-sided polygonal elements Xe such as Nn Nn p¼1 p¼1 M ¼ A Mp ¼ A qtAp Ip ; ð66Þ where Mp is the lump mass matrix of pth node; Ip is the identity matrix of size 2; N n is the total number of nodes of the problem domain; Ap is the area surrounding 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 pth node as shown in Fig 4; q 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 [73] 4.2 Shape functions of the nES-FEM The general shape functions of n-sided polygonal elements in the nES-FEM were presented by Liu and Nguyen Thoi Trung in Ref [28] However, in actual computation of the nES-FEM, it is not necessary to use such shape functions to compute the ~ s Instead, as shown in Section 3.4, we only need to evaluate the shape function values at Gauss smoothed stiffness matrix K ~ s This computational process is very simple and perpoints along boundary segments of smoothing domains to compute K formed in 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 smoothing domains by linear interpolation from the available shape function values of n field nodes and central point Fig Model of the 2D deformable solid backed by a closed box filled with water 336 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig A discretization using n-sided polygonal elements for solid domain and triangular elements for fluid domain of the 2D deformable solid backed by a closed box filled with water Table Convergence of first coupled eigenmodes Method FEM T3-FEM T3 FEM Q4-FEM Q4 NS/nESFEM Reference solution Comsol solution Degrees of freedom of solid domain 48 242 840 1248 7.6201 6.4495 5.5874 5.2576 5.1920 6.0990 5.5927 5.3329 5.5118 5.3454 5.2765 5.4258 5.3128 5.2676 Fig Convergence of the first coupled eigenmode by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nES-FEM 337 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig 10 Comparison of eight coupled eigenmodes of the fluid–solid system by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nESFEM Table Values of seven first coupled eigenmodes Method FEM T3-FEM T3 FEM Q4-FEM Q4 NS/nESFEM Reference solution Comsol solution Mode sequence number 6.0990 5.5927 5.3329 5.2576 5.1920 11.6333 10.7703 10.0501 10.0322 10.0479 15.3598 15.2919 15.2270 15.2196 15.2063 19.8868 18.5403 16.8509 17.0497 17.1889 27.8838 26.1624 23.1079 23.7424 23.5445 30.7824 30.6069 30.1753 30.3495 29.8955 38.1610 35.9586 30.6115 32.0158 30.0462 Fig and Table give explicitly the shape function values at different points of the smoothing domain associated with ðkÞ the edge 1–6 The number of support nodes for the smoothing domain is (from #1 to #9) We have segments Csi on CðkÞ s (1A, A6, 6B, B1) Each segment needs only one Gauss point, and therefore, there are a total of Gauss points (g1, g2, g3, g4) used for the entire smoothing domain XsðkÞ associated with edge k (1–6), and the shape function values at these 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 points In other words, these points carry no additional independent field variable Therefore, the total degrees of freedom (DOFs) of a nES-FEM model will be exactly the same as the standard FEM using the same set of nodes Numerical examples In this section, two numerical examples are performed to show the advantageous properties of the proposed coupling NS/ nES-FEM for 2D fluid–solid interaction problems The numerical results of coupled NS/nES-FEM will be compared with those of the coupled FEM-T3/FEM-T3 using triangular elements for both fluid and solid domains, and of the coupled FEM-Q4/FEMQ4 using quadrilateral elements In addition, to illustrate the convergent property of the numerical methods, the reference solution 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 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 the material constants are given by: density of the material qs ¼ 2500 kg=m2 ; elastic module E ¼ 2:1 109 N=m2 ; poisson’s ratio t ¼ 0:3 A closed box filled with water is attached to the solid and has the dimension of 10 m m The following data 338 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig 11 Shape of eight coupled eigenmodes of the fluid–solid system by NS/nES-FEM (a)Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6; (g) Mode 7; (h) Mode were used in the fluid, q ¼ 1000 kg=m2 and speed of air c ¼ 1500 m=s2 The model is shown in Fig and a discretization of the model using n-sided polygonal elements for solid domain and triangular elements for 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 coupling FEM-Q8/FEM-Q8 with 729 degree of freedom (DOFs) for fluid and 1290 DOFs for solid is used as reference solution Table and Fig display the convergence of the first coupled eigenmode by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nES-FEM The results show that with the same DOFs, the NS/nES-FEM gives more accurate results and converges better than the reference coupled methods Fig 10 and Table compare seven values of coupled eigenmodes by different coupled methods Again, it is seen that the results by the NS/nES-FEM is closest to the reference solution and comsol solution Besides, it can be seen that the reference solution matches very well with comsol solution Hence, it is suitable to use the results by the FEM-Q8/FEM-Q8 as the reference solution Furthermore, Fig 11 shows the shape and value of eight first coupled eigenmodes by the NS/nES-FEM It is seen that the shapes of eigenmodes express correctly the real physical modes 339 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig 12 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 NS/nES-FEM Fig 13 Forced frequency response 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 by the NS/nES-FEM Table Values of seven first coupled and uncoupled eigenmodes of solid Method Without coupling With coupling Mode sequence number 2.9891 5.3329 7.57730 10.0501 13.5226 15.2270 15.2688 16.8509 20.2439 23.1079 27.3707 30.1753 30.2838 30.6115 5.1.2 Forced vibration analysis The forced frequency response analysis for the fluid–solid system by the NS/nES-FEM is now investigated First, the force applied to structure is a harmonic vertical point load Fx; xị ẳ dc xịixeixt , where dc ðxÞ is the Dirac function at the point x Considering the case t ¼ 0, the point x is put at point A(2.0, 5.0) as shown in Fig 8, and x=2p is changed from Hz to 17 Hz 340 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig 14 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 NS/ nES-FEM and by the nES-FEM for solid domain without couping with the fluid domain Fig 15 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 NS/nES-FEM and FEM-Q4/FEM-Q4 where the value of three first coupled eigenfrequencies appears as shown in Fig 11 The displacement responses measured at the loaded point A(2.0, 5.0) is shown in Fig 12 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 11 Similarly, if 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 13 Again, it is seen clearly that the peaks of the three first responses occur exactly at the T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 341 Fig 16 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 NS/ nES-FEM and FEM-Q4/FEM-Q4 Fig 17 Model of 2D deformable water dam Fig 18 A discretization using n-sided polygonal elements for the dam and triangular elements for fluid domain 342 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 Fig 19 Convergence of the error of first coupled eigenmode by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nES-FEM Fig 20 Shape of eight coupled eigenmodes of the fluid–solid system by the NS/nES-FEM (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6; (g) Mode 7; (h) Mode values of three first eigenfrequencies These results imply that the forced frequency response analysis by the NS/nES-FEM can provide sufficient information about the eigenfrequencies of the fluid–solid interaction system, where the peaks of the response occur T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 343 Fig 21 A discretization of noncoincient nodes at the interactive boundary using n-sided polygonal elements for the dam and triangular elements for fluid domain Fig 22 Convergence of the strain energy of solid domain by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nES-FEM In addition, the forced frequency response analysis also shows clearly the difference of eigenfrequencies between the solid system without couping with the fluid and the fluid–solid interaction system as shown in Table and Fig 14 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 use the fluid–solid interaction system to model accurately the solid system coupling with the fluid Fig 15 compares the forced frequency response analysis between the NS/nES-FEM and FEM-Q4/FEM-Q4 It is seen that the displacement responses of the NS/nES-FEM is closest to those of the reference solution by FEM-Q8/FEM-Q8 In addition, as depicted in Fig 15, at lower frequencies, both the coupled NS/nES-FEM and FEM-Q4/FEM-Q4 can provide quite good results However, with the increase in the frequency, the results obtained from the coupled NS/nES-FEM still agree well with the reference results and provide accurate eigenfrequencies prediction in these frequencies range; while the deviation between the coupled FEM-Q4/FEM-Q4 and the reference is becoming larger as the frequency becomes higher Fig 16 compares the transient response between the NS/nES-FEM and FEM-Q4/FEM-Q4 Again, it is seen that the displacement responses of the NS/nES-FEM is closest to those of the reference solution by FEM-Q8/FEM-Q8 This numerical example hence shows that the coupled NS/nES-FEM provides more accurate results in eigenfrequencies, frequency responses and transient responses than those of many existing coupled FEM/FEM models 344 T Nguyen-Thoi et al / Applied Mathematics and Computation 232 (2014) 324–346 5.2 2D deformable water dam A deformable water dam is fixed at the dam’s foundation and has dimensions as shown in Fig 17 The dam has the material constants as: density of the material qs ¼ 2500 kg=m2 ; elastic module E ¼ 2:1 109 N=m2 ; poisson’s ratio t ¼ 0:3, and The water is attached to the dam and has the dimension of 10 m m The following data were used in the fluid, q ¼ 1000 kg=m2 and speed of air c ¼ 1500 m=s2 A discretization using n-sided polygonal elements for solid domain and triangular elements for fluid domain is shown in Fig 18 5.2.1 Free vibration analysis First, the eigenmodes analysis for the fluid–solid system is investigated The solution of coupled FEM-Q8/FEM-Q8 with 697 degree of freedom (DOFs) for fluid and 1570 DOFs for solid is used as reference solution Fig 19 shows the convergence of the error of first coupled eigenmode by different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and NS/nES-FEM The results show that with the same DOFs, the NS/nES-FEM is the best one Fig 20 shows the shape and value of eight first coupled eigenmodes by the NS/nES-FEM It is again seen that the shapes of eigenmodes express correctly the real physical modes 5.2.2 Static analysis In this example, the strain energy of solid domain is studied For this case, the meshes using independent densities of fluid domain and solid domain on the interface boundaries are used as shown in Fig 21 The fluid domain is subjected to a pressure load P = 1.5 GPa at the point (9, 1) and a two-step sequential analysis for static analysis as mentioned at the end of Section 3.6 is performed The solution of coupled FEM-Q8/FEM-Q8 with 697 degree of freedom (DOFs) for fluid and 1570 DOFs for solid is still used as reference solution Fig 22 shows the convergence of the strain energy of the solid domain with different independent mesh dentisities of fluid domain and solid domain It can be seen that the results of the two-step sequential analysis NS/nES-FEM are also better than those of FEM-T3/FEM-T3 and FEM-Q4/FEM-Q4 This result hence also verifies the advantage of the present approach NS/nES-FEM for the static analysis compared to the other reference approaches Conclusion This paper attempts to combine the NS-FEM-T3 and nES-FEM to give the coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems based on the pressure–displacement formulation In the present method, the NS-FEM-T3 using three-node triangular elements is used for the fluid domain and the gradient of pressure is smoothed, while the nES-FEM using n-sided polygonal elements is used for the solid domain and the gradient of displacement is smoothed This gradient smoothing technique can provide proper softening effect, which will effectively relieve the overly stiff behavior of the FEM model and thus improve significantly the solution of coupled system Using the NS-FEM-T3 for the fluid domain helps combine the simple advantage of discretizing the domain by three-node triangular elements with the advantage of node-based smoothing technique While using the nES-FEM for the solid domain helps combine the advantage of the edge-based smoothing technique with the practical advantage of discretizing the domain by n-sided polygonal elements which are suitable for many material microstructures in polycrystalline alloys and biomechanics In addition, a simple lump mass matrix for the nES-FEM is also proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nES-FEM Some numerical examples have been presented to illustrate the effectiveness of the coupled NS/nES-FEM compared with some existing methods for 2D fluid–solid interaction problems The numerical examples show that the coupled NS/nES-FEM provides more accurate results 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