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...

DOI: 10.1142/S0219876213400033

AN APPLICATION OF THE ES-FEM IN SOLID DOMAIN FOR DYNAMIC ANALYSIS OF 2D FLUID–SOLID

INTERACTION PROBLEMS

T NGUYEN-THOI∗,†,,∗∗, P PHUNG-VAN†, T RABCZUK§,

H NGUYEN-XUAN∗,†and C LE-VAN¶

∗ Department of Mechanics , Faculty of Mathematics &

Computer Science , University of Science, VNU-HCM

227 Nguyen Van Cu , District 5 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

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 interac- tion 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 gradi- ent smoothing technique can provide proper softening effect, and thus improve signifi- cantly 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.

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1 Introduction

The need of computing the dynamic behavior of two-dimensional (2D) fluid–solidsystem arises in many important engineering problems The dam-reservoir interac-tion during earthquakes and fluid storage containers subjected to dynamic loads areexamples of this class of problems However, predicting the response of fluid–solidcoupled systems is generally a difficult task In most practical problems, it is notpossible to obtain closed form analytical solutions for the coupled systems As aresult, much effort has been performed in order to develop the different numericalmethods 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) andthe 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/FEMwith 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, the3-node linear triangular element (FEM-T3) are preferred by many engineers due toits simplicity and efficiency of adaptive mesh refinements However, the FEM-T3element possesses “overly stiff” property which causes the following certain draw-backs: (1) They overestimate excessively the stiffness of the problem which leads

to poor accuracy in solutions; (2) They are subjected to locking in the problemswith bending domination and incompressible materials In order to overcome thesedisadvantages, some new finite elements were proposed Allman [1984, 1988] intro-duced rotational degrees of freedom at the element nodes to achieve an improvementfor the overly stiff behavior Elements with rotational degrees of freedom were alsoconsidered by Bergan and Felippa [1985] and Cook [1991] Piltner and Taylor [2000]combined the rotational degrees of freedom and enhanced strain modes to give atriangular element which can achieve a higher convergence in energy and deal withthe nearly incompressible plane strain problems However, using more degrees offreedom at the nodes limits the practical application of those methods Dohrmann

et al [1998] presented a weighted least-squares approach in which a linear

displace-ment field is fit to an eledisplace-ment’s nodal displacedisplace-ments The method is claimed to

be computationally efficient and avoids the volumetric locking problems However,

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more nodes are required on the element boundary to define the linear ment field.

displace-Recently, in order to “soften” the system using FEM-T3, Liu and Thoi [2010] incorporated the gradient smoothing technique of meshfree methods

Nguyen-[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 formbased on smoothing domains created from the entities of the element mesh such ascells/elements, or nodes, or edges

Among these S-FEM models, the ES-FEM-T3 [Liu et al (2009)] using

triangu-lar elements shows some following excellent properties for the 2D solid mechanicsanalyses: (1) The numerical results are often found super-convergent and very accu-rate; (2) The method is stable and works well for dynamic analysis; (3) The imple-mentation of the method is straightforward and no penalty parameter is used TheES-FEM-T3 has been developed forn-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 togive a so-called the face-based smoothed finite element method (FS-FEM) [Nguyen-

Thoi et al (2009); Nguyen-Thoi et al (2009)].

This paper hence attempts to extend the ES-FEM-T3 to the dynamic yses of 2D fluid–solid interaction problems based on the pressure-displacementformulation In this coupled method, both solid and fluid domain is discretized

anal-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 smoothingdomains associated with the edges of triangles is used to smooth the gradient

of displacement This gradient smoothing technique can provide proper softeningeffect, which will effectively relieve the overly stiff behavior of the standard FEMmodel and thus improve significantly the solution of coupled system Some numericalexamples have been presented to illustrate the effectiveness of the proposed cou-pled method compared with some existing methods for 2D fluid–solid interactionproblems

2 Governing Equations for Fluid–Solid Interaction Problems

The fluid–solid interaction problem is schematically sketched in Fig 1 It consists of

a fluid domain, Ωf, and a solid domain, Ωs The interaction boundary between the

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fluid domain and the solid domain is denoted,∂Ω sf; two remaining fluid boundariesare given by prescribed pressure,p = ¯p on ∂Ω p, and a prescribed normal pressure

gradient nf ∇p = ¯ w on ∂Ω z; the remaining solid boundaries are given by prescribed

displacement, us= ¯ u on∂Ω u, and prescribed force vector, nsσs= ¯ ts on∂Ω t.For the fluid–solid system, the solid is described by the differential equation ofmotion 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 undergoessmall translations

Coupling conditions at the interaction boundary between the solid and fluiddomains ensure the continuity in displacement and pressure between the domains.Hence, the governing equations and boundary conditions were described in general

where for the fluid,p(t) is dynamic pressure; q f(t) is the added fluid mass per unit

volume;c0 is the speed of sound; ∇ = [∂/∂x ∂/∂y] T and∇2=∇ · ∇ = ∂2/∂x2+

∂2/∂y2; nf = [n fx n fy] is the boundary normal vector pointing outward from thefluid domain; and for the solid,σ s= [σ x σ y σ xy]T is the stress; us= [u sx u sy]T is

the displacement; bs= [b sx b sy]T is the body force;ρ sis the density of the material;

nsis the boundary normal matrix pointing outward from the solid domain written as

where Ds(3× 3) is a symmetric positive definite (SPD) matrix of material constants.

3 A Coupled FEM/ES-FEM for the Fluid–Solid

Interaction Problems

The weak form of the differential equation is derived by multiplying the first term

in Eq (1) with a weight function,v f ∈ H1, and integrating over the fluid domain,

Due to∂Ω f =∂Ω sf ∪ ∂Ω p ∪ ∂Ω z, Eq (9) is rewritten

f ] = Nf, we obtain the finite element formulation for the

fluid domain from Eq (11) as

in which fsrepresents 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

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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 ∈ H1, and integrating over the solid domain,

Ωs,



Ωs

vT s

of the solid domain becomes

Supposing the solid domain Ωsis discretized intoNnod

where vector dscontains the approximate displacement values at nodes; cscontains

the chosen test values at nodes; and Nscontains the finite element shape functions

for the solid domain

Similarly as in the fluid domain, by choosing Nnod

s linear independent vectors

cs such that [vs1 vs2 · · · v sNnod

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∂Ω sf, thefluid particles and the solid moves together in the normal direction of the bound-

ary Introducing the normal vector n = [nx n y] = [n fx n fy] = [−n sx −n sy], thecontinuous boundary condition in displacement can be written

where uf = [u fx u fx]T is the displacement of the fluid particles andp is the fluid

pressure Using Eq (28), the force vector ff in Eq (26) can be expressed in thefluid pressure by

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

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 theFEM-T3, and hence the displacement field in the ES-FEM-T3 is also ensured to becontinuous on the whole problem domain However, being different from the FEM-

T3 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

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Γ Ω

inner edge k

: centroid of triangles : field node

Fig 2 (Color online) Triangular elements and the smoothing domains associated with edges in ES-FEM-T3.

the smoothed stiffness matrix and symbolized ˜ K In this process, the finite element

mesh in the solid domain is divided into smoothing domains Ω(k) s based on edges

of elements such that Ωs=Ned

endpoints of the edge to centroids of adjacent elements as shown in Fig 2

Applying the edge-based smoothing operation, the compatible displacement dient ∇ sus in Eq (22) is used to create a smoothed displacement gradient ˜ ∇ suson

gra-the smoothing domain Ω(k) s associated with edgek such as:

Ω(k) s dΩ is the area of Ω(k) s and applying a divergence theorem, one

can obtain the smoothed displacement gradient ˜ ∇ su(k) s that is constant over thedomain Ω(k) s as follows

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where Γ(k) s is the boundary of the domain Ω(k) s as shown in Fig 2, and n(k) s (x) is

the outward normal matrix on the boundary Γ(k) s and has the form

In the ES-FEM-T3, the trial displacement function us(x) is the same as in

Eq (23) of the FEM-T3 and therefore the force vectors ff and fb in the

ES-FEM-T3 are calculated in the same way as in the FEM-ES-FEM-T3

Substituting us(x) in Eq (23) into Eq (39), the smoothed displacement gradient

˜

∇ su(k) s on the smoothing domain Ω(k) s associated with edgek can be written in the

following matrix form of nodal displacements

s = 3 for boundary edges andN (k)

s = 4 for inner edges as shown in Fig 2), and

(Gaussian point) of the boundary segment of Γ(k) si , whose length and outward unit

normal are denoted asl (k)

Equation (44) implies that only shape function values at some particular pointsalong segments of boundary Γ(k) si are needed and no derivatives of the shape function

are required The smoothed stiffness matrix ˜ Ks of the system is then assembled by

a similar process as in the FEM

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,

the smoothed stiffness matrix ˜ Ksis computed based on the edge-based smoothing

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 theFEM, the system of equations for the 2D fluid–solid interaction problems using thecoupled FEM-T3/ES-FEM-T3 will be expressed in the following form

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-T3can be expressed as follows:

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and C is the damping matrix Using the Rayleigh damping, matrix C is assumed

to be a linear combination of ˜ K and M,

whereα and β are the Rayleigh damping coefficients.

Many existing standard schemes can be used to solve the second-order timedependent problems, such as the Newmark method, Crank–Nicholson method, etc.[Smith and Griffiths (1998)] In this paper, the Newmark method is used Whenthe current state at t = t0 is known as (x0, ˙x0, ¨x0), we aim to find a new state

(x1, ˙x1, ¨x1) att1=t0+θ∆t where 0.5 ≤ θ ≤ 1 Note that the Newmark method is

a one step implicit method for solving the transient problem, and by choosing theparameterθ 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 ofsolutions [Smith and Griffiths (1998)] The formulation of the method is expressed

as follows:



α + 1θ∆t

˜

A general solution of such a homogenous equation can be written as

wheret 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

5 Numerical Examples

In this section, two numerical examples are performed to show the advantageousproperties 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-T3will be compared with those of the coupled FEM-T3/FEM-T3 using standard trian-gular elements for both fluid and solid domains, and of the coupled FEM-Q4/FEM-Q4 using quadrilateral elements for both fluid and solid domains In addition, toillustrate the convergent property of the numerical methods, the reference solution

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