A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems

A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid–solid interaction problems

A smoothed coupled NS/nES-FEM for dynamic analysis of 2D

fluid–solid interaction problems

T Nguyen-Thoia,b,⇑, P Phung-Vanb, S Nguyen-Hoanga, Q Lieu-Xuanc

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 iscombined to give the smoothed coupled NS/nES-FEM for dynamic analysis of two-dimensional (2D) fluid–solid interaction problems based on the pressure–displacementformulation In the present method, the NS-FEM-T3 is used for the fluid domain and thegradient of pressure is smoothed, while the nES-FEM is used for the solid domain andthe gradient of displacement is smoothed This gradient smoothing technique can provideproper softening effect, which will effectively relieve the overly stiff behavior of the FEMmodel and thus improve significantly the solution of coupled system Some numericalexamples have been presented to illustrate the effectiveness of the coupled NS/nES-FEMcompared with some existing methods for 2D fluid–solid interaction problems

Ó2014 Elsevier Inc All rights reserved

1 Introduction

The need of computing the dynamic behavior of two-dimensional (2D) fluid–solid system arises in many important neering problems The dam–reservoir interaction during earthquakes and fluid containers subjected to dynamic loads areexamples of this class of problems However, predicting the response of fluid–solid coupled systems is generally a difficulttask 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

engi-Numerical analysis of fluid–solid interaction problems involves the modeling of fluid domain, solid domain, and the action between these two domains The finite element method (FEM), the boundary element method (BEM) and the mesh-

numerical solution of the fluid–solid interaction problems can be performed using only FEM, or a coupled BEM/FEM with a

In numerical computation using the conventional FEM for 2D solid mechanics problems, the use of triangular and rilateral elements is well-established However, there is significant bottleneck in generating quality meshes using polygonalelements for complex geometries The use of elements with an arbitrary number of sides will provide greater flexibility and

quad-0096-3003/$ - see front matter Ó 2014 Elsevier Inc All rights reserved.

⇑ 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).

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

better accuracy to solve problems that arise in solid mechanics and biomechanics Since material microstructure in talline alloys and piezoelectrics, and bone can be described through polygonal sub-domains, the use of n-sided polygonalfinite elements (nFEM) in such applications is a natural choice However so far, although there have been many researches

polycrys-[19–27]about the nFEM using polygonal elements, these nFEM still possesses some among four following main tages: (1) the construction of shape functions is complicated; (2) the numerical integration on the polygonal elements is dif-ficult; (3) they overestimate excessively the stiffness of the problem which leads to poor accuracy in solutions; (4) they aresubjected to locking in the problems with bending domination and incompressible materials

disadvan-In the 2D fluid–solid interaction problems, the geometrical domain of fluid is usually simpler than that of the solid main Hence, the three-node linear triangular elements (FEM-T3) are preferred due to its simplicity, robustness, and effi-ciency of adaptive mesh refinements However, the FEM-T3 element also possesses ‘‘overly stiff’’ property which causesthe following certain drawbacks: (1) they overestimate excessively the stiffness of the problem which leads to poor accuracy

do-in solutions; (2) they are subjected to lockdo-ing do-in the problems with benddo-ing domdo-ination and do-incompressible materials

In order to overcome these disadvantages of both triangular element and n-sided polygonal element, Liu and Nguyen Thoi

In the S-FEM models, the finite element mesh is used similarly as in the FEM models However these S-FEM models uate the weak form based on smoothing domains created from the entities of the element mesh such as cells/elements, ornodes, or edges These smoothing domains can be located inside the elements (CS-FEM and nCS-FEM) or cover parts of adja-cent elements (NS-FEM, nNS-FEM, ES-FEM and nES-FEM) These smoothing domains are linear independent and hence en-sure stability and convergence of the S-FEM models They cover parts of adjacent elements, and therefore the number ofsupporting nodes in smoothing domains is larger than that in elements This leads to the bandwidth of stiffness matrix inthe S-FEM models to increase and the computational cost is hence higher than those of the FEM However, also due to con-tributing of more supporting nodes in the smoothing domains, the S-FEM models often produce the solution that is muchmore accurate than that of the FEM Therefore in general, when the efficiency of computation (computation time for thesame accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM models are more efficient

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 neededand only the compatibility of the nodal shape functions on the boundaries of these smoothing cells is required in the formu-lation This gives tremendous freedom to compute these shape function values for the S-FEM, and they can be easily obtained

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 ofbenchmark and practical mechanics problems The S-FEM models have also been further investigated and applied to various

[63,64], and some other applications[65,66], etc

the upper bound property in strain energy; (2) it is immune naturally from the volumetric locking; (3) it achieves accurate 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 Thesesix properties are very promising to apply the NS-FEM-T3 effectively in the more complicated non-linear problems The

prop-erties for the 2D solid mechanics analyses: (1) the numerical results are often found super-convergent and much more rate than those of nFEM using polygonal elements with the same sets of nodes; (2) there are no spurious non-zeros energymodes found and hence the method is also stable and works well for dynamic analysis; (3) the implementation of the meth-

accu-od is straightforward and no penalty parameter is used, and the computational efficiency is better than nFEM using the samesets of nodes; (4) the smoothed stiffness matrix can be computed directly from the shape function values at Gauss pointsalong boundary segments of smoothing domains, without constructing the shape functions explicitly as in the FEM Thiscomputational 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 T3 is used for the fluid domain and the gradient of pressure is smoothed, while the nES-FEM is used for the solid domain andthe gradient of displacement is smoothed This gradient smoothing technique can provide proper softening effect, which willeffectively relieve the overly stiff behavior of the FEM model and thus improve significantly the solution of coupled system

NS-FEM-Using the NS-FEM-T3 for the fluid domain helps combine the simple advantage of discretizing the domain by three-node

domain helps combine the flexible and practical advantage of discretizing the domain by n-sided polygonal elements with

effectiveness of the coupled NS/nES-FEM compared with some existing methods for 2D fluid–solid interaction problems

by n-sided polygonal elements which are suitable for many material microstructures in polycrystalline alloys and

ap-proach uses only the strain smoothed techniques in the analysis which is more simple than two reference apap-proaches

[14,15]using different complicated coupled techniques The coupling ES-FEM/BEM[14]uses the smoothed finite elementmethod (ES-FEM) for the solid domain and the boundary element method (BEM) for the fluid domain While the immersed

2 Governing equations for fluid–solid interaction problems

For the fluid–solid system, the solid is described by the differential equation of motion for a continuum body assumingsmall deformations and the fluid is described by the wave equation in which the fluid is inviscid, irrotational and only under-goes small translations

Coupling conditions at the boundary between the solid and fluid domains ensure the continuity in displacement and

the body force;qsis the density of the material; nsis the boundary normal matrix pointing outward from the solid domainwritten as

3 A coupled NS/nES-FEM for the fluid–solid interaction problems

ðr vfÞTrp dV ¼ c2Z

@Xf

vfnfrp dS þ c2Z

X f

vf

@qf

@tdV : ð11Þ

NT f

NT f

@qf

@tdX:

ð15Þ

Xs

vT

contains the finite element shape functions for the solid domain

3.3 FEM for the coupled fluid–solid system[71]

boundary condition in displacement can be written

and acceleration in the fluid domain

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-T3are also identical to those in the FEM-T3, and hence the displacement field in the NS-FEM-T3 is also ensured to be continuous

Applying the node-based smoothing operation, the pressure gradientrp in Eq.(9)is used to create a smoothed pressure

con-stant smoothing function

fx nðkÞ fy

is the outward normal vector on

f

written in the following matrix form of nodal displacements

NfIðxÞnðkÞfyðxÞdC:

ð42Þ

: centroid of triangle : field node

Fig 2 Triangular elements and smoothing domainsXðkÞ

(shaded area) associated with the nodes in the NS-FEM-T3.

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

In the nES-FEM, the domain discretization is still based on polygonal elements of arbitrary number of sides However,

Γ

: central point of elements (I, O, K) : field node

(CK,KD,DO,OC)

Ω

C(CD)

(CKDO)

edge k

Γ(k)

D O

con-stant smoothing function

Z

CðkÞs

ðnðkÞ

NsIðxÞnðkÞ

syðxÞdC:

ð52Þ

no derivatives of the shape function are required This gives tremendous freedom in shape function construction In this

3.6 A coupled NS/nES-FEM for the fluid–solid interaction problems

the FEM, the system of equations for the 2D fluid–solid interaction problems using the coupled NS/nES-FEM will beexpressed in the following form

Note that the present dynamic approach requires the concident nodes at the interactive boundary of the fluid and soliddomains, 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 tinuity of the displacement fields and traction fields should be automatically satisfied in the formulation through the con-cident nodes on the interactive boundary While in the static analysis, the boundary force of the solid domain affected to the

two discrete problems in fluid domain and solid domain, and the mesh density of fluid domain and solid domain can be formed independently And the fluid domain can be analyzed in advance to find the values of pressure at nodes of fluid do-

polynomial function of interactive forces at the interactive boundary and then transformed into the forces at nodes on

the values of the displacement field at nodes of solid domain

4 Dynamic analysis for the fluid–solid interaction problems

such as free and forced vibrations analyses

interaction problems using the coupled NS/nES-FEM can be expressed as follows:

Many existing standard schemes can be used to solve the second-order time dependent problems, such as the Newmark

4.1 Lump mass matrix in the nES-FEM

How-ever, this computational process will be rather difficult and cumbersome due to the sub-division of polygonal elements into

Fig 6 A scheme for determining the forces at nodes on the solid domain by approximating a polynomial function of interactive forces at the interactive

: central point of n-sided polygonal element

B

g49

: field node

g37

8

s (k)

n

n(k) s

s (k)

n

s

Fig 5 Gauss points of the smoothing domains associated with edges for n-sided polygonal elements in the nES-FEM.

smoothing cells, and also due to the sub-division of the shape function of polygonal elements into linear piecewise shape

M ¼ ANn

p¼1Mp¼ ANn

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

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

per-formed in 3 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 shapefunction values of n field nodes at step 1

 Step 3: Evaluate the shape function values at Gauss points along boundary segments of smoothing domains by linearinterpolation from the available shape function values of n field nodes and central point

Table 1

Shape function values at different sites on the smoothing domain boundary associated with the edge 1–6 in Fig 5

Site Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Description

Fig 8 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 2

Convergence of first coupled eigenmodes.

Fig 5andTable 1give explicitly the shape function values at different points of the smoothing domain associated with

si onCðkÞ s

(1A, A6, 6B, B1) Each segment needs only one Gauss point, and therefore, there are a total of 4 Gauss points (g1, g2, g3, g4) used

eval-uation 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 independentfield variable Therefore, the total degrees of freedom (DOFs) of a nES-FEM model will be exactly the same as the standardFEM using the same set of nodes

FEM-Q4/FEM-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  1 m The solid is fixed supported at two ends and

Fig 10 Comparison of eight coupled eigenmodes of the fluid–solid system by 3 different coupled methods: FEM-T3/FEM-T3, FEM-Q4/FEM-Q4 and FEM.

NS/nES-Table 3

Values of seven first coupled eigenmodes.