Smoothed FE-Meshfree method for solid mechanics problems

Acta Mech https://doi.org/10.1007/s00707-018-2124-4

O R I G I NA L PA P E R

Guangsong Chen · Linfang Qian · Jia Ma · Yicheng Zhu

Smoothed FE-Meshfree method for solid mechanics problems

Received: 26 August 2017 / Revised: 3 December 2017 © Springer-Verlag GmbH Austria, part of Springer Nature 2018

Abstract This paper presents a smoothed FE-Meshfree (SFE-Meshfree) method for solving solid mechanics problems. The system stiffness matrix is calculated via a strain-smoothing technique with the composite shape function, which is based on the partition of unity-based method, combing the classical isoparametric quadrilateral function and radial-polynomial basis function. The corresponding Gauss integration in the element is replaced by line integration along the edges of the smoothing cells, so no derivatives of the composite shape functions are needed during the field gradient estimation process. Several numerical examples including an automobile mechanical component are employed to examine the presented method. Calculation results indicate that SFE-Meshfree can obtain a high convergence rate and accuracy without introducing additional degrees of freedom to the system. In addition, it is also more tolerant with respect to mesh distortion. The volumetric locking problem is also explored in this paper under a selective smoothing integration scheme.

1 Introduction The finite element method (FEM) [1] is commonly applied in engineering problems in solid mechanics, fluid mechanics, heat transfer, and other fields due to its high accuracy, convenience and flexibility. Inherent disadvantages such as the discontinuous stress field at the inter-element boundaries, especially for lowerorder elements, can render its ineffectiveness under some situations. Besides, the accuracy of some classic isoparametric elements is highly sensitive to mesh distortions, and the volumetric locking phenomenon may not be avoided when the Poisson’s ratio approaches to 0.5 for incompressible solids [2,3]. During the last two decades, many researchers focused their attention on the meshfree method. It does not need meshes to discretize the computational domain, being immune from mesh distortion effects and effective for solving some complex practical problems such as large deformation, fracture propagation simulation and impact-induced failure. The major advancements in meshfree methods include smooth particle hydrodynamics (SPH) [4], element-free Galerkin method (EFG) [5], reproducing kernel particle method (RKPM) [6], meshfree local Petrov–Galerkin method (MLPG) [7,8], radial point interpolation method (RPIM) [9,10] and finite point method [11]. Although the meshfree methods come without the drawbacks of FEM, they still exhibit limitations [12, 13]: such as difficulties in essential boundary condition implementation, high computational cost, and overly complex trial function construction processes. Hybrid schemes, including the combination of meshfree and FEM methods, have been proposed to make full use of the advantages of both while mitigating their respective shortcomings [14,15]. By combining the existing FEM technology and the strain-smoothing technique of meshfree methods [16], Liu and his co-workers proposed the cell-based smoothed FEM (CS-FEM) [17]. In this method, an element as in the FEM may be further subdivided into several smoothing cells which are then G. Chen (B) · L. Qian · J. Ma · Y. Zhu School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China E-mail: [email protected]

G. Chen et al.

individually subjected to strain-smoothing operations under constant strain; the integrations over the weaker cells become line integrations along the cell boundaries. Therefore, no derivatives of shape functions are needed in computing the field gradients to form the stiffness matrix. Without excessive effort in modeling or computation, CS-FEM possesses the following properties [17]: (1) The stiffness of SFEM is softer than that of FEM, so its displacement and strain energy are preferable; (2) it exhibits a much higher degree of tolerance to mesh distortion; (3) field gradients are computed directly using only the shape functions themselves without requiring derivatives; (4) shape functions can be constructed much more easily, which allows for explicit interpolations of field variables; and (5) many existing FEM algorithms can be modified easily and applied to CS-FEM. The past ten years have witnessed an array of methods based on CS-FEM including the CS-SFEM for both 2D and 3D problems [17,18], node-based SFEM (NS-FEM) for both 2D and 3D [19,20], edge-based SFEM (ES-FEN) for 2D and 3D [21,22], face-based SFEM (FS-FEM) for 3D [23], and other hybrids such as αFEM [24] and βFEM [25]. Another common and extensively studied hybrid method is the partition of unity (PU)-based method. One of its main advantages is the construction of high-order global approximation of any degree without requiring extra nodes. Many other methods have been developed based on the PU concept: PU-based FEM (PUFEM) [26,27], numerical manifold methods [28,29], HP-clouds [30], generalized FEM (GFEM) [31], particle-partition of unity methods [32], eXtended FEM (XFEM) [33], and local maximum entropy shape functions coupled with the extrinsic enrichments (XLME) [34]. However, the serious problem called “linear dependence (LD)” will arise when both the PU functions and the local approximation are taken as explicit polynomials [26]. In order to solve the LD problem, Rajendran and Zhang [35,36] developed a new PU-based FE-Meshfree element which employs the classical shape functions of isoparametric elements combined with meshfree shape functions to yield hybrid shape functions, called composite shape functions. The shape function of the classical isoparametric quadrilateral element is used for the construction of PU and the meshfree shape functions for local approximation. The shape functions of FE-Meshfree elements possess the much-desired Kronecker-delta property, which is crucial in the implementation of essential boundary conditions as in FEM, inter-element compatibility properties, and all the completeness properties, necessary to ensure the reproducibility of not only the linear polynomial terms but also the higher-order ones included in the local approximation. Based on the formulation of PU-based FE-Meshfree, Xu and Rajendran [37] developed an FE-Meshfree TRIA3 element and used it for linear and geometry nonlinear analyses, which exhibited better performance than classical linear triangular as well as linear quadrilateral elements. To improve the properties of FE-Meshfree elements, Tang and Yang [38–40] developed new FE-Meshfree elements with continuous nodal stress. According to PU-based FE-Meshfree, they developed four-node quadrilateral, three-node triangular and hexahedral FEMeshfree elements. By adopting the least-squares approach for local approximation instead of the least-squares approach used by Rajendran and Zhang [35], Cai et al. [41] proposed a new PU based on a three-node triangular finite element which yields the stiffness matrix with better computational efficiency. To prevent singularity in the moment matrix when the high-order basis function is employed, Xu and Rajendran [42] develop the FE-Meshfree QUAD4 element with radial-polynomial basis functions. Inheriting the advantages of meshfree method, FE-Meshfree elements provide higher-order global approximations with better accuracy and faster convergence rate than FEM without adding extra nodes. Based on the radial-polynomial basis functions, PUbased FE-Meshfree QUAD4 and triangular elements with continuous nodal stress were developed by Yang et al. [43,44]. Instead of radial basis functions, Yang et al. [45] also used mean value coordinates to construct a new FE-Meshfree QUAD4 element which shows better stability without any uncertain parameters. The PU-based FE-Meshfree method has been applied to many fields so far, such as statics, free vibration [46], forced vibration [47], geometric non-linearity [48,49], acoustic problems [50] and so on. However, it is time consuming to construct the composite shape functions at each node if the radial basis function is utilized. In addition, the estimation process of the trail functions and the corresponding derivative also takes much time during the integration, especially when the high-order Gauss integration scheme is needed. Furthermore, it is also highly susceptible to the volumetric locking problem. In this study, we attempted to combine advantages of both FEM and meshfree methods by establishing the smoothed FE-Meshfree (SFE-Meshfree) method. The composite shape function using radial-polynomial basis function is constructed for interpolation without deriving the shape function. The strain, which is smoothed via cell smoothing technique, is used to integrate the stiffness matrix of the elements avoided time-consuming calculation of derivatives of the composite shape function. The SFE-Meshfree method can be regarded as an extension of the FE-Meshfree method with strain-smoothing technique. Also, it can be regarded as the development of SFEM by using a novel FE-Meshfree element. Compared to the FE-Meshfree method or SFEM, SFE-Meshfree can give better accuracy and convergence rate.

Smoothed FE-Meshfree method for solid mechanics problems

The remainder of this paper is organized as follows: Sect. 2 reviews the formulation of the composite shape function and radial-polynomial basis function. Section 3 presents the basic formulations used in linear analysis by employing the strain-smoothing technique with composite shape function in details; it is followed by numerical tests demonstrating the convergence characteristic and accuracy of the present interpolation in Sect. 4, and finally some conclusions are drawn in the last section.

2 Composite shape functions based on radial-polynomial basis function Here, we only briefly outline the composite shape functions for the FE-Meshfree QUAD4 element with radialpolynomial basis functions proposed by Xu and Rajendran [42]. The displacement interpolation can be written as follows: u (x) = N [u 1 (x) , u 2 (x) , u 3 (x) , u 4 (x)]T ,

(1)

in which the shape function matrix N is defined as   N = N1 N2 N3 N4 , 1 N1 = (1 − ξ ) (1 − η) , 4 1 N2 = (1 + ξ ) (1 − η) , 4 1 N3 = (1 + ξ ) (1 + η) , 4 1 N4 = (1 − ξ ) (1 + η) , 4

(2) (3) (4) (5) (6)

where N1 , N2 , N3 and N4 are the well-known isoparametric shape functions used in the classic QUAD4 element; u i (x) i = 1, 2, 3, 4 denotes the nodal displacement function. ξ ∈ [−1, 1] and η ∈ [−1, 1] represent the natural coordinates. These unknown nodal displacement functions are interpolated by the radial basis function r j (x) and polynomial basis function pk (x) as follows: u i (x) =

n 

r j (x) a j +

j=1

m 

pk (x) bk = rT (x) a + pT (x) b,

(7)

k=1

where m is the number of polynomial terms and n denotes the number of nodes in the support of node i including node i. For node i = 1, the supported domain 1 is marked by a dashed line in Fig. 1, which includes nodes {1, 2, 3, 4, 5, 6, 7, 8, 9}. Other nodal supported domains are 2 = {1, 2, 3, 4,9, 10, 11, 12} for node i = 2, 3 = {1, 2, 3, 4, 11, 12, 13,14, 15} for node i = 3 and 4 = {1, 2, 3, 4, 5, 6, 14,15, 16} for node i = 4. The element supported domain is defined as 1 = 1 ∪ 2 ∪ 3 ∪ 4 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}. The vectors aT = [a1 , a2 , . . . , an ] and bT = [b1 , b2 , . . . , bm ] should be determined. pT (x) = [ p1 (x) , p2 (x) , . . . , pm (x)] and rT (x) = [r1 (x) , r2 (x) , . . . , rn (x)] denote the row vectors corresponding to the polynomial and the radial basis functions, respectively. For 2D problems, if we consider three, four and six polynomial terms, respectively, thus p (x) can be expressed as follows:  p (x) = 1  p (x) = 1  p (x) = 1

T

,

x

y

x

y

xy

x

y

x2

T

(8) , xy

y

 2 T

(9) .

(10)

The radial basis function vector r (x) is expressed as  r (x) = r1 (x, y)

r2 (x, y)

···

r N (x, y)

T

,

(11)

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Ω1

16

15

14

13

5

4

3

12

1 6

1

2

11

7

8

9

10

Fig. 1 Nodal supported and element supported domain Table 1 Typical radial basis functions Name

Expression

2   r j (x, y) = exp −c d j /dc q  r j (x, y) = d 2j + (cdc )2 q  r j (x, y) = 1 d 2j + (cdc )2   r j (x, y) = d 2j ln d j

Gaussian (Exp) Multi-quadrics (MQ) Inverse multi-quadrics (IMQ) Thin plate spline (TPS)



Parameters c c, q c, q

in which r j (x, y) is the function of distance d j , which quantifies the distance between the interpolation point x and a node xi . For a 2D problem, d can be given as

 2  2 (12) x − x j + y − yj . dj = Numerous radial functions have been proposed and investigated as listed in Table 1. In this paper, we use the multi-quadrics (MQ) radial basis function due to its preferable performance compared to the other methods. The parameters c = 2.01 and q = 0.0001 are suggested by Xu and Rajendran [42] for the MQ radial basis function. As the strain-smoothing technique is used, the choices of numerical parameters in Table 1 are discussed in the following numerical examples. Generally, the number of polynomial terms will be three, four and six. Actually, almost the same results can be obtained regardless of the number of polynomial terms. Thus, here the three-polynomial term is adopted. Enforcing Eq. (7) to pass through all the nodes in the nodal support of node i, the following equations can be obtained: ui = Ra + Pb,

(13)

where ui is a vector of corresponding nodal displacements of all the nodes in the nodal support of node i, R and P represent the radial basis matrix and polynomial basis matrix, respectively: ⎤ ⎡ r1 (x1 , y1 ) r2 (x1 , y1 ) · · · rn (x1 , y1 ) ⎢ r1 (x2 , y2 ) r2 (x2 , y2 ) · · · rn (x2 , y2 ) ⎥ ⎥, (14) R=⎢ .. .. .. .. ⎦ ⎣ . . . . ⎡

r1 (xn , yn )

p1 (x1 , y1 ) ⎢ p1 (x2 , y2 ) P=⎢ .. ⎣ .

p1 (xn , yn )

r2 (xn , yn )

···

p2 (x1 , y1 ) p2 (x2 , y2 ) .. .

··· ··· .. .

p2 (xn , yn )

···

rn (xn , yn )

⎤ pm (x1 , y1 ) pm (x2 , y2 ) ⎥ ⎥. .. ⎦ .

pm (xn , yn )

(15)

Smoothed FE-Meshfree method for solid mechanics problems

There are altogether (n + m) coefficients to be determined, whereas only n equations are available. The polynomial term is an extra requirement that guarantees unique approximation. The constraints imposed can be expressed as follows [51]: Pa = 0.

(16)

a = Sa u, b = Sb u,

(17) (18)

The coefficients vector a and b can be solved:

in which   Sb = PT R−1 P PT R−1 , Sa = R

−1

[I − PSb ] = R

(19) −1

−R

−1

PSb .

(20)

Eventually the nodal displacement function in Eq. (7) can be expressed as: u i (x, y) = i ui , i = 1, 2, 3, 4, i = r(x, y)Sa + p(x, y)Sb ,

(21) (22)

1×N

ui = [u 1 u 2 u 3 · · · u N ]T ,

(23)

where ui is the corresponding nodal displacement vector. i is the array of shape functions of the nodes in the nodal support of node i, i = [i1 i2 i3 · · · iN ].

(24)

The number of nodes in the support of any node must be sufficient for the node in the boundary. First-order support is sufficient for the three-term polynomial (Fig. 2). Under this definition, no singularity problem arises in the numerical calculations reported in this paper. The arrays i , i = 1, 2, 3, 4 are to be assembled into a matrix 4×L where L is the number of nodes in the e-th element support e . Substituting Eqs. (13), (17) and (18) into Eq. (1) allows us to rewrite the element displacement field as follows: u(x) = N ( 

u )=(N

1×4 4×L L×1

 ) u = ψ

1×4 4×L

L×1

u ,

1×L L×1

(25)

5

Ω5 6

Ω6

Fig. 2 Nodal supports for boundary nodes 5 and 6 considering first-order nodal connectivity, i.e., nodes of all elements connected to a given node

G. Chen et al.

where u L×1 denotes the x-direction displacement of all the nodes in an element support and the composite shape function array, ψ1×L , is: ψ = [ψ1 , ψ2 , ψ3 , . . . , ψ L ] = N  . 1×4 4×L

1×L

(26)

Similarly, the element displacement field in the y-direction is defined as follows: v(x) = ψ

v ,

(27)

1×L L×1

where v L×1 denotes the y displacements of all nodes in an element support. The strain-smoothing technique over the smoothed cells associated with elements is employed here, so line integration is performed along the edges of the smoothing cells in the calculation process of the system stiffness matrix. Therefore, there is no need to compute the derivatives of composite shape function when the field gradients are estimated. The composite shape function ψ is characterized by the Kronecker-delta property, inter-element compatibility property, and higher-order completeness properties, i.e., reproducibility of all the Cartesian terms appearing in the assumed basis Eq. (1). 3 Strain-smoothing technique with composite shape function A 2D static elasticity problem can be described by equilibrium equation in the problem domain  bounded by : σi j, j + bi = 0 in .

(28)

It is subjected to the following boundary conditions: σi j n j = ti on t ,

(29)

u i = u¯ i on u ,

(30)

and

where  = t ∪ u , t ∩ u = ∅, σi j and σi j, j are the stress tensor and corresponding derivative components, respectively. bi is the body force component, ti is the traction on t , n i is the outward unit normal vector, and u¯ i denotes the boundary displacements on u . The variational weak form can be given as:   δ∇s (u)i j Di jkl ∇s (u)kl d − δu i ti d = 0. (31) 

t

Similar to FEM, the domain discretization of SFEM is based on elements. The Galerkin weak form given in Eq. (31) is applied, and the integration is performed based on the element. Depending on the stability and accuracy requirements, the elements can be further subdivided into n SC smoothing cells as shown in Fig. 3 for n SC = 1 and n SC = 4. Strain-smoothing operation and stiffness evaluation are performed simultaneously on each cell. The assembly of the stiffness matrix of each cell yields an element stiffness matrix. In the SFEM, a smoothing operation is performed to the gradient of displacement on the smoothing cell C , which may be the entire or a portion of an element:  ˜ h (xC ) = 1 ∇u (32) ∇u h (x)d, A C C  where AC = C d and C is the smoothing cell. The operation is very similar to the mean dilatation procedure to deal with the incompressibility in nonlinear mechanics. By using the composite shape function based on PU [26], the displacement can be approximated as follows: u h (x) =

L  I =1

ψI u I ,

(33)

Smoothed FE-Meshfree method for solid mechanics problems

nSC=1

4

4 3

3

1

1

2

nSC=4

2

Fig. 3 Quadrilateral mesh and smoothing cells

where L denotes the number of nodes in an element supported domain. Substituting u h into Eq. (32) yields the smoothed gradients of displacement: ˜ h (xC ) = 1 ∇u AC



n  1  u (x) n (x)d = ψ I (x) n (x)du I , AC C C h

(34)

I

where C is the boundary of the smoothing cell C . Note that the smoothing operation makes the domain integration become boundary integration around the smoothing cell in Eq. (32). For 2D elasticity problems, the smoothed strain can be expressed as follows: ε˜ h (xC ) =

n 

B˜ I (xC )d I ,

(35)

I =1

 T where d I = u I v I is the nodal displacement vector for 2D problems. B˜ I (xC ) represents the smoothed strain matrix, which can be given as ⎤ ⎡ 0 b˜ I 1 (xC ) B˜ I (xC ) = ⎣ (36) 0 b˜ I 2 (xC ) ⎦ . ˜b I 2 (xC ) b˜ I 1 (xC ) It is easy to relate the smoothed strain matrix B˜ I to its counterpart B I = ∇s N I (x) in FEM:  1 ˜ BI = B I (x) d. A C C

(37)

B˜ I is the averaged value of the standard B I over the cell C . If one Gaussian point is used for line integration along each edge iC of C , Eq. (37) can be transformed into the following algebraic form: b˜ I k (xC ) =

M 

  C C ψ I xiGP n ik li

(k = 1, 2) ,

(38)

i=1

where xiGP denotes the midpoint (Gaussian point) of the boundary segment of iC , the length and outward unit normal of which can be denoted as liC and n iC , respectively; M is the edge number of C . The smoothed element stiffness matrix can be obtained by assembling each of the smoothing cells in the element: ˜e = K

n SC  C

˜ TC DB ˜ C AC , B

(39)

G. Chen et al.

˜ C matrix is constructed based on the cell and n SC is the number of cells subdivided from the The smoothed B element of interest. The smoothing operation in a cell ensures satisfaction of the equilibrium equation for each point within the cell; thus, its definition is the “equilibrator”. ˜ e yields the system stiffness matrix. The discrete The assemblage of each of the element stiffness matrix K governing equation is ˜ = f, Kd

(40)

˜ is the smoothed system stiffness matrix and f denotes the nodal force vector: where K   ψTI bd + ψTI td. fI = 



(41)

The smoothed stress σ˜ can be obtained in the same manner as ε˜ h which keeps constant over a given smoothing cell. For linear elastic problems, σ˜ = D˜εh is calculated at the smoothing cell level; the stress can be weighted per the respective area of each cell. 4 Numerical examples This section provides examples of the SFE-Meshfree method in comparison with FE-Meshfree, SFEM, FEM and analytical solutions for a series of numerical problems. All the numerical examples are implemented by using the MATLAB software programming. The convergence rate of the proposed method is measured by two standards, namely the displacement norm and strain energy norm. The displacement norm is defined as follows:  ndof  h  i=1 u i − u i εd = , (42) ndof i=1 |u i | where ndof denotes the total number of DOFs of the system. The strain energy of the numerical solution E num and the total strain energy of the exact solution E exact are defined as follows: E num = E exact =

1 T u Ku, 2 Ne  1 2

i=1

i

(43) εiT Dεi d.

(44)

The strain energy norm can be given as εe =

|E num − E exact | . E exact

(45)

4.1 Cantilever beam subjected to tip-shear force In this example, a cantilever beam with length L and height D is investigated as a benchmark problem. The system is subjected to a parabolic traction at the free end as shown in Fig. 4. The beam is assumed to have a unit thickness so that plane stress condition can be met. The analytical displacement solution is described in detail by Timoshenko and Goodier [52].    Py D2 2 ux = , (46) (6L − 3x) x + (2 + v) y − 6E I 4   1 P (47) 3vy 2 (L − x) + (4 + 5v) D 2 x + x 2 (3L − x) , uy = − 6E I 4 where the moment of inertia I of the beam is given by I = D 3 /12.

Smoothed FE-Meshfree method for solid mechanics problems

y D

A

P x

L Fig. 4 Cantilever beam problem

Fig. 5 Domain discretization of the beam using 4-node element with regular elements

The corresponding stresses are: P (L − x) y , I σ y = 0,   P D2 2 −y . τx y = − 2I 4 σx =

(48) (49) (50)

The parameters are given as follows: E = 3.0 × 107 kPa, v = 0.3, D = 12 m, L = 48 m, and P = −1000 kN. During the computational process, the nodes on the left boundary are constrained using the exact displacements obtained from Eqs. (46) and (47). The distributed parabolic shear stresses, expressed as Eqs. (48)–(50), are acted on the right boundary. Different number of elements with 4 smoothing cells are employed to make analyses of the beam. Figure 5 gives an example of the discretization with 24 × 6 meshes. The selection of the parameters c and q in MQ has a great influence on the performance of radial functions and radial-polynomial functions. Dinis et al. [53,54] found that higher accuracy can be obtained when c approaches 0 and q is close to 1.0. Liu and Gu [10] obtained favorable results with c being 1.0 and q being 1.03 for free vibration problem. Xu and Rajendran [42] chose a combination of the parameters c (1 × 10−4 ) and q (2.01) for a study on FE-Meshfree with radial-polynomial basis functions. We have conducted several simulations to obtain the optimal parameters for our proposed method. Since c is less sensitive than q, here we set parameter c to 1, 2, 3, and 4 and parameter q from 0.98 to 1.65. As shown in Fig. 6, system performance reached optimal for these two combinations, namely c equals 1, and q equals 1.25 and c equals 2, q equals 1.03, which differs markedly from Xu’s results due to the use of strain-smoothing technique in our method. For these values of c and q, almost the same results reported in Table 2 can be obtained under the following mesh conditions, such as 2 × 2, 4 × 4, 8 × 8, 16 × 16, or 32 × 32 meshes. Thus, we set the parameter combination: c = 2, q = 1.03 for all the subsequent simulations. We use two types of discretization for the quadrilateral elements to investigate the effects of element shape when distortions are severe: One with regular elements and the other with irregular interior nodes, the corresponding coordinates can be given as x  = x + x · rc · αir , y  = y + y · rc · αir ,

(51) (52)

G. Chen et al.

Fig. 6 Influence of parameters c and q of MQ: a displacement error norm; b strain energy error norm

Table 2 Displacement norm and strain energy norm Mesh 2×2 4×4 8×8 16 × 16 32 × 32

c = 1, q = 1.25

c = 2, q = 1.03

εd

εe

εd

εe

1.158 × 10−1 1.342 × 10−2 1.497 × 10−3 4.180 × 10−4 1.120 × 10−4

1.040 × 10−1 8.483 × 10−3 8.612 × 10−4 3.889 × 10−4 1.070 × 10−4

9.566 × 10−2 1.069 × 10−2 1.457 × 10−3 4.619 × 10−4 1.246 × 10−4

8.725 × 10−2 6.631 × 10−3 1.285 × 10−3 4.923 × 10−4 1.327 × 10−4

Fig. 7 Domain discretization of the beam using 4-node element with irregular elements (αir = 0.3)

where x and y are the initial regular element sizes along the x and y axes, respectively. rc denotes a random number between −1.0 and 1.0, and αir is a prescribed irregularity factor between 0.0 and 0.5. The larger the value αir is chosen, the more irregular the shape of generated elements in the patch becomes. The domain is discretized using 24 × 6 regular and relatively irregular 4-node element with αir = 0.3 as shown in Fig. 7. Figure 8 gives the displacement v along x (y = 0) by regular and irregular mesh, and the corresponding errors are shown in Fig. 9. It can be observed that the computed displacements of the SFEMeshfree method are in good agreement with the analytical solution for both regular and irregular elements. Also, the errors of the displacement are much smaller than those obtained by FEM. Here, four different techniques are employed to check the convergence rates of the proposed method with the same meshes 2 × 2, 4 × 4, 8 × 8, 16 × 16, and 32 × 32. The deflection of point A due to a tip-shear force is summarized in Table 3. It can easily be observed that the proposed method is more accurate than FEM, SFEM, or FE-Meshfree with smaller meshes (2 × 2, 4 × 4). Almost the same results can be obtained and all converge to the analytical solution for the 8 × 8, 16 × 16, or 32 × 32 meshes. Compared with FEM, the utilization of composite shape function and strain-smoothing techniques in the present method both enables the stiffness matrix of the system to be softer, which may sometimes cause “over-softening phenomenon” and hence lead to overestimation of the results evident in Table 3. However, as the number of mesh increases, the present method can eventually converge to the exact solutions. Figures 10 and 11 show the displacement norm and

Smoothed FE-Meshfree method for solid mechanics problems

Fig. 8 Comparison between computed and exact displacements of a beam along y = 0

Fig. 9 The Displacement error between computed and exact displacements of a beam along y = 0 Table 3 Deflection of point A in tip-shear force problem Method

Mesh 2×2

4×4

8×8

16 × 16

32 × 32

FEM SFEM FE-Meshfree SFE-Meshfree Reference solution [52]

− 3.551 − 4.216 − 7.666 − 8.222 − 8.900

− 6.432 − 6.961 − 8.824 − 8.838

− 8.114 − 8.317 − 8.892 − 8.909

− 8.688 − 8.746 − 8.898 −8.904

−8.846 −8.861 −8.900 − 8.901

strain energy norm, from which we can see that all the four methods can converge to a certain value while the grid is refined; however, the convergence rate of the proposed method ranks highest. 4.2 Cook’s skew beam problem Figure 12 shows the Cook’s skew beam problem to assess the distortion tolerance of the presented method with different types of elements. The shear force is uniformly distributed on the right side and the total shear force P = 1 N. The Cook’s skew beam is also modeled using 2 × 2, 4 × 4, 8 × 8, 16 × 16, and 32 × 32 meshes. Figure 13 gives an example of the discretization with 8 × 8 meshes, Poisson ratio v = 1/3, and Young’s modulus E = 1 N/m2 . The deflection results of point A of all the four methods (FEM, SFEM, FE-Meshfree

G. Chen et al.

Fig. 10 Displacement error norm of the cantilever problem

Fig. 11 Strain energy error norm for the cantilever problem

P=1 A

44

44

48 Fig. 12 Cook’s skew beam

16

Smoothed FE-Meshfree method for solid mechanics problems

Fig. 13 Domain discretization of Cook’s skew beam Table 4 Computed deflection at point A for Cook’s skew beam Method

Mesh

FEM SFEM FE-Meshfree SFE-Meshfree Reference solution [55]

2×2

4×4

8×8

16 × 16

32 × 32

14.42 15.98 20.40 21.34 23.96

19.62 20.58 23.19 23.55

22.50 22.90 23.75 23.87

23.51 23.66 23.89 23.93

23.80 23.88 23.94 23.95

Table 5 Tip deflection at point A for Cook’s skew beam as v → 0.5 with 32 × 32 meshes v

FEM

SFEM

FE-Meshfree

SFE-Meshfree

Reference solution

0.49 0.499 0.4999 0.49999 0.499999 0.4999999

24.06 24.07 24.07 24.07 24.07 24.07

24.09 24.10 24.10 24.10 24.10 24.10

24.16 24.17 24.17 24.17 24.17 24.17

24.17 24.18 24.18 24.18 24.18 24.18

24.18 24.19 24.19 24.19 24.19 24.19

and SFE-Meshfree) are listed in Table 4, from which it can easily be observed that the proposed method yields better convergence compared with the other ones. The tip deflection is computed under plain stress condition by varying the Poisson ratio as v = 0.49, 0.499, 0.4999, 0.49999, 0.499999, 0.4999999 with 32 × 32 elements, and the results are summarized in Table 5. The reference solution is carried out by using the ABAQUS with 64 × 64 6-node elements. It is observed that the present method shows better performance.

4.3 Infinite plate with a circular hole Figure 14 shows a plate with a central circular hole of 1 m radius subjected to a unidirectional tensile load of 1.0 N/m2 at infinity in the x-direction. Due to its symmetry, only the upper right quadrant of the plate is modeled here. The plane strain condition is considered with E being 1.0×103 N/m2 and v being 0.3. Symmetric

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y

y

5

r

θ a

σ =1

o

x

5

σ =1

x Fig. 14 Infinite plate with a circular hole and its quarter model

conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction-free. The exact solution of the stress can be given as [52]: σ11 σ22 σ12

  3a 4 a2 3 cos 2θ + cos 4θ + 4 cos 4θ, = 1− 2 r 2 2r   a2 1 3a 4 =− 2 cos 2θ − cos 4θ − 4 cos 4θ, r 2 2r   2 a 1 3a 4 =− 2 sin 2θ + sin 4θ + 4 sin 4θ, r 2 2r

(53) (54) (55)

where (r, θ ) represent the polar coordinates and θ is measured counterclockwise from the positive x-axis. Traction boundary conditions are imposed on the right (x = 5.0) and top (y = 5.0) edges based on the exact solution (Eqs. (53)–(55)). The displacement components corresponding to the stresses listed above are   a r a a3 u1 = (K + 1) cos θ + 2 ((K + 1) cos θ + cos 3θ) − 2 3 cos 3θ , 8G a r r   a r a a3 u2 = (K − 1) sin θ + 2 ((1 − K ) sin θ + sin 3θ) − 2 3 sin 3θ , 8G a r r

(56) (57)

where G = E/2 (1 + V ) and K is defined in terms of Poisson’s ratio by K = 3 − 4v for plane strain cases. The domain is discretized using 625 regular quadrilateral elements shown as Fig. 15. Each element is divided into four smoothing cells. From Figs. 16 and 17, it is easily observed that all the displacements and stresses are in good agreement with the analytical solutions. The results of displacement norm and strain energy norm, listed in Table 6, also indicate that the proposed method outperforms the others. To check the availability of the proposed method for incompressible material, the Poisson’s ratio is enforced to gradually approach 0.5. The selective integration scheme is applied [56,57] in which the material property matrix D for isotropic materials can be written as follows: D = D1 + D2 ,

(58)

where D1 denotes the μ-part of D and D2 is the λ-part, in which μ is the shearing modulus, λ represents the 2vμ Lamé parameter and can be obtained by 1−2v . v is the Poisson ratio. For plane strain, D can be given as ⎡

λ + 2μ D=⎣ λ 0

λ λ + 2μ 0

⎤ ⎡ 0 2 0 ⎦ = μ⎣0 μ 0

0 2 0

⎤ ⎡ 0 1 0⎦ + λ⎣1 1 0

1 1 0

⎤ 0 0⎦, 0

(59)

Smoothed FE-Meshfree method for solid mechanics problems

Fig. 15 Domain discretization of infinite plate with circular hole using QUAD4 elements

Fig. 16 Computed and exact displacement of infinite plate with a circular hole: a displacement u of nodes along bottom side; b displacement v of nodes along left side

Fig. 17 Computed and exact stresses of infinite plate with circular hole: a stress σ y of nodes along bottom side; b stress σx of nodes along left side

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Table 6 Displacement norm and strain energy norm

εd εe

SFE-Meshfree

FE-Meshfree

SFEM

FEM

1.500 × 10−3

2.249 × 10−3

1.230 × 10−2

1.401 × 10−2 4.561 × 10−3

5.338 × 10−4

7.518 × 10−4

4.178 × 10−3

Fig. 18 Error in displacement with various Poisson’s ratios

where

⎡

2 D1 = μ ⎣ 0 0

0 2 0

⎤ 0 0⎦, 1

⎡

1 D2 = λ ⎣ 1 0

1 1 0

⎤ 0 0⎦. 0

(60)

The selective integration scheme is employed, thus n SC equals 4 on D1 (related to the μ-part) and n SC equals 1 on D2 (related to the λ-part). As shown in Fig. 18, the SFEM-Meshfree and FE-Meshfree methods both show volumetric locking phenomena, while the selective SFE-Meshfree is immune.

4.4 Semi-infinite plate The semi-infinite plate subjected to a uniform pressure within a finite range (− a ≤ x ≤ a) shown in Fig. 19 is studied here to explore the resulting plane strain conditions. The analytical stresses are given by [52]: p [2 (θ1 − θ2 ) − sin 2θ1 + sin 2θ2 ] , 2π p = [2 (θ1 − θ2 ) + sin 2θ1 − sin 2θ2 ] , 2π p = [cos 2θ1 − cos 2θ2 ] , 2π

σ11 =

(61)

σ22

(62)

σ12

(63)

The directions of θ1 and θ2 can be found in Fig. 19. The corresponding displacements are expressed as follows:    p 1 − v 2 1 − 2v r1 u1 = , (64) [(x + a) θ1 − (x − a) θ2 ] + 2y ln πE 1−v r2     p 1 − v 2 1 − 2v 1 y (θ1 − θ2 ) + 2H arctan u2 = πE 1−v c   + 2 (x − a) ln r2 − 2 (x + a) ln r1 + 4a ln a + 2a ln 1 + c2 , (65)

Smoothed FE-Meshfree method for solid mechanics problems

y

p a

−a

θ1

θ2 r1

H

O′

x

r2

A

Fig. 19 Semi-infinite plane subjected to uniform pressure

Fig. 20 Domain discretization of semi-infinite plate using 4-node elements

where H = ca is the distance from the origin to point O  , the vertical displacement is assumed to be zero, and c denotes a coefficient. Due to the symmetry about the y-axis, this problem was modeled as a 5a × 5a square with a = 0.2 m, c = 100 and p = 1 MPa, in which the left and bottom sides are constrained using the exact displacements given by Eqs. (64) and (65) while the right side is subjected to tractions computed from Eqs. (61)–(63). Figure 20 gives the discretization of the domain using the 4-node quadrilateral. Four smoothing cells are used for each element. The computed displacements along the free surface (y = 0) are shown in Fig. 21 while the stress distributions along the diagonal line of the semi-infinite plane (y = −x) are given in Fig. 22. All numerical results obtained via SFE-Meshfree are in good accordance with the analytical solutions. We also investigated the volumetric locking phenomena under SFE-Meshfree, selective SFE-Meshfree, and FE-Meshfree methods. Figure 23 shows the immune properties from the volumetric locking of the proposed method with a selective integration scheme.

4.5 An automotive part: connecting rod Finally, a static analysis of an automotive part is studied. The connecting rod with a relatively complex configuration is illustrated in Fig. 24, in which the boundary conditions as well as the applied load ( p = 1 MPa) can also be clearly observed. The rod is discretized using 723 irregularly distributed nodes as shown in Fig. 25.

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Fig. 21 Computed and exact displacements of semi-infinite plane: a u; b v

Fig. 22 Computed and exact stresses of semi-infinite plane: a σx ; b σ y ; c τx y

Smoothed FE-Meshfree method for solid mechanics problems

Fig. 23 Displacement error norm with different Poisson’s ratio

y

7.5

Cran k p in

W ri st p in 12.5 10

35 25

45

62.5 80

7.5

100 117.5 162.5

5

x

p

10 17.5

Fig. 24 Geometric model and boundary conditions an automotive connecting rod

Fig. 25 Domain discretization of connecting bar using 4-node elements

The plane stress problem is considered with the material constants E = 10 GPa and v = 0.3. Since the closed form solutions are not available, a reference solution is computed utilizing the commercial software ABAQUS with 24149 elements. Figure 26 gives the displacement distributions along x-axis. It can be observed that they

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Fig. 26 Displacement u distribution along the middle line from the present method and FEM via ABAQUS

Fig. 27 Normal stress σx distribution along the middle line from the present method and FEM via ABAQUS

are in good accordance with those obtained by ABAQUS. Figure 27 shows the stress distribution along the x-axis. The results are still as smooth as in the previous example. 5 Conclusions In this study, the SFE-Meshfree method in combination with FE-Meshfree and strain-smoothing techniques is employed to solve solid mechanics problems. Some conclusions can be summarized as follows: 1. SFE-Meshfree allows the use of elements generated by the traditional 4-node quadrilateral without any modification. 2. In FE-Meshfree, the field gradient estimation process is complicated due to the utilization of the radial basis function, while for SFE-Meshfree, gradients are computed directly using only shape functions themselves at some particular points along segments of the cells. Also, the shape function values for the discrete points of the element can be defined in a trivial, simple manner. 3. SFE-Meshfree is immune from volumetric locking phenomena per the selective smoothing integration scheme, while the original FE-Meshfree shows limitations on it. 4. Unlike the conventional FEM, which utilizes isoparametric elements, there is no coordinate transformation performed in SFE-Meshfree and thus no limitations should be imposed on the shape of its elements. Even severely distorted elements are allowed, making the domain discretization considerably more flexible than FEM.

Smoothed FE-Meshfree method for solid mechanics problems

Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 11472137.

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