Comput Mech (2014) 54:353–368 DOI 10.1007/s00466-014-0989-1
ORIGINAL PAPER
Quadratically consistent nodal integration for second order meshfree Galerkin methods Qinglin Duan · Bingbing Wang · Xin Gao · Xikui Li
Received: 30 October 2013 / Accepted: 27 January 2014 / Published online: 19 February 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Robust and efficient integration of the Galerkin weak form only at the approximation nodes for second order meshfree Galerkin methods is proposed. The starting point of the method is the Hu-Washizu variational principle. The orthogonality condition between stress and strain difference is satisfied by correcting nodal derivatives. The corrected nodal derivatives are essentially linear functions which can exactly reproduce linear strain fields. With the known area moments, the stiffness matrix resulting from these corrected nodal derivatives can be exactly evaluated using only the nodes as quadrature points. The proposed method can exactly pass the quadratic patch test and therefore is named as quadratically consistent nodal integration. In contrast, the stabilized conforming nodal integration (SCNI) which prevails in the nodal integrations for meshfree Galerkin methods fails to pass the quadratic patch test. Better accuracy, convergence, efficiency and stability than SCNI are demonstrated by several elastostatic and elastodynamic examples. Keywords Meshfree/meshless · EFG · Nodal integration · Hourglass · Hu-Washizu variational principle 1 Introduction Meshfree Galerkin methods, such as the element-free Galerkin (EFG) method [1] and the reproducing kernel particle Q. Duan (B) · B. Wang · X. Gao · X. Li State key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, People’s Republic of China e-mail:
[email protected] Q. Duan · B. Wang · X. Gao · X. Li Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China
method (RKPM) [2], attract intensive studies from the community of computational mechanics in the past twenty years due to their superiorities against the traditional finite element method (FEM) such as the super convergence, the convenience to construct high order approximation and the ease to deal with large deformations, etc. These benefits are essentially provided by the meshfree approximations, e.g. the moving least-square (MLS) approximation [1], the reproducing kernel particle approximation [2], the maximum entropy approximation [3], etc. The common feature shared by these meshfree approximations is that they are based on scattered data approximation techniques which, in the sense of approximation, do not need explicit nodal connectivity to construct elements. More importantly, the smoothness of meshfree approximations is much better than the element-based Lagrangian interpolation employed in FEM. For example, the smoothness of the MLS approximation using Gaussian weight [4] is C ∞ . In contrast, the FEM interpolation only has C 0 continuity. The high order smoothness usually leads to super convergence, especially for stress and strain fields. However, the adverse effect of meshfree approximation also needs to be dealt with. It is well known that most meshfree approximants do not possess Kronecker delta property and this complicates the imposition of essential boundary conditions. The developed strategies are coupling methods [5,6], corrected collocation method [7] and penalty method [8], etc. Nitsche’s method [9] is simply employed in this paper and this issue is not the concern of this study. Interested readers are redirected to [9,10]. More difficult issue caused by the adverse effect of meshfree approximation is the stable and efficient numerical integration of the Galerkin weak form. Since most of the meshfree approximants are non-polynomial rational functions, more sampling points are required to accurately integrate the weak form. For example, it is shown in [11] that
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at least 16 quadrature points per background triangle cell are required for second order EFG method to result stable solutions to elastostatic problems. This severely impairs the efficiency of meshfree computations. Probably, this is the main reason why meshfree methods, after 20 years development, are still an academic endeavor rather than an industrial technology. Efforts to accelerate meshfree computations can be found in Dolbow and Belytschko [12], Griebel and Schweitzer [13], and also in the methods of stress-point integration [14–16], etc. The major academic development towards this issue is nodal integration [17–21], which evaluates the integrals of the Galerkin weak form only at the approximation nodes. Beissel and Belytschko [17] initiated this kind of study and they found that direct nodal integration is not stable. For this reason, a least-square term is introduced into the weak form as a stabilization mechanism to remove the spurious oscillations. However, the magnitude of such stabilization term is controlled by an artificial numerical parameter and its selection depends on numerical experiments. Nagashima [18] proposed a nodal integration scheme for EFG by performing a Taylor’s expansion to the stiffness matrix and similar technique was used by Liu et al. [19] to develop nodal integration for radial point interpolation method (RPIM). Such technique is first introduced by Liu et al. [22] in FEM and also in RKPM [23]. The merit of this technique is that the stabilization terms are introduced in a rational manner and no artificial parameter is needed. However, Duan and Belytschko [16] found that it has a poor stabilization effect for nodal integration of EFG using second order approximation and spurious oscillations present, especially in the resulting stress fields. Better approach to stabilize nodal integration can be developed by satisfying linear patch test condition. Bonet and Kulasegaram [20] presented an integration correction which enables the method of corrected smooth particle hydrodynamics (CSPH) with nodal integration to pass linear patch test. However, the integration correction needs to be solved in an iterative manner. In addition, least-square stabilization similar to that in [17] is still required. These shortcomings are circumvented by the method of stabilized conforming nodal integration (SCNI) developed by Chen et al. [21]. They derived an integration constraint (IC) according to the satisfaction of linear patch test and further developed a strain smoothing technique to meet IC without any artificial parameter. Linear patch test is exactly passed, that is, the linear exactness is achieved by SCNI. Note that meshfree methods with direct high order Gauss integration cannot pass linear patch test exactly. Therefore, SCNI, with reduced number of sampling points, provides even better accuracy than Gauss integration and thus efficiency is greatly improved. So far, SCNI has already developed into a major nodal integration technique for meshfree Galerkin method and has been applied to various problems, e.g. see Wang and Chen [24].
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However, Puso et al. [25] found that SCNI may result sawtooth mode at the domain boundaries. In addition, the strain smoothing in SCNI only meets the linear patch test condition. Thus, it is not adequate for second order meshfree Galerkin method which requires the quadratic patch test should be exactly passed. To remedy this, Duan et al. [11] proposed a general framework to correct nodal derivatives for arbitrary order approximations based on the divergence theorem between nodal shape function and its derivatives. The corresponding high order patch tests required by the order of the approximation can be exactly passed by replacing the original derivatives with such corrected ones in the computation of stiffness matrix. Particularly, under such framework, they designed a three-point integration scheme using background triangle cells which can exactly pass quadratic patch test with second order MLS approximation. Such scheme is named as quadratically consistent 3-point (QC3) integration. Further, a one-point integration scheme [26] is also proposed by introducing the Taylor’s expansion technique into QC3. However, these schemes are not nodal integrations. In addition, the replacement of the original derivatives with corrected ones in these schemes is lack of theoretical foundation. The purpose of this paper is to develop a nodal integration scheme with derivative correction for second order meshfree Galerkin methods which can exactly pass quadratic patch test. The final formulation for derivative correction is similar to that used in [26]. However, the development of the method is based on the Hu-Washizu three-field variational principle and this fixes the issue “ lack of theoretical foundation ” mentioned above. In addition, dynamic problems are also tested. The outline of this paper is as follows. A brief summary of the EFG method is given in Sect. 2. The existing stabilized conforming nodal integration is reviewed in Sect. 3. The derivation of the formulation for nodal derivative correction based on the Hu-Washizu three-field variational principle is given in Sect. 4. The proposed quadratically consistent nodal integration (QCNI) is described in Sect. 5. Numerical results of the proposed QCNI and the existing SCNI methods are compared in Sect. 6 for both elastostatic and elastodynamic examples, followed by the conclusions in Sect. 7.
2 Element-free Galerkin (EFG) method EFG was invented by Belytschko et al. [1] about twenty years ago and so far it has already developed into one of the most popular and successful meshfree Galerkin methods. Consider a two dimensional elastostatic problem in the domain ⊂ R2 with a set of nodes X I , the displacement T u (x) = u (x) v (x) at an arbitrary point x is approximated in a form similar to that in FEM
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uh (x) = N (x) U =
N I (x) U I
355
(1)
I
where U is the unknown vector of nodal displacement paraT meters and U I = u I v I . N (x) is the matrix of nodal shape functions N (x) = N1 (x) N2 (x) · · · Nn (x) (2) where n is the number of nodes, N I (x) = N I (x) I2 and I2 is the 2D unit matrix. The nodal shape function N I (x) is constructed by MLS and can be written as N I (x) = p (X I ) w I (x) α (x) T
(3)
where p (x) is a vector of base functions which usually includes a complete basis of the polynomials to a given order, w I (x) a weight function and α (x) the unknown vector. In this study, the following normalized Gaussian weight is employed ⎧ 2 2 s¯ 1 ⎪ ⎨ e− α −e− α for s¯ ≤ 1 2 1 (4) w I (x) = w (¯s ) = − α 1−e ⎪ ⎩ 0 for s¯ > 1 where s¯ = s/r, r is the radius of the support, s = |x − X I | the distance from the point x to the node X I and the parameter α = 1/3 is used. Note that MLS approximation using such Gaussian weight has C ∞ continuity. The unknown vector α (x) can be determined by the so called reproducibility condition, i.e. the consistency condition p (X I ) N I (x) (5) p (x) = I
where A (x) =
p (X I ) pT (X I ) w I (x)
A, i (x) =
p (X I ) pT (X I ) w I, i (x)
(10)
I
The equilibrium equation and boundary conditions for an elastostatic problem on a 2D domain bounded by is ∇ ·σ +b=0 σ · n = t¯
in
(11)
on t
(12)
u = u¯
on u
(13)
where σ is the Cauchy stress, b the body force. t¯ and u¯ are, respectively, the prescribed traction and displacement on boundaries t and u , n the unit normal to the boundary. EFG uses the classical displacement variational principle to construct the weak form δ (u), i.e. (14) δ (u) = δεT Dεd − δuT t¯d − δuT bd
t
where D is the material modulus, prefix δ denotes a variation and the strain is
T T ∂u ∂v ∂u ε = εx x ε yy γx y = ∂ x ∂ y ∂ y + ∂∂vx = BU = BI UI (15) I
with ⎡ ∂N
I
∂x
⎢ B = B1 B2 · · · Bn and B I = ⎢ ⎣0
0 ∂ NI ∂y ∂ NI ∂x
∂ NI ∂y
Substitution of Eq. (3) into Eq. (5) leads to A (x) α (x) = p (x)
with
(6)
(7)
⎤ ⎥ ⎥ ⎦
(16)
By taking the variation, the following discretized equation can be obtained from the weak form Eq. (14) KU = f
(17)
where K = BT DBd, f = NT bd + NT t¯d
(18)
I
The nodal MLS shape functions N I (x) can be obtained from Eq. (3) after the unknown vector α (x) is solved from Eq. (6). As you see, the procedure to compute the MLS shape functions is very simple and this is probably one of the main reasons why EFG is widely used in various applications. Computation of the derivatives of the MLS shape functions is also straightforward, i.e. directly taking the derivative of Eq. (3) (8) N I, i (x) = pT (X I ) w I, i (x) α (x) + w I (x) α, i (x) where subscripts preceded by commas denote partial derivatives with respect to spatial coordinates. The unknown α, i (x) in Eq. (8) can be solved by differentiating Eq. (6) A (x) α, i (x) = p, i (x) − A, i (x) α (x)
(9)
t
As you see, the weak form and the final discretized equation in EFG are identical in form to those in FEM. However, these two methods use different nodal shape functions: the non-polynomial rational functions constructed by the MLS approximation are employed in EFG whereas the polynomial functions constructed by the element based Lagrangian interpolation are used in FEM. Note that the non-polynomial rational functions are very difficult to be exactly integrated and this is one of the major drawbacks of meshfree Galerkin methods.
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3 Stabilized conforming nodal integration (SCNI)
with
Although, by using more sampling points, meshfree Galerkin methods are able to obtain stable and reasonable numerical results, their low efficiency is unacceptable. On the other hand, nodal integration is appealing since it requires only minimum evaluations of the weak form and thus accelerates meshfree computations tremendously. In addition, nodal integration avoids the projection between approximation nodes and sampling points since these two sets of points coincide. This leads to a more truly-meshless like method. So far, the dominant technique in nodal integration is SCNI proposed by Chen et al. [21]. The method is based on strain smoothing which at node x L is written as ε˜ i j (x L ) = εi j (x) (x; x − x L ) d (19)
where ε˜ i j (x L ) is the smoothed strain at node x L , εi j (x) = u i, j (x) + u j,i (x) /2 is the strain and (x; x − x L ) is the distribution function at node x L given by φ (x − x L ) J =1 φ (x − x J ) A J
(x; x − x L ) = n
(20)
where A J = J d is the area of the representative domain J of node J . Such partition can be generated by several techniques, for example, the Voronoi diagram. The function φ (x − x L ) in Eq. (20) is defined as 1, x ∈ L (21) φ (x − x L ) = 0, x ∈ / L Substitution of Eq. (21) into Eq. (20) gives 1/A L , x ∈ L
(x; x − x L ) = 0, x∈ / L
(22)
Substituting Eq. (22) into Eq. (19) leads to 1 ∂u i (x) ∂u j (x)
(x; x − x L ) d ε˜ i j (x L ) = + 2 ∂x j ∂ xi ∂u i (x) ∂u j (x) 1 d + = 2 AL ∂x j ∂ xi L 1 = u i (x) n j + u j (x) n i d (23) 2 AL L
where L is the boundary of L . The last equal sign in Eq. (23) is due to the divergence theorem. Eq. (23) implies the smoothed strain at node x L can be constructed by smoothed (corrected) nodal derivatives as follows T ε˜ (x L ) = ε˜ x x (x L ) ε˜ yy (x L ) γ˜x y (x L ) ˜ I (x L ) U I B = B˜ (x L ) U = (24) I
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B˜ (x L ) = B˜ 1 (x L ) B˜ 2 (x L ) · · · B˜ n (x L ) and ⎤ ⎡ ˜ ∂ N I (x L ) 0 ⎥ ⎢ ∂x ∂ N˜ I (x L ) ⎥ B˜ I (x L ) = ⎢ ∂y ⎦ ⎣0
(25)
∂ N˜ I (x L ) ∂x
∂ N˜ I (x L ) ∂y
where the smoothed nodal derivatives are ∂ N˜ I (x L ) 1 = N I (x) n x d ∂x AL
(26)
L
1 ∂ N˜ I (x L ) = ∂y AL
N I (x) n y d
(27)
L
Chen et al. [21] showed that replacing the strain in the weak form Eq. (14) with such smoothed strain, i.e. using the smoothed nodal derivatives defined by Eqs. (26, 27) to compute stiffness matrix, leads to a good stabilization effect for direct nodal integration. In addition, such strain smoothing can exactly meet linear patch test condition (see [21] for details) and thus SCNI results even better numerical performance than high order Gauss integration. However, as you see, the strain field in each integration sub-domain L is characterized only by ε˜ (x L ). This means SCNI can only reproduce a constant strain field in L . However, for second order meshfree Galerkin methods, a linear strain field should be exactly reproduced. Thus, SCNI is not adequate for such methods. The purpose of this paper is to develop a nodal integration scheme which can exactly reproduce a linear strain field in each nodal representative domain L and thus fits second order meshfree Galerkin methods. The development of the scheme is based on Hu-Washizu variational principle which is described in the next section.
4 Derivative correction based on Hu-Washizu variational principle The Hu-Washizu three-field weak form for elastostatic problem can be written as δ∗ u, ε˜ , σˆ
δ ε˜ T σ˜ − σˆ + δ σˆ T ε − ε˜ + δεT σˆ d =
− t
δuT t¯d −
δuT bd
(28)
where the displacement u, the interpolated (or assumed) strain ε˜ and the assumed Cauchy stress σˆ are three independent variables, σ˜ = D˜ε the Cauchy stress obtained from the constitutive function. By rearrangement of the terms, Eq. (28)
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can be written as δ u, ε˜ , σˆ = ∗
357
T
δ ε˜ D˜εd −
T¯
⎤
σˆ T ε − ε˜ d⎦
σˆ
δu td − t
⎡
+δ⎣
δu bd T
(29)
the variational structure of Eq. (29) can remain as simple as the classical one and can be written as T ∗∗ T¯ δ (u) = δ ε˜ D˜εd − δu td − δuT bd (31)
where the displacement u is the only one independent variable. To make this happen, we will approximate the interpolated strain ε˜ by the same formulation as shown in Eq. (24) and then determine the corrected nodal derivatives according to Eq. (30) such that the orthogonality condition is satisfied. This is presented in the following. Suppose the whole computational domain is divided into a set of nodal representative domain L for the purpose of nodal integration. To meet Eq. (30), we can let this equation be satisfied at each sub-domain L , i.e. σˆ T ε − ε˜ d = 0 (32) L
Substitution of Eq. (15) and Eq. (24) into Eq. (32) gives ⎤ ⎡
⎥ ⎢ T B I − B˜ I d⎦ U I = 0 (33) ⎣ σˆ L
∂ N I ∂ N˜ I − ∂ xi ∂ xi
d = 0 (xi = x, y for i = 1, 2) (35)
Note that the first three terms on the r.h.s. of Eq. (29) have the same form as the classical displacement variational principle, i.e. Eq. (14), except that the strain ε is replaced by the interpolated strain ε˜ . Obviously, if the interpolated strain ε˜ can be somehow constructed from the displacement u and meets the following orthogonality condition σˆ T ε − ε˜ d = 0 (30)
t
L
T
I
Eq. (33) will hold provided, for each node I , the following equation holds
(34) σˆ T B I − B˜ I d = 0 L
By further substituting Eq. (16) and Eq. (25) into Eq. (34), it is not difficult to know that the following equation should be satisfied in order to meet the orthogonality condition
The interpolation space for the assumed Cauchy stress σˆ in Eq. (35) should be carefully selected. The most straightforward and reasonable choice is the space one order lower than the space for the displacement u in a consistent manner, i.e. choose linear space for σˆ if u is from quadratic space, etc. With this consideration in mind, we choose the space for σˆ as follows: if the displacement is approximated by the MLS shape functions with the base p (x), the assumed Cauchy stress σˆ will be chosen from the space spanned by the base q (x) = p,x (x) ∪ p,y (x)
(36)
For example, if displacement is approximated by quadratic T MLS approximation with the base p (x) = 1 x y x 2 x y y 2 , T we have q (x) = 1 x y according to Eq. (36). Obviously, Eq. (35) will always hold provided this equation holds for the base q (x), i.e. ∂ NI ∂ N˜ I q (x) − d = 0 (xi = x, y for i = 1, 2) ∂ xi ∂ xi
L
(37) This equation can be rewritten as ∂ N˜ I ∂ NI q (x) d = q (x) d ∂ xi ∂ xi
L
(38)
L
Further, integration by parts for the r.h.s. of Eq. (38) yields ∂ N˜ I q (x) d = N I (x) q (x) n i d ∂ xi L L − N I (x) q,i (x) d (39) L
This is the equation we will use for derivative correction and by doing this the orthogonality condition is satisfied. It is noted that if we replace N˜ I,i (x) on the l.h.s of Eq. (39) with N I,i (x), this equation is essentially the divergence theorem between the nodal shape function and its derivatives. This theorem is the starting point of the derivation for the QC3 method in [11] and the QC1 method in [26]. However, the development from the Hu-Washizu variational principle given here is more rigorous in mathematics since the corrected nodal derivatives as the unknowns to be determined naturally present on the l.h.s of Eq. (39) whereas they do not present in the divergence theorem. Furthermore, the interpolated strain ε˜ approximated by the corrected nodal derivatives also naturally presents in Eq. (31)
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which is exactly the weak form used for subsequent discretization. Therefore, the Hu-Washizu variational principle is the ideal theoretical foundation for such derivativecorrection methods. Note that Eq. (39) applies to arbitrary order approximations. Especially, for linear approximation, T we have p (x) = 1 x y and q (x) = [1] and . Then, Eq. (39) reduces to Eqs. (26, 27) which are the equations to compute the smoothed derivatives in SCNI. This demonstrates that SCNI is only consistent to linear approximation and inconsistent to higher order one. Obviously, SCNI is a special case of the framework given here for derivative correction. In other words, SCNI can be formulated from the HuWashizu variational principle in a rational manner, instead of using strain smoothing technique.
5 Quadratically consistent nodal integration (QCNI) In this section, we will develop a nodal integration scheme using the derivative correction described in the last section for EFG method with second order approximation. To begin with, the partition of the whole solution domain into nodal representative domains is described. Apparently, the Voronoi diagram is a nature choice for this purpose. However, to take full advantage of the commercial mesh generators, the partition based on triangle mesh is used in this study. As shown in Fig. 1, the approximation nodes are connected to form a triangle mesh and then, by joining the centroids of the triangles surrounding a node L, a polygon L is constructed and is used as the nodal representative domain of node L. Note that the mid-edge points are used in this partition method for boundary nodes such as the node K in Fig. 1.
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According to the discussion in the last section, to meet the orthogonality condition in the nodal representative domain L , Eq. (39) has to be satisfied. Note that, for second order meshfree Galerkin method, we have p (x) = T T and q (x) = 1 x y . Therefore, 1 x y x 2 x y y2 Eq. (39) actually contains three equations for the nodal derivatives with respect to x and y, respectively. As an example, the three equations for x-derivatives are
N˜ I,x d =
L
N I (x) n x d
(40)
L
N˜ I,x xd =
L
N I (x) xn x d −
L
N˜ I,x yd =
L
N I (x) d
(41)
L
N I (x) yn x d
(42)
L
In the nodal integration scheme, i.e. the node L is the only one quadrature point in L , we only have one unknown N˜ I,x (x L ) and thus cannot make these three equations satisfied at the same time. To this end, we introduce the derivatives of the function N˜ I,x by means of Taylor’s expansion such that N˜ I,x x (x L ) and N˜ I,x y (x L ) are introduced and can serve as the other two unknowns. The Taylor’s expansion for N˜ I,x is N˜ I,x (x) = N˜ I,x (x L ) + (x − x L ) N˜ I,x x (x L ) + (y − y L ) N˜ I,x y (x L ) + H.O.T
(43)
To make the two domain integration terms in Eq. (41) has the same integration accuracy, N I (x) is also expanded as N I (x) = N I (x L ) + (x − x L ) N I,x (x L ) 1 + (y − y L ) N I,y (x L ) + (x − x L )2 N I,x x (x L ) 2 + (x − x L ) (y − y L ) N I,x y (x L ) 1 + (y − y L )2 N I,yy (x L ) + H.O.T (44) 2 Substitution of Eqs. (43-44) into Eqs. (40-42) leads to ⎡
Fig. 1 Schematic diagram of the partition to form the nodal representative domains
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A
⎢ Ax + I x ⎣ L L y Ay L + I L ⎧ ⎪ NI ⎪ ⎪ ⎪ ⎪ ⎪ L ⎪ ⎪ ⎪ ⎨ NI = ⎪ ⎪ L ⎪ ⎪ ⎪ ⎪ ⎪ NI ⎪ ⎪ ⎩ L
⎫ ⎤⎧ ˜ ⎪ N I,x (x L ) ⎪ ⎪ ⎪ ⎬ ⎨ y xy ˜ I Lx x L + I Lx x I L x L + I L ⎥ N (x ) I,x x L ⎦ ⎪ xy y yy ⎪ ⎪ ⎭ ⎩ N˜ I,x y (x L ) ⎪ I Lx y L + I L I L y L + I L ⎫ ⎪ (x) n x d ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ L (x) xn x d − F (45) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (x) yn x d ⎪ ⎪ ⎭ I Lx
y
IL
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359
with y I L N I,y (x L )
FL = AN I (x L ) + I Lx N I,x (x L ) + 1 1 yy xy + I Lx x N I,x x (x L )+ I L N I,x y (x L )+ I L N I,yy (x L ) 2 2 (46) where A = d is the area of L and the area moments are I Lx
=
L
(x − x L ) d, L
(x − x L )2 d, L
yy IL
=
(y − y L ) d
xy
(47)
IL =
(x − x L ) (y − y L ) d L
=
L
I Lx x =
y IL
(y − y L ) d 2
proposed QCNI scheme should fit the second order meshfree Galerkin methods better than SCNI, which will be validated by the numerical results given in the next section. Since the orthogonality condition is satisfied by these corrected nodal derivatives, the simpler weak form, i.e. Eq. !(31), can be used. Once N˜ I,x (x L ) , N˜ I,x x (x L ) , N˜ I,x y (x L ) and ! N˜ I,y (x L ) , N˜ I,yx (x L ) , N˜ I,yy (x L ) are solved, the linear functions N˜ I,x and N˜ I,y are obtained. Substituting them into Eq. (31) leads to the component of the stiffness matrix contributed by L L K IJ
=
B˜ TI D B˜ J d
L
(48) =
L
The boundary integration terms in Eq. (45) is evaluated by two Gauss points per edge of the polygon L as shown in Fig. 1. Note that the node L is not the centroid of the polygon L . Therefore, the first order area moments given by Eq. (47) are non-zero and these terms present in Eq. (45). This is different to the QC1 integration scheme given in [26] where the centroids of background triangle cells are used as quadrature points. By solving Eq. (45), the corrected nodal derivative N˜ I,x (x L ) and its derivatives, i.e. N˜ I,x x (x L ) and N˜ I,x y (x L ), are obtained. Follow the same derivation, the equation for y-derivatives can be written as ⎫ ⎤⎧ ˜ ⎡ y ⎪ N I,y (x L ) ⎪ IL A I Lx ⎪ ⎪ ⎨ ⎬ ⎢ Ax L + I x I x x L + I x x I y x L + I x y ⎥ N˜ (x ) I,yx L L L L L L ⎦ ⎣ ⎪ y xy y yy ⎪ ⎪ ⎩ N˜ I,yy (x L ) ⎪ ⎭ Ay L + I L I Lx y L + I L I L y L + I L ⎫ ⎧ ⎪ ⎪ N I (x) n y d ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ N d xn (x) I y (49) = ⎪ ⎪ ⎪ ⎪ L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L⎪ ⎪ ⎪ N d − F yn (x) ⎪ I y ⎪ ⎪ ⎪ ⎭ ⎩ L
where FL is given by Eq. (46). It is stressed that the system matrix of Eq. (49) is the same as that of Eq. (45). ! Note that N˜ I,x (x L ) , N˜ I,x x (x L ) , N˜ I,x y (x L ) obtained from Eq. (45) determines a linear field for the function N˜ I,x in L as shown by Eq. (43) and { N˜ I,y (x L ) , N˜ I,yx (x L ) , N˜ I,yy (x L )} obtained from Eq. (49) determines a linear field for the function N˜ I,y as well. Then, by the approximation, i.e. Eq. (24), a linear strain field can be reproduced in L by using only one quadrature point, i.e. the node L. In contrast, the existing SCNI scheme described in Sect. 3 can only reproduce a constant strain field in L . This implies that the
L
⎡ ⎤ N˜ J,x 0 N˜ I,x 0 N˜ I,y D⎣0 N˜ J,y ⎦ d 0 N˜ I,y N˜ I,x ˜ N J,y N˜ J,x (50)
Since N˜ I,x and N˜ I,y in Eq. (50) are known linear functions, L K I J can be exactly integrated at node L with the help of the known area moments defined by Eqs. (47, 48). This demonstrates the nodal integration of the stiffness matrix. Similarly, substitution of Eq. (44) into Eq. (31) leads to the computation of the body force on L fb L
=
" " N bd = AN b" T
L
T
xL
+
I Lx
" ∂(NT b) "" ∂ x "x L
" " ∂(NT b) "" 1 x x ∂ 2 (NT b) "" + I ∂ y "x L 2 L ∂ x 2 "x L " " 2 T 1 yy ∂ 2 (NT b) "" x y ∂ (N b) "" +I L + I ∂ x∂ y "x L 2 L ∂ y 2 "x L y +I L
(51)
which is also evaluated only at the approximation nodes. To check the stability of the proposed QCNI method, an eigenvalue analysis of its stiffness matrix is performed and the results are plotted in Fig. 2. Only three zero eigenvalues which corresponds to regular rigid body modes exist. The fourth smallest eigenvalue is non-zero which is associated with two deformation modes shown in Fig. 2d. No spurious zero energy modes (hourglass modes) are detected.
6 Numerical examples Numerical results of the following five examples are presented in this section. Material parameters are E=2.1 × 1011 Pa, υ = 0.3, ρ = 7.8×103 kg/m3 and plane stress conditions are assumed. The errors in displacement and energy
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Fig. 2 Mode shapes corresponding to the four smallest energy modes in the eigenvalue analysis of the QCNI method (λ denotes eigenvalues): a x-translation (λ = 0.0); b y-translation (λ = 0.0); c rotation (λ=0.0); d two deformation modes (λ = 0.113)
are respectively evaluated by
E disp
E eng
# $ n h e T u h − ue $ I =1 u I − u I I I % = n eT e I =1 u I u I #
T
$ $ εh − εe D εh − εe d $ $ $ =$ $ % εeT Dεe d
6.1 Patch tests
(52)
(53)
where the superscripts e and h denote the exact and the numerical solutions, respectively. T Quadratic base p (x) = 1 x y x 2 x y y 2 is employed in MLS approximation. Solutions are reported for the proposed QCNI, the existing SCNI which is the dominant nodal integration technique in meshfree fields and full integration (FI) which employs 16 quadrature points in each background triangle cell. FI scheme is also used to evaluate the errors in energy given by Eq. (53).
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Patch tests on a 2 × 2 domain with 5 × 5 nodes are first investigated. For the linear patch test, all the boundary displacements are prescribed as & 0.1 + 0.1x + 0.2y u¯ = (54) 0.05 + 0.15x + 0.1y The exact solution is u = u¯ in the absence of body forces. For the quadratic patch test, the following body force is applied & −0.2D (1, 1) − 0.1D (1, 2) − 0.2D (3, 3) (55) b= −0.12D (1, 1) − 0.1D (1, 2) − 0.69D (3, 3) where D is the elastic modulus. The displacement is prescribed at the boundaries as & 2 2 ¯u¯ = 0.1x 2+ 0.1x y + 0.2y 2 (56) 0.05x + 0.15x y + 0.1y ¯¯ and the exact solution is u = u. Tables 1 and 2 show the results of the linear and quadratic patch tests, respectively. FI results relatively large errors in both displacement and energy and it fails for both tests. SCNI
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Table 1 Results of the linear patch test FI
SCNI
QCNI
E disp
0.57E-5
0.71E-12
0.59E-12
E eng
0.50E-4
0.66E-11
0.51E-11
Table 2 Results of the quadratic patch test FI
SCNI
QCNI
E disp
0.75E-5
0.62E-2
0.63E-12
E eng
0.37E-4
0.36E-1
0.23E-11
and its exact solution is a2 T κ +1 r cos θ + ux = [(κ + 1) cos θ + cos 3θ ] 4μ 2 r & a4 (57) − 3 cos 3θ r a2 T κ −3 r sin θ + uy = [(1 − κ) sin θ + sin 3θ ] 4μ 2 r & a4 (58) − 3 sin 3θ r where the parameters μ and κ in the case of plane stress are given by μ=
exactly passes the linear patch test but fails for the quadratic one. This is due to the fact that it can only reproduce constant strain field and cannot reproduce linear strain field. The proposed QCNI exactly passes both tests. This demonstrates that the quadratic exactness, which is consistent to the approximation order, is achieved by QCNI. In contrast, SCNI cannot provide quadratic exactness for second order EFG method. Numerical results on patch tests indicate that, for quadratic EFG method, QCNI should have a better numerical performance than SCNI in terms of accuracy, convergence, etc. This will be further demonstrated by the following examples.
6.2 Plate with a hole This example is an infinite plate with a hole of radius a centered at the origin and loaded at infinity by σx x = T, σ yy = σx y = 0. Figure 3a is a schematic diagram of this example
E , 2 (1 + υ)
κ=
3−υ 1+υ
(59)
The exact stress field is & 3a 4 a2 3 cos (2θ ) + cos (4θ ) + 4 cos (4θ ) σx x = T 1 − 2 r 2 2r (60) 2 & a 1 3a 4 σ yy = −T cos (2θ ) − cos (4θ ) + 4 cos (4θ ) r2 2 2r (61) 2 & a 1 3a 4 σx y = −T sin (2θ ) + sin (4θ ) − 4 sin (4θ ) r2 2 2r (62) As shown in Fig. 3b, due to two-fold symmetry, only the first quadrant with a = 1 is modeled. u x = 0 along the line x = 0 and u y = 0 along the line y = 0 are imposed. The tractions are prescribed on the other two boundaries according to the exact solutions. Convergence and efficiency of the three tested methods are compared in Figs. 4 and 5, respectively. SCNI is convergent. However, even if the quadratic MLS approximation is
Fig. 3 Plate with a hole problem: a schematic diagram; b solution domain
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Fig. 4 Convergence of the plate with a hole problem: a displacement; b energy
Fig. 5 Computational efficiency of the plate with a hole problem: a displacement; b energy
employed, its convergence rates are only comparable to those of linear FEM. Furthermore, its accuracy is much lower than the other two methods. The proposed QCNI which employs only approximation nodes as quadrature points performs, in accuracy and convergence, as good as FI which employs 16 quadrature points in each background triangle cell. Therefore, efficiency is greatly improved by the proposed method as shown in Fig. 5. Apparently, QCNI is the most efficient method. Figure 6 shows the σ yy distributions of this example. Quite obvious oscillations present in the result of SCNI. In contrast, the proposed QCNI results very smooth stress field and no spurious oscillations present. The stress field obtained by FI is reasonable, but it is not as smooth as the one given by QCNI. In fact, the stress plots of QCNI and the exact solution are identical. Numerical results of this example demonstrate that the existing SCNI, although it is convergent, is not suitable for second order meshfree Galerkin method. It cannot exploit
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the advantages of quadratic approximation such as the higher convergence in comparison to the linear approximation. In addition, it even introduces spurious oscillations in the stress fields. In contrast, the proposed QCNI shows second order convergence, higher accuracy and very smooth stress distributions. Furthermore, its efficiency is much higher than FI. There is no doubt that in these three integration schemes QCNI is the most suitable one for second order meshfree Galerkin methods. 6.3 Cantilever beam As shown in Fig. 7, a cantilever beam with length L = 10 m and height D = 1 m is next examined. The beam is subjected to a parabolic traction at the free end. The exact solution to this problem is ' ( Py D2 2 (63) ux = − (6L − 3x) x + (2 + υ) y − 6E I 4
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Fig. 6 σ yy Stress fields of the plate with a hole problem obtained by: a FI; b SCNI; c QCNI; d Exact solution
σx x = −
P (L − x) y P , σ yy = 0, σx y = I 2I
D2 − y2 4
(65)
Fig. 7 Schematic diagram of the cantilever beam example
uy =
P D2 x 3υy 2 (L − x)+(4+5υ) +(3L − x) x 2 6E I 4 (64)
and the exact stress field is given by
where P is the integration of the applied traction along the boundary of the free end and I = D 3 /12 is the moment of inertia. The non-zero displacements in accordance with Eqs. (63, 64) are prescribed at x = 0 as the essential boundary conditions. The traction given by the theoretical stress field is applied at x = L. Four irregular grids, i.e. 6 × 51, 9 × 81, 11 × 101 and 17 × 161 are used for convergence study and the results are plotted in Fig. 8. For this example, the developed QCNI shows even higher accuracy and convergence
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Fig. 8 Convergence of the cantilever beam problem: a displacement; b energy
Fig. 9 Computational efficiency of the cantilever beam problem: a displacement; b energy
rates than FI in energy. Note that FI almost loses its convergence in energy at fine discretizations. This should be a consequence of its failure in passing patch tests as shown in the first example. In contrast, QCNI is convergent within the whole tested range of discretizations and its convergence rates even exceed the theoretical prediction for quadratic FEM. This is also observed in [11] and is probably due to the higher order smoothness of meshfree approximation than that of FEM. SCNI, once again, displays lower accuracy and convergence rates than the other two methods. Especially, its accuracy in displacement and energy for the finest discretization is only on the order of 10−3 and 10−2 , respectively. In contrast, the corresponding errors of the proposed QCNI is on the order of 10−7 and 10−4 . A huge improvement in accuracy is achieved by QCNI. Furthermore, QCNI costs only a little more CPU time than SCNI. This can be observed from Fig. 9 which compares the efficiency of the three methods. Obviously, QCNI is the most efficient method.
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Figure 10 shows the resulting stress fields. Once again, spurious oscillation presents in the result of SCNI. In contrast, the proposed QCNI can result very smooth stress distribution which is identical to the theoretical solution. These results confirm, once again, the proposed QCNI fits second order meshfree Galerkin method much better than SCNI in terms of accuracy, convergence, efficiency and stability.
6.4 Natural frequencies of a rectangular plate Extension of the developed QCNI to dynamic analysis is straightforward. We only mention that, in our implementation, Newmark algorithm is used for the discretization in time domain. A rectangular plate with sizes a = 3 m and b = 2 m is first considered to investigate the accuracy of QCNI in the computation of frequencies. The four boundaries are fixed in the normal directions and free in the tangent directions. The exact frequencies of the plate in free vibration
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Fig. 10 σx y stress fields of the cantilever beam problem obtained by: a FI; b SCNI; c QCNI; d Exact solution
is given by
2 ωmn
Eπ 2 = 2ρ(1 + υ)
6.5 Manufactured dynamic problem '
m2 n2 + a2 b2
( (66)
where m and n are integers. For example, the first natural frequency corresponds to m = 1 and n = 0. A 13 × 9 irregular grid is used for frequency analysis. Table 3 lists the top five frequencies and the corresponding errors with respect to the theoretical solution given by Eq. (66). Apparently, the proposed QCNI is much more accurate than SCNI. Roughly speaking, its errors are only one percent of those of SCNI. On the other hand, QCNI is generally as accurate as FI, but more efficient. The CPU time consumed by FI, QCNI and SCNI in the construction of mass and stiffness matrixes is 0.2, 0.06 and 0.02 S, respectively. Obviously, taking both the accuracy and the computational speed into consideration, the proposed QCNI is the most efficient method.
Finally, a manufactured problem is examined to further investigate the numerical performance of the proposed QCNI for dynamic problems. A 2 × 2 square plate without initial displacement and velocity is considered. The displacement on the four boundaries is prescribed as 1 2 2 2 (67) u = xg (t) − (1 + υ) L + υy + x 3 1 2 2 2 v = yg (t) (1 + υ) L − υx − y (68) 3 where L is the side length of the plate and ' ( 2 − βt2 g (t) = α 1 − e
(69)
The following body force is applied & 1 2 2 2 bx = x −2Eg (t)+ρ g¨ (t) −(1 + υ) L +υy + x 3 (70)
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Table 3 Comparison of the top 5 frequencies and their errors (in parentheses) ω1 /Hz
ω2 /Hz
ω3 /Hz
ω4 /Hz
ω5 /Hz
FI
3369.85 (1.32 × 10−5 )
5055.01 (6.17 × 10−5 )
6078.37 (5.55 × 10−4 )
6740.56 (1.43 × 10−4 )
8437.26 (1.51 × 10−3 )
SCNI
3358.97 (3.21 × 10−3 )
5018.60 (7.14 × 10−3 )
6021.17 (8.86 × 10−3 )
6654.17 (1.27 × 10−2 )
8280.32 (1.71 × 10−2 )
QCNI
3369.60 (6.12
× 10−5 )
(7.29 × 10−5 )
(1.07 × 10−4 )
(2.18 × 10−4 )
8423.09 (1.67 × 10−4 )
Exact
3369.80
5054.33
6074.35
5054.70
6075.00
6738.13 6739.60
8424.50
Fig. 11 Convergence of the manufactured dynamic problem: (a) displacement; (b) energy Table 4 Consumed CPU time 12 × 12 (s)
18 × 18 (s)
22 × 22 (s)
33 × 33 (s)
FI
6.589
17.787
30.905
88.758
SCNI
0.440
1.001
1.683
4.296
QCNI
0.541
1.191
2.023
6.079
& 1 b y = y 2Eg (t) + ρ g¨ (t) (1 + υ) L 2 − υx 2 − y 2 3 (71) such that Eqs. (67, 68) are the exact displacement solutions and the corresponding exact stress fields are
(72) σx x = Eg(t) x 2 − L 2
σ yy = Eg(t) L 2 − y 2 (73) σx y = 0
(74)
Four regular grids, i.e. 12 × 12, 18 × 18, 22 × 22 and 33 × 33, with the time step size t = 0.01 s are employed. The accuracy and convergence after 200 time steps, i.e. at t = 2 s, are compared in Fig. 11 and the consumed CPU time is listed in Table 4. The proposed QCNI is evidently more accurate and converges faster than FI in both displacement and energy. Furthermore, the CPU time consumed by QCNI is only about a fifteenth of that consumed by FI. Thus, computational efficiency of second order meshfree Galerkin
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methods is greatly improved by the proposed QCNI method. The existing SCNI method, although it consumes the least CPU time, is not as accurate as the other two methods. In fact, QCNI consumes comparable CPU time with SCNI, but improves the accuracy and convergence rates tremendously. Figure 12 shows the resulting stress fields. Again, SCNI present spurious oscillations whereas the proposed QCNI shows smooth stress field which is identical to exact solution.
7 Conclusion A framework for derivative correction based on Hu-Washizu three-field variational principle is presented. Through rational derivations in the proposed framework, a highly efficient nodal integration scheme, named as QCNI, is developed for second order meshfree Galerkin methods. In each nodal representative domain, i.e. the background integration cell, QCNI can exactly reproduce a linear strain field which is consistent to the quadratic approximation of the displacement. In contrast, inconsistency presents in the existing SCNI for second order meshfree Galerkin methods, that is, the quadratic approximation can reproduce a linear strain field, however the integration scheme cannot. The major findings in the numerical tests of QCNI, SCNI and FI are
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Fig. 12 σx x stress fields of the manufactured dynamic problem obtained by: a FI; b SCNI; c QCNI; d Exact solution
(1) The proposed QCNI can exactly pass linear and quadratic patch tests and is the most accurate and efficient methods with highest convergence rates for both elastostatic and elastodynamic problems; furthermore, it can result very smooth stress field which is identical to exact solutions; (2) The existing SCNI can only pass linear patch test and fails for the quadratic one; it is convergent, but converges much slower than QCNI; its accuracy is also much lower than the other two methods; in addition, spurious oscillation present in its stress fields; (3) Full integration cannot exactly pass both the linear and quadratic patch tests; it gives reasonable accuracy and stress fields in most of the examples, however it consumes a lot more CPU time than the other two methods; furthermore, its convergence rate reduces
in fine discretizations and may be lost in some cases. In conclusion, the proposed QCNI is the most suitable integration scheme for second order meshfree Galerkin methods in consideration of accuracy, efficiency, convergence and stability. It is noted that QCNI is rationally established in the framework of derivative correction based on Hu-Washizu three-field variational principle, therefore its extension to higher order is possible and seems straightforward. In this way, the required quadrature points in meshfree computation can be reduced to minimum. To further improve the meshfree efficiency, computation of the nodal shape functions should be accelerated. The explicit forms of the MLS shape functions in EFG [27] and the explicit evaluation of the integrals in RKPM [28] for speeding up
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the computation of nodal shape functions are worth of consideration. Acknowledgments The authors are pleased to acknowledge the support of this work by the National Natural Science Foundation of China through contract/grant numbers 11102036, 11232003 and 11372066, the National Key Basic Research and Development Program (973 Program, No. 2010CB731502), the Fundamental Research Funds for the Central Universities through contract/grant number DUT12LK08 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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