DSpace at VNU: COMPUTATION OF LIMIT LOAD USING EDGE-BASED SMOOTHED FINITE ELEMENT METHOD AND SECOND-ORDER CONE PROGRAMMI...
Trang 1Vol 10, No 1 (2013) 1340004 (15 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0219876213400045
COMPUTATION OF LIMIT LOAD USING EDGE-BASED SMOOTHED FINITE ELEMENT METHOD AND SECOND-ORDER CONE PROGRAMMING
C V LE∗
Department of Civil Engineering International University, VNU-HCM, Vietnam
lvcanh@hcmiu.edu.vn
H NGUYEN-XUAN
Department of Mechanics Faculty of Mathematics and Computer Science University of Science, VNU-HCM, Vietnam
H ASKES
Department of Civil and Structural Engineering The University of Sheffield, Sheffield S1 3JD, UK
T RABCZUK
Institute of Structural Mechanics Bauhaus-University Weimar Marienstrasse 15, 99423 Weimar
T NGUYEN-THOI
Department of Mechanics Faculty of Mathematics and Computer Science University of Science, VNU-HCM, Vietnam
Received 17 January 2011 Accepted 10 June 2011 Published 18 January 2013
This paper presents a novel numerical procedure for limit analysis of plane problems using edge-based smoothed finite element method (ES-FEM) in combination with second-order cone programming In the ES-FEM, the discrete weak form is obtained based on the strain smoothing technique over smoothing domains associated with the edges of the elements Using constant smoothing functions, the incompressibility condition only needs to be enforced at one point in each smoothing domain, and only one Gaussian point is required, ensuring that the size of the resulting optimization problem is kept to a minimum The discretization problem is transformed into the form of a second-order cone
∗Corresponding author.
Trang 2programming problem which can be solved using highly efficient interior-point solvers Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plane stress and strain problems.
Keywords: Collapse load; limit analysis; SFEM; ES-FEM; SOCP.
1 Introduction
Upper bound limit analysis has been widely used to provide upper-bound estimates
of the load required to cause collapse of a body or structure One of the key conditions in the upper-bound computational limit analysis is that the flow rule (or incompressibility condition in plane strain) is required to hold everywhere in the problem domain This requirement can be easily met using constant strain finite elements However, it is well-known that the low-order displacement finite elements exhibit volumetric locking phenomena in the kinematic formulations It
is, therefore, desirable to develop an efficient method that can overcome the locking problem while allowing the flow rule to be met easily
Recently, Le et al [2010d]proposed a numerical kinematic formulation using the cell-based smoothed finite element method (CS-FEM) to furnish good (approx-imate) upper-bound solutions They have shown that when smoothed strains (using one cell version CS-FEM1) are used in the kinematic formulation the volumetric locking can be removed, the flow rule only needs to be enforced at any one point
in each smoothing cell, and it is guaranteed to be satisfied everywhere in the prob-lem domain Following this line of research, the main objective of this paper is to develop a computational limit analysis procedure which combines the edge-based smoothed finite element method (ES-FEM) with second-order cone programming (SOCP) to approximate upper-bound solutions for plane problems
The strain smoothing technique, which was originally proposed by Chenet al.
[2001] to stabilize a direct nodal integration in mesh-free methods, has been applied to the FEM settings to formulate various smoothed finite element meth-ods (SFEM) [Liu and Nguyen-Thoi (2010)], including CS-FEM [Liuet al (2007a); Liu et al (2007b); Nguyen-Xuan et al (2008)], node-based SFEM (NS-FEM) [Liuet al (2009)], and the ES-FEM [Liuet al (2009);Nguyen-Xuanet al (2009)] and the face-based SFEM (FS-FEM) [Nguyen-Thoi et al (2009)] Each of these smoothed FEM has different characters and properties, and has been used success-fully in solid mechanics [Nguyen-Xuan et al (2008); Nguyen-Thoi et al (2010); Nguyen-Xuan et al (2010)] Theoretical study of these SFEM was also reported
in Liu et al (2007b) and Nguyen-Xuan et al [2008] In general, these SFEM are
able to remove locking problems while accurate solutions can be obtained with minimal computational effort.Tranet al [2010]has applied the ES-FEM to limit and shakedown analysis problems using Koiter’s theorem, in which fictitious elastic stresses are assumed However, in this paper the ES-FEM is formulated associ-ated with Markov’s kinematic theorem, and the resulting optimization problem is cast in the form of a SOCP problem so that a large-scale problem can be solved
Trang 3efficiently [Leet al (2009); Le et al (2010a); Le et al (2010b); Le et al (2010c) and
references therein] Therefore, the present approach is different from the previous ES-FEM approach given inTranet al [2010]
The paper is organized as follows The next section briefly describes the edge-based smoothed finite element method The kinematic limit analysis formulation
is then recalled and ES-FEM based discretization problem is then formulated as
a SOCP in Sec 3 Numerical examples are provided in Sec 4 to illustrate the performance of the proposed procedure
2 Brief of the ES-FEM
In ES-FEM, basing on the mesh of elements, we further discretize the problem domain into smoothing domains based on edges of the elements such that Ω ≈
Ned
k=1
Ωk and Ωi ∩ Ω j = for i = j, in which Ned is the total number of edges of all elements in the entire problem domain Moreover, ES-FEM shape functions are identical to those in the FEM However, instead of using compatible strains, the ES-FEM uses strains smoothed over local smoothing domains These local smoothing domains are constructed based on edges of elements as shown in Fig.1
A strain smoothing formulation is now defined by the following operation
˜
k =
Ωk
h (x)φ k(x)dΩ =
Ωk
∇ suh (x)φ k (x)dΩ, (1) where∇ suhare the compatible strains of the approximate fields, uh , and φ k(x) is
a distribution (or smoothing) function that is positive and normalized to unity
Ωk
For simplicity, the smoothing function φ k is taken as
φ k(x) =
1/A k , x ∈ Ω k
1
2
3
element j
edge-based smoothing cell
centroid
Ωk
Γk
4
element i
Fig 1 (Color online) Smoothing cell Ωkconnected to edgek of triangular elements.
Trang 4where A k is the area of the smoothing domain Ωk, and is calculated by
A k =
Ωk
dΩ = 1 3
N k e
j=1
where N k is the number of elements around the edge k (N k = 1 for the boundary
edges and N e k = 2 for interior edges) and A j e is the area of the jth element around the edge k.
Since interior edges are formed by two neighboring elements, the smoothed strains in the smoothing domain Ωk can be determined by
˜
k = ˜Bk1dk1+ ˜Bk2dk2 = ˜Bkdk , (5)
where d k1 and d k2 are nodal displacement vectors of element e1 and element e2,
respectively, d k is the displacement vector of the nodes associated with edge k, and
˜
Bkj (j = 1, 2) are the strain-displacement matrices defined by
˜
Bkj =
˜
N 1,x kj 0 N˜n,x kj 0
0 N˜1,y kj 0 N˜n,y kj
˜
N 1,y kj N˜1,x kj N˜n,y kj N˜n,x kj
with
˜
N I,α kj = A j e
3A k
Γkj
where ˜N I,α kj is the smoothed version of shape function derivative N I,α kj , n α is the normal vector and Γkj is boundaries of element j associated with edge k.
It is worth noting that the ES-FEM is different from the standard FEM by two key points: (1) FEM uses the compatible strain on the element, while ES-FEM uses the smoothed strain on the smoothing domain; and (2) the assembly process of FEM
is based on elements, while that of ES-FEM is based on smoothing domain Ωk
3 Limit Analysis Based on ES-FEM
Consider a rigid-perfectly plastic body of area Ω∈ R2 with boundary Γ, which is
subjected to body forces f and to surface tractions g on the free portion Γtof Γ The
constrained boundary Γuis fixed and Γu ∪ Γ t= Γ, Γu ∩ Γ t= Let ˙u = [ ˙u ˙v ] T be
plastic velocity or flow fields that belong to a space Y of kinematically admissible
velocity fields [Ciria et al (2008); Christiansen (1996)], where ˙u and ˙v are the velocity components in x- and y-direction, respectively.
The external work rate associated with a virtual plastic flow ˙u is expressed in
the linear form as
F ( ˙u) =
Ω
fTu dΩ +˙
Γt
Trang 5Upon defining C = { ˙u ∈ Y | F ( ˙u) = 1}, the collapse load multiplier λ+ can be determined by the following mathematical programming
λ+= min
˙
u∈C
Ω
where strain rates ˙ are given by
˙
=
˙ xx
˙ yy
˙γ xy
with L is the differential operator
L =
∂
∂y
∂
∂y
∂
∂x
The plastic dissipation D( ˙) is defined by
D( ˙) = max
ψ(σ)≤0 σ : ˙ ≡ σ : ˙ (12)
in which σ represents the admissible stresses contained within the convex yield
surface andσ represents the stresses on the yield surface associated to any strain rates ˙ through the plasticity condition.
In the framework of a limit analysis problem, only plastic strains are considered and they are assumed to obey the normality rule
˙
where the plastic multiplier ˙µ is non-negative and the yield function ψ(σ) is convex.
In this study, the von Mises failure criterion is used
ψ(σ) =
σ2
xx + σ2
yy − σ xx σ yy + 3σ2
xy − σ0 plane stress
1
4(σ xx − σ yy)2+ σ2
xy − σ0 plane strain,
(14)
where σ0is the yield stress
Then the power of dissipation can be formulated as a function of strain rates
asCapsoni and Corradi [1997]
D( ˙) = σ0
˙
Trang 6Θ =
1 3
plane stress
−11 −1 01 0
plane strain
(16)
Note that condition (13) acts as a kinematic constraint which confines the vectors
of admissible strain rates For plane strain problems, the yield surface ψ(σ) is
unbounded, and the incompressibility condition χ T = 0, where χ = [ 1 1 0 ]˙ T,
must be introduced to ensure that the plastic dissipation D( ˙) is finite [Andersen
et al (1998);Christiansen and Andersen (1999)]
In the ES-FEM formulation, the plastic dissipation is expressed as
D ES−FEM=
Ned
k=1
Ωk
σ0
˙˜
k Θ ˙˜ h
in which ˙˜ h
k can be obtained from Eq (5).
When three-node triangular elements are used, ˙˜ h
kare constant over the
smooth-ing domain Ωk, and hence Eq (17) can be rewritten as
D ES−FEM=
Ned
k=1
σ0A k
˙˜
k Θ ˙˜ h
Consequently the optimization problem (9) associated with the ES-FEM can now be rewritten as
λ+ = min
Ned
k=1
σ0A k
˙˜
k Θ ˙˜ h k
s.t
˙
uh= 0 on Γu
which will search the nodal velocities vector
The above limit analysis problem is a nonlinear optimization problem with equal-ity constraints In fact, the problem can be reduced to the problem of minimizing
a sum of norms as
λ+ = min
Ned
k=1
σ0A k ρ β , β = 1, 2
s.t
˙
uh= 0 on Γu
Trang 7whereρ β are additional variables defined by
ρ1
ρ2
ρ3
=√1
3
21 √0 0
3 0
˙˜ h
k for plane stress, (21)
ρ1
ρ2
=
( ˙˜ h xxk − ˙˜ h
yyk)
˙˜γ h xyk
Introducing auxiliary variables t1 , t2, , t Ned, optimization problem (20) can
be cast as a SOCP problem
λ+ = min
Ned
j=1
σ0A j t j
s.t
˙
uh= 0 on Γu
F ( ˙u h) = 1
ρ β i ≤ t i i = 1, 2, , Ned,
(23)
where the third constraint in problem (23) represents quadratic cones
4 Numerical Examples
In this section, the performance of the proposed solution procedure is illustrated via
a number of benchmark problems in which analytical and other numerical solutions are available All examples are considered in either plane stress or plane strain state and the von Mises criterion is exploited Because distorted meshes does not necessarily result in better solutions [Leet al (2010d)], and for convergence study,
regular meshes will be used in examples
The first example deals with a square plate with a central circular hole which is
subjected to biaxial uniform loads p1 and p2as shown in Fig.2(a), where L = 10 m.
Owing to symmetry, only the upper-right quarter of the plate is modeled, see Fig.2(b) Symmetry conditions are enforced on the left and bottom edges
Analytical solutions are available [Gaydon and McCrum (1954)] for the case
when p2 = 0, namely λ = 0.8 p1
σ0 Therefore, to enable objective validation, the procedure was first applied to this case Numerical solutions obtained for different
models with variation of N are compared with those obtained using CS-FEM, as
shown in Table 1 It can be seen that solutions obtained by using ES-FEM are more accurate than those obtained by using CS-FEM2, CS-FEM4 and FEMQ4,
and are slightly higher than results obtained by using CS-FEM1 (CS-FEMk– SFEM
Trang 8L/5
2
p
1
p
1
p
2
p
N = 10
Fig 2 A square plate with a circular hole: (a) geometry and loading, (b) finite element mesh (N is the number of elements along the horizontal symmetry axis, and thus a measure for the mesh density).
Table 1 Collapse load multiplier of the plate with variation ofN (p2 = 0).
N × N
Formulations 6× 6 12× 12 24× 24 48× 48 Analytical solution CS-FEM1 0.8151 0.8047 0.8017 0.8006 0.800 CS-FEM2 0.8216 0.8078 0.8035 0.8018
CS-FEM4 0.8226 0.8085 0.8038 0.8019 FEMQ4 0.8238 0.8090 0.8041 0.8021 Present method 0.8217 0.8077 0.8030 0.8013
four-noded quadrilateral element with k smoothing cells or subcells, whereby k =
1, 2 and 4).
When 288 T3 elements (N = 12) are used, the proposed method provides a
solu-tion of 0.8077 compared with 0.7980 obtained inTranet al [2010], where ES-FEM was used in combination with a primal-dual algorithm for shakedown analysis of structures proposed by Vu et al [2004] It is evident that the method presented
byTranet al [2010]provides a lower solution than the actual collapse multiplier, and therefore it does not guarantee an upper-bound on the actual collapse load On the contrary, the proposed method can provide strict upper bounds on the actual collapse multiplier, as shown in Table 1 Moreover, the present solution procedure with the use of SOCP is more efficient and robust since just less than 30 secondsa
were taken to solve the optimization problem with up to 32,834 variables and 21,123 constraints The convergence rate is also illustrated in Fig.3 It is evident that all
numerical solutions converge to the exact solution as the mesh size h tends to zero.
aThe code is written using MATLAB and was run using a 2.8 GHz Pentium 4 PC running Microsoft XP.
Trang 9−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
log10(mesh size h)
FEMQ4 CS−FEM1 CS−FEM2 CS−FEM4 ES−FEM
~1.13
~1.15
~1.59
~1.16
~1.36
Fig 3 (Color online) Convergence rate withp2 = 0 (values indicated in the figure are approximated
slopes and the results of CS-FEM and FEMQ4 were obtained in Le et al [2010]).
It can also be observed that convergence rate of the ES-FEM is higher than that
of CS-FEM2, CS-FEM4, and FEM Although the ES-FEM converges slower than CS-FEM1, it guarantees stable solutions when Koiter’s kinematic theorem is used [see Transet al (2010)] while the CS-FEM1 does not Moreover, the ES-FEM is
based on triangular meshes that can be generated more easily than quadrilateral meshes used in the CS-FEM
Next, the geometric effect of the circular hole was examined by considering
various values of ratio R/L The obtained solutions were reported in Table 2 It can be observed that in case when 288 elements are used, p1 = 1 and p2 = 0, the present solutions are in good agreement with those obtained previously
Table 3 compares the best solutions obtained using the present method with solutions obtained previously by different limit analysis approaches (kinematic or
Table 2 Collapse multiplier:p1 = 1 andp2 = 0.
R/L Heitzer [1999] Tran [2008] [Trans et al (2010)] Present method
Trang 10Table 3 Collapse load multiplier with different loading cases andN = 48 compared with
previously obtained solutions.
Loading cases Approach Authors p2 =p1 p2 =p1/2 p2 = 0 Kinematic da Silva and Antao [2007] 0.899 0.915 0.807 (upper bound) Le et al [2010d] 0.895 0.911 0.801
Present method 0.896 0.911 0.801
Mixed formulation Zouain et al [2002] 0.894 0.911 0.803 Analytical solution Gaydon and McCrum [1954] — — 0.800
Static Chen et al [2008] 0.874 0.899 0.798 (lower bound) Gross-Weege [1997] 0.882 0.891 0.782
Nguyen-Dang and Palgen [1979] 0.704 — 0.564
Fig 4 (Color online) Collapse mechanisms (original shape in green) for loading casep2 = 0.
static) using FEM, CS-FEM or element-free Galerkin simulations The present solu-tions are in excellent agreement to those obtained by Leet al (2010d) when p2= 0
and p2 = p1 /2 The collapse mechanism for the case when p2 = 0 is shown in Fig.4, where the deformation is calculated by multiplying the computed collapse velocity by a suitable time scale and then adding it to the original grid The plastic dissipation distribution is also displayed in Fig.5
This classical plane strain problem was originally investigated by Prandtl [1920], which consists of a semi-infinite rigid-plastic von Mises medium under a punch