Dual La Efg Canhle

7/29/2019 Dual La Efg Canhle

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Stabilized displacement and equilibrium meshfree modelsfor computation of collapse loads

Canh V. Le

Department of Civil Engineering, International University, VNU HCMC, Vietnam

SUMMARY

This paper describes dual limit analysis formulations based on stabilized displacement and equilibriummeshfree models. Displacement and stress fields are approximated using moving least squarestechnique. A stabilized conforming nodal integration is applied, ensuring that the size of the resultingoptimization problem is kept to a minimum, and that equilibrium only needs to be enforced at thenodes while numerical instability problems can b e eliminated. The resulting optimizations are the castin the form of a standard second order cone programming so that solutions can be obtained rapidly.

key words: limit analysis; mesh-free methods; stabilized displacement and equilibrium models;

second order cone programming.

1. INTRODUCTION

The load required to cause collapse of a body or structure can be estimated using eitherupper or lower bound theorems. Displacement and stress fields can be approximated by usingfinite element method. However, solutions obtained from finite element based computationallimit analysis procedures can be very sensitive to mesh geometry, particularly for problemswhich contain strong singularities in the stress and/or displacement fields [1]. It is thereforeworthwhile to explore a range of alternative methods. In recent years so-called meshfreemethods have been developed to provide a flexible alternative approach to FEM. Meshfreemethods have been applied successfully to a wide range of computational problems, provingpopular due to its rapid convergence characteristics and its ability to obtain highly accuratesolutions for problems involving stress discontinuities. The EFG method have been developedfor limit analysis of plates [24], and it has been shown that EFG-based limit analysisprocedures can provide accurate solutions with minimal computational effort. Following thisline of research, the main objective of this paper is to develop dual meshfree-based formulationsfor limit analysis of plane strain problems.

Once the displacement or stress fields are approximated and the bound theorems applied, theunderlying limit analysis problem becomes a problem of optimization involving either linear ornonlinear programming. In this paper, both upper and lower bound formulations will be castin the form of a second order cone programming so that the resulting optimization problemcan be solved using efficient interior-point solvers.


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STABILIZED DISPLACEMENT AND EQUILIBRIUM MESHFREE MODELS 1

2. DUALITY AND BOUNDS IN LIMIT ANALYSIS

Let X denote an appropriate space of a statically admissible stress state, whereas Y is anappropriate space of a kinematically admissible velocity state. For smooth fields and u,the classical form of the equilibrium equation can always be transformed to a more precisevariational form as

a(, u) = Wext(u), u Y (1)

where the internal work rate a is rewritten as a function of and u

a(, u) =

(LT ) u d +

t

(n ) u d (2)

and the external work rate Wext is

Wext(u) = f u d + t g u d (3)The static principle of limit analysis can be now expressed as

= max

s.t

{a(, u) = Wext(u), u Y B, B = { X| ((x)) 0 x }

(4)

in which the so-called yield function () is convex.Due to the fact that both a and Wext are linear functions of u, the equation (4) can be cast

as [5]

= maxB

minuC

a(, u) (5)

where the set C is defined by C = {u Y | Wext(u) = 1}If expressing the plastic dissipation rate in terms of and u as

Wint(u) = maxB

a(, u), (6)

the kinematic principle of limit analysis can be written

= minuC

Wint(u)

= minuC

maxB

a(, u) (7)

It is clear from equation (5) and (7) that strong duality holds as

maxB

minuC

a(, u) = minuC

maxB

a(, u) (8)

In summary, the exact collapse load multiplier can be obtained by solving one of the following

optimisation problems

= max { | B : a(, u) = Wext(u), u Y} (9)

= maxB

minuC

a(, u) (10)

= minuC

maxB

a(, u) (11)

= minuC

Wint(u) (12)


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2 CANH V. LE

In any numerical procedure for limit analysis problems, the problem spaces must

be discretised by numerical methods. For the static approach (9) or the kinematicformulation (12), only one field need be discretised; that is the stress or displacement field,respectively. On the other hand, the mixed formulations (10, 11) require the approximationof both stress and displacement fields, and therefore mixed finite elements can be used [58].In the present work, static and kinematic forms, which respectively provide lower and upperbounds on the actual collapse multiplier, are of particular interest.

3. MESHFREE DISCRETIZATION

3.1. Moving least squares approximation and smoothing technique

Using the moving least squares (MLS) technique [9], the function of the field variable fh(x)

in the domain can be approximated at point x as

fh(x) =n

I=1

I(x)fI (13)

in which

I(x) = pT(x)A1(x)BI(x) (14)

A(x) =n

I=1

wI(x)p(xI)pT(xI) (15)

BI(x) = wI(x)p(xI) (16)

where n is the number of nodes; p(x) is a set of basis functions; wI(x) is a weight functionassociated with node I.

The smoothing technique presented in [10] will be adapted in order to stabilize problemsinvolving first derivatives as follows

fh,(xJ) =1

aJ

J

fh(x)n(x) d (17)

where J is the boundary of the representative domain J.

Now, introducing a moving least squares approximation of the function fh, the smoothedversion of fh, can be expressed as

fh,(xJ) =n

I=1

I,(xJ)fI (18)

with

I,(xJ) =1

aJ

J

I(x)n(x)d (19)


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3.2. Smoothed upper bound formulation

Using the moving least squares technique, the approximated displacement uh(x) can beexpressed as

uh(x) =

uh

vh

=

nI=1

I(x)

uIvI

(20)

Hence the upper-bound limit analysis problem for plane strain problems can be formulated as

+ = minn

J=1

0 AJ

hT(xJ) h(xJ)

s.t

F(uh) = 1uh = 0 on u

T

h

(xJ) = 0 J = 1, 2, . . . , n

(21)

in which hij is the smoothed value of strains hij at node J

=

1 1 01 1 0

0 0 1

; = [ 1 1 0 ]T (22)

3.3. Smoothed lower bound formulation

Stress fields can be approximated by

h(x) =

hxxhyy

hxy =

n

I=1

I(x)xxIyyI

xyI (23)

With the use of the smoothed value h, equilibrium equation can be rewritten as

A11 + A23 = 0 (24)

A13 + A22 = 0 (25)

where

A1 =

. . . . . . . . . . . .1,x(xj) 2,x(xj) . . . n,x(xj)

. . . . . . . . . . . .

nn

(26)

A2 = . . . . . . . . . . . .

1,y(xj) 2,y(xj) . . . n,y(xj). . . . . . . . . . . .

nn

(27)

1 =[

xx1 . . . xxn]T

(28)

2 =[

yy1 . . . yyn]T

(29)

3 =[

xy1 . . . xyn]T

(30)


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4 CANH V. LE

The lower bound limit analysis formulation can now be expressed as

= max

s.t

A11 + A23 = 0A13 + A22 = 0 B

(31)

4. SOLUTION OF THE DESCRETE PROBLEM

In this section, above described optimization problems will cast in the form of a standardsecond-order cone programming so that they can be solved using highly efficient primal-dualsolvers.

4.1. Upper bound programming

If defining a vector of additional variables is introduced as

=

1

2

=

hxx(xJ) hyy(xJ)hxy(xJ)

(32)

and introducing auxiliary variables t1, t2, . . . , tn, the problem (21) can be cast as a second-order cone programming (SOCP) problem

+ = minn

j=1

0Ajtj

s.t

F(uh) = 1

u

h

= 0 on uT h(xJ) = 0 J = 1, 2, . . . , n

||||i ti i = 1, 2, . . . , n

(33)

4.2. Lower bound bound programming

If introducing a vector of additional variables as

=

1

2

3

=

0

1

2(xx yy)

xy

(34)

the von Mises failure criterion for plane strain problems can be rewritten as

B L={ R3 | 1 23L2 =

22 +

23

}(35)

Hence the lower bound limit analysis formulation can now be expressed as

= max

s.t

A11 + A23 = 0A13 + A22 = 0k Lk, k = 1, 2, . . . , n p

(36)


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where np is the number of yield points.

5. NUMERICAL EXAMPLE

The performance of the limit analysis procedure described will now be tested by examining theclassical plane strain problem, as shown in Figure 1. For a load of 20, the analytical collapsemultiplier is = 2 + = 5.142. The strong discontinuity at the edge of the indentor presents asevere challenge to many numerical analysis procedures. Problems were setup using MATLABand the Mosek version 5.0 optimization solver was used to obtain all solutions.

H1 1

B

02W

Figure 1. Prandtl problem: geometry and loading

Due to symmetry, only half the domain needs to be considered. A rectangular region ofdimensions B = 5 and H = 2 was used and the indentor (or punch) was represented by auniform vertical load. Finally, appropriate boundary conditions were imposed, all as indicatedin Figure 2.

0u

y (v)

0vu

0vu

x (u)

Figure 2. Prandtl problem: nodal layout, Voronoi cells and displacement boundary conditions

A convergence study for the clamped square plate is shown in Figure 3. It can be observedthat when increasing the number of nodes the numerical solutions appear to converge towardsthe actual solution. While lower bound solutions converge from below, upper bound solutionsconverge from above.


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6. Conclusions

Displacement and equilibrium EFG-based models for limit analysis of plane strain problemshave been proposed. The solutions obtained using the proposed numerical procedures showgood agreement with results available in the literature. Advantages of applying EFG to limitanalysis problems are that problem size is reduced, and accurate solutions can be obtainedusing a relatively small number of nodes. The combination of the stabilized conformingnodal integration technique (SCNI) and second order cone programming (SOCP) optimizationalgorithm leads to an efficient and robust method.

REFERENCES

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4. C. V. Le, H. Askes, and M. Gilbert. Adaptive Element-Free Galerkin method applied to the limit analysisof plates. Computer Methods in Applied Mechanics and Engineering, 199:24872496, 2010.

5. E. Christiansen. Limit analysis of collapse states. Handbook of Numerical Analysis, Elsevier ScienceB.V., 1996.

6. E. Anderheggen and H. Knopfel. Finite element limit analysis using linear programming. InternationalJournal of Solids and Structures, 8:14131431, 1972.

7. E. Christiansen. Computation of limit load. International Journal for Numerical Methods in Engineering,17:15471570, 1981.

8. R. Casciaro and L. Cascini. A mixed formulation and mixed finite elements for limit analysis. InternationalJournal for Numerical Methods in Engineering, 18:211243, 1982.

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13. A. Makrodimopoulos and C. M. Martin. Upper bound limit analysis using simplex strain elementsand second-order cone programming. International Journal for Numerical and Analytical Methods inGeomechanics, 31:835865, 2006.

14. C.V. Le, H. Nguyen-Xuan, H. Askes, S. Bordas, T. Rabczuk, and H. Nguyen-Vinh. A cell-based smoothedfinite element method for kinematic limit analysis. International Journal for Numerical Methods inEngineering, 83:16511674, 2010.

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Dual La Efg Canhle