A primal–dual algorithm for shakedown analysis of structures

Comput. Methods Appl. Mech. Engrg. 193 (2004) 4663–4674

www.elsevier.com/locate/cma

A primal–dual algorithm for shakedown analysis of structures

D.K. Vu, A.M. Yan, H. Nguyen-Dang *

Division of ASMA-Fracture Mechanics University of Liege, Bat. B52, Chemin des Chevreuils 1, Liege 4000, Belgium

Received 16 October 2002; received in revised form 19 March 2004; accepted 22 March 2004

Abstract

The paper describes a primal–dual algorithm for shakedown analysis of structures with the use of kinematically

admissible finite elements. Starting from Koiter kinematic theorem, a primal–dual optimization condition is established

by adopting static terms as complementary variables. Newton-like iterations are performed to give both upper and

quasi-lower bounds of the load factor that converge rapidly to the accurate solution of shakedown limit.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Shakedown analysis; Limit analysis; Duality; Optimization

1. Introduction

Shakedown analysis provides the loading variation limit of a structure, under which the plasticdeformation occurred in first loading cycles will cease developing and the behaviour of the structure is

again elastic. In engineering practice the shakedown limit load may be used as a design loading

parameter. Application of shakedown analysis may be found in safety assessment and design of boilers,

pressure vessels and piping, metal forming, offshore structures, foundations, pavement and railway rails,

etc.

In classical limit analysis the loading mode is considered simple, namely, monotonic and proportional

mechanical load. However in practice, structures are often subjected to the action of varying mechanical

and thermal loading. These loads may be repeated (cyclic) or varying arbitrarily in certain ranges. In thiscase, loads that are lower than plastic collapse limit may cause failure of structures due to either excessive

deformation (incremental plasticity or ratcheting) or a local low-cycle fatigue break (alternating plas-

ticity). On the other hand if variable loads belong to a certain loading domain (bounds), after finite time

or cycles of loading, the plastic strain happened in a structure may cease to develop and the accumulated

dissipative energy in the whole structure remains bounded. This occurrence is traditionally referred to as

(elastic) ‘‘shakedown’’. Shakedown analysis aims to estimate whether the structure under a given loading

* Corresponding author. Tel.: +32-4-3668240; fax: +32-4-3668311.

E-mail address: [email protected] (H. Nguyen-Dang).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2004.03.011

mail to: [email protected]

4664 D.K. Vu et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 4663–4674

domain can attain such an elastic stabilisation, and to find the critical limits of load variation. In thespecial case when having only one loading vertex, shakedown analysis reduces to limit analysis. The

detailed description and discussion on shakedown analysis can be found in [4,8].

Theoretically shakedown limit may be found by an elastic–plastic finite element calculation, but in

most cases the task is extremely cumbersome. Alternatively, the shakedown limit load factor is for-

mulated by a direct method using static or kinematic theorem. Based on Melan static theorem, we look

for an optimal residual stress field to find a maximum loading factor [6]. Based on Koiter kinematic

theorem, we minimize the loading factors by optimizing the kinematical fields [4]. The two inadaptation

loading limits (the incremental plasticity or ratcheting limit and the alternating plasticity or low-cyclefatigue limit) may be obtained separately. The calculation of the former, which is most important in

practice, can be transformed into an equivalent limit analysis [7,8]. Alternatively, a unique shakedown

limit, which is the minimum of the two inadaptation limits, may be calculated by an optimization

algorithm [2,5,7–9].

However, in almost all the existing algorithms the duality of the static lower bound and the kinematic

upper bound of the load factor was not practically used in numerical calculations. The optimization

variables are either purely static or purely kinematic. This fact impeded further improvement of the cal-

culating efficiency. Besides, although Newton method has shown its high competence in limit analysis, itwas not applied effectively in shakedown. The application of the duality was explored in limit analysis by

Zouain et al. [10] in the case where the plastic incompressibility condition could be automatically satisfied

with the finite elements used. Recently, Andersen et al. [1] developed an excellent primal–dual interior-point

algorithm for minimizing a sum of Euclidean norms. They showed that the application of the duality

combining with Newton method may lead to very accurate results in limit analysis with high efficiency. The

present work constitutes a new development along this direction for computing the shakedown limit load of

structures.

2. Shakedown analysis as a problem of non-linear programming

Consider a structure made of elastic–perfectly plastic material and subjected to n time-dependent loads.

By using Koiter’s kinematical theorem the upper bound of the shakedown limit load factor can be for-

mulated. If kinematically admissible elements and von Mises yield criterion are used, this upper bound can

be discretized as

aþ ¼ minXmk¼1

XNG

i¼1

ffiffiffi2

pwikv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_eTikD _eik þ e2

q

s:t::

Pmk¼1

_eik ¼ Biq 8i ¼ 1;NG;

Dv _eik ¼ 0 8i ¼ 1;NG; 8k ¼ 1;m;Pmk¼1

PNG

i¼1

wi _eTikrEik ¼ 1;

8>>>>>><>>>>>>:ð2:1Þ

where _eik and rEik denote the vector of deformation rate and vector of the fictitious elastic stress at Gauss

point i and load vertex k; q is the nodal displacement vector; Bi is the deformation matrix; m ¼ 2n; NG

denotes the total number of Gauss points of the whole structure with integration weight wi at Gauss point i;e is a small regularization value. Generally speaking, the multiplier aþ in (2.1) is an approximation of the

truly upper bound of the shakedown load multiplier [11].

D.K. Vu et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 4663–4674 4665

For the sake of simplicity, let us rewrite the upper bound limit (2.1) in the form:

aþ ¼ minXNG

i¼1

Xmk¼1

ffiffiffi2

pkv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieTikeik þ e2

q; ðaÞ

s:t::

Pmk¼1

eik � bBiq ¼ 0 8i ¼ 1;NG; ðbÞ

1

3Dveik ¼ 0 8i ¼ 1;NG; 8k ¼ 1;m; ðcÞPNG

i¼1

Pmk¼1

eTiktik � 1 ¼ 0; ðdÞ

8>>>>>>><>>>>>>>:ð2:2Þ

where we define a new strain rate vector (the dot mark denoting time derivative is omitted for simplicity):

eik ¼ wiD1=2 _eik; a new fictitious elastic stress field: tik ¼ D�1=2rE

ik; and a new deformation matrix:bBi ¼ wiD1=2Bi. In these definitions D1=2 and D�1=2 are symmetric matrices such that: D�1=2 ¼ D1=2

� ��1and

D ¼ D1=2D1=2.

By using dual theory, the dual form of (2.2) can be proved [12] to be

a� ¼ maxcik ;bi ;a

a ðaÞ

s:t::

cik þ bi þ tikak k6ffiffiffi2

pkv; ðbÞPNG

i¼1

bBTi bik ¼ 0; ðcÞ

8><>:ð2:3Þ

where cik; bi; a are Lagrange multipliers, and specially cik; bi can be interpreted, respectively, as residual and

hydrostatic stress vectors related to the compatibility and incompressibility conditions. By using these

Lagrange multipliers, the stationary conditions of (2.2) can be written as follows:ffiffiffi2

pkveikffiffiffiffiffiffiffiffiffiffieTikeik

p � cik þ bi þ atikð Þ ¼ 0; ðaÞ

Dveik ¼ 0; ðbÞPmk¼1

eik � bBiq ¼ 0; ðcÞ

PNG

i¼1

ðbBTi biÞ ¼ 0; ðdÞ

PNG

i¼1

Pmk¼1

eTiktik � 1 ¼ 0; ðeÞ

ð2:4Þ

It should be pointed out that the formulation (2.3) is exactly the discretized form formulated by usingMelan’s static theorem, and that the lower bound in (2.3) is only a quasi-lower bound and has no strict

bounding characteristic. The reason is that local equilibrium is not guaranteed and von Mises’ yield

condition is stressed only at Gauss points [11,12].

3. Primal–dual algorithm for shakedown analysis

The shakedown limit load factor can be found by solving the system (2.4). Unfortunately, solving di-rectly this system is not a good idea because it leads to a system of equations much bigger than that in the

case of purely elastic computation. The resulted system thus requires large amount of computer memory as

well as computational effort to solve. Trying to keep our problem size as small as possible, we use here the


penalty method to handle the equality conditions (2.4) with Lagrange multipliers playing intermediateroles. A similar technique, which showed great efficiency in large-scale problems, was successfully applied in

limit analysis by Andersen et al. [1].

Numerically speaking, the stationary conditions (2.4) are very difficult to satisfy due to their singular

property. Here we lack an appropriate criterion to stop optimization process or to assess the exactness of

the solution. However, if a strictly lower bound with a strictly upper bound is found at the same time, we

will possess a very useful tool to control the obtained results. Although those strict bounds are hard to find,

in the following algorithm we will try to build an approximation while employing Newton method and

making use of (2.3) to solve (2.4).

Primal–dual algorithm 3.1:

1. Initialize the displacement and strain rate vectors q0 and e0 so that the normalization condition is satisfied:XNG

i¼1

Xmk¼1

tTike0ik ¼ 1: ð3:1Þ

Set all stress vectors to null:

c0ik ¼ 0; b0i ¼ 0 8i ¼ 1;NG; k ¼ 1;m: ð3:2ÞSet up values for penalty parameter c and for e. Set up convergence criteria.

2. Calculate the incremental vectors dq; deik of displacement, deformations and ðdaþ aÞ at the current val-ues of q; e by solving the following system:

deik ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieTikeik þ e2

p cM�1ik � gik þ dbi þ tik dað Þ �cM�1

ik f ik;PNG

i¼1

Pmk¼1

tTik eik þ deikð Þ ¼ 1;

dq ¼ �qþ bS�1 f̂1 þ aþ dað ÞbS�1 f̂2;

8>>><>>>: ð3:3Þ

where

bS ¼XNG

i¼1

bBTibK�1

ibBi;

f̂1 ¼XNG

i¼1

bBTibK�1

i

Xmk¼1

eik

�Xmk¼1

cM�1ik eik � c

Xmk¼1


q cM�1ik Dveik

!;

f̂2 ¼XNG

i¼1

bBTibK�1

i

Xmk¼1


q cM�1ik tik;

bKi ¼ I

"þ c

Xmk¼1


q cM�1ik

#;

cMik ¼1

2Mik

�þMT

ik

�þ


qcDv;

Mik ¼ I

"� cik þ bi þ atikð ÞeTikffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

eTikeik þ e2p #

;

f ik ¼ eik �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieTikeik þ e2

qcikð þ bi þ atikÞ;

gik ¼ cik þ cDveik;

hi ¼ bi þ cXmk¼1

eik

� bBiq

!:

ð3:4Þ


3. Perform a line-search to find k̂q such that

k̂q ¼ min FP qð þ kdq; eþ kdeÞ; ð3:5Þ

where FP is the penalty function:

FP ¼XNG

i¼1

ffiffiffi2

pkvXmk¼1


q8<: þ c2

Xmk¼1

eTikDveik þc2

Xmk¼1

eik

� Biq

!T Xmk¼1

eik

� Biq

!9=;: ð3:6Þ

4. Update the displacement and strain rate vectors as

q ¼ qþ k̂q dq; eik ¼ eik þ k̂q deik: ð3:7Þ5. Calculate the incremental vectors of stress vectors cik, bi

dcik ¼ �cDvdeik � gik; dbi ¼ �cXmk¼1

deik

� bBi dq

!� hi: ð3:8Þ

Perform a line-search to find k̂s such that

k̂s ¼ max k

s:t: : cikðk þ bi þ tikaÞ þ k dcikð þ dbi þ tik daÞk6ffiffiffi2

pkv:

ð3:9Þ

Update stress vectors cik, bi and shakedown limit a (with a chosen parameter s : 0 < s6 1):

cik ¼ cik þ sk̂s dcik; bi ¼ bi þ sk̂s dbi; a ¼ aþ sk̂s da: ð3:10Þ6. Check the convergence criteria: if they are all satisfied then stop, otherwise repeat steps 2–5.

Note that the algorithm may fail due to some reasons such as unsuccessful initialization step and failure in

computing the matrix inversion bS�1, but this rarely happens. As can be seen from the above primal–dual

algorithm 3.1, three parameters has been used in the computation process, they are c, e and s. Detail

parametric studies showed that within the range c ¼ 107–1010, e ¼ 10�8–10�11, s ¼ 0:5–0:9 the algorithm is

quite stable. Regardless of these possible numerical obstacles, we can show that the algorithm converges toan accurate solution set ð�q;�e; �aÞ. The convergence property of this primal–dual algorithm is proved by the

following theorem (the proof of this theorem is presented in Appendix A):

Theorem 3.1. If a feasible set ðq0; e0Þ can be found such that:PNG

i¼1

Pmk¼1 e

0Tik tik 6¼ 0, the primal–dual algorithm

3.1 leads to solution of (2.2).

Since limit analysis may be considered as a special case of shakedown when the structure is loaded with only

one monotonic load, i.e. ½l0; l0�P0, the above algorithm is expected to give accurate solution in limitanalysis. Numerical examples hereafter shows that such requirement is fairly satisfied.

4. Numerical examples

4.1. Cylindrical shell under internal pressure and temperature variation

Consider an axially restrained cylindrical shell of radius R and thickness t (Fig. 1). The cylindrical shell issubjected to a constant internal pressure p and a uniformly distributed temperature h which varies within

Fig. 1. (a) Cylindrical shell and (b) FEM mesh.

0 2 4 6 8 100.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Iteration

Sha

kedo

wn

Load

Fac

tor

Upper boundLower bound

6 8 10 12 140.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Nc (c=10Nc)

Sha

kedo

wn

Load

Fac

tor


(a) (b)

10 12 14 16 18 20 220.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Ne (ε2=10-Ne)

Sha

kedo

wn

Load

Fac

tor


0.5 0.6 0.7 0.8 0.90.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Sha

kedo

wn

Load

Fac

tor


(c) (d) τ

Fig. 2. Shakedown analysis of cylindrical shell: (a) shakedown analysis, (b) influence of penalty parameter c, (c) influence of parameter

e, (d) influence of parameter s.



the range ½h0; h0 þ Dh�. If one does not consider the possible failure caused by axial instability, theshakedown load can be computed by using the following condition [9]:

s2p þ s2h ¼ 1 sp ¼ffiffiffi3

ppR

2r0tsh ¼

EahDh2r0

: ð4:1Þ

In the above formulation: ah is the thermal expansion coefficient, E is the Young modulus, r0 is the yield

stress of material.

Numerical investigations are realized with 10 axial symmetric 8-node elements as shown in Fig. 1b with

the geometrical data and material properties: R ¼ 500 mm, t ¼ 10 mm, r0 ¼ 116:2 MPa, ah ¼ 1:5e–5 �C�1,

E ¼ 151 000 MPa. When sh ¼ 0, the formulation (4.1) leads to p ¼ pp ¼ 2:684 MPa, on the other hand

Dh ¼ Dhh ¼ 102:605 �C when sp ¼ 0. The numerical result for the reference loads p ¼ pp, Dh ¼ Dhh is

presented in Fig. 2a.Note that the analytical solution (shakedown load factor) in this case is a ¼

ffiffi2

p

2� 0:70711 (obtained by

using formulation (4.1)). Both quasi-lower bound and upper bound converge to the analytical solution with

high precision. Parametric studies show the influence of three parameters used: c, e and s (Fig. 2b–d).

4.2. Pipe-junction subjected to internal pressure

The problem was examined by Staat and Heitzer [5] and Heitzer [3]. In their studies, 125 solid 27-node

hexahedron elements were used to model one fourth of the structure because of its symmetric property.In our analysis, the FEM mesh is presented in Fig. 3: the mesh contains 720 solid 20-node hexahedron

elements with 12,479 DOFs. The dimensions of the pipe-junction are chosen as described in the work of

Staat and Heitzer [5]. The structure is subjected to internal pressure p varying within ½0; p0�, the ends of thejunction are closed. Numerical integration is realized with 2 · 2 · 2 Gauss points.

Numerical analysis after 24 iterations leads to the plastic collapse pressure of 0.14434r0 (upper bound)

and 0.14424r0 (quasi-lower bound) compared with lower bound of 0.134r0 obtained by Staat and Heitzer.

Shakedown analysis gives 0.11044r0 as upper bound and 0.10983r0 as quasi-lower bound. The alternating

Fig. 3. FEM mesh and geometrical properties.

0 5 10 15 20 250. 08

0. 1

0. 12

0. 14

0. 16

0. 18

0. 2

Iteration


0

limpIter. Upper bound Lower bound

1 0 .18317 0.891472 0.15565 0.119703 0.14715 0.132654 0.14524 0.136935 0.14469 0.139946 0.14449 0.141367 0.14440 0.142658 0.14436 0.143489 0.14434 0.14362

10 0.14434 0.1438111 0.14434 0.1440112 0.14434 0.1441113 0.14434 0.1442014 0.14434 0.1442315 0.14434 0.14424

. . . . . . . . .23 0 .14434 0.14424

σ

(a) (b)

Fig. 4. Limit analysis of pipe-junction (c ¼ 109, e ¼ 10�11, s ¼ 0:9).


limit calculated by using elastic analysis is 0.10983r0. Staat and Heitzer [5] obtained a lower bound of

0.0952r0. In this example the structure fails due to alternating plasticity, thus the shakedown load factor

could be evaluated more precisely if we use a set of 3 · 3 · 3 Gauss points in numerical integration. The

shakedown limit load factor in this case is 0.10031r0 (upper bound) and 0.09939r0 (quasi-lower bound).

The alternating limit calculated by using elastic stresses is 0.09939r0. The result depicted in Fig. 4a is

obtained with 3 · 3 · 3 Gauss points. It is important to note that whatever the number of Gauss points, all

the two bounds converge to alternating limit.

5. Conclusions

The accuracy, rapid convergence together with the upper and quasi-lower bounds obtained at every

iteration marks the distinction between the algorithm presented in this paper and the other algorithms

currently available. Starting from the kinematic theorem of Koiter, a new primal–dual algorithm for

shakedown analysis of structures has been developed in this work. The penalty and Lagrange methods are

used to handle the incompressibility, compatibility and normalization conditions. Consequently, themethod may be implemented with any displacement-based finite elements. The residual stresses and

hydrostatic pressure are introduced as complementary variables conjugated respectively with the com-

patibility and incompressibility conditions. Newton iterative method is used to solve the resulting system of

equations. The problem size is reduced to the size of linear elastic analysis, thus there exists no limit in

practical application. Line search is performed to improve the current kinematic solution, plastic admissible

condition is used to regulate the static variables. At every iteration, the upper bound is given by the ob-

tained kinematic field and the quasi-lower bound is given by the obtained static field. The numerical

examples show high calculating efficiency: both upper and quasi-lower bounds converge to the accuratesolution after few iterations. Although no proof is found to guarantee that the quasi-lower bound increases

after every iteration, it is an interesting feature observed in all of our examinations. The very small dual gap

between upper bound and quasi-lower bound shows that the algorithm is reliable and it provides a useful

tool to estimate the accuracy of the obtained solution.


Acknowledgements

The work is funded by the European Community under the Brite-EuRam project LISA BE97-4547,

contract no. BRPR-CT97-0595.

Appendix A. Proof of theorem 3.1

By using the penalty method, the problem (2.2) may be formulated as

aþ ¼ffiffiffi2

pkvðmin FPÞ

s:t::

FP ¼PNG

i¼1

Pmk¼1


pþ c2

Xmk¼1

eTikDveik þc2

Xmk¼1

eik � Biq

!T Xmk¼1

eik � Biq

!8<:9=;;

XNG

i¼1

Xmk¼1

eTiktik ¼ 1:

8>>>>><>>>>>:ðA:1Þ

In order to simplify the proof, let us define here some notations of vectors and matrices.

The matrix Lik defines the relation between the global and local strain rate vectors e and eik:

eik ¼ Like and e ¼XNG

i¼1

Xmk¼1

LTikeik: ðA:2Þ

The relation between local and global fictitious stress vectors which define load domain:

tik ¼ Likt and t ¼XNG

i¼1

Xmk¼1

LTiktik: ðA:3Þ

The global matrix V:

V ¼XNG

i¼1

Xmk¼1

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieTikeik þ e2

p LTikLik: ðA:4Þ

The global matrix DGv:

DGv ¼XNG

i¼1

Xmk¼1

LTikDvLik: ðA:5Þ

The global matrix H:

H ¼XNG

i¼1

Xmk¼1

LTikHikLik: ðA:6Þ

this matrix H is symmetric positive definite and may be written in another form:

H ¼ ðIþ cDGvV�1Þ; ðA:7Þ

where I stands for identity matrix.

The global compatibility condition:

Fe ¼ bBq: ðA:8Þ


The global matrix cM:

cM ¼XNG

i¼1

Xmk¼1

LTikcMikLik: ðA:9Þ

The relation between local cik and global c stress vectors:

cik ¼ Likc and c ¼XNG

i¼1

Xmk¼1

LTikcik: ðA:10Þ

The relation between local bi and global b stress vectors:

b ¼ b1 b2 . . . bi . . . bNG½ �T: ðA:11Þ

By using these notations, the non-linear problem (A.1) has a compact form as

aþ ¼ffiffiffi2

pkvðmin FPÞ

s:t::eTt ¼ 1;

FP ¼ eTVeþ c2eTDGveþ c

2Fe� bBq� �T

Fe� bBq� �:

8<: ðA:12Þ

The gradient of objective function FP is written:

gradðFPÞ ¼

oFPoeoFPoq

8>><>>:9>>=>>; ¼ Veþ cDGveþ cFTðFe� bBqÞ

�cbBTðFe� bBqÞ� �

ðA:13Þ

and the Newton directions are

dq ¼ ðaþ daÞbS�1 bAt� qþ bS�1 bAðcM �HÞVe; ðaÞde ¼ ðaþ daÞcWt� eþcWðcM �HÞVe; ðbÞ

(ðA:14Þ

where the matrices bA, cM and cW are defined asbA ¼ bBT bK�1FcM�1V�1; ðaÞ

A^

¼ cM�1V�1FT bK�1bB; ðbÞ

cW ¼ cAT^ bS�1 bA � cbG þ bQ; ðcÞbG ¼ cM�1V�1FTK�1FcM�1V�1; ðdÞbQ ¼ cM�1V�1 ðeÞ

8>>>>>>>>>><>>>>>>>>>>:ðA:15Þ

and

ðdaþ aÞ ¼ tTcWZe

tTcWtðA:16Þ

with the symmetric matrix Z:

Z ¼ VHþ cFT½I� bBðbBTbBÞ�1bBT�F: ðA:17Þ


Firstly we show that cW, Z are positive definite matrices. To this end, let us consider two matrices P and Pe

defined as

P ¼ I� BðBTBÞ�1BT; ðaÞ

Pe ¼ ð1þ eÞI� BðBTBÞ�1BT; ðbÞ

�ðA:18Þ

By applying the Bartlett–Sherman–Morrison–Woodbury formula 1 to Pe, one gets to

P�1e ¼ ½ð1þ eÞI� BðBTBÞ�1

BT��1 ¼ 1

ð1þ eÞ Iþ1

eð1þ eÞBðBTBÞ�1

BT: ðA:19Þ

The equalities (A.19) show that P�1e is a positive definite matrix for all positive values of e, it means that Pe

is positive definite with e > 0 and that P is positive semidefinite. Because H is positive definite by definition

we have Z positive definite. Furthermore the Bartlett–Sherman–Morrison–Woodbury formula permits us

to write the inverse matrix of cW in the following form:

cW�1 ¼ ð bQ � cbGÞ�1 � ð bQ � cbGÞ�1cA^

ðbS þ bAð bQ � cbGÞ�1cA^

Þ�1 bAð bQ � cbGÞ�1; ðA:20Þ

where

ð bQ � cbGÞ�1 ¼ ½cM�1V�1 � ðcM�1V�1FTÞcbK�1ðFcM�1V�1Þ��1

¼ VcM þ VcMðcM�1V�1FTÞbKc

"� ðFcM�1V�1ÞVcMðcM�1V�1FTÞ

#�1

ðFcM�1V�1ÞVcM¼ ðVcM þ cFTFÞ: ðA:21Þ

At lastcW�1 ¼ VcM þ cFT½I� BðBTBÞ�1BT�F ¼ VcM þ cFTPF: ðA:22Þ

According to (A.22) and the fact that V, cM are positive definite, the matrix cW is positive definite and thus

in formulation (A.16) we always have

tTcWt > 0 8t : ktk 6¼ 0: ðA:23ÞIn order to prove that the algorithm converges to a solution set or in other words the value of objective

function FP decreases after each iteration, one must ensure that (A.14) represents a descent direction:

s ¼ �

oFPoeoFPoq

8>><>>:9>>=>>;½ deT dqT �P 0: ðA:24Þ

In addition, equality in (A.24) must be possible only at solution.

By making use of formulations (A.13) and (A.14), the left-hand side of condition (A.24) may be written

in the following form:

s ¼ �fVeþ cDGveþ cFTfFe� bBqggTfcW½cM � bH�Veþ ðaþ daÞcWt� egþ cfFe� bBqgTbBfbS�1 bA½cM �H�Veþ ðaþ daÞbS�1bBT bK�1FcM�1V�1t� qg:

1 ðAþ BCDTÞ�1 ¼ A�1 � A�1BðC�1 þDTA�1BÞ�1DTA�1.


This formulation leads to

s ¼ feTV½cM �H�cW þ ðaþ daÞtTcW � eTgcW�1fcW½cM �H�Veþ ðaþ daÞcWt� eg

� ð�eTVcM�1 þ eTcW�1Þeþ cfFe� bBqgTfFe� bBqg:Using (A.16) and (A.22) one finally gets

s ¼(

� Zeþ tTcWZe

tTcWtt

)TcW(� Zeþ tTcWZe

tTcWtt

)þ cfbBTFe� bBTbBqgT½bBTbB��1fbBTFe� bBTbBqg:

ðA:25ÞEq. (A.25) means that s is non-negative with arbitrary set of ðe; q; tÞ such that the fictitious stress vector is

not equal to zero ktk 6¼ 0. The value of s reaches zero if and only if we have at the same time:

�Zeþ tTcWZe

tTcWtt ¼ 0;

bBTFe� bBTbBq ¼ 0

8><>:or equivalently:

VHeþ cFTfFe� bBqg � tTcWZe

tTcWtt ¼ 0;bBTfFe� bBqg ¼ 0:

8<: ðA:26Þ

It is clear that the equalities (A.26) are exactly the stationarity conditions for the objective function FP.By starting from a feasible set ðq0; e0Þ, which fulfills the normalization condition, Eqs. (A.25) and (A.26)

guarantee that the objective function is reduced progressively to optimal value. h

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A primal–dual algorithm for shakedown analysis of structures