Theory Manual Volume 1

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Theory Manual Volume 1LUSAS Version 14 : Issue 2


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LUSASForge House, 66 High Street, Kingston upon Thames,

Surrey, KT1 1HN, United Kingdom

Tel: +44 (0)20 8541 1999Fax +44 (0)20 8549 9399Email: [email protected]://www.lusas.com

Distributors Worldwide


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Table of Contents

i

Table of Contents

1 Introduction ..................................................................................................11.1 General....................................................................................................1

2 Analysis Procedures ...................................................................................32.1 Basic Finite Element Equations ..............................................................32.2 Linear Static Analysis And Equation Solution.........................................82.3 Nonlinear Static Analysis ......................................................................222.4 Linear Step By Step Dynamics .............................................................482.5 Nonlinear Dynamics..............................................................................652.6 Eigenvalue Extraction ...........................................................................672.7 Natural Frequency Analysis ..................................................................79

2.8 Eigenvalue Extraction For Buckling ......................................................832.9 Eigenvalue Extraction Including Damping ............................................842.10 Modal Analysis....................................................................................862.11 Steady State Field Analysis ................................................................952.12 Transient Field Analysis....................................................................1002.13 Thermo-Mechanical Coupling...........................................................1072.14 Fourier Analysis ................................................................................1132.15 Analysis With Superelements ...........................................................1182.16 Activation and Deactivation Of Elements..........................................126

3 Geometric Nonlinearity ...........................................................................1293.1 Introduction..........................................................................................1293.2 Virtual Work Equation .........................................................................1303.3 Lagrangian Geometric Nonlinearity ....................................................130

3.4 Eulerian Geometric Nonlinearity .........................................................1363.5 Co-Rotational Geometric Nonlinearity ................................................1473.6 General Comments.............................................................................154

4 Constitutive Models.................................................................................1574.1 Linear Elastic Models..........................................................................1574.2 Elasto-Plastic Models..........................................................................1624.3 Plasticity Models For Beams And Shells ............................................1994.4 Interface Models..................................................................................2064.5 Damage...............................................................................................2134.6 Viscoelastic Models.............................................................................2164.7 Multi-Crack Concrete Model................................................................2224.8 Resin Cure Model ...............................................................................2354.9 Shrinkage............................................................................................240

4.10 CEB-FIP Creep and Shrinkage Model ..............................................2414.11 Generic Polymer Material..................................................................2524.12 Joint Models ......................................................................................2684.13 Field Models......................................................................................2774.14 Composite Models ............................................................................2804.15 Rubber Models..................................................................................283


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Theory Manual Volume 1

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4.16 Volumetric Crushing Model ...............................................................2854.17 Delamination Damage Model............................................................289

5 Load And Boundary Conditions .............................................................2935.1 General Load Types............................................................................2935.2 Constraint Equations...........................................................................3005.3 Transformed Freedoms.......................................................................3025.4 Slidelines .............................................................................................3055.5 Thermal Surfaces................................................................................385

6 Post-Processing Facilit ies ......................................................................4036.1 Nodal Extrapolation.............................................................................4036.2 Wood-Armer Reinforcement ...............................................................4046.3 Strain Energy And Plastic Work Calculations .....................................4106.4 Composite Failure Criteria...................................................................411

7 References................................................................................................415


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Notation

ii i

Notation

Standard matrix notation is used whenever possible throughout this manual and the

expressions are defined as follows:

Basic Expressions

Vector

Matrix or second order tensor

Fourth order tensor

: Matrix scalar product

| | Determinant of a matrix

|| || Norm of a vector

trb g Trace of a matrix

bgT Transpose of a vector of matrix

bg1 Inverse of a matrix

db g Variation

b g Virtual variation

ch Rate

b g Increment

b g Summationdiag ,

Diagonal matrix with terms given


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Notation

v

c Cohesion in friction based material models

c Wave speed

Da

Maximum distance between two adjacent contact nodes

E Youngs Modulus

fi Slideline interface force on contact node i

F Yield surface

g Initial gap for nonlinear joint models

G Shear Modulus

Gf Fracture energy in concrete model

h Transfer coefficient in field analysis

I1 First stress invariant

I1,I2,I3 Strain invariants

I I1 2, Modified strain invariants

J Volume ratio (det F)

J2 Second deviatoric stress invariantJ3 Third deviatoric stress invariant

k Interface stiffness coefficient

k Bulk modulus

K Thermal conductivity

Kc Spring stiffness when in contact for nonlinear joint models

K1 Spring stiffness after liftoff for nonlinear joint models

l Length of local contact segment

lo Initial chord length of beam element

ln Current chord length of beam element


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vi

M Moment

N Stress resultant

p Number of required eigenvalues

P Participation factor in spectral response analysis

P Axial force

q Field variable flux in field analysis

q Number of starting iteration vectors for subspace iteration

Q Rate of internal heat generation in field analysis

Qhg Hourglass constant

Q1,Q2 Constants for shock wave smoothing

rz Contact zone radius

Sd Spectral displacement in special response analysis

Sv Spectral velocity in spectral response analysis

Sa Spectral acceleration in spectral response analysis

t Thickness of local contact segment

T Period of oscillation for transient and dynamic analysis

T Temperature in structural applications

T Torque

u Axial stretch

V Element volume

w Crack width in concrete model

W Work

X Normal penetration distance

Radial overlap constant

Coefficient of thermal expansion

Softening Parameter in concrete model


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Notation

vi i

Constant used in dynamic recurrence algorithms


Shear retention factor in concrete model


d Displacement norm used for convergence

Residual norm used for convergence

w Work norm used for convergence

1 Root mean square of residuals convergence criterion

2 Maximum absolute residual convergence criterion

i Error estimate in subspace iteration

Step length multiplier for line search

e Lodes Angle

e Angle between old and new displacement vector in arch-lengthmethod

e Angle of orthotropy

e Angle defining crack directions in concrete model

e Local slope at a node

Strain hardening parameter

Load factor

Plastic strain rate multiplier

i Eigenvalue (ith)

i Principal stretches

Eigenvalue shift in subspace iteraction

Friction coefficient


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viii

p p

, Ogden constants

Poissons ratio

Modal Damping ratio

Model coefficient for CQC combination in spectral response

analysis

Mass density

Effective stress

Interface stiffness scale factor

Friction angle in friction based models

Structural damping in harmonic response analysis

Field variable in field analysis

Potential energy

Circular frequency in transient and dynamic analysis

Circular frequency of load in harmonic response analysis

1 Local spin at the centroid

Vectors

a Nodal displacement vector

ei Unit vectors forming the co-rotated base axes

E Green-Lagrange strain vector

f Vector of master slideline surface forces

f Vector of nodal body forces

g g g , , Covariant base vectors

Mm Vector of master slideline surface mass


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Notation

ix

n Vector of unit segment normal

P Global internal force vector

P Local internal force vector

~q Euler parameters

R External force vector

ri Unit vectors defining the beam cross section at a gauss point

s Deviatoric Cauchy stress

S Second Piola-Kirchhoff stress vector

t Vector of surface tractions

t qi i, Unit vectors defining the beam cross section at a node

x Vector of unit segment tangent

Y Generalised displacement vector in spectral response analysis

Logarithmic strain vector

Displacement gradient vector (geometric nonlinearity)

Unscaled pseudovector (co-rotational formulation - section 3.5.1)

Pseudovector of rotation

Lagrangian multiplier vector

i Incompatible modes for enhanced elements

Cauchy stress vector

dJ

Jaumann variation of Cauchy stress

i i, Eigenvector


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x

Residual force vector

1

Local spin at the centroid

Matrices/Tensors

A Matrix of slopes

B Strain-displacement matrix

B0

Linear strain-displacement matrix

B1 Displacement dependent strain-displacement matrix

C Damping matrix in dynamic analysis

C Matrix of constrain constants

C Green deformation tensor

C Compliance matrix of material moduli

D Rate of deformation tensor

D Material modulus matrix

F Deformation gradient matrix

G Matrix of shape functions

K Stiffness matrix

KT

Tangent stiffness matrix

K

Stress stiffness matrix

M Mass matrix

N Shape function array


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Notation

xi

Ni

Principal directions of the Lagragian triad

Q Vector of constraint constants

R Rotation tensor

Sb g Skew symmetric matrix of the vector

$S Matrix containing Second Piola-Kirchoff stresses

T Transformation matrix for co-rotational formulation

U Right stretch tensor

Angular acceleration tensor

Matrix of eigenvalues

Local engineering strain tensor

Density matrix

$ Matrix containing Cauchy stresses

$$ Matrix containing Cauchy stresses

Biot stress tensor

Matrix of eigenvectors

Kirchhoff stress tensor

Angular velocity tensor


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xi i


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Introduction

1

1Introduction

1.1General

The objective of this manual is to outline the theories on which the LUSAS finiteelement analysis system is formulated.

The manual aims to provide sufficient information for the user to understand the basicconcepts of the procedures and to provide references to more detailed information.Therefore, by itself, the manual does not provide comprehensive coverage of all

topics, and in this sense it is only a reference volume.

The manual also supplies many useful hints on how to select the best options for each

analysis type e.g. the most effective element and load incrementation scheme. Assuch, the manual should also be a useful aid to a reader who is unfamiliar with aparticular area of finite element analysis.


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2


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Analysis Procedures

3

2Analysis

Procedures

2.1Basic Finite Element EquationsThis section provides a brief introduction to the formulation of the equations of bothstatic and dynamic equilibrium. These equations form the basis for the evaluation of

the response of a structure using the finite element method.

2.1.1Static Equilibrium

Consider the equilibrium of a general three dimensional body subject to forces

t - surface forces

f - body forces

F - concentrated loads

The body will be displaced from its original configuration by an amount u, which

give rise to strains and corresponding stresses .

The governing equations of equilibrium may be formed by utilising the principle of

virtual work. This states that, for any small, virtual displacements u imposed on thebody, the total internal work must equal the total external work for equilibrium to bemaintained, i.e.

T

v

T

v

T

s

Tdv u f dv u t ds du F z z z = + + (2.1-1)

where are the virtual displacements corresponding to the virtual displacements .

In finite element analysis, the body is approximated as an assemblage of discrete

elements interconnected at nodal points. The displacements within any element arethen interpolated from the displacements at the nodal points corresponding to thatelement, i.e. for element e

u N ae e e( ) ( ) ( )

= (2.1-2)


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where N

e( )

is the displacement interpolation or shape function matrix and a

e( )

is thevector of nodal displacements.

The strains ( )e

within an element may be related to the displacements a e( )

by

( ) ( ) ( )e e e

B= a (2.1-3)

where B is the strain-displacement matrix.

For linear elasticity, the stresses within the finite element are related to the strainsusing a constitutive relationship of the form

( ) ( ) ( ) ( ) ( )

( )

e e e

o

e

o

e

D= + (2.1-4)

where D e( )

is a matrix of elastic constants, and oe( )

and oe( )

are the initial

stresses and strains respectively, e.g. due to thermal effects.

Therefore, using (2.1-2), (2.1-3) and (2.1-4), the virtual work equation (2.1-1)may be

discretised to give

a B D B dv a

a

N f dv N t ds

B D dv F

T e T e e

ve

n

T

e T e

ve

n

s

e T e

se

n

e T

o

e e

o

e

ve

n

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

z

z

z

z

=

= =

=

=

+

+

L

N

MMMM

O

Q

PPPP

1

1 1

1

(2.1-5)

where Nse( )

are the interpolation functions for the surfaces of the elements and n is

the number of elements in the assemblage.

By using the virtual displacement theorem, the equilibrium equations of the elementassemblage becomes

Ka R= (2.1-6)

where Kis the structure stiffness matrix, defined as

K B D B dve T e e

ve

n

= z=

( ) ( ) ( )

1

(2.1-7)


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Analysis Procedures

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and R is the structure force vector, defined as

R R R R Rb s o c= + + (2.1-8)

and Rb is the force vector due to the element body loads

R N f dvbe T e

ve

n

= z=

( ) ( )

1

(2.1-9)

Rs is the force vector due to the element surface tractions

R N t dvs se T e

se

n

=

z=

( ) ( )

1

(2.1-10)

Ro is the force vector due to the initial stresses and strains

R B D dvoe T

o

e e

o

e

ve

n

= z=

( ) ( ) ( ) ( )( )

1

(2.1-11)

Rc is the force vector due to concentrated loads

R Fc= (2.1-12)

Equation (2.1-6) may be utilised for situations where the applied loading is

independent of time or when the load level changes very slowly. If rapid changes inthe load level occur, inertia and damping forces must be included in the equilibriumequations.

2.1.2Dynamic Equations

Using d'Alembert's principle, the inertia and damping forces may be included as part

of the body load vector. Assuming the accelerations and velocities are approximatedusing the same interpolation functions as displacement:

velocity

& &( ) ( ) ( )

u N ae e e

= (2.1-13)

acceleration

&& &&( ) ( ) ( )

u N ae e e

= (2.1-14)

The body force vector (2.1-9)then becomes


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6

R N f N a c N a dvb e T e e e e eve

n

= z=( ) ( ) ( ) ( ) ( ) ( )&& &

1

(2.1-15)

where is the density and c is the damping constant for the material.

Substitution of (2.1-15)in (2.1-6)leads to the dynamic equilibrium equations

Ma Ca Ka R t&& & ( )+ + = (2.1-16)

where Mis the mass matrix defined as

M N N dve T e e

v

e

n

= z=

( ) ( ) ( )

1

(2.1-17)

and Cis the damping matrix defined as

C N c N dve T e e

ve

n

= z=

( ) ( ) ( )

1

(2.1-18)

Note the influence of geometric nonlinearity on the basic equilibrium equations willbe considered in Section 3.

2.1.3Model Pre-Solution Output Summary

As an initial check on the finite element model the following quantities are computed

and output for all analysis types:

summation of nodal loads in global directions

summation of moments about the origin

element volume and mass summaries

the centre of mass and moments of inertia

mass and geometric property summaries

Brief descriptions of the computations involved are presented in the followingsections.

2.1.3.1Resultant loads

The moments Mabout the origin have been approximated by the discrete equation

M P x x Mi i ii

n

= +=

1

(2.1-19)


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8

where the skew-symmetric matrix, S x i( ) is given by

S x

z y

z x

y x

i

i i

i i

i i

( ) =

L

N

MMM

O

Q

PPP

0

0

0

(2.1-23)

Expansion and summation of (2.1-22)results in

I

I I I

I I

Io

xx xy xz

yy yz

zzSYM

=

L

N

MMM

O

Q

PPP

(2.1-24)

The parallel axes theorem is used to compute the moments of inertia about global

directions with the origin at the centroid of the structure, IG

.

I I M S G S GG o s

T= ( ) ( ) (2.1-25)

An eigenvalue analysis can then be carried out on IG

to compute the principal

moments of inertia (the eigenvalues) and the directions about which they act (theeigenvectors). Note that for axisymmetric elements the mass properties are computed

for the complete revolution.

2.2Linear Static Analysis And Equation SolutionThis section provides a brief description of the equations of linear static equilibrium

and outlines the techniques available to solve these equations. For the frontal solutionmethod some modelling considerations are presented together with a description ofthe 'diagonal decay' solution monitor. Frontwidth optimisation techniques which maybe used to reduce both the memory and CPU usage are also outlined. Finally adescription of the iterative solver is given which also includes brief details of the

'preconditioning' process.

2.2.1Governing Equations

The finite element equilibrium equations for linear static analysis are (2.1-6)

Ka R= (2.2-1)

where the load vector R is replaced by a matrix R which may contain several load

cases.


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Analysis Procedures

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These equations can be solved directly using Gaussian reduction [H6] or iteratively

using a conjugate gradient technique [M6],[C10] and [H3]. The frontal solver utilises

Gaussian elimination to invert the stiffness matrix K, followed by a back-substitution

process to evaluate the displacements a . This technique may be performed on

several load cases simultaneously, reducing the analysis costs. The iterative solver is

effective for extremely large and comparatively well conditioned problems whereonly one load case is to be solved. The success of this method depends very much onthe condition of the stiffness matrix and the preconditioning stage is an imperativepart of the solution process.

2.2.2Frontal Solver

The Gaussian reduction solution technique requires the structure stiffness matrix to be

non-singular. For structural applications, this is equivalent to stating that for anydisplacement field, the strain energy stored by the finite element system must begreater than zero.

Therefore, the structure must be supported so that no rigid body displacements orrotations are possible. Failure to comply with this criterion will result in LUSAS erroror warning messages.

The stiffness matrix may also be poorly conditioned. This is the result of a largevariation in magnitude of diagonal stiffness terms, which may be due to

large, stiff elements being connected to small, less stiff elements

elements with highly disparate stiffnesses, e.g. a beam element may havea bending stiffness that is orders of magnitude less than it's axial stiffness.

Poor conditioning may result in round-off error, which is a loss of accuracy in theevaluation of the terms during the reduction process. This in turn leads to inaccuraciesin the predicted displacements and stresses.

LUSAS monitors the round-off error by evaluating the diagonal decay during the

reduction process. This criterion [I2] is based on the assumption that initially largediagonal terms accumulate errors proportional to their size. As reduction progresses,the diagonal term is reduced, amplifying the errors until they become a maximumwhen the diagonal term is the pivot. An indication of probable errors may be obtainedby examining the change in magnitude of the diagonal term, i.e. the diagonal decay isdefined as

D

K

Ki

iij

j

m

ii

m=

LNM OQP=( )( )

( )

2

1 (2.2-2)

where


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10

Di

is the diagonal decay for the ithequation,

Kiij( )

is the value of the diagonal term of the ithequation at the jthmodification

m is the last modification i.e. Kiim( )

is the pivot.

If D Di kl> (default 1.0E+05) then a warning is output and the solution in the regionof this node should be examined carefully.

If D Di kl> (default 1.0E+12) then the solution is terminated as the matrix hasbecome singular, probably due to insufficient restraints.

The specific form of Gaussian reduction implemented in LUSAS is the frontalsolution technique [I2]. This technique uses sophisticated 'housekeeping' techniques

which are designed to minimise the number of operations performed and the memoryrequired during the equation solution.

Instead of assembling the complete structural stiffness matrix, the elements are

considered in turn according to a prescribed order, with assembly and eliminationprocesses proceeding simultaneously. Whenever a new element is called in, itsstiffness contribution is added to the currently active structure stiffness by either

adding the contribution to the current equations, if the degrees of freedomare already active, or

creating new equations, if the degrees of freedom are not active.

When all the stiffness contributions for a degree of freedom have been assembled, theequation is removed from the list of active equations using Gaussian elimination. The

list of active nodes is called the front and the number of variables in the front is thefrontwidth.

The size of the frontwidth greatly influences the CPU time and disk storage required

for the solution of the equations. Therefore, the elements should be assembled suchthat the frontwidth or root mean square of the frontwidth is a minimum. This may beachieved by either

Careful ordering of elements in the mesh (fig.2.2-1), or

Using the automatic element reordering algorithm (see Section 2.2.2.1).

The frontal solution technique is also used for equation solution in the nonlinear,dynamic, eigenvalue and field analysis procedures.

2.2.2.1Frontwidth optimisation

Four algorithms are available for frontwidth optimisation

(i) STANDARD OPTIMISER

This optimiser is based on the minimisation of the growth of the element front. Thefollowing steps are used to optimise the frontwidth


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Analysis Procedures

11

1. For each element, form a vector containing all elements which are neighbours to

the current element.

2. Use each element in turn as a starting element. The current front then contains oneelement

Update the current front by

adding to the front the neighbours of the elements that form the front,

removing elements that are no longer active from the front.

If the current frontwidth for this starting element exceeds the maximumfrontwidth obtained using a previous starting element, the algorithm

selects the next starting element (return to step 2.). Otherwise return to

step 2.aunless the complete mesh has been considered.

If the maximum frontwidth for the current starting element is less than thepreviously obtained value, update optimum starting element andmaximum frontwidth.

3. Form element assembly order using the starting element that provides the smallestfrontwidth.

The procedure is illustrated for a simple example in fig.2.2-2. As the algorithm useselement rather than nodal data, it is computationally efficient. Also, the simple rulesutilised to choose the starting element and update the front, ensure that the algorithmis very stable, i.e. it will always produce reasonable element assembly order.However, for certain cases the element order may be significantly greater than the

true optimum. This is illustrated by the examples in fig.2.2-3. For the long, thin meshthe algorithm produces a maximum frontwidth close to the optimum value. Whereasfor the square mesh, the maximum frontwidth is significantly greater than theoptimum value.

(ii) AKHRAS AND DHATT OPTIMISER

This frontwidth optimisation technique is based on the procedure presented by Akhrasand Dhatt [A5]. The method was developed by considering various topological

characteristics of band matrices corresponding to typical well-ordered networkproblems and then employing the simplest of these properties as criteria for

developing an optimisation algorithm. Consider the mesh shown in fig.2.2-4, with 12interconnected nodes and an optimum bandwidth of 4, in order to discuss theseproperties of interest.

The node connectivity matrix of the network is defined by column 2 of table 2.2-1. Itcontains the complete information regarding the connectivity of the network. Thematrix has, among others, the following properties:

Sum Consider the rows having the same number of terms. It is noticed thatthe sum of the terms in each row is arranged in an increasing order as shown incolumn 3. For example, rows 1,3,10 and 12 have the same number of terms,and the sum of terms in these rows 12,16, 36 and 40, is arranged in an

ascending order.


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12

Ponderation Column 4 gives a ponderation vector which is obtained by

dividing the row sum by the number of terms in the row. It is observed thesevalues are arranged in an ascending order from the first row to the last one.

Span The sum of the minimum term and the maximum term for each row isarranged in ascending order (column 5).

Each of these criteria may not be necessarily appropriate for the optimal bandwidthorganisation of a general sparse matrix or network. However, it is assumed that if the

nodes of a mesh are numbered in such a way that the above mentioned criteria arerespected, the corresponding matrix will have a minimum bandwidth, but this may not

be the absolute minimum.

The LUSAS implementation of the method works on elements rather than on nodes toreduce the size of the problem.

The present optimisation algorithm employs these criteria in an iterative manner and

it is divided into two phases.

Phase 1 The elements are relabelled on the span criterion and the ponderationcriterion respectively. The relabelling process is repeated a number of timesrespecting these two criteria until no more reduction in bandwidth is obtained.

Phase 2 The elements are relabelled such that the ponderation criterion andthe sum criterion are respected simultaneously. This process is repeated until

no further reduction in bandwidth is realised. Usually the process terminatesafter a finite number of iterations.

Node Node connecti vity matrix Sum Ponderation Span

1 1 2 4 5 12 3 6

2 2 1 3 4 5 6 21 3.5 7

3 3 2 5 6 16 4 8

4 4 1 7 2 5 8 27 4.5 9

5 5 1 2 3 4 6 7 8 9 45 5 10

6 6 2 3 5 8 9 33 5.5 11

7 7 4 5 8 10 11 45 7.5 15

8 8 4 5 6 7 9 10 11 12 72 8 16

9 9 5 6 8 11 12 51 8.5 17

10 10 7 8 11 36 9 18

11 11 7 8 9 10 12 57 9.5 19

12 12 8 9 11 40 10 20

TABLE 2.2-1


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Analysis Procedures

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Note that this optimiser will function with two or more unconnected meshes (e.g. in a

slideline analysis).

(iii) REVERSE CUTHILL-MCKEE OPTIMISER

Cuthill and McKee [C15] noted that the ordering of a sparse symmetric matrix forminimum profile and wavefront is equivalent to the labelling of an undirected graph.The terminology of 'graph theory' provides a means of assessing various node

ordering schemes [J3]. The Reverse Cuthill-McKee optimiser works by re-orderingnode numbers directly (rather than element numbers) in an attempt to optimise theelement solution order. The optimiser may be used to minimise any of the followingcriteria (see figure 2.2-5 for descriptions of terms):

Maximum wavefront

RMS wavefront

Bandwidth

Profile

The basic steps taken in this optimisation method are as follows:

1. Choose a node to be relabelled as 1. This node should be one with the lowestconnectivity.

2. The nodes connected to the new node are relabelled 2,3 etc., in order of increasingdegree (connectivity).

3. The sequence is then extended by relabelling the nodes which are directlyconnected to the new node 2 and which have not been previously relabelled. Thenodes are again listed in order of increasing degree.

4. The last operation is repeated for the new nodes 3,4 etc., until renumbering iscomplete.

5. Finally, the Reverse Cuthill-McKee method requires that the ordering of therelabelled nodes is reversed.

Steps 1.to 5.are repeated for all the nodes with the lowest degree of connectivity.

The final sequence chosen is the one which best satisfies the criterion to beminimised.

When the optimum node sequence has been established the element solution order ismodified to reflect the new sequence. To illustrate how this is done reference will bemade to the finite element mesh shown in figure 2.2-6. The following steps are taken

[S12,S13]:

1. The element topology is taken as: Element label Node labels Element number

2 5 9 7 1

3 8 3 2 2

4 7 9 6 3


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Analysis Procedures

15

2.2.2.2Constraints and slidelines using the Frontal Solver

When the finite element analysis contains constraint equations and/or slidelines thesolution frontwidth is affected since these are included into the solution in the form ofconstraint and/or contact elements. These elements are inserted into the frontal

solution according to the following steps

1. If frontwidth optimisation is required then

obtain optimised solution order with all elements (normal, constraint andcontact).

obtain a list of normal elements in optimised solution order otherwise

obtain a list of normal elements in presented/numbered solution order2. Insert contact elements into the solution order after the elements to which they are

connected.

3. Insert constraint elements into the solution order only when the constraintequation can be eliminated (i.e. after all the required variables have been

assembled).

Penalty slidelines with constantly changing connectivity will constantly be generatingnew contact elements and removing old ones. Therefore

with SOLUTION ORDER AUTOMATIC (optimisation is required) thefrontwidth will be re-optimised when necessary following the proceduredescribed above.

with SOLUTION ORDER the contact elements will be inserted

immediately after the normal elements as appropriate (as 2.above).

with SOLUTION ORDER PRESENTED, the contact elements will be

inserted at the end of the normal elements according to their higherelement numbers. This would cause an un-optimised order when slidelinesare in use.

Note that LUSAS Modeller does not set up the connectivities required to include theconstraint elements or the contact elements - hence differing frontwidths may beobtained when computed in LUSAS Solver.

2.2.3Iterative Solver

The advantage of the iterative solver implemented in LUSAS is related to the solution

of extremely large but well conditioned problems where size precludes the use of thefrontal technique. The iterative solver cannot be used in analyses involvingsuperelements, constraint equations or eigenvalue extraction and is very inefficient ifmore than one load case is to be solved. The conjugate gradient method with an

incomplete Cholesky preconditioner is employed for an iterative solution in LUSAS.


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16

2.2.3.1The Conjugate Gradient method

The conjugate gradient method solves the equations defined as (2.2-1)by consideringthe minimisation of the functional:

f a a Ka R a cT T

( ) = +1

2 (2.2-3)

where the first term on the r.h.s. of (2.2-3)is the strain energy, the second term thepotential energy of the applied loads and c is an arbitrary constant corresponding tosome reference value. The gradient of f(a) is the vector:

= =f a Ka R a( ) ( ) (2.2-4)

where (a) is called the residual. In structural finite element applications the residualcan be viewed as the out of balance force vector which should be zero at equilibrium.Since the functional f(a) decreases most rapidly in the direction opposite of thegradient, it is advantageous to search in this direction for the minimiser of f(a). A

first-degree iteration is commonly written:

a a pk k k k+ = +1 (2.2-5)

where pk

is the search direction for the kth iteration and k>0 is a step length.

Hence, from the point ak, the approximation proceeds in the direction pk for a

distance ak along the surface f(a), attempting to choose ak and pkat each step so

as to approach the minimum point of f(a).

This forms the basis of iterative solution methods and in LUSAS the conjugategradient method is used which is an algorithm that seeks to accelerate 'the method ofsteepest descent'. Full details of this method can be found in [M6], [C10] and [N3] butthe basic iteration procedure can be summarised as follows:

ao given (value at start of increment), o oR Ka= = , p

o o=

for k k= 01, ,..., max

KK

T

K

K

T

K

p K p= (2.2-6)

a a pk k k k+ = +1 (2.2-7)

k k k k

K p+

= 1

(2.2-8)


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Analysis Procedures

17

test for convergence - > || |||| ||k

T

R+

1 < (default =1e-6)

if no convergence continue iterating

k

k

T

k

k

T

k

= + +1 1 (2.2-9)

p pk k k k+ +

= +1 1

(2.2-10)

2.2.3.2Preconditioning

In the analysis of structural problems very ill-conditioned equations may arise. Thecondition number of a symmetric matrix K can be related to its extreme

eigenvalues

( )K N=

1

(2.2-11)

If the condition number is large, then a small change in Kor R may cause a gross

change in the solution a , and Ka R= is said to be ill-conditioned. Rounding errors

and approximations make the solution a unreliable if the condition number is very

large. A further disadvantage of a large condition number is that it usually reduces therate of convergence of iterative methods.

Preconditioning is a method of clustering the eigenvalues more closely together andthereby improving the efficiency of the iterative procedure. The basic idea of anypreconditioning method is to replace the system

Ka R= (2.2-12)

by

~~ ~Ka R= (2.2-13)

where~K H K H

T=

1,

~a H aT

= ,~

R H R= 1

The aim of preconditioning is to make Ka better conditioned matrix than K. In

LUSAS an incomplete Cholesky factorisation of K is used for the preconditioningprocess and more details may be found in [M6],[C10] and [N3].


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18

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Max. Frontwidth = 13 (1 freedom/node)

(a) Poor Solution Order

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Max. Frontwidth = 6 (1 freedom/node)

(b) Efficient Solution Order

Fig.2.2-1 EXAMPLE ILLUSTRATING POOR AND EFFICIENT SOLUTION

ORDERS


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Analysis Procedures

19

1

1 4

2

1 4

2 3

5 6

9

8

7

1 4

2 3

5 6

9

7

12

11

10

8

3

Current front

Non-assembled

elements

Assembled elements

Neighbours of element 1

added to front

Neighbours of elements 2,3

and 4 added to front

Neighbours of elements 7,8

and 9 added to front

Frontwidth achieved = 7

Optimum frontwidth = 6

Fig.2.2-2 EXAMPLE ILLUSTRATING STANDARD FRONTWIDTH

OPTIMISATION ALGORITHM


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20

1

2 3

4

5 6 10

11

129

8

7

32

33

31

(a) Long Thin Mesh

1

2

4

3

49

48

47

46

45

44

43424140393837

(b) Square Mesh

Fig.2.2-3 EXAMPLES COMPARING THE MINIMUM FRONTWIDTHS

OBTAINED WITH THE STANDARD OPTIMISER WITH THE OPTIMUM

VALUES

1 4 7 10

2 5 8 11

3 6 9 12

FIG.2.2-4 WELL-ORDERED NETWORK OF 12 NODES


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22

8

7

4

2

5

8

3

6

9

7 2

3

FIG.2.2-6 CUTHILL-MCKEE OPTIMISATION EXAMPLE

2.3Nonlinear Static Analysis

Nonlinearities may arise in several forms including large deflections, large straining,nonlinear stress-strain laws, deformation dependent boundary conditions, anddeformation dependent loading.

(i) MATERIALLY NONLINEAR ANALYSIS

This type of analysis should be utilised if the material stress-strain relationship issignificantly nonlinear. Consider for example the idealised stress-strain relationshipfor a steel bar (fig.2.3-1). This is linear in the elastic range so that an elastic analysis

would predict the correct deformed configuration provided the yield stress is notexceeded. If yielding occurs then the stiffness of the bar decreases resulting in a

nonlinear stress-strain law. Therefore incremental loading is required to trace thecomplete material response. This is illustrated for a simple two bar arrangement(fig.2.3-2).

LUSAS has a number different material models which permit modelling of a varietyof physical materials including ductile metals, concrete, foam and soils.

(ii) GEOMETRICALLY NONLINEAR ANALYSIS

In this analysis the changing effect of structural deformation on the structural stiffnessand on the position of applied loads is considered. A simple problem that illustrates

this is a simply supported beam with uniformly distributed loading (fig.2.3-3). Thelinear solution would predict the familiar simply supported bending moment and zeroaxial force. However, in reality, as the beam deforms so the angle of inclination of the

beam at the supports introduces an axial component of force. This force may becomesignificant if the deformations and consequently the angle of inclination become

large.


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24

Force

Bar 2

Yield

stress

Displacement

Stress-Strain Relationship for Bar 2

Force

Bar 1 yields

Displacement

Bar 2 yields Ultimateload

Combined Response of System

Fig.2.3-2 NONLINEAR RESPONSE OF A TWO BAR SYSTEM

Problem Definition Axial forces

Axial Forces Induced By Large Deflections

Fig.2.3-3 GEOMETRICALLY NONLINEAR RESPONSE OF A SIMPLY

SUPPORTED BEAM


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Analysis Procedures

25

Another simple example is the bar assemblage of fig.2.3-4. As the load is increased,

the response exhibits softening until 'snap-through' occurs, after which the responseshows stiffening.

Another form of nonlinearity often associated with large deformations is that offollower forces (or non-conservative loading). With large deformations, some loads

vary in both their spatial location and orientation. Failure to represent these changesmay lead to errors with certain load types, e.g. pressure loading on a surface, wherethe pressure should always act normal to the deformed surface. Non-conservativeloading is modelled in LUSAS by continuously updating the load vector (UpdatedLagrangian or Eulerian formulations only [Section 3]).

(iii) DEFORMATION DEPENDENT BOUNDARY CONDITIONS

In this analysis the boundary conditions are modified during the course of the analysis

depending upon the deformed shape of the structure. This is illustrated by theexample shown in fig.2.3-5. The mass is subjected to a pressure P and is initiallysupported by a single spring. As the load is increased, contact is established with asecond spring, which alters the load-deformation response of the structure.

Nonlinear boundary conditions are imposed by using joint elements or via theslideline facility. In the above case a joint element with an initial gap and zerostiffness is used to connect the mass to the spring. Once the gap has been closed theelement is given a stiffness so that it forms a rigid link.

Load

(a) Problem Definition


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26

Displacement

at apex

Load

Elasticresponse

Geometrically nonlinear response

(b) Load-Displacement Relationship

Fig.2.3-4 GEOMETRICALLY NONLINEAR RESPONSE OF A BAR

ASSEMBLAGE

Load

Gap

Spring

Supports K2K1

(a) Problem definition


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Analysis Procedures

27

Displacement

Load

Contact

K1+ K

2

K1

(b) Load-Deflection Response

Fig.2.3-5 RESPONSE OF A SPRING-MASS SYSTEM WITH NONLINEAR

SUPPORT CONDITIONS

2.3.1Time Stepping And The Tangent Modulus Matrix

To trace the structural response of materially or geometrically nonlinear problems, atime or load stepping procedure must be used. If a significant degree of nonlinearityoccurs during a load step, the stresses integrated through the volume of the structure

will not satisfy equilibrium with the external forces. Consequently, a residual forcevector q(a) will remain, defined by

( )a P Rt t t t

= + +

(2.3-1)

t t T t t

vP B dv

+ += z (2.3-2)

andt t

R+

is the external force vector.

Therefore, a correction procedure is required to restore equilibrium. The simplest

corrector may be derived by using a Taylor series expansion to obtain the

approximate solution, i.e. assuming that for iteration i-1 we have evaluated ai1

then

a a a

aa

i i i

i

i

+ = + +1 1

1 K (2.3-3)


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28

where a

i

is the change in displacement for the ith

iteration and the higher orderterms are neglected. Since we require a zero residual for iteration i, (2.3-3)becomes

aa

ai

i

i=

LNM

OQP

1

1

1( ) (2.3-4)

a K ai

T

i=

1 1( ) (2.3-5)

where KT is known as the tangent stiffness matrix. The next displacement

approximation is then obtained using

a a a

i i i

= +

1

(2.3-6)

This equilibrium iteration procedure is known as Newton-Raphson iteration and isillustrated in fig.2.3-6, which also shows the physical significance of the tangentmodulus as the tangent to the stress-strain relationship at the current configuration.This also implies that, for zero deformation, the tangent modulus corresponds to the

elastic modulus.


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Analysis Procedures

29

Load (R)

Displacement (a)

Da1 Da2

R

y(a1)

P1

P2

y(a2)

KT1

KT2

Fig.2.3-6 Illustration Of Newton-Raphson Iteration For A One Degree Of Freedom

Response

2.3.2Iteration Procedures

2.3.2.1Newton iteration

Although Newton-Raphson iteration is stable and converges quadratically (providedthe initial estimate is close enough to the solution), it has the disadvantage that thetangent stiffness matrix needs to be inverted during each iteration. Also, it may fail to

converge when extreme material nonlinearities are present in a structure. For this casemodified Newton iteration may be more effective.


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Analysis Procedures

31

Displacement

Load

Stiffness reformulation

t+tR

tR

(c) KT2 Method

Fig.2.3-7 Common Forms Of Modified Newton Iteration

2.3.2.2Line searches

The line search technique is designed to improve the convergence rate of both full andmodified Newton iteration. It involves modifying the iterative displacement updatefor iteration i to obtain

a a ai i i= +

1 (2.3-7)

where i is the step length and is selected to minimise the potential energy. i

corresponds to the parameter ETA output in the nonlinear logfile. The stationaryvalue may thus be obtained as

i i

i

i i

i

i

ia

a( ) ( )

= =1

0 (2.3-8)

The first term is the gradient of the potential energy, and partially differentiating

(2.3-7)with respect to enables (2.3-8)to be written as

i i Ta a( ) = 0 (2.3-9)

Equation (2.3-9)is excessively strict and complete satisfaction would be numericallyexpensive. Therefore, (2.3-9)is replaced by

i i T i i T

a a a a( ) ( )toline< 1 1

(2.3-10)


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32

Where EPSLN (the nonlinear logfile parameter) is computed fora a

i i1 and the

tolerance factor (corresponding to the parameter toline in the NONLINEAR

CONTROL data chapter) is usually set to between 0.4 and 0.8. If equation (2.3-10)isnot satisfied with the initial step length of unity, then a simple linear interpolation(fig.2.3-8) is used to calculate a new step length. Typically, for line search step j+1

the step length is j+1 evaluated using

j

i

j

i

T i

T i

j

i i

a

a+

=

L

NMM

O

QPP1

1

1( )

(2.3-11)

The process is repeated until the convergence criterion (2.3-10)is satisfied or until a

preset maximum number of line searches per iteration (corresponding to nalps) is

exceeded. Line searches are not carried out if the computed length is close to unity orclose to zero. If the step length is close to unity, little would be achieved by the linesearch. If the step length is close to zero, little progress would be made with the

solution, and the new iterative direction provided by re-solution would be beneficial.

Step

length

Potential

energy

Satisfactory zone

Exact solution a/a= 0

Fig.2.3-8 Line Search Procedure


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Analysis Procedures

33

Displacement

Load

Iteration within each load step

t+tR

tR

t+2 tR

Fig.2.3-9 Constant Load Level Incremental/Iterative Procedure

2.3.2.3Convergence

When using incremental/iterative solution algorithms, a measure of the convergenceof the solution is required to define when equilibrium has been achieved. Theselection of appropriate convergence criteria is of utmost importance. An excessivelytight tolerance may result in unnecessary iterations and consequent waste of computer

resources, whilst a slack tolerance may provide incorrect answers.

Assigning tolerance values is very much a matter of experience. In general, sensitivegeometrically nonlinear problems require a tight convergence criteria in order tomaintain the solution on the correct equilibrium path, whereas a slack tolerance isusually more effective with predominantly materially nonlinear problems in which

high local residuals may have to be tolerated.

There are many ways of monitoring convergence and within LUSAS the followingcriteria are available

Euclidian residual norm,

Euclidian displacement norm,

Euclidian incremental displacement norm, work norm,

root mean squares of the residuals,

maximum absolute residual.


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34

A description of each of the above follows, in which the bracketed parameter

equivalis the ent variable in the nonlinear logfile.

(i) EUCLIDIAN RESIDUAL NORM (rdnrm)

The Euclidian residual norm is defined by the norm of the residuals as a

percentage of the norm of the external forces R and is written as

= || ||

|| ||

2

2

100R

(2.3-12)

where R contains the external loads and reactions. Owing to the inconsistency of theunits of displacement and rotation, usually only translational degrees of freedom are

considered although all freedoms may optionally be included. For problems involvingpredominantly geometric nonlinearity, a tolerance of

< 0.1 (2.3-13)

is suggested. Where plasticity predominates, a more flexible tolerance of

0.1 < < 5.0 (2.3-14)

is suggested.

(ii) EUCLIDIAN DISPLACEMENT NORM (dpnorm)

The Euclidian displacement norm d is defined by the norm of the iterative

displacements a as a percentage of the norm of the total displacements a and iswritten as

d d

a

a=

|| ||

|| ||

2

2

100 (2.3-15)

As in the residual norm, usually only translation degrees of freedom are consideredbut all freedoms may be optionally included. The criterion is a physical measure ofhow much the structure has moved during the current iteration. However, if

convergence is slow then false convergence may be obtained with this criterion.Typical values are

01 10

0 001 01

. . ( )

. . ( )

< > 0.001.

Some comments on the implementation and use of this algorithm are given in [Z6]

and a summary of these observations is included here:

The input parameters required for the automatic time stepping algorithm are

the first step size t1 , the step length parameter , the maximum step size

tmax and the minimum step size tmin .

t1 should be given a value which is smaller than the expected optimum stepsize. This is particularly important for transient problems for which the mostimportant dynamic events take place at the beginning.

depends on the particular time integration scheme and accuracy

requirements. Good accuracy can normally be achieved with =0.05,corresponding to about 20 time steps for one full characteristic period. Forlinear dynamics using the trapezoidal rule, this corresponds to a period ofelongation of about 1%.

It may also be necessary to prescribe bounds for the step size. For instance, theloading may be changing very rapidly, while the dynamic response isrelatively slow, e.g. for earthquake problems. In such cases it may be

necessary to limit the step size by an upper value tmax in order to ensure thatthe load intensity is sampled at sufficiently close intervals. In other situations itmay be necessary to ensure that the step size does not go below a prescribed

lower limit tmin in order to prevent excessive computational costs.

Sometimes the incremental displacements may become very small or vanish,such as when passing through maximum amplitudes where the velocities tendto zero. In such cases the current frequency is not computed and the currentstep size will be used for the next time step.

To ensure a smooth response, the current time step will not increase by morethan twice the previous time step, nor twice the mean time step value.However, there is no control over the maximum decrease that can be applied.

This is to ensure that the algorithm can react promptly to sudden changes indynamic response.

2.4.1.6Starting conditions

Neither the explicit nor the implicit dynamic algorithms require any special startingprocedure but the initial conditions need to be defined.


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60

For the explicit central differenceprocedure these are

d0 0=

v vn =1 2/

f f0

=

a M f01

0=

(2.4-11)

These equations indicate that the initial velocity and force vectors need to be defined.The initial acceleration will then be automatically computed.

For the Hilber-Hughes-Taylor implicitprocedure these are

d d0=

v v0=

f f0

=

a M f C v K d 01

0 0 0=

( ) (2.4-12)

These equations indicate that the initial displacement, velocity and force vectors needto be defined. The initial acceleration will then automatically be computed if the

vector on the right hand side of equation (2.4-12d) is non-zero, otherwise the initialaccelerations will be assumed to be zero.

2.4.2Load Conditions For Dynamic Analysis

An arbitrary load-time curve may be modelled by defining the loading conditions ateach time step (fig.2.4-4a).

The transient dynamic analysis may be started after a static analysis has been carriedout with the currently defined load case and boundary restraints. This permits thespecification of an initial deformed configuration due to some static loading.

If the dynamic loads are to be applied to an initially unloaded structure then thedisplacements at time t=0 should be zero. This condition is generally assumed within

the dynamics algorithms (fig.2.4-4b).

Ramp loading may be approximated by initially applying zero load to the structure.The load is then applied on the second time interval thus increasing the load to the

required value during one time step (fig.2.4-4c).


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Analysis Procedures

61

Impulse loading may be approximated by applying the load pulse during one time

step and then removing it in the next time step. This is illustrated in fig.2.4-4d.Alternatively, if the pulse is applied during the first time step then the structure may

be initially fully restrained with the pulse load, and then released with the loadsimultaneously removed (fig.2.4-4e).

2.4.3Damping

One of the most common models of damping used in finite element systems isproportional damping. In this model no attempt is made to construct dampingelements, but instead the arbitrary assumption is made that the damping is a linear

combination of the mass and stiffness matrices. This is usually called Rayleigh

damping and defines the damping matrix Cas

C a M b K R R

= + (2.4-13)

There is no real physical justification for this and in reality this assumption is madefor mathematical convenience as it simplifies the solution procedure [N2]. It shouldbe noted that the damping distribution is rarely known in sufficient detail to warrantany other more complicated model. Proportional damping will serve to give eachmode of vibration the correct order of magnitude of damping and, provided that thecomponent of the response of interest is not controlled by the damping, it is oftensufficient. It ceases to be reliable when the damping within the system becomes

significant, say above 10% critical, or where the damping is concentrated in a smallregion of the structure. This is especially the case if some damping has been includedinto the structure with the specific purpose of controlling the dynamic response.

The values aR and bR may be chosen to give the correct order of magnitude for the

damping at two frequencies [N2]. If the required damping ratio at a circular frequency

of r is r and the required damping ratio at a circular frequency of s is s

then the values of aR and bR are

aR

r s s r r s

r s

=

22 2

( ) (2.4-14)

bR

r r s s

r s

=

22 2

( )

(2.4-15)

Figure 2.4-5 indicates how proportional damping varies with frequency for various

values of specified damping. Figure 2.4-5(a) shows that for the case where r s< the damping increases rapidly with frequency. This assumption is often used whenproportional damping is being employed to provide numerical damping since it gives


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Analysis Procedures

67

2.5.2General ConsiderationsWithin LUSAS both materially nonlinear and geometrically nonlinear analysis may

be undertaken. The iteration procedure used for solving (2.5-2) is input using theNONLINEAR CONTROL command. The convergence checks used in static analysismay be used in an identical manner in dynamic analysis.

Note The iterative equations with implicit time integration are exactly the same formas those in static analysis except that inertia effects are included in the stiffness andforce matrices. Therefore the iterative procedures used in static nonlinear control, i.e.Newton-Raphson and modified Newton-Raphson may be utilised in an identical

manner. Also, the inertia effects tend to smooth the solution such that convergence ofthe iterative process should always occur and will be at a faster rate than for static

analyses.

The time step is chosen using the same considerations as for linear dynamic analysis.However, the eigenvalues continually change due to the nonlinear behaviour so carehas to be taken to account for this phenomenon. If convergence difficulties occur inthe equilibrium iterations then the time step is generally too large.

Damping can be included in the same way as described for linear dynamic analyses.

2.6Eigenvalue ExtractionIn LUSAS, two methods are available to extract the eigenvalues and eigenvectors

from large equation systems

Subspace iteration

Guyan reduction

Both techniques have been designed for structural applications in which the lowesteigenvalues and eigenvectors are typically required. The techniques efficientlyevaluate the lowest p eigenvalues and eigenvectors of large sparse systems of order nwhere n >> p. In general, subspace iteration is more effective than Guyan reduction.

2.6.1Subspace Iteration

The subspace iteration procedure [B1] is utilised in natural frequency, linear bucklingand field analyses to solve generalised eigen-problems of the form

K M = (2.6-1)

where is a diagonal matrix containing the eigenvalues i and contains the

corresponding eigenvectors i .


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70

With successive iteration

k+1

tends to eigenvalues of the full system

and

Xk+1

tends to eigenvectors of the full system

Subspace iteration is repeated until convergence is obtained. This is measured by

| |,

ik+1

i

k

i

k+1

=rtol i p1 (2.6-8)

where rtol is typically 10 6

.

2.6.1.3Sturm sequence check

The subspace algorithm is very stable but does not necessarily converge to the lowestp eigenpairs. Therefore the Sturm sequence check has been incorporated into themethod. This is an extremely stable procedure which indicates the number of

eigenvalues below a given eigenvalue and is utilised to test for missing values, e.g. ifp eigenvalues have been found and the Sturm sequence check indicates that r > peigenvalues exist below the highest eigenvalue, (r-p) eigenvalues have been missed. Ifsome eigenpairs are missing, the analysis must be repeated with a different number ofstarting iteration vectors and/or a different convergence tolerance.

2.6.1.4Error estimate

Having calculated all the required eigenpairs, the solution is completed by calculatingerror estimates of the precision to which the eigenvalues and eigenvectors have been

evaluated. The specific error quantities for each eigenpair are as follows

ii

K

K=

|| ||

|| ||

Q - MQ

Qi i

i

2

2

(2.6-9)

where i and i are the ith eigenvector and eigenvalue of the solution. Thisestimate effectively states that, for an accurate solution, the norm of the out of balanceforces divided by the elastic nodal forces should be small.

2.6.2Guyan ReductionFinite element approximations to low frequency natural vibrations may often beobtained by considering only those freedoms whose contribution is of most

significance to the oscillatory structural behaviour.


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Analysis Procedures

71

This characteristic may be utilised in the condensation of the full discrete model to a

reduced system in which the remaining equations adequately encompass the requiredvibrational modes. Such a procedure is often termed Guyan Reduction or Guyan

Condensation [G1], and may be used to significantly reduce the overall problem size.

In typical Guyan reduced eigenvalue extractions, the mass contribution of those

freedoms whose inertia effect is considered relatively insignificant (designated slavefreedoms) is progressively condensed from the system, leaving the reduced equationsystem dependent on those remaining freedoms (designated masters).

The effective selection of the master freedoms is therefore central to the accuracy ofthe simulated structural response.

2.6.2.1Theory

The Guyan reduction procedure [G1] is utilized in natural frequency, linear bucklingand field analyses to solve generalized eigen-problems of the form

K M = (2.6-10)

where is a diagonal matrix containing the eigenvalues i and contains the

corresponding eigenvectors i .The solution of (2.6-10)yields

( )K M a = 0 (2.6-11)

Equation (2.6-11) may be written in partitioned form as

K KK K

M MM M

aa

mm ms

ms

T

ss

mm ms

ms

T

ss

m

sLNM OQP

LNM OQPFHG IKJRST UVW= 0 (2.6-12)

where the subscripts (m) and (s) refer to the "master" and "slave" degrees of freedomrespectively.

For Guyan reduction/condensation it is assumed that, for the lower frequencies, theinertia forces on the slave degrees of freedom are insignificant compared to the elasticforces transmitted by the master degrees of freedoms. Hence we may discard all massterms except that relating directly to the master freedoms

K K

K K

M a

a

mm ms

ms

T

ss

mm m

s

L

NM

O

QP

L

NM

O

QP

F

HG

I

KJ

R

ST

U

VW=

0

0 0

0 (2.6-13)

and from the lower partition

K a K ams

T

m ss s+ = 0 (2.6-14)


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72

and hence

a K K as ss msT

m= 1( ) (2.6-15)

In terms of the full system, and the (m*m) unit matrix I

a=RST

UVW= =

RST

UVW

a

aX a

I

K K

m

s

m

ss ms

T

~1 (2.6-16)

Substitution into (2.6-12)and premultiplication by~

XT

allows the condensed system

to be written as

K Mc m c c m = (2.6-17)

where the condensed stiffness and mass matrices are given by the expressions

K X K X c

T

=~ ~

, and M X M Xc

T

=~ ~

(2.6-18)

Hence progressive condensation of the equations associated with the slave freedomsresults in a reduced equation system in terms of the master freedoms which has thesolution

( )K M ac c m

= 0 (2.6-19)

The fully condensed stiffness and mass matrices are (m*m) symmetric and are

generally full. The condensed system is solved using either implicit QL or Jacobiiteration for the master eigenvalues.

2.6.2.2Manual selection of master freedoms

The master and slave freedoms may be selected by hand. The selection of thesemaster freedoms is central to the accuracy of the solution and consideration should be

taken of the following points

The master freedoms must accurately represent all the significant modes ofvibration.

Master freedoms should exhibit high mass to stiffness ratios, hence rotationalfreedoms are usually inappropriate masters.

Master freedoms should, where appropriate, be as evenly spaced as possible.

The ratio of master to slave freedoms should generally be within the range1:10 to 1:2.

Poor selection of the master freedoms will have a detrimental effect on theaccuracy of the solution.


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Analysis Procedures

73

2.6.2.3Automatic master selection

The manual selection of master freedoms has a number of disadvantages

For large structures the data input is laborious.

Poor selection of the master freedoms will detrimentally affect the accuracy ofthe solution.

The automatic masters selection procedure utilised follows the algorithm of Henshall

and Ong [H8]. Since it is required to retain only those equations which have a highmass to stiffness ratio, the diagonal components of the stiffness and mass matrices are

scanned, and the equation with the highest K Mii ii ratio is condensed out.

Scanning of the progressively condensed matrices is repeated until the fullycondensed matrices contain only those components associated with the specifiednumber of master freedoms. Furthermore, since the remaining equations will havebeen automatically sorted they will contain only those components associated with

the lowest stiffness to mass ratios as required.

2.6.3Comparison Between Subspace Iteration and

Guyan Reduction

The subspace iteration and Guyan reduction algorithms have several similarities

Both methods start by examining which degrees of freedom will provide thegreatest contribution to the structural response, i.e. the q degrees of freedomwith the highest mass to stiffness ratios.

A transformation matrixX

is formed, and a reduced set of q equations areformed.

The reduced set of equations are solved using either the implicit QL or Jacobiiteration to obtain the eigenvalues and eigenvectors of the reduced system.

The eigenvectors of the full system are obtained using the transformation

matrixX

.

Further, for certain special cases, Guyan reduction and the first subspace iteration areequivalent and provide identical solutions [B1].

The major advantage of the subspace iteration technique is that the starting basis is

improved with each update of the subspace, whereas the Guyan reduction procedureis a "one off" solution. Consequently, a poor choice of initial master degrees offreedom may only necessitate more iterations to convergence in subspace iteration,whereas it would be detrimental to accuracy in Guyan reduction. Therefore, subspaceiteration is, in general, the more effective algorithm.


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Eigenvectors mustbe normalized to the mass if a spectral or harmonic response is to

be performed using the eigenvectors from a natural frequency analysis.

2.6.6Evaluation of the Highest Eigenvalues

Subspace iteration may be used to solve for the highest eigenvalues by swapping K

and Mof equations (2.6-3)and (2.6-6)in the iteration process. This facility is useful

for estimating the critical time step values for explicit transient field and dynamic

structural analyses. However, it demands that the specific heat matrix C and the

mass matrix Mmust be positive definite. The solution algorithm will then converge

on the smallest eigenvalues of this new system which are the inverse of the largest

eigenvectors of the original system, i.e. we solve

M K = (2.6-24)

where is diagonal and contains terms of the form i , where i i= 1 and

max( )min( )

i

i

=1

(2.6-25)

2.6.7Eigenvalue Bracketing

Eigenvalue bracketing involves the use of an inverse iteration method which allows

all eigenmodes that lie within a specified frequency or eigenvalue range to beextracted. Most often only the lower frequencies of vibration are required for aparticular structure. Consequently most eigenvalue solution algorithms have beenwritten for this specific purpose. For example, the subspace iteration technique is veryeffective in computing the lowest frequencies of vibration [B1]. Unfortunately it canonly be used to compute modes within a certain range if reasonable approximations tothe lower modes are already available [B4].

The inverse iteration method can be shown to converge to any eigenmode provided areasonable shift is applied. This method does not make use of any other eigenmodes

and for this reason is very effective in computing a few eigenmodes within a specifiedrange. Natural frequency, buckling and stiffness matrix eigenvalues can be extractedusing this method and eigenmodes which have the same eigenvalue can also beobtained.

The basic problem is to find eigenvalues , and eigenvectors u, (eigenmodes) for

K M u =m r 0 (2.6-26)


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within a certain specified range. Both K and M are symmetric matrices but they can

be singular.

The inverse iteration method with shifts can be shown to converge to the eigenmodethat is closest to the shift defined [B1]. Defining

= +0 (2.6-27)

where 0 is a defined constant called the shift point, equation (2.6-26) can be

rewritten as

K M M u = 0 0c hn s (2.6-28)

which can be solved for and hence can be extracted. Using this equation withinverse iteration only allows a single eigenmode to be extracted for each shift pointand it therefore becomes an inefficient and cumbersome process if many modes are tobe extracted within a desired frequency range without knowing approximately wherethe eigenvalues exist within the range defined.

The inverse iteration method also has known defects that include:

Awkwardness of procedure in the presence of zero eigenvalues.

Slow convergence rates for closely spaced eigenmodes.

Deterioration of accuracy in the higher modes as more eigenmodes are found.

This method has thus been modified to overcome these problems and to enableeigenmodes within a defined range to be extracted efficiently. The method utilised

here is based on the Inverse Power Method using an iteration algorithm from [B1].This method is general enough to extract a nominal number of eigenmodes in theregion of a defined shift point.

There is a possibility that the inverse power method may miss eigenmodesparticularly if they are clustered in groups within a small frequency band. This

problem has been overcome in the LUSAS implementation by the use of Sturmsequence checks in order to make sure that this is not the case. Additional passes overthe shiftpoints are also carried out which will also alleviate this problem.

2.6.7.1Inverse iteration algorithm

The inverse iteration method utilises the following iterative algorithm to compute oneeigenvalue and eigenvector. It is used in the general algorithm described above to

compute the eigenmodes.

Start with an initial u0 and compute f M u0 0= .

Then for iterations n=1, ... until convergence:


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The n=1,...,Ns shiftpoints are distributed evenly across the range with their locations

given by

sn n Ns= + min max min( . )*( )0 5 (2.6-37)

Each shiftpoint has a defined range which is given as

sn Ns< 0.55 * ( - ) /max min (2.6-38)

The value 0.55 allows the range of each shiftpoint to overlap the neighbouring rangeby 10%.

3. For each shiftpoint a maximum of Nn eigenmodes are computed. If lesseigenmodes exist near a shiftpoint then the process moves onto the next shiftpoint.If more modes exist near a shiftpoint then the process automatically updates theshiftpoint so that further eigenmodes can be computed as efficiently as possible.

4. The process is terminated only when the desired number of eigenmodes have beencomputed or when Ne eigenmodes have been located, whichever occurs first.

2.6.7.3Computing multiple Eigenmodes

The method of computing multiple eigenmodes from a particular shiftpoint is asfollows:

1. Find the first eigenmode from an arbitrary starting vector (e.g. a vector of 1's).

2. Use the second last trial vector obtained during step 1.as the starting vector for

the second eigenmode. This vector will be swept as in equation (2.6-30) and

provides a good starting point because it will be quite "rich" in the eigenvectorthat is next closest to the shiftpoint. Two iteration vectors are required for the

convergence checks so that this information will always be available.3. Continue until an eigenvalue is found that lies outside the computed range of theshift point.

4. At this stage use a new arbitrary selected starting vector and compute oneeigenmode.

5. If the eigenvalue found in step 4.lies outside the range of the shifting point go to

a new shifting point since all eigenmodes within the range have been computed.

If the eigenvalue does not lie outside the range repeat steps 2, 3 and 4. until the

eigenvalue found in step 4. does lie outside the range. This ensures that all

eigenvalues within the range are computed even for eigenmodes that have the same

eigenvalue. The maximum number of times that steps 2., 3.and 4.will be repeated

is equal to the maximum number of eigenvectors that have a common eigenvalue.

2.6.7.4Convergence checking

A number of convergence checks are utilised in order to enhance the efficiency of thealgorithm. These convergence checks determine if:


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Analysis Procedures

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The inverse iteration algorithm has converged; in this case the eigenvalue and

eigenvector have been computed to sufficient accuracy.

The inverse iteration algorithm is converging rapidly; in this case the iterationsequence is continued and a simple convergence check is applied until a

converged solution has been attained.

The inverse iteration algorithm is not converging; in this case tests are carriedout to determine if the shift point should be moved to enable a betterconvergence rate to be achieved. If this needs to be done then the inverse

iteration algorithm must be restarted utilising this new shift point in order tocompute the eigenmode.

2.6.7.5Recommendation on usage

The major advantages of this method are obtained when a few eigenvalues of higher

frequency are required for a well-banded structural system; these higher modes can beextracted without computing all the lower modes (although some are computed and

swept from the solution). In many cases too many lower modes would have to becomputed with subspace iteration and the method will therefore becomeuneconomical. Another advantage that inverse iteration has over the subspaceiteration method is that only the frequency (or eigenvalue) range of interest needs tobe known and there is no need to estimate how many eigenmodes lie below thatrange.

Inverse iteration does not require that the matrices are non-singular. This is important

for buckling analyses since it is possible that the stress-stiffness matrix may besingular. If this is the case then a special alternative buckling procedure is utilisedwith subspace iteration to ensure correct results. With inverse iteration this is not the

case and therefore knowledge of what these matrices may be like is not required.

2.7Natural Frequency AnalysisThe equations of dynamic equilibrium of an elastic discretised body can be written in

matrix form as (2.1-16)

M a Ca Ka R t&& & ( )+ + = (2.7-1)

For natural frequency analysis, we consider the damping Cand the external force

R(t) both to be zero. Each freedom is assumed to excite harmonic motion in phasewith all other freedoms

a a t a a t = = sin && sin and 2 (2.7-2)

where is the circular frequency. Then (2.7-2) yields

(K M- ) a =i 0 where i= i2

(2.7-3)


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If a structure that has current strain E due to displacements a is subjected to a small

displacement increment a the incremental strain E is defined as

E B a B a a Ao

= + +1

12( ) (2.7-10)

where Bo

, B1, A and are defined in detail in section 3.3.1.

To remove the effect of eigenvector displacements on the strain-displacementrelationship we require the second order term to be zero. Therefore, the displacements

are scaled by the system variable EIGSCL (default value 1.0E-20) before evaluating

the strain increment E. Once the stresses have been evaluated using

S D E= (2.7-11)

they are scaled up using the inverse of EIGSCL.

Updated Lagrangian and Eulerian Analyses

For Updated Lagrangian and Eulerian formulations the strain-displacementrelationship is evaluated in the current configuration. Therefore the stress incrementmay be evaluated directly using

=D ao

B (2.7-12)

where Bo

is the strain-displacement matrix in the current configuration.

2.7.3Centripetal Stiffening Effects

In rotating machinery, load correction terms that arise from the effects of rotation maysignificantly influence the natural frequencies of vibration. Within LUSAS the loadcorrection terms due to centripetal acceleration are considered. The load correctionterms due to Coriolis and angular acceleration result in non-symmetric damping andstiffness matrices respectively and are ignored [Section 5.1.3].

The centripetal load correction terms result in a modified stiffness matrix of the form

K dv dv N dvv v v

= + +z z zB D B G S G NT T T$ (2.7-13)where is the skew-symmetric angular velocity tensor.

The first two terms are the standard linear and geometrically nonlinear contributionsand the third term represents the additional stiffness due to the centripetalacceleration.


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Occasionally the computed stress stiffness matrix may be non-positive definite

causing both negative and positive eigenvalues. To overcome this problem (2.8-2)may be recast in the following form

=K K

1 (2.8-4)

If a shift of 1 1 = is applied, then

K K K+ =

d i (2.8-5)

where now we seek the smallest positive eigenvalue g. The critical load 1 factor is

given by

1

1

1

1=

( ) (2.8-6)

Notes

In certain situations the eigenvalues may be very closely spaced, causingconvergence problems in the iterative solution.

When shifting is used, if 1 is large then 1 will be close to 1.0 and an error

in 1 can yield a large error in 1 . Thus, the applied load must be reasonably

close to the buckling load to yield a small value of 1 .

Negative eigenvalues may indicate bifurcation in tension or bifurcation that

would occur if the loading is reversed in sign.

2.9Eigenvalue Extraction Including Damping

To obtain a solution to the damped natural frequency problem, an eigensolver must becapable of dealing with real, non-symmetric matrices. When non-proportionaldamping is considered such matrices are generated and the solver furnishes solutionsthat consist of complex numbers. The algorithm adopted in LUSAS to solve the non-symmetric eigenvalue problem is referred to as the Arnoldi Method which is a

variation of the Lanczos method for non-symmetric problems.

The problem of finding natural frequencies and mode shapes in a given structureamounts to solving the differential equation

M a Ca Ka R t&& & ( )+ + = (2.9-1)


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where a is the displacement vector, M is the mass matrix, K is the stiffness

matrix and C is the damping matrix, with R t( ) = 0 for natural (as opposed to

forced) oscillations. For undamped motion, C= 0 and the resulting generalised

eigenvalue problem takes the form

Ka M a= (2.9-2)

where both Kand Mare real, symmetric matrices such that M K1

is symmetric,

and thus the standard eigenvalue problem

M Ka a

=1

(2.9-3)

is readily solved by a variety of methods, for example the Lanczos method.

For damped motion, Cis a real non-zero matrix that may or may not be symmetric,

and may also be non-proportional i.e. the product matrix =

C M K1

is non-

symmetric. Examples of this include localised Rayleigh damping where

C K Mi i= + , with the constants i and i dependent on the location in

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