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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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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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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
Constant used in dynamic recurrence algorithms
Shear retention factor in concrete model
Constant used in dynamic recurrence algorithms
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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