GSA
Maths Applied to Structural Analysis
Stephen Hendry
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“Engineering problems are under-defined,
there are many solutions, good, bad and
indifferent. The art is to arrive at a good
solution.
This is a creative activity, involving
imagination, intuition and deliberate
choice.”
Ove Arup
CCTV - Beijing
Kurilpa Bridge - Brisbane
Dragonfly Wing
Design Process – The Idea
Royal Ontario Museum - Toronto
Design Process – The Geometry
Design Process – The Analysis
Design Process – The Building
An Early Example
In 1957 Jørn Utzon won the £5000 prize in a
competition to design a new opera house
Sydney Opera House
Sydney Opera House
• One of the first structural projects to use a
computer in the design process (1960s)
• Early application of matrix methods in
structural engineering
• Limitations at the time meant that shells were
too difficult
• Structure designed using simpler beam
methods
Sydney Opera House
Structural Analysis
Structural analysis types
• Static analysis – need to know how a structure
responds when loaded.
• Modal dynamic analysis – need to know the
dynamic characteristics of a structure.
• Modal buckling analysis – need to know if the
structure is stable under loading
Computers & Structural Analysis
• Two significant developments
– Matrix methods in structural analysis (1930s)
– Finite element analysis for solution of PDEs
(1950s)
• Computers meant that these methods could
become tools that could be used by engineers.
• Structural analysis software makes use of
these allowing the engineer to model his
structure & investigate its behaviour and
characteristics.
Static Analysis
• The stiffness matrix links the force vector and
displacement vector for the element
𝐟𝑒 = 𝐊 𝑒 𝐮𝑒
• Assemble these into the equation that
governs the structure
𝐟=𝐊𝐮
• Solve for displacements
𝐮 = 𝐊 −𝟏 𝐟
Static Analysis
• Challenge is that the matrix 𝐊 can be large…
• … but it is symmetric & sparse
• GSA solvers have gone through several
generations as the technology and the
engineer’s models have evolved
– Frontal solver
– Active column solver
– Conjugate gradient solver
– Sparse direct
– Parallel sparse solver
Modal Dynamic Analysis
• We create a stiffness matrix and a mass matrix
for the element
𝐊 𝑒 , 𝐌𝑒
• Assemble these into the equation that
governs the structure
𝐊φ − λ𝐌φ = 𝟎
• Solve for eigenpairs (‘frequency’ & mode
shape)
λ, φ , 𝑓 =
1
2π
λ
Modal Buckling Analysis
• We create a stiffness matrix and a geometric
stiffness matrix for the element
𝐊 𝑒 , 𝐊𝑔,𝑒
• Assemble these into the equation that
governs the structure
𝐊φ + λ𝐊 𝒈 φ = 𝟎
• Solve for eigenpairs (load factor & mode
shape)
λ, φ
Aquatic Centre, Beijing
© Gary Wong/Arup
Comparison of Static Solvers
11433 nodes
22744 elements
65634 degrees of freedom
Solver
Solution
time (s)
No. terms
% non-zero
terms
Active column
216
62229172
1.445
Sparse
12
1403012
0.036
Parallel sparse
4
734323
0.017
Modelling Issues
What is the Right Model
• Need to confidently capture the ‘real’
response of the structure
• Oversimplification
– Over-constrain the problem
– Miss important behaviour
• Too much detail
– Response gets lost in mass of results
– More difficult to understand the behaviour
Emley Moor Mast
• Early model where dynamic effects were
important
– Modal analysis
• Model stripped down to a lumped mass –
spring system (relatively easy in this case)
Emley Moor Mast
Emley Moor Mast
One-dimensional geometry
𝑘1 + 𝑘2
−𝑘2
λ
𝑚1
−𝑘2
𝑘2 + 𝑘3
𝑚2
⋱
φ1
φ2 −
⋮
⋱
φ1
φ2 = 0
⋮
Over-constraining
Modal analysis – restrained in y
& z to reduce the problem size
‘Helical’ structure – response
dominated by torsion &
restraint in y suppressed this
Graph Theory
Graph Theory & Façades
Graph Theory & Façades
• Many structural models use beam elements
connected at nodes.
• Graph theory allows us to consider these as
edges and vertices.
• Use planar face traversal (BOOST library) to
identify faces for façade.
Graph Theory & Façades
• Problem: graph theory sees the two graphs
below as equivalent.
• The figure on the left is invalid for a façade…
• … so additional geometry checks are required
to ensure that these situations are trapped.
Graph Theory & Façades
Current Developments
Current development work
• Model accuracy estimation
– Structure – what error can we expect in the
displacement calculation
– Elements – what error can we expect in the
force/stress calculation
• How can we run large models more efficiently
Solution Accuracy
Model Accuracy – Structure
• Ill-conditioning can limit the accuracy of the
displacement solution
• ‘Model stability analysis’ – looks at the
eigenvalues/eigenvectors of the stiffness
matrix
𝐊φ − λφ = 0
– Eigenvalues at the extremes (low/high stiffness)
are indication that problems exist
– Eigenvectors (or derived information) give location
in model
Model Accuracy – Structure
• For each element calculate ‘energies’
1 𝑇
𝑣𝑒 = 2φ𝑒 φ𝑒
𝑠𝑒 = 12φ𝑒 𝑇 𝐊 𝑒 φ𝑒
• For small eigenvalues, large values of 𝑣𝑒
indicate where in the model the problem
exists.
• For large eigenvalues, large values of 𝑠𝑒
indicate where in the model the problem
exists.
Model Accuracy - Structure
Model Accuracy – Elements
• Force calculation depends on deformation of
element, for bar
𝐴𝐸
𝑓=
𝑢2 − 𝑢1
𝑙
• If 𝑢1 & 𝑢2 are large and 𝑢1 ≈ 𝑢2 then the
difference will result in a loss of precision
Model Accuracy – Elements
• Remove rigid body displacement to leave the
element deformation
𝑧𝑧
𝑢𝐷 = 𝑢 −
𝑢𝑅𝑖 𝑢. 𝑢𝑅𝑖
𝑖=𝑥
• Number of significant figures lost in force
calculation
𝑢
𝑛 = log
𝑢𝐷
Solver Enhancements
Domain Decomposition
• Method of splitting a large model into ‘parts’.
• Used particularly to solve large systems of
equations on parallel machines.
Domain Decomposition
• For many problems in structural analysis the
concept of domain decomposition is linked
with repetitive units
– Analyse subdomains (in parallel)
– Assemble instances of subdomains into model
– Analyse complete model
• Exploit both repetition & parallelism
• Substructure & FETI/FETI-DP methods
Substructuring & FETI methods
• Substructuring – parts are connected at
boundaries.
• FETI (Finite Element Tearing & Interconnect) –
parts are unconnected. Lagrange multipliers
used to enforce connectivity.
• FETI-DP – parts are connected at ‘corners’ and
edge continuity is enforced by Lagrange
multipliers.
A Historic Example – COMPAS
A Historic Example – COMPAS
• Historically substructuring was used to allow
analysis of ‘large’ models on ‘small’
computers.
• Tokamak has repetition around doughnut
Split model into one repeating
‘simple slices’ and …
… a set of ‘slices with ports’
• Used PAFEC to do a
substructuring analysis on
Cray X-MP
Substructure Identification
Substructuring
•
•
•
•
Make it easy for the engineer!
Use GSA to create component(s).
In GSA master model – import component(s).
Create parts
– Instances of components
– Defined by component + axis set
• Maintain a map between elements in
assembly and elements in part/component.
Substructuring & Static Analysis
• Basic equations for part (substructure) are
partitioned into boundary and internal
degrees of freedom
𝑓𝑏
𝐊 𝑏𝑏 𝐊 𝑏𝑖 𝑢𝑏
=
𝑢
𝐊 𝑖𝑏 𝐊 𝑖𝑖
𝑓𝑖
𝑖
• Reduce part to boundary nodes only
−1
𝐊 𝑏𝑏 = 𝐊 𝑏𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 𝐊 𝑖𝑏
𝑓𝑏 = 𝑓𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝑓𝑖
• Include only boundary nodes in assembly.
Substructuring & Static Analysis
• Solve for displacements of assembly.
𝑢 = 𝐊 −1 𝑓
• Calculate the displacements inside the part
𝑢𝑏 = 𝐓𝒃 𝑢
𝑢𝑖 = 𝐊 𝑖𝑖 −1 𝑓𝑖 − 𝐊 𝑖𝑏 𝑢𝑏
• Element forces calculated at element level.
𝑓𝑒 = 𝐊 𝑒 𝑢𝑒
Substructuring & Modal Analysis
• Substructuring cannot be applied directly to
modal analysis.
• Craig-Bampton method and component mode
synthesis give an approximate method
Craig-Bampton Method
• For each substructure
– Assume a fixed boundary
– Select the number of modes required to represent
the dynamic characteristics of this component
• The component can be represented in the
assembly by
– Boundary nodes and displacements
– A matrix of modal mass and modal stiffness, with
modal displacements as variables
Craig-Bampton Method
• Each substructure is represented in the
assembly as a hybrid system
𝐌𝑟𝑟
𝐌𝑚𝑟
𝐌𝑟𝑚 𝑢𝑟
+
μ
𝑞𝑚
𝐊 𝑟𝑟 𝐊 𝑟𝑚 𝑢𝑟
0
=
𝑞
𝐊 𝑚𝑟
κ
𝑚
0
• Similarly for buckling analysis
Key Drivers
• Engineer
– Understanding and optimising the
behaviour/design of their structures
– Need for more detail in the computer models
• Software developers
– Problem size (see above)
– Parallelism – making efficient use of multiple cores
– Confidence in the results
Conclusions
• Modern structural analysis software depends
on maths – which engineers may not
understand in detail.
• Continual need for better/faster/more
accurate methods to solve linear equations
and eigenvalue problems.
• Dialogue between engineers and
mathematicians can be mutually beneficial.
• Any novel ideas for us to make use of?
www.arup.com
www.oasys-software.com