Stephen Hendry Engineering problems are underdefined 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 ID: 830423
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Slide1
GSA Maths Applied to Structural Analysis
Stephen Hendry|
Slide2“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
Slide3CCTV - Beijing
Slide4Kurilpa Bridge - Brisbane
Slide5Dragonfly Wing
Slide6Design Process – The Idea
Royal Ontario
Museum - Toronto
Slide7Design Process – The Geometry
Slide8Design Process – The Analysis
Slide9Design Process – The Building
Slide10An Early Example
In 1957 Jørn Utzon won the £5000 prize in a competition to design a new opera house
Slide11Sydney Opera House
Slide12Sydney Opera HouseOne of the first structural projects to use a computer in the design process (1960s)
Early application of matrix methods in structural engineeringLimitations at the time meant that shells were too difficultStructure designed using simpler beam methods
Slide13Sydney Opera House
Slide14Structural Analysis
Slide15Structural analysis typesStatic 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
Slide16Computers & Structural Analysis
Two significant developmentsMatrix 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.
Slide17Static Analysis
The stiffness matrix links the force vector and displacement vector for the element
Assemble these into the equation that governs the structure
Solve for displacements
Static Analysis
Challenge is that the matrix
can be large…… but it is symmetric & sparseGSA 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
,
Assemble these into the equation that governs the structure
Solve for
eigenpairs
(‘frequency’ & mode shape)
,
Modal Buckling Analysis
We create a stiffness matrix and a geometric stiffness matrix for the element
,
Assemble these into the equation that governs the structure
Solve for
eigenpairs
(load factor & mode shape)
Aquatic Centre, Beijing
© Gary Wong/Arup
Slide22Comparison
of Static Solvers
Solver
Solution time (s)
No.
t
erms
% non-zero terms
Active column
216
62229172
1.445
Sparse
12
1403012
0.036
Parallel
sparse
4
734323
0.017
11433 nodes
22744 elements
65634 degrees of freedom
Slide23Modelling Issues
Slide24What is the Right ModelNeed to confidently capture the ‘real’ response of the structure
OversimplificationOver-constrain the problemMiss important behaviourToo much detailResponse gets lost in mass of results
More difficult to understand the behaviour
Slide25Emley Moor MastEarly model where dynamic effects were important
Modal analysisModel stripped down to a lumped mass – spring system (relatively easy in this case)
Slide26Emley Moor Mast
Slide27Emley Moor Mast
One-dimensional geometry
Slide28Over-constraining
Modal analysis – restrained in
y
&
z
to reduce the problem size
‘Helical’ structure – response dominated by torsion & restraint in
y
suppressed this
Slide29Graph Theory
Slide30Graph Theory & Façades
Slide31Graph Theory & FaçadesMany 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.
Slide32Graph Theory & FaçadesProblem: 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.
Slide33Graph Theory & Façades
Slide34Current Developments
Slide35Current development workModel accuracy estimationStructure – what error can we expect in the displacement calculation
Elements – what error can we expect in the force/stress calculationHow can we run large models more efficiently
Slide36Solution Accuracy
Slide37Model Accuracy – Structure
Ill-conditioning can limit the accuracy of the displacement solution‘Model stability analysis’ – looks at the eigenvalues/eigenvectors of the stiffness matrix
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’
For small eigenvalues, large values of
indicate where in the model the problem exists.
For large eigenvalues, large
values of
indicate where
in the model the
problem
exists.
Model Accuracy - Structure
Slide40Model Accuracy – Elements
Force calculation depends on deformation of element, for bar
If
&
are large and
≈
then the difference will result in a loss of precision
Model Accuracy – Elements
Remove rigid body displacement to leave the element deformation
Number of significant figures lost in force calculation
Solver Enhancements
Slide43Domain DecompositionMethod of splitting a large model into ‘parts’.
Used particularly to solve large systems of equations on parallel machines.
Slide44Domain DecompositionFor many problems in structural analysis the concept of domain decomposition is linked with repetitive units
Analyse subdomains (in parallel)Assemble instances of subdomains into modelAnalyse complete modelExploit
both repetition & parallelismSubstructure & FETI/FETI-DP methods
Slide45Substructuring & FETI methodsSubstructuring – 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.
Slide46A Historic Example – COMPAS
Slide47A Historic Example – COMPAS
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
Historically
substructuring
was used to allow analysis of ‘large’ models on ‘small’ computers.
Tokamak
has repetition around doughnut
Slide48Substructure Identification
Slide49SubstructuringMake 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 setMaintain a map between elements in assembly and elements in part/component.
Slide50Substructuring & Static Analysis
Basic equations for part (substructure) are partitioned into boundary and internal degrees of freedom
Reduce part to boundary nodes only
Include only boundary nodes in assembly.
Substructuring & Static Analysis
Solve for displacements of assembly.
Calculate the displacements inside the part
Element forces calculated at element level.
Substructuring & Modal AnalysisSubstructuring cannot be applied directly to modal analysis.
Craig-Bampton method and component mode synthesis give an approximate method
Slide53Craig-Bampton Method
For each substructure Assume a fixed boundarySelect the number of modes required to represent the dynamic characteristics of this componentThe component can be represented in the assembly by
Boundary nodes and displacementsA matrix of modal mass and modal stiffness, with modal displacements as variables
Slide54Craig-Bampton Method
Each substructure is represented in the assembly as a hybrid system
+
Similarly for buckling analysis
Key DriversEngineerUnderstanding and optimising the behaviour/design of their structures
Need for more detail in the computer modelsSoftware developersProblem size (see above)
Parallelism – making efficient use of multiple coresConfidence in the results
Slide56ConclusionsModern 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?
Slide57www.arup.com
www.oasys-software.com