gsa maths applied to structural analysis stephen hendry |
TRANSCRIPT
GSA Maths Applied to Structural Analysis
Stephen Hendry|
“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
• 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 & 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
, • 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
Comparison of Static Solvers
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
11433 nodes22744 elements65634 degrees of freedom
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
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
– 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
Model 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
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
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
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
• 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 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
+
• 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.comwww.oasys-software.com