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GSA    Maths Applied to Structural Analysis GSA    Maths Applied to Structural Analysis

GSA Maths Applied to Structural Analysis - PowerPoint Presentation

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GSA Maths Applied to Structural Analysis - PPT Presentation

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

amp analysis structure model analysis amp model structure modal structural element matrix accuracy graph elements stiffness theory design large

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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

Slide3

CCTV - Beijing

Slide4

Kurilpa Bridge - Brisbane

Slide5

Dragonfly Wing

Slide6

Design Process – The Idea

Royal Ontario

Museum - Toronto

Slide7

Design Process – The Geometry

Slide8

Design Process – The Analysis

Slide9

Design Process – The Building

Slide10

An Early Example

In 1957 Jørn Utzon won the £5000 prize in a competition to design a new opera house

Slide11

Sydney Opera House

Slide12

Sydney 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

Slide13

Sydney Opera House

Slide14

Structural Analysis

Slide15

Structural 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

Slide16

Computers & 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.

Slide17

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

 

Slide18

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

 

Slide19

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)

,

 

Slide20

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)

 

Slide21

Aquatic Centre, Beijing

© Gary Wong/Arup

Slide22

Comparison

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

Slide23

Modelling Issues

Slide24

What 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

Slide25

Emley Moor MastEarly model where dynamic effects were important

Modal analysisModel stripped down to a lumped mass – spring system (relatively easy in this case)

Slide26

Emley Moor Mast

Slide27

Emley Moor Mast

 

One-dimensional geometry

Slide28

Over-constraining

Modal analysis – restrained in

y

&

z

to reduce the problem size

‘Helical’ structure – response dominated by torsion & restraint in

y

suppressed this

Slide29

Graph Theory

Slide30

Graph Theory & Façades

Slide31

Graph 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.

Slide32

Graph 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.

Slide33

Graph Theory & Façades

Slide34

Current Developments

Slide35

Current 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

Slide36

Solution Accuracy

Slide37

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

 

Slide38

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.

 

Slide39

Model Accuracy - Structure

Slide40

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

 

Slide41

Model Accuracy – Elements

Remove rigid body displacement to leave the element deformation

Number of significant figures lost in force calculation

 

Slide42

Solver Enhancements

Slide43

Domain DecompositionMethod of splitting a large model into ‘parts’.

Used particularly to solve large systems of equations on parallel machines.

Slide44

Domain 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

Slide45

Substructuring & 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.

Slide46

A Historic Example – COMPAS

Slide47

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

Slide48

Substructure Identification

Slide49

SubstructuringMake 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.

Slide50

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.

 

Slide51

Substructuring & Static Analysis

Solve for displacements of assembly.

Calculate the displacements inside the part

Element forces calculated at element level.

 

Slide52

Substructuring & Modal AnalysisSubstructuring cannot be applied directly to modal analysis.

Craig-Bampton method and component mode synthesis give an approximate method

Slide53

Craig-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

Slide54

Craig-Bampton Method

Each substructure is represented in the assembly as a hybrid system

+

Similarly for buckling analysis

 

Slide55

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

Slide56

ConclusionsModern 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?

Slide57

www.arup.com

www.oasys-software.com