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