GSA Maths Applied to Structural Analysis Stephen Hendry

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 (1960 s) • 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 (1930 s) – Finite element analysis for solution of PDEs (1950 s) • 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 •

Static Analysis •

Modal Dynamic Analysis •

Modal Buckling Analysis •

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

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 •

Model Accuracy – Structure •

Model Accuracy - Structure

Model Accuracy – Elements •

Model Accuracy – Elements •

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 •

Substructuring & Static Analysis •

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 •

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