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GSA Maths Applied to Structural Analysis Stephen Hendry |

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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

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CCTV - Beijing

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Kurilpa Bridge - Brisbane

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Dragonfly Wing

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Design Process – The Idea Royal Ontario Museum - Toronto

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Design Process – The Geometry

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Design Process – The Analysis

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Design Process – The Building

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

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Sydney Opera House

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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

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Sydney Opera House

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Structural Analysis

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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

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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.

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Static Analysis

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Modal Dynamic Analysis

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Modal Buckling Analysis

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Aquatic Centre, Beijing © Gary Wong/Arup

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Comparison of Static Solvers SolverSolution time (s) No. terms% non-zero terms Active column Sparse Parallel sparse nodes elements degrees of freedom

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Modelling Issues

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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

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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)

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Emley Moor Mast

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One-dimensional geometry

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Over-constraining Modal analysis – restrained in y & z to reduce the problem size Helical structure – response dominated by torsion & restraint in y suppressed this

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Graph Theory

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Graph Theory & Façades

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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.

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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.

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Graph Theory & Façades

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Current Developments

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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

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Solution Accuracy

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Model Accuracy – Structure

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Model Accuracy - Structure

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Model Accuracy – Elements

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Solver Enhancements

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Domain Decomposition Method of splitting a large model into parts. Used particularly to solve large systems of equations on parallel machines.

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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

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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.

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A Historic Example – COMPAS

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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

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Substructure Identification

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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.

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Substructuring & Static Analysis

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Substructuring & Modal Analysis Substructuring cannot be applied directly to modal analysis. Craig-Bampton method and component mode synthesis give an approximate method

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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

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Craig-Bampton Method

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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

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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?

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