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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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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

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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 The stiffness matrix links the force vector and displacement vector for the element 𝐟 𝑒 = 𝐊 𝑒 𝐮 𝑒 Assemble these into the equation that governs the structure 𝐟=𝐊 𝐮 Solve for displacements 𝐮= 𝐊 −𝟏 𝐟

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**Static Analysis Challenge is that the matrix 𝐊 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

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**Modal Dynamic Analysis**

We create a stiffness matrix and a mass matrix for the element 𝐊 𝑒 , 𝐌 𝑒 Assemble these into the equation that governs the structure 𝐊φ−λ𝐌φ=𝟎 Solve for eigenpairs (‘frequency’ & mode shape) λ,φ , 𝑓= 1 2π λ

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**Modal Buckling Analysis**

We create a stiffness matrix and a geometric stiffness matrix for the element 𝐊 𝑒 , 𝐊 𝑔,𝑒 Assemble these into the equation that governs the structure 𝐊φ+λ 𝐊 𝒈 φ=𝟎 Solve for eigenpairs (load factor & mode shape) λ,φ

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**Aquatic Centre, Beijing**

© Gary Wong/Arup

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**Comparison of Static Solvers**

11433 nodes 22744 elements 65634 degrees of freedom Solver Solution time (s) No. terms % non-zero terms Active column 216 1.445 Sparse 12 0.036 Parallel sparse 4 734323 0.017

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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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**Emley Moor Mast One-dimensional geometry**

𝑘 1 + 𝑘 2 −𝑘 2 −𝑘 2 𝑘 2 + 𝑘 ⋱ φ 1 φ 2 ⋮ − λ 𝑚 𝑚 ⋱ φ 1 φ 2 ⋮ =0

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**Modal analysis – restrained in y & z to reduce the problem size**

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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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.

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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**

Ill-conditioning can limit the accuracy of the displacement solution ‘Model stability analysis’ – looks at the eigenvalues/eigenvectors of the stiffness matrix 𝐊φ−λφ=0 Eigenvalues at the extremes (low/high stiffness) are indication that problems exist Eigenvectors (or derived information) give location in model

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**Model Accuracy – Structure**

For each element calculate ‘energies’ 𝑣 𝑒 = φ 𝑒 𝑇 φ 𝑒 𝑠 𝑒 = φ 𝑒 𝑇 𝐊 𝑒 φ 𝑒 For small eigenvalues, large values of 𝑣 𝑒 indicate where in the model the problem exists. For large eigenvalues, large values of 𝑠 𝑒 indicate where in the model the problem exists.

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**Model Accuracy - Structure**

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**Model Accuracy – Elements**

Force calculation depends on deformation of element, for bar 𝑓= 𝐴𝐸 𝑙 𝑢 2 − 𝑢 1 If 𝑢 1 & 𝑢 2 are large and 𝑢 1 ≈ 𝑢 2 then the difference will result in a loss of precision

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**Model Accuracy – Elements**

Remove rigid body displacement to leave the element deformation 𝑢 𝐷 =𝑢− 𝑖=𝑥 𝑧𝑧 𝑢 𝑅𝑖 𝑢. 𝑢 𝑅𝑖 Number of significant figures lost in force calculation 𝑛=log 𝑢 𝑢 𝐷

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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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**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

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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**

Basic equations for part (substructure) are partitioned into boundary and internal degrees of freedom 𝐊 𝑏𝑏 𝐊 𝑏𝑖 𝐊 𝑖𝑏 𝐊 𝑖𝑖 𝑢 𝑏 𝑢 𝑖 = 𝑓 𝑏 𝑓 𝑖 Reduce part to boundary nodes only 𝐊 𝑏𝑏 = 𝐊 𝑏𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝐊 𝑖𝑏 𝑓 𝑏 = 𝑓 𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝑓 𝑖 Include only boundary nodes in assembly.

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**Substructuring & Static Analysis**

Solve for displacements of assembly. 𝑢= 𝐊 −1 𝑓 Calculate the displacements inside the part 𝑢 𝑏 = 𝐓 𝒃 𝑢 𝑢 𝑖 = 𝐊 𝑖𝑖 −1 𝑓 𝑖 − 𝐊 𝑖𝑏 𝑢 𝑏 Element forces calculated at element level. 𝑓 𝑒 = 𝐊 𝑒 𝑢 𝑒

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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 Each substructure is represented in the assembly as a hybrid system 𝐌 𝑟𝑟 𝐌 𝑟𝑚 𝐌 𝑚𝑟 μ 𝑢 𝑟 𝑞 𝑚 + 𝐊 𝑟𝑟 𝐊 𝑟𝑚 𝐊 𝑚𝑟 κ 𝑢 𝑟 𝑞 𝑚 = 0 0 Similarly for buckling analysis

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**Key Drivers Engineer Software developers**

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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**www.arup.com www.oasys-software.com**

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