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 𝐟𝑒 = 𝐊 𝑒 𝐮𝑒 • Assemble these into the equation that governs the structure 𝐟=𝐊𝐮 • Solve for displacements 𝐮 = 𝐊 −𝟏 𝐟 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 Modal Dynamic Analysis • We create a stiffness matrix and a mass matrix for the element 𝐊 𝑒 , 𝐌𝑒 • Assemble these into the equation that governs the structure 𝐊φ − λ𝐌φ = 𝟎 • Solve for eigenpairs (‘frequency’ & mode shape) λ, φ , 𝑓 = 1 2π λ Modal Buckling Analysis • We create a stiffness matrix and a geometric stiffness matrix for the element 𝐊 𝑒 , 𝐊𝑔,𝑒 • Assemble these into the equation that governs the structure 𝐊φ + λ𝐊 𝒈 φ = 𝟎 • Solve for eigenpairs (load factor & mode shape) λ, φ 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 𝑘1 + 𝑘2 −𝑘2 −𝑘2 𝑘2 + 𝑘3 𝑚1 λ 𝑚2 ⋱ φ1 φ2 − ⋮ ⋱ φ1 φ2 = 0 ⋮ 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 𝐊φ − λφ = 0 – 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’ 1 𝑇 𝑣𝑒 = 2φ𝑒 φ𝑒 𝑠𝑒 = 12φ𝑒 𝑇 𝐊 𝑒 φ𝑒 • For small eigenvalues, large values of 𝑣𝑒 indicate where in the model the problem exists. • For large eigenvalues, large values of 𝑠𝑒 indicate where in the model the problem exists. Model Accuracy - Structure Model Accuracy – Elements • Force calculation depends on deformation of element, for bar 𝐴𝐸 𝑓= 𝑢2 − 𝑢1 𝑙 • If 𝑢1 & 𝑢2 are large and 𝑢1 ≈ 𝑢2 then the difference will result in a loss of precision Model Accuracy – Elements • Remove rigid body displacement to leave the element deformation 𝑧𝑧 𝑢𝐷 = 𝑢 − 𝑢𝑅𝑖 𝑢. 𝑢𝑅𝑖 𝑖=𝑥 • Number of significant figures lost in force calculation 𝑢 𝑛 = log 𝑢𝐷 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 • Basic equations for part (substructure) are partitioned into boundary and internal degrees of freedom 𝑓𝑏 𝐊 𝑏𝑏 𝐊 𝑏𝑖 𝑢𝑏 = 𝑢 𝐊 𝑖𝑏 𝐊 𝑖𝑖 𝑓𝑖 𝑖 • Reduce part to boundary nodes only −1 𝐊 𝑏𝑏 = 𝐊 𝑏𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 𝐊 𝑖𝑏 𝑓𝑏 = 𝑓𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝑓𝑖 • Include only boundary nodes in assembly. Substructuring & Static Analysis • Solve for displacements of assembly. 𝑢 = 𝐊 −1 𝑓 • Calculate the displacements inside the part 𝑢𝑏 = 𝐓𝒃 𝑢 𝑢𝑖 = 𝐊 𝑖𝑖 −1 𝑓𝑖 − 𝐊 𝑖𝑏 𝑢𝑏 • Element forces calculated at element level. 𝑓𝑒 = 𝐊 𝑒 𝑢𝑒 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 𝐌𝑟𝑟 𝐌𝑚𝑟 𝐌𝑟𝑚 𝑢𝑟 + μ 𝑞𝑚 𝐊 𝑟𝑟 𝐊 𝑟𝑚 𝑢𝑟 0 = 𝑞 𝐊 𝑚𝑟 κ 𝑚 0 • 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.com www.oasys-software.com