Presentation on theme: "GSA Maths Applied to Structural Analysis"— Presentation transcript:
1GSA Maths Applied to Structural Analysis Stephen Hendry|
2“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
12Sydney Opera HouseOne of the first structural projects to use a computer in the design process (1960s)Early application of matrix methods in structural engineeringLimitations at the time meant that shells were too difficultStructure designed using simpler beam methods
15Structural 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
16Computers & Structural Analysis Two significant developmentsMatrix 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.
17Static AnalysisThe stiffness matrix links the force vector and displacement vector for the element𝐟 𝑒 = 𝐊 𝑒 𝐮 𝑒Assemble these into the equation that governs the structure𝐟=𝐊 𝐮Solve for displacements𝐮= 𝐊 −𝟏 𝐟
18Static Analysis Challenge is that the matrix 𝐊 can be large… … but it is symmetric & sparseGSA solvers have gone through several generations as the technology and the engineer’s models have evolvedFrontal solverActive column solverConjugate gradient solverSparse directParallel sparse solver
19Modal Dynamic Analysis We create a stiffness matrix and a mass matrix for the element𝐊 𝑒 , 𝐌 𝑒Assemble these into the equation that governs the structure𝐊φ−λ𝐌φ=𝟎Solve for eigenpairs (‘frequency’ & mode shape)λ,φ , 𝑓= 1 2π λ
20Modal Buckling Analysis We create a stiffness matrix and a geometric stiffness matrix for the element𝐊 𝑒 , 𝐊 𝑔,𝑒Assemble these into the equation that governs the structure𝐊φ+λ 𝐊 𝒈 φ=𝟎Solve for eigenpairs (load factor & mode shape)λ,φ
24What is the Right ModelNeed to confidently capture the ‘real’ response of the structureOversimplificationOver-constrain the problemMiss important behaviourToo much detailResponse gets lost in mass of resultsMore difficult to understand the behaviour
25Emley Moor Mast Early model where dynamic effects were important Modal analysisModel stripped down to a lumped mass – spring system (relatively easy in this case)
28Modal analysis – restrained in y & z to reduce the problem size Over-constrainingModal analysis – restrained in y & z to reduce the problem size‘Helical’ structure – response dominated by torsion & restraint in y suppressed this
31Graph Theory & FaçadesMany 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.
32Graph Theory & FaçadesProblem: 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.
35Current development work Model accuracy estimationStructure – what error can we expect in the displacement calculationElements – what error can we expect in the force/stress calculationHow can we run large models more efficiently
37Model Accuracy – Structure Ill-conditioning can limit the accuracy of the displacement solution‘Model stability analysis’ – looks at the eigenvalues/eigenvectors of the stiffness matrix𝐊φ−λφ=0Eigenvalues at the extremes (low/high stiffness) are indication that problems existEigenvectors (or derived information) give location in model
38Model Accuracy – Structure For each element calculate ‘energies’𝑣 𝑒 = φ 𝑒 𝑇 φ 𝑒𝑠 𝑒 = φ 𝑒 𝑇 𝐊 𝑒 φ 𝑒For 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.
43Domain Decomposition Method of splitting a large model into ‘parts’. Used particularly to solve large systems of equations on parallel machines.
44Domain DecompositionFor many problems in structural analysis the concept of domain decomposition is linked with repetitive unitsAnalyse subdomains (in parallel)Assemble instances of subdomains into modelAnalyse complete modelExploit both repetition & parallelismSubstructure & FETI/FETI-DP methods
45Substructuring & 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.
47A Historic Example – COMPAS Historically substructuring was used to allow analysis of ‘large’ models on ‘small’ computers.Tokamak has repetition around doughnutSplit model into one repeating ‘simple slices’ and …… a set of ‘slices with ports’Used PAFEC to do a substructuring analysis on Cray X-MP
49Substructuring Make it easy for the engineer! Use GSA to create component(s).In GSA master model – import component(s).Create partsInstances of componentsDefined by component + axis setMaintain a map between elements in assembly and elements in part/component.
50Substructuring & Static Analysis Basic equations for part (substructure) are partitioned into boundary and internal degrees of freedom𝐊 𝑏𝑏 𝐊 𝑏𝑖 𝐊 𝑖𝑏 𝐊 𝑖𝑖 𝑢 𝑏 𝑢 𝑖 = 𝑓 𝑏 𝑓 𝑖Reduce part to boundary nodes only𝐊 𝑏𝑏 = 𝐊 𝑏𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝐊 𝑖𝑏𝑓 𝑏 = 𝑓 𝑏 − 𝐊 𝑏𝑖 𝐊 𝑖𝑖 −1 𝑓 𝑖Include only boundary nodes in assembly.
51Substructuring & Static Analysis Solve for displacements of assembly.𝑢= 𝐊 −1 𝑓Calculate the displacements inside the part𝑢 𝑏 = 𝐓 𝒃 𝑢𝑢 𝑖 = 𝐊 𝑖𝑖 −1 𝑓 𝑖 − 𝐊 𝑖𝑏 𝑢 𝑏Element forces calculated at element level.𝑓 𝑒 = 𝐊 𝑒 𝑢 𝑒
52Substructuring & Modal Analysis Substructuring cannot be applied directly to modal analysis.Craig-Bampton method and component mode synthesis give an approximate method
53Craig-Bampton Method For each substructure Assume a fixed boundarySelect the number of modes required to represent the dynamic characteristics of this componentThe component can be represented in the assembly byBoundary nodes and displacementsA matrix of modal mass and modal stiffness, with modal displacements as variables
54Craig-Bampton MethodEach substructure is represented in the assembly as a hybrid system𝐌 𝑟𝑟 𝐌 𝑟𝑚 𝐌 𝑚𝑟 μ 𝑢 𝑟 𝑞 𝑚 +𝐊 𝑟𝑟 𝐊 𝑟𝑚 𝐊 𝑚𝑟 κ 𝑢 𝑟 𝑞 𝑚 = 0 0Similarly for buckling analysis
55Key Drivers Engineer Software developers Understanding and optimising the behaviour/design of their structuresNeed for more detail in the computer modelsSoftware developersProblem size (see above)Parallelism – making efficient use of multiple coresConfidence in the results
56ConclusionsModern 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?