FE-Modelling To create a simplified model of the geometry, material, loading and boundary conditions that produce results with as small deviations as possible from the real behaviour.
Finite Element Method FEM
Physical
phenomenon
Differential
equation
Finite element
equations
Mathematical model
Approximation
Deformations
Heat transfer
Fluid- or gas flow
Diffusion
etc. ...
Engineering Computations FEM
Modelling Geometry Material Loading, Boundary conditions
Software Static Dynamic (Implicit, Explicit) Multiphysic (Coupled problems)
Execution Serial / parallel
Example
FEM Work Flow
Pre-processor FE-solver Post-processor
Geometry
From CAD-software ...
Geometry
... or FE-preprocessor
Material
Isotropic / anisotropic Linear
Elastic Time dependent, (viscoelastic / creep)
Non-linear Elastic-plastic Time dependent, (viscoplastic / creep)
Loading and Boundary Conditions
Loads Point- , Line-, Surface- and Body-loads
Boundary displacements Prescribed displacements, constraint equations ...
Contact Deformable bodies in contact with other deformable- or
rigid bodies Friction
Coulumb, stick-slip ...
Contact conditions - example
Contact conditions - example
Solvers Implicit integration in time
Direct solution of linear equation system Equilibrium after each time step Large time steps, (unconditional stable)
Explicit integration in time Solution by determining solution in next time step by
only considering the past. (Central difference, Forward Euler method)
Contain only matrix and vector multiplications, (lumped mass och linear element approximations)
No equilibrium Very small time-steps, (conditional stable)
Solvers
Implicit method Static or dynamic
ABAQUS/Standard NASTRAN Marc Cosmos ...
Explicit method Dynamic for rapidly changing events (Short time
durations) ABAQUS/Explicit Marc LS-Dyna ...
Computer Systems
Single processor Serial Cheap (and expensive)
Multiprocessor (Fast internal communication) Serial / Parallel Very expensive (If more than a few processors)
Cluster (Fairly slow intercommunication) Serial / Parallel Cheap
Lunarc Platon: 216 nodes 8 CPU cores/node 24 GB ram/node
Alarik: 208 nodes 16 CPU cores/node 32-128 GB ram/node
Parallel FE-methods
Explicit codes Typical 14 times faster on 16 processors (Suitable for
parallel execution)
Implicit codes Typical 2 times faster on 4 processors (Not suitable for
parallel execution) Very large equation systems (> million dofs) may yield
good performance.
Domain Decomposition Methods - basic concept
Study first a single domain FE-equation system
Partition system into boundary dofs and interior dofs
(1)fKu =
(2), TIBBII
B
I
B
IIIB
BIBB KKff
uu
KKKK
=
=
Boundary dofs, uB Interior dofs, uI
Domain Decomposition
Eliminate the interior dofs uI in first eq. in (2)
which can be written
with
Assemble and solve the reduced system in parallell or on singel processor.
IBIITIBBBB KKKKK
1~ =
( ) )3(,11 IIITIBBBIBIITIBBB fKKfuKKKK =
IIITIBBB fKKff
1~ =
(4)~~ BBB fuK =
boundary stiffness matrix boundary load matrix
Domain Decomposition Start with a domain and divide into
subdomains
Geometry Partitioned geometry
FE-mesh
Example -TetraPak Masters Thesis
Experiment
Kopplade problem
Fluid - strukturinteraktion
Kopplade problem temperatur-frskjutning
Brandfrlopp temp.~ tid
Andra problemomrden
Strmning - CFD (Computational fluid dynamics)
MAX IV - Synchrotron light source External loads Internal loads
MAX IV - Synchrotron light source
Max. vibration RMS 26 nm (26 10-9 m), 1s Diameter of human hair, ~75 m (75 10-6 m) 3000 times
Linear accelerator ~ 500m
Storage ring Perimeter ~700m
Mockup
MAX IV - investigation of vibration reducing measures
External loads (roads, E22) Cut-off
wall Stabilised soil, ~ 4m
Bedrock
Clay ~12-17m
Shaped terrain
Internal loads
Peter Persson and Kent Persson
Some results of measures Soil stabilisation
Reduction of vibrations from external loading ca 25%. Reduction of vibrations from internal lloading ca 50%.
Cut-off wall Reduction of vibrations from external loading up to ca 45%.
Shaped terrain Reduction of vibrations from external loading up to 40%. ( Need long distance between source and building)
MAX IV vibrations from office building
Ola Flodn and Kent Persson
MAX IV vibrations from office building
MAX IV example vibrations from office building
Assignment Dynamic analysis of a wing
Assignment Dynamic analysis of a wing
Calculate eigenvalues and eigenmodes Calculate steady-state response Calculate response for impulse load (transient) Determine the number of modes needed for the impulse
load (modal reduction)
Eigenmodes
Acceleration vs Time
FE-ModellingFinite Element Method FEMEngineering Computations FEMFEM Work FlowGeometryGeometryMaterialLoading and Boundary ConditionsContact conditions - exampleContact conditions - exampleSolversSolversComputer SystemsLunarcParallel FE-methodsDomain Decomposition Methods- basic conceptDomain DecompositionDomain DecompositionExample -TetraPakMasters ThesisExperimentBildnummer 21Kopplade problemKopplade problem temperatur-frskjutningAndra problemomrdenMAX IV- Synchrotron light sourceMAX IV- Synchrotron light sourceMockupMAX IV - investigation of vibration reducing measuresSome results of measuresMAX IV vibrations from office buildingMAX IV vibrations from office buildingMAX IV example vibrations from office buildingAssignmentDynamic analysis of a wingAssignmentDynamic analysis of a wingEigenmodesAcceleration vs Time