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