FEA convergence

MCEN90026 SOLID MECHANICS • FINITE ELEMENT ANALYSIS

CONVERGENCE AND LIMITATIONS JONATHAN MERRITT 2013

 THE UNIVERSITY OF MELBOURNE

 TODAY'S LECTURE LECTURE

•  Different levels of approximation in FEA •  Monotonic convergence (beam demonstration) •  Criteria for monotonic convergence •  The Mysterious Case of the Infinitely Sharp Corner •  Limitations of linear FEA (Black Hawk Helicopter example)

LEVELS OF APPROXIMATION 01   Actual

physical problem

•  Geometry •  Material •  Loading •   Boundary conditions 02  Mathematical model / mechanical •  Kinematics •  Three-dimensional •  Kirchoff plate •  Beam •  Etc… •  Material •  Loading • BCs 03  Finite

idealization

element solution

•  Approximate solution to mathematical model

MONOTONIC CONVERGENCE

Under the correct conditions, FEA solutions will converge  monotonically  to the exact solution of the problem-governing equations (As just pointed out, the solution to the problem-governing equations is not always identical to the solution of the real system)

ABAQUS BEAM EXAMPLE PARAMETERS p  = 0.2 N/mm 50 mm

u  2000 mm

LOADING

50 mm

CROSS-SECTION

•  Cantilever beam loaded with a constant force •  Measuring tip displacement u  •  Units: mm, N, MPa •  Young’s modulus E   = 200 GPa •  Poisson’s ratio ν  = 0.33 •  Exact solution (mathematical idealisation): u  = −(pL4 )/(8EI ) •  Using B21 beam elements

BEAM EXAMPLE OF MONOTONIC CONVERGENCE −3.8 FEA Exact

     )     m     m      (     t −4.2     n     e     m     e     c −4.4     a      l     p     s      i      d     p −4.6      i      T

−4

−4.8 0

5

10

15

20

Number of elements

25

30

35

CRITERIA FOR MONOTONIC CONVERGENCE

01  Elements

must be  complete . They must be able to represent

•  All rigid body displacements •  All constant strain states 02  Elements

of the mesh must be compatible

•  No voids •  No overlaps

RIGID BODY DISPLACEMENTS

The rigid body displacements of an element involve  zero   strain. For example, for a plane stress element

CONSTANT STRAIN STATES

The element must also be able to represent all constant strain states This can be understood by considering that, as the mesh becomes finer, the variation in strain within each element becomes small compared with the average strain that the element is experiencing

INFINITELY SHARP CORNER

Sometimes convergence is not possible because the governing equations contain a singularity

INFINITELY SHARP CORNER PARAMETERS 10

30

von Mises stresses here t  = 1

30

•  Units: (unspecified) •  L-plate with an edge shear of  t  = 1 •  Young’s modulus E   = 30000 •  Poisson’s ratio ν  = 0.3 •  CPS4 elements with thickness z  = 5

10

VON MISES STRESSES AT THE SHARP CORNER 24      )     a      P      M      (     s     s     e     r     t     s     s     e     s      i      M     n     o    v

22 20 18 16 14 12 10 1

2

3

4

5

Elements per edge

6

7

8

FILLETING THE CORNER

10

30

von Mises stresses here

t  = 1

30

•  Same geometry, but with r  = 10 mm fillet •  Using both CPS4 and CPS8 elements

10

VON MISES STRESSES AT THE FILLETED CORNER

     )     a      P      M      (     s     s     e     r     t     s     s     e     s      i      M     n     o    v

10 CPS4 – Linear element CPS8 – Quadratic element

8

6

4 0

2

4

6

8

10

Elements per edge

12

14

NON-LINEARITIES

•  Material •   Non-linear elasticity •  Plasticity •  Etc …

•  Contact •   Elements may contact each other and introduce additional forces

•  Geometric •  Bucking •  Etc …

CENTRIFUGAL STIFFENING OF A HELICOPTER ROTOR BLADE

•  Example: Sikorski Aircraft Corporation UH-60A Blackhawk •  Used by the NASA Aeromechanics research group • http://rotorcraft.arc.nasa.gov

•  4 titanium spar rotor blades with fibre-glass skins and “Nomex” honeycomb core filler •  Nominal aircraft weight: 7,500 kg

BLACK HAWK ROTOR BLADE CONSTRUCTION

Fibreglass cover

Titanium spar Nomex   core

SPAR MODEL 

centrifugal load at 26 rad/s lift load = 2.217 N/mm

t  = 3.429 mm 45 mm

u  200 mm 8300 mm

LOADING

•  Titanium spar (E  = 115 GPa, •  Units: mm, N, MPa, T/mm 3 •  Using B32 elements

CROSS-SECTION

ν 

= 0.33, ρ = 4.65 g/cm3 )

ROTOR BLADE IN ABAQUS

Demonstration that  geometric non-linearity  is necessary for the lift and centrifugal load effects on the blade to interact

END

Slides created using Free Software: • LATEX typesetting system • Beamer slides • Ti k Z graphics language • Inkscape • QCAD • ImageMagick • Blender

Other software used: •  Abaqus (commercial FEA) •  Code Aster (open source FEA) •  gmsh open source meshing •  Octave •  Matlab •  Python + SciPy + Matplotlib