Robust non-algebraic Reissner-Mindlin plate finite elements

N = 6 x 6

N = (6 x 6)

W exact = 0.01160000

C

h

a = 1

Kirchhoff

‘ Lim { 3D ; R - M } ’ = ‘ Kirchhoff model ’

Morgenstern,1959;Gol’denveizer,1965;
Babuska & Pitkaranta,1990.

Reduced / Selective numerical integration, Zienkiewicz et el., 1971, 1976.

Simply Supported
Square Plate

P

Pa / D

2

Over-Stiff FEM equations - Much slow convergence and poor accuracy

C

2

LARGE Stiff W = small

FEM - analysis

Compatible

exact asymptotic

[ 2 x 2 ] Gauss –
Legendre / Exact
integration

a priori
independent

Uniform

Stable

0.01192

0.00000

[ 2 x 2 ] +

Rank Deficiency

[ 1 x 1 ]

A/S rule: “Accuracy x Stability = Constant”

Low-order Algebraic Interpolation

The most “STATIC” area in FEM is Shape Functions of Algebraic type.

First

NO Internal Nodes

Field Inconsistency &
Excessive-Stiffness =
Delayed Convergence

Only ONE level of the
Energy ( displacement )

2

6

3

4

5

log (a/h)

: EXACT integr.

element stiffness matrix

exact

non-algebraic

[2x2]

0

R – M shear locking problem with DoF:

8-node
FE

4-node Bilinear: [ 2 x 2 ] + [1 x 1 ]

~ 0

[ 2 x 2 ] : Uniform / Stable

0. 200

(Ex.: max = )

How can we control the Energy levels ?

h <<1

1

Non-Stable

to get Accurate/Stable solution: Uniform int. [ 2 x 2 ]

SR

Mesh: 6 x 6

1.192

SR

another
way

1.160

Crime

No Crime

Wc x 100

Quadratic – Serendipity : [ 3 x 3 ] + [ 2 x 2 ]

Need

3D

to Key

infinite set of

NO Control

introduce - Shape Functions Variability

A Continuation Method: D.F. Davidenko; V.I. Shalashilin

Convergence with N increase

Convergence Improvement

exact

Infinite Set of Energy
( displacement )
levels of

Q.: How we can MORE improve solution?

Convergence
from below

wc x 100

1.254

1.273

C - above

the Best / below

ACCURACY

0.200

NO introducing Rank Deficiency &
ill-Conditioning

same coarse mesh

?

Slow

same mesh

2

3

4

5

6

log(a/h)

0

Kirchhoff exact

Scheme Better than Scheme

Slow

Fast

0.2

1.160

1.054

from below

from above

to Single level Selection
- Parametric Study

Infinite Set
of levels

?

“True thin”

wc

0

exact

wc

0

No DCP

may be

What the Energy (displacement) approximating level
is TRUE for THIN plates ?

~ 0

~ 0

Infinite Set of Energy ( displacement ) levels of

Unique Choice of : = ?

plate mechanics

Scheme S1 ( )

DCP – Degenerated / inflexion Critical Point : Structural Stability of Set

Catasrophe / Singularity Theory : Fold Catasrophe

0.011600

8.3

10.7

Wc 0 ( )

DCP – Search

No

No

0

Search
Space

max

min

Reaction to Small Perturbation

Energy level

min

1

0

U

[K]

STABLE

K

RM

ms

t

0

Variational Crime =
energy unbalance
Instability

[ssm] := Singular

DCP

round-off error

continuously

inflexion p.

critical point

inflexion &

ENERGY Consistency of Field functions via APPROXIMATIONS

Shape Functions for Deflection and Rotations

to Select
K - solution
from
R-M family :

URM UK

Kirchhoff case is a member from the Reissner – Mindlin family

a priori independent

Slow varying -

- Fast varying

R - M Energy / Stiffness Parametric family

- control

find

Kirchhoff

another way

&

Problem

APPR

ENER

Fold Catasrophe

contrast

No DCP

P (K) = 0

Seek !

=:Unique

S1( )

Fold

Perturbation

Rank Deficiency

9.2

9.8

Consistency via MultiScale

1

x

y

1

~

~

8 – node Kirchhoff – Reissner – Mindlin thin Plate FE

Uniqueness of Critical Point of Inflexion: – finding

Selection of K – solution from Reissner – Mindlin family

‘ Energy via Deflection ’ & FEM analysis data

simply
supported

K

=

Seeking

C

36 FE

at Center

Compatible

= [ 9.55 ; 9.75 ]

h

Fold Catastrophe

Degenerated

Cr-Point

Reduced Integration

Parametric
Approach

instable

Round – off Error

stable

Consistency

P (K)=0

Rank Deficiency

Square Plate : a x a x h Simply Supported (SS - soft),
loaded at the Center by a concentrated force P
KRM: Constructed Kirchhoff-Reissner-Mindlin FE with
agreed C0 – deflection and rotations

wcor

Scheme:

s

Mesh: 6 x 6

P

P

8 – node

corner

SS

36 FE

36 FE

Compound Scheme

: Displacement – based FEM

R – Control via Shape Functions OR via Variational Principle

{

8 - node

dispose

K-RM solution: Sch.

stable = + / -

-

+

36 FE

Quality = ( Accuracy + Stability ) + Robustness

Robustness = Stability towards: Round-off error & Problem parameters

+

robust = + / -

+

+

-

Large parameter (Stiff Problem ):

Method Stability : to Zero Energy Modes = Mechanisms

496

2

0.03

relative error %

same mesh : 6 x 6

N

CSR

C

S S

Mixed Boundary Conditions

P

c

Wc

log (a/h)

SR

S3 ( )

Compound

S/Quadr : [ 3x3 ]b + [ 2x2 ]s

S2 : [ 2 x 2 ] = FI

‘ Exact ’

0.0078

0.0075

0.0074

0.0020

0.0007

0.0083

0.0079

0.0078

0.0021

Ref. : Tseitlin A.I., 1971.

*

*

Mesh : 6 x 6

Compound: KRM

SR

8 – node FE

1

locking

N – convergence

N

CSR

inducing

CSR

-

clamp

SR

SR

Compound

Compound

0.0069

0.0068

0.00653

0.00651

0.0059

S / Quadr

0.00262

S2

Wc

WA

0.0015

0.00077

- 0.0038

- 0.0043

- 0.0036

- 0.0015

0.00085

- 0.0008

- 0.0005

Ref. : Jiang Z., 1992

1 / 3

1 / 3

1

1

Compound

at Corners

KRM

4

0

log ( a / h )

W

Mesh: 6 x 6

SR

clamp – Free

SS - Free

Point Singular Support

SR

locking

S / Quadr

S2

_

+

CSR

CSR

at centre

Change

inducing Reactions

+

Physical Stability

SS 0

Mesh: (6x6) of 8-node FE

x

y

F

0

1

1

RM:

K:

0

+

-

mid-edge

Corner

essential

natural

CSR

variational boundary conditions (SS-soft)

the principle of virtual work (displacements)

KRM-FE

RM

K

Torsion of Thin plate : 3 Node – Supported plate,
loaded at the Corner by a concentrated force F

K

Corner

h=0.00001

( Boundary Oscillations = Instability )

Compound Scheme
No Locking and ZEM

10 x

= W

Torsion of Thin Plate

exact

h=0.00001

Instability / Zero Energy Modes & Control by Stabilization

Scheme Selective Reduced Integration

Scheme with 4 Corner Stabilizing FE

KRM FE

Stabilization

= 80

o

= 70

o

= 80

= 70

o

o

w

w

w

w

36 SR FE

( 32 SR + 4 KRM ) FE

b/2

a/b=1

P

ZEM

Corner Shear Reaction

No CSR

Stable

0

y

x

0

0

0.223

0.446

0.341

0.682

b/2

CSRs

P

b/2

+

CSR

inducing sign

-

+

-

CSR

C

C

C

C

Г

Г

Г

Г

С

Г

+

exact

h=0.00001

C

h=0.00001

w

w

0

0

Selective Reduced FEs

ZEM – Amplitude

Bending ZEM

36 SR

Stabilization by K – RM FEs at CORNERS

Torsion ZEM

LARGER

To Corners

36 SR

LIFT

LIFT

SR

VERSUS

1

0.18

0.11

F

F

DOWN

Corner

36 SR

4 Stabilizing K – RM FEs

LIFT

DOWN

Checking FEM Solution

32 SR

ZEM

- 0.3577

Interpolation

Wc = - 0. 7153

Wc = Wc / 2

W

C

a

P

P

P

P

P

X

Y

EXACT

h=0.00001

Kirchhoff

Kirchhoff

Reproducing

32 SR + 4 K – RM

DoF – Numerical Values

Pure Torsion

Pure Torsion

X

Y

?

!

4 – Point Singular Thin Plate Bending & Stabilization by RM Shear FEs

36 Selective Reduced

32 SR + 4 RM Shear

Oscillations

Oscillations

SR: NO Stability

h = 0.00001; Mesh: 6 x 6

Free

SS 0

UZ= -.004015 UKT= -.004727
UZ= -.002307
UZ= -.001067
UZ= -.000080
UZ= .000928
UZ= .000453
UZ= .000000 UKT= .000000
UZ= .000453
UZ= .000928
UZ= -.000080
UZ= -.001067
UZ= -.002307
UZ= -.004015 UKT= -.004727
X=0.5
UZ= .000000 UKT= .000000
UZ= .003395
UZ= .006811
UZ= .008794
UZ= .010798
UZ= .012394
UZ= .014011 UKT= .015456
UZ= .012394
UZ= .010798
UZ= .008794
UZ= .006811
UZ= .003395
UZ= .000000 UKT= .000000

32 SR + 4 RM Shear FEs

Singular: Mixed b.c.

Break

X

Y

Discontinuity

1

-

No Oscillations

Point

Jiang & Liu, exact

State of Equilibrium

h = 0.00001 ; Mesh : 6 x 6

Crime

ZEM

Stabilization

Rank Deficiency

SR

Corners

at Sides

WILD

LARGE ZEM