Robust non-algebraic Reissner-Mindlin plate finite elements презентация

Ilya Yu. Kolesnikov

Geophysical Center of the RAS

Robust Non-Algebraic Reissner-Mindlin Plate Finite Elements

thin plates

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

of Boundary Nodes

NO Internal Nodes

(SR) with decomposition of shear stiffness matrix [ssm]: [2x2]b+[1x1+1x1+1x1+2x2]s for Laplace Operator

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

Operator

0

2

3

4

5

6

log (a/h)

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

(Uniform) Integration (MS): [2x2] for Helmholtz Operator

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”

Scheme

Scheme :

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

RM-straight line up to K-normal’ )

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

of FEM – data

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

– Shear Locking Problem & ROBUSTNESS

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

Deflection at with varying (a/h) ratios

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 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

-

5

6

0.0155

0.0047

C

P

C

A

A

P

free

SS

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

/ Zero Energy Modes

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

FE

Oscillations Stabilization

oscillations

robust

CSR

Selective Reduced Integration : Zero Energy Modes

( Boundary Oscillations = Instability )

Compound Scheme
No Locking and ZEM

10 x

= W

Torsion of Thin Plate

exact

h=0.00001

– Torsion

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

SR + 4 K – RM

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

SR + 4 K – RM

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

?

!

.083 UZ= 675.4
X= .167 UZ= -.00892
X= .250 UZ= 675.4
X= .333 UZ=1350.9
X= .417 UZ= 675.4
X= .500 UZ= .0000
X= .583 UZ= 675.4
X= .667 UZ=1350.9
X= .750 UZ= 675.4
X= .833 UZ= .00218
X= .917 UZ= 675.4
X= 1.000 UZ=1350.9
X=0.5
Y= .000 UZ= .0000
Y= .083 UZ= 675.4
Y= .167 UZ=1350.9
Y= .250 UZ= 675.4
Y= .333 UZ= -.00197
Y= .417 UZ= 675.4
Y= .500 UZ=1350.9
Y= .583 UZ= 675.4
Y= .667 UZ= -.000018
Y= .750 UZ= 675.4
Y= .833 UZ=1350.9
Y= .917 UZ= 675.4
Y= 1.000 UZ= .0000

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

For Optimal Nodes NO Runge Phenomenon.

For Complete Interpolation Bases NO Gibbs Phenomenon.

For Arbitrary Number of Boundary Nodes NO Internal Nodes.

1D, 2D, 3D Interpolations for Uniformly Spaced Nodes

Expansions into the Shape Functions series