lecture 13: finite element method vii

Department of Civil and

Environmental Engineering

Lecture 13: Finite element method VII

Anton Tkachuk

CTU, Prague

06.06.2018

What is Locking?

Enhanced assumed strain formulation in 2D (Simo & Rifai 1990)

B-bar formulation (Hughes 1980)

Reduced integration and hourglass stabilization (Belytschko & Bachrach 1986)

Practical importance of locking-free finite elements

Baustatik und Baudynamik

Universität Stuttgart

2

What is Locking?

1HERRMANN & TOMS (1964). A REFORMULATION OF THE ELASTIC FIELD EQUATION, IN TERMS OF DISPLACEMENTS, VALID FOR ALL

ADMISSIBLE VALUES OF POISSON’S RATIO. JOURNAL OF APPLIED MECHANICS, 31(1), 140-141.2HERRMANN, TAYLOR & GREEN (1967). FINITE ELEMENT ANALYSIS FOR SOLID ROCKET MOTOR CASES.3CLOUGH & WILSON (1999). EARLY FINITE ELEMENT RESEARCH AT BERKELEY. PRESENTED AT 5TH US NATIONAL CONF. COMP. MECH.

First evidences: 1960’s - solid elements for nearly incompressible materials

[en:U

ser:

Pbro

ks13,

Wik

ime

dia

Com

mons, C

C B

Y-S

A 3

.0]

Solid-fuel rocket

• propellant materials had nearly incompressible properties (v ≈ 0.5)

• standard displacement FEM gives poor result for v > 0.4

• standard FDM also locked

• naive mixed methods (displacement/pressure) worked satisfactory

• could be extended to elastic and thermo-viscoelastic materials

Relative error for finite difference analysis

of pressurized cylinder: standard (dashed)

and modified (solid) formulation1

Finite element model of segment of

solid-fuel rocket using 4-node

element with constant pressure2,3

locking is triggered by underlying BVP



3

What is Locking?

First evidences: 1970’s – shear deformable beam/plate/shell at thin limit

1ZIENKIEWICZ, TAYLOR & TOO (1971). REDUCED INTEGRATION TECHNIQUE IN GENERAL ANALYSIS OF PLATES AND SHELLS. INTERNATIONAL

JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Displacement at the center W vs. slenderness t/a for a

simply supported square plate under uniform load q0.1

a – edge length, t – thickness, D – plate rigidity

• shear deformable plate and shell finite

elements perform badly for high ratios t/a

• selective reduced integration of shear

terms in the stiffness improves behavior

• influence of ‘parasitic’ shear is identified

‘Parasitic’ shear stresses induced in a linear

element under bending mode. (a) Constraint

mode, (b) true mode.1



4

What is Locking?

Definition

Locking means the effect of a reduced rate of convergence for coarse meshes in

dependence of a critical parameter

Symptoms of locking

• solution is too stiff (displacements are to small)

• ‘parasitic’ strain mode

• oscillation of stresses/member forces

• dependency on a critical parameter

Importance

BISCHOFF, RAMM, TKACHUK & KOOHI (2015). COMPUTATIONAL MECHANICS OF STRUCTURES, LECTURE NOTES, WINTER TERM 2015/16

• good accuracy with coarse meshes

• improved behavior for large aspect ratio

• elimination of spurious stress/member force oscillations

• improved modeling capabilities for nearly incompressible materials (rubber), plastic flow, etc.



5

What is Locking?

Examples of locking types

Locking types

volumetric in-plane shear transverse shear membrane trapezoidal

element

types

axisymmetric, 2D

plane strain and

3D solids

solid elements shear deformable

beams and shells

curved

beam/shell

solid and solid-

shell element

critical

parameter

Poison ratio

v→0.5

aspect ratio of

element t/le→0slenderness t/a→0 thickness-to-

curvature

t/R→0

thickness-to-

curvature

t/R→0

benchmark thick sphere

under pressure,

Cook membrane

bending of

cantilever

beam/plate

bending of

cantilever

beam/plate

Scordelis-Lo

roof, pinched

ring

Scordelis-Lo

roof, pinched

ring

un-locking

methods

IM, EAS, RI,

B-bar

IM, EAS, RI IM, EAS, RI, ANS IM, EAS, RI,

DSG

DSG

material geometric

IM – incompatible mode method, covered in Lecture notes for L12?

EAS – enhanced assumed strain method,

RI – reduced integration,

B-bar – B-bar

ANS – assumed natural strain,

DSG – discrete strain gap method, out of scope of this course, see ref.1

1BLETZINGER, BISCHOFF & RAMM (2000). A UNIFIED APPROACH FOR SHEAR-LOCKING-FREE TRIANGULAR AND RECTANGULAR SHELL

FINITE ELEMENTS. COMPUTERS & STRUCTURES

covered in Lecture 13



6

Enhanced assumed strain (EAS) formulation in 2D (Simo & Rifai 1990)

'. . . two wrongs do make a right in California' G. STRANG (1973)

'. . . two rights make a right even in California' R. L. TAYLOR (1989)

Motivation

• compatible

• variationally based locking-free element

• no zero energy modes

• less sensitive to distortions than IM

SIMO & RIFAI (1990). A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES. IJNME

Analysis of ‘parasitic’ strains for square finite element

Displacement field Strain field

Material law

Stress field



7SIMO & RIFAI (1990). A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES. IJNME


Deformation modes of a bilinear Q1, choice of EAS trial functions

Re-parametrized strain field

with standard strain-displacement matrix and trial function matrix of the enhanced strains

(explained later)transformation





Variational basis is the Hu-Washizu principle with introduced re-parametrized strain field

Elimination of the stresses by orthogonality condition

Introducing the discretization and results in

with the abbreviations

First variation leads to

(covered in Lecture 4)




Transformation of natural to global EAS strains


With the fundamental lemma of variational calculus, we obtain the linear system of equations

By static condensation we obtain

additional strains are referred to the -coordinate system (covariant transformation)

transformation of the trial function matrix

where contains the trial functions in local coordinates

with




Stress recovery

Note about PLANE182 family


recovery of the assumed strains and compatible strains

EAS strains

stress recovery

Enriched choice of more enhanced strain terms

for improved behavior for distorted elements



11

Enhanced assumed strain (EAS) formulation in 3D

ANDELFINGER & RAMM (1993). EAS‐ELEMENTS FOR TWO‐DIMENSIONAL, THREE‐DIMENSIONAL, PLATE AND SHELL STRUCTURES AND THEIR

EQUIVALENCE TO HR‐ELEMENTS. IJNME

Enriched choice of more enhanced strain terms

Basic enhanced strain matrix



12

B-bar formulation

HUGHES (1980). GENERALIZATION OF SELECTIVE INTEGRATION PROCEDURES TO ANISOTROPIC AND NONLINEAR MEDIA. IJNME

Note default formulation for PLANE182

Starting point: standard stiffness matrix

and modified volumetric part for nearly-incompressible applications

with typical strain-displacement sub-matrix for node I

Volumetric-deviatoric split

with standard deviatoric part

with



13

B-bar formulation

Analysis of ‘parasitic’ strains for square finite element

modified volumetric part

standard strain-displacement matrix

volumetric-deviatoric split

standard deviatoric part

application of in-plane bending deformation



14

Reduced integration and hourglass stabilization

Reduced integration

Rang of the numerically integrated stiffness matrix

Local eigenvalues for square element: reduced integration and full integration

whereas correct rank is

spurious zero energy modes (ZEM)



15


dimensions:

• length = 25 mm

• breadth = 5 mm

• height = 5 mm

material:

• linear elastic steel

boundary conditions:

• A = fixed support

• B = prescribed boundary motion

(6 mm vertical displacement)

cantilever beam

uniform reduced integration 1 GP

uniform reduced integration 1GP with hourglass

stabilization

AB

IN COOPERATION WITH ZAHID MIR



16


BELYTSCHKO & BACHRACH (1986). EFFICIENT IMPLEMENTATION OF QUADRILATERALS WITH HIGH COARSE-MESH ACCURACY. CMAME

The stiffness matrix consists of a one-point quadrature matrix and stabilization matrix

The stabilization matrix is constructed by

with the strain-displacement

matrix for node I

with with hourglass mode

with as free parameters

with

and are nodal coordinate vectors

with strain-displacement vectors



17

Comparison cost of different formulations of Q1 family

Assumptions

• no optimization of operation and no use of matrix symmteries is assumed.

• matrix summation A and B (nxm) costs n*m [flops]

• matrix multiplication A (nxm) and B (mxk) is n*m*k [flops] (naïve algorithm)

• operation count for chain matrix multiplication

A.B.C = (A.B).C = (nxm).(mxk).(kxl) = n*m*k + n*l*k = (l+m)*n*k

• operation count for matrix inversion A (nxn) is assumed 4*n^3 [ops]

• calculation of Jacobian - 4 items each cost 16 opearation = 64 [ops]

• cost of material procedure (if necessary) is neglacted

• multiplications with transformation matrix T0 at EAS is included

Element type

locking cost

volumetric shear absolute relative to Q1 2x2

Q1 2x2 (full) integration 5 (ok) yes yes 3040 1.00

Q1RI 1x1 3 (2 ZEM) no no 476 0.16

Q1RI 1x1 + stab* 5 (ok) no no 3536 1.16

Q1E4 5 (ok) no no 5156 1.70

Q1E5 5 (ok) no no 5656 1.86

Q1E7 5 (ok) no no 7907 2.60

*USED IN NON-LINEAR EXPLICIT TIME INTEGRATION WHERE THE STIFFNESS MATRIX NOT USED



18


1

X

Y

Z

JUN 9 2016

10:41:26

DISPLACEMENT

STEP=1

SUB =6

FREQ=9.10488

DMX =2.19216

1

X

Y

Z

JUN 9 2016

10:30:51

DISPLACEMENT

STEP=1

SUB =1

FREQ=1.64421

DMX =1.76658

1

X

Y

Z

JUN 9 2016

10:32:53

DISPLACEMENT

STEP=1

SUB =2

FREQ=3.2095

DMX =2.46029

1

X

Y

Z

JUN 9 2016

10:35:20

DISPLACEMENT

STEP=1

SUB =3

FREQ=5.658

DMX =2.04025

1

X

Y

Z

JUN 9 2016

10:37:52

DISPLACEMENT

STEP=1

SUB =4

FREQ=6.11137

DMX =2.25608

1

X

Y

Z

JUN 9 2016

10:39:28

DISPLACEMENT

STEP=1

SUB =5

FREQ=8.77062

DMX =2.13391

REFERENCE PLANE 42, full int. PLANE 182, Bbar PLANE 182, RI PLANE 182, EAS

mode no. frequency frequency err. in % frequency err. in % frequency err. in % frequency err. in %

1 1.6442 2.5929 57.6998 1.7763 8.0343 1.5955 2.9619 1.7760 8.0161

2 3.2095 4.3352 35.0740 3.4551 7.6523 3.1539 1.7324 3.4380 7.1195

3 5.6580 7.5946 34.2276 6.1776 9.1835 5.6011 1.0057 6.1568 8.8158

4 6.1114 7.6241 24.7521 6.3439 3.8044 6.0568 0.8934 6.3571 4.0204

5 8.7706 11.3730 29.6719 9.2298 5.2357 7.9390 9.4817 9.3127 6.1809

6 9.1049 12.5480 37.8159 10.2120 12.1594 8.6722 4.7524 10.2220 12.2692

Eigenvalue analysis of an annular plate

- comparison of 18x3 element mesh of

PLANE 42, full integration with locking!

PLANE182, full integration + B-bar formulation

PLANE182, uniform reduced int. + hourglass stab.

PLANE182, EAS formulation

- reference solution: 180x30 Q2 elements, mixed u/p formulation



19


When does volumetric locking come into play?

Plastic flow for J2 plasticity (von Mises) is isochoric

Elasto-plastic problem for a plate with a hole*

plane strain!

*PROBLEM DATA TAKEN FROM P.96 „FINITE ELEMENT METHOD FOR SOLID AND STRUCTURAL MECHANICS“ ZIENKIEWICZ AND TAYLOR



20


symmetry

pilot node prescribes

displacement

on the upper edge

Model definition with ANSYS

maximum value



21


equivalent plastic strains for different formulations!

different picture for CST! Q1 does not completely catch it!



22

Important elements that are out of scope of this lecture

• methods against membrane, trapezoidal

• U-p formulation1

-H (Abaqus), SOLID18X family (ANSYS)

• averaged nodal pressure tetrahedron2

C3D4H (Abaqus), SOLID285 (ANSYS), elform=13 (LS-Dyna)

• nodally integrated tetrahedrons3

• linked interpolation against locking (2-node Timoshenko beam4, plate elements5,6)

• isogeometric (NURBS-based) beams7,8, shells9,10 and solids11

REFERENCES ARE NOT COMPLETE, COLLECTED TO MY TASTE

1BOFFI, D., BREZZI, F., & FORTIN, M. (2013). MIXED FINITE ELEMENT METHODS AND APPLICATIONS (VOL. 44). BERLIN: SPRINGER.2BONET, J., & BURTON, A. J. (1998). A SIMPLE AVERAGE NODAL PRESSURE TETRAHEDRAL ELEMENT FOR INCOMPRESSIBLE AND NEARLY

INCOMPRESSIBLE DYNAMIC EXPLICIT APPLICATIONS. COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING.3PUSO, M. A., & SOLBERG, J. (2006). A STABILIZED NODALLY INTEGRATED TETRAHEDRAL. IJNME.4SECTION 12.7, ZIENKIEWICZ & TAYLOR (2005) THE FINITE ELEMENT METHOD5GREIMANN, L. F., & LYNN, P. P. (1970). FINITE ELEMENT ANALYSIS OF PLATE BENDING WITH TRANSVERSE SHEAR DEFORMATION. NUCLEAR

ENGINEERING AND DESIGN, 14(2), 223-230.6AURICCHIO, F., & TAYLOR, R. L. (1994). A SHEAR DEFORMABLE PLATE ELEMENT WITH AN EXACT THIN LIMIT. CMAME.7ECHTER, R., & BISCHOFF, M. (2010). NUMERICAL EFFICIENCY, LOCKING AND UNLOCKING OF NURBS FINITE ELEMENTS. CMAME8BOUCLIER, R., ELGUEDJ, T., & COMBESCURE, A. (2012). LOCKING FREE ISOGEOMETRIC FORMULATIONS OF CURVED THICK BEAMS. CMAME9ECHTER, R., OESTERLE, B., & BISCHOFF, M. (2013). A HIERARCHIC FAMILY OF ISOGEOMETRIC SHELL FINITE ELEMENTS. CMAME.10CASEIRO, J. F., VALENTE, R. A. F., REALI, A., KIENDL, J., AURICCHIO, F., & DE SOUSA, R. A. (2014). ON THE ASSUMED NATURAL STRAIN

METHOD TO ALLEVIATE LOCKING IN SOLID-SHELL NURBS-BASED FINITE ELEMENTS. COMPUTATIONAL MECHANICS, 53(6), 1341-1353.11TAYLOR, R. L. (2011). ISOGEOMETRIC ANALYSIS OF NEARLY INCOMPRESSIBLE SOLIDS. IJNME

Department of Civil and

Environmental Engineering

https://www.ibb.uni-stuttgart.de

[email protected]

Lecture 13: Finite element method VII

Anton Tkachuk

CTU, Prague

06.06.2018

lecture 13: finite element method vii

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