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

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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

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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

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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.

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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

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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

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

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

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

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8SIMO & RIFAI (1990). A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES. IJNME

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

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)

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91SIMO & RIFAI (1990). A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES. IJNME

Transformation of natural to global EAS strains

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

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

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101SIMO & RIFAI (1990). A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES. IJNME

Stress recovery

Note about PLANE182 family

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

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

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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

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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

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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

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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)

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Reduced integration and hourglass stabilization

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

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Reduced integration and hourglass stabilization

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

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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

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18

Practical importance of locking-free finite elements

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

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Practical importance of locking-free finite elements

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

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Practical importance of locking-free finite elements

symmetry

pilot node prescribes

displacement

on the upper edge

Model definition with ANSYS

maximum value

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Practical importance of locking-free finite elements

equivalent plastic strains for different formulations!

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

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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

tkachuk@ibb.uni-stuttgart.de

Lecture 13: Finite element method VII

Anton Tkachuk

CTU, Prague

06.06.2018