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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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
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ime
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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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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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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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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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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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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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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
Lecture 13: Finite element method VII
Anton Tkachuk
CTU, Prague
06.06.2018
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