Common pitfalls while using FEM Chair for Computational Engineering Faculty of Civil Engineering, Cracow University of Technology e-mail: [email protected]With thanks to: R. de Borst (Delft University of Technology) R.L. Taylor (University of California at Berkeley) M. Radwa´ nska, Z. Waszczyszyn, A. Winnicki, A. Wosatko (Cracow Univ. of Technol.) SOKI, BIM, 2020 Contents Power of FE technology What is locking? In-plane shear locking Volumetric locking What is localization? Sources: Books of Hughes, Cook, Zienkiewicz & Taylor, Belytschko et al Figures taken from: R.D. Cook, Finite Element Method for Stress Analysis, J. Wiley & Sons 1995. C.A. Felippa, Introduction to Finite Element Methods, University of Colorado, 2001. http://caswww.colorado.edu/Felippa.d/FelippaHome.d/Home.html R. Lackner, H.A. Mang. Adaptive FEM for the analysis of concrete structures. Proc. of EURO-C 1998 Conference, Balkema, Rotterdam, 1998. SOKI, BIM, 2020
Transcript
Common pitfalls while using FEM
Chair for Computational Engineering
Faculty of Civil Engineering, Cracow University of Technology
With thanks to:R. de Borst (Delft University of Technology)R.L. Taylor (University of California at Berkeley)M. Radwanska, Z. Waszczyszyn, A. Winnicki, A. Wosatko (Cracow Univ. of Technol.)
SOKI, BIM, 2020
Contents
Power of FE technology
What is locking?
In-plane shear locking
Volumetric locking
What is localization?
Sources:Books of Hughes, Cook, Zienkiewicz & Taylor, Belytschko et al
Figures taken from:R.D. Cook, Finite Element Method for Stress Analysis,J. Wiley & Sons 1995.C.A. Felippa, Introduction to Finite Element Methods,University of Colorado, 2001.http://caswww.colorado.edu/Felippa.d/FelippaHome.d/Home.html
R. Lackner, H.A. Mang. Adaptive FEM for the analysis of concrete structures.
Proc. of EURO-C 1998 Conference, Balkema, Rotterdam, 1998.
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Modelling process
From: T. Kolendowicz Mechanika budowli dla architektw
Set of assumptions: model of structure, material and loadingPhysical model: representation of essential featuresMathematical model: set of equations (algebraic, differential, integral) +limiting (boundary, initial) conditionsProblems can be stationary (static) or nonstationary (dynamic)Mathematical models can be linear or nonlinear
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Understanding a structure
tension
compression
Stress flow in panels
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Numerical model
Discretization (e.g. FEM)
Simplest case: set of linear equations
Ku = f
K - stiffness matrix
u - vector of degrees of freedom
f - loading vector
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Discontinuity of derivatives
Contour plots of σxx
Without smoothing With smoothing
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Smoothing of selected component
σh – function obtained from FE solutionσ∗ – function after smoothing
Difference between these two fields is a discretization error indicator ofZienkiewicz and Zhu
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Where FE mesh should be finer (Felippa)
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Variants of mesh refinement (Cook)
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Adaptive mesh refinement
Example from Altair Engineering http://www.comco.com
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Mesh generation
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Discretization error monitored
Adaptive mesh refinement
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Advanced problems solved using FEMMechanics:
I Extreme load cases, e.g. impactI Physical nonlinearities, e.g. damage, cracking, plasticityI Geometrical nonlinearities, i.e. large displacements and/or strains,
e.g. spongeI Contact problems (unilateral constraints)
Brazilian test, plane strain, one quarter, elasticity
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Brazilian test, elasticity
Deformation, vertical stress σyy and stress invariant Jσ2
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Brazilian split test
Elasticity, mesh sensitivity of stresses
Stress σyy for coarse and fine meshes
Stress under the force goes to infinity (results depend on mesh density) -solution at odds with physics
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Brazilian split test
Ideal Huber-Mises-Hencky plasticity
Final deformation and stress σyy
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Brazilian split test
Ideal Huber-Mises-Hencky plasticity
Final strain εyy and strain invariant Jε2
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Brazilian split test
Ideal Huber-Mises-Hencky plasticity
0 0.2 0.4 0.6 0.8 1
Displacement
0
200
400
600
800
Forc
e
0 0.2 0.4 0.6 0.8 1
Displacement
0
200
400
600
800
Forc
e
This is correct!
For four-noded element load-displacement diagram exhibits artificialhardening due to so-called volumetric locking, since HMH flow theorycontains kinematic constraint - isochoric plastic behaviour which cannotbe reproduced by FEM model.
Eight-noded element does not involve locking.
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Limitations of finite elements
I Various kinds of locking(overstiff response)
I Zero-energy deformation modes
I Kinematic constraints (e.g.incompressibility)
I Ill-posed problems (e.g. due to softening)
Locking is a result of two many constraints in comparison with thenumber of degrees of freedom.
I Special arrangement of elements (e.g. crossed-diagonal)
I Selective integration or B approach of Hughes
I Mixed formulations (e.g. pressure discretization)
I Enhanced Assumed Strain (EAS) apprach of Simo
Sometimes locking does not prevent convergence, but affects accuracyfor coarse meshes.
Be careful with CST, Q4, T4 i H8
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In-plane shear locking (Cook)
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In-plane shear locking
Only at the element centre γxy = 0
Incompatible quadrilateral Q6
u =∑4
i=1 Niui + (1− ξ2)g1 + (1− η2)g2
v =∑4
i=1 Nivi + (1− ξ2)g3 + (1− η2)g4
γxy =∑4
i=1∂Ni
∂y ui +∑4
i=1∂Ni
∂x vi−
− 2yb2 g2 − 2x
a2 g3
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In-plane shear locking
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Incompressibility locking
For plane strain or 3D when ν → 0.5Pressure related to volumetric strain grows to infinity (isochoricdeformation is impossible).
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Deviatoric-volumetric split
G =E
2(1 + ν), K =
E
3(1− 2ν)
(GKdev + KKvol)u = f
When ν → 0.5, KKvol acts as a penalty constraint and locks thesolution, unless Kvol is singular.
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Mixed formulation
Linear elasticity
σij = 2Gui,j + λuk,kδij , λ =2νG
1− 2ν
Incompressibilityuk,k = 0
Modification of theory
σij = 2Gui,j − pδij , p =1
3σii − extra unknown
Incompressibility or compressibility
uk,k +p
λ= 0
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Mixed formulation
Strong formLTσ + b = 0
∇Tu +p
λ= 0
Weak form∫V
(Lδu)TσdV =
∫V
(δu)TbdV +
∫S
(δu)TtdS ∀δu
∫V
δp(∇Tu +
p
λ
)dV = 0 ∀δp
Discretization of displacements and pressure
u = Nu u, p = Np p
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Mixed formulationTwo-field elements [
K GGT M
] [up
]=
[ffp
]If M = 0 (incompressibility) then eliminate u:(1) → u → (2) discrete Poisson equation → pIf M 6= 0 (compressibility) then eliminate p:(2) → p → (1) standard → u
Constraint ratior =
nequncon
Optimal r = 2, e.g. Q4p1 - constant pressure element (B, SI)
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Localization of deformation
Active process takes place in a narrow band
From: D.A. Hordijk Local approach to fatigue of concrete, Delft University of Technology, 1991
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Definition of localization
I Strain localization is a constitutive effect.
I It is a precursor to failure in majority of materials.
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Forms of localization
From: M.S.A. Siddiquee,FEM simulations of deformation and failureof stiff geomaterials based on elementtest results, University of Tokyo, 1994
From: P.B. Lourenco,Computational strategies for masonrystructures,Delft University of Technology, 1996
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Cause of localization
From: D.A. Hordijk Local approach to fatigue of concrete,
Delft University of Technology, 1991
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Pathological mesh sensitivity of numerical solution
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Enhanced continuum description - no pathology
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Continuum vs discontinuumDisplacement and strain distribution in one dimension
