Common pitfalls while using FEM
Transcript of Common pitfalls while using FEM
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. Radwanska, Z. Waszczyszyn, A. Winnicki, A. Wosatko (Cracow Univ. of Technol.)
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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)
Multiphysics:I ANSYS simulations 1 2 3 4I ADINA simulations 1 2 3 4
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Let’s solve a simple problem
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.
Q4-FI Q4-RI(NDOF=4×2=8, NCON=4×3=12) (NDOF=4×2=8, NCON=1×3=3)
Locking (overstiff response) Singularity (hourglass modes)
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Remedies to locking
I Higher-order interpolation
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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