Common pitfalls while using FEM

Common pitfalls while using FEM

Chair for Computational Engineering

Faculty of Civil Engineering, Cracow University of Technology e-mail: JPamin@CCE.pk.edu.pl

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.)

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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.

Modelling process

From: T. Kolendowicz Mechanika budowli dla architektw

Set of assumptions: model of structure, material and loading Physical model: representation of essential features

Mathematical model: set of equations (algebraic, differential, integral) + limiting (boundary, initial) conditions

Problems 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

Numerical model

Discretization (e.g. FEM)

Simplest case: set of linear equations Kˇu = f K - stiffness matrix

ˇu - vector of degrees of freedom f - loading vector

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Discontinuity of derivatives

Contour plots of σ_xx

Without smoothing With smoothing

Smoothing of selected component

σ^h – function obtained from FE solution σ^∗ – function after smoothing

Difference between these two fields is a discretization error indicator of Zienkiewicz and Zhu

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Where FE mesh should be finer (Felippa)

Variants of mesh refinement (Cook)

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Adaptive mesh refinement

Example from Altair Engineering http://www.comco.com

Mesh generation

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Discretization error monitored

Adaptive mesh refinement

Advanced problems solved using FEM

Mechanics:

I Extreme load cases, e.g. impact

I Physical nonlinearities, e.g. damage, cracking, plasticity

I Geometrical nonlinearities, i.e. large displacements and/or strains, e.g. sponge

I Contact problems (unilateral constraints)

Multiphysics:

I ANSYS simulations 1 2 3 4

I ADINA simulations 1 2 3 4

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Let’s solve a simple problem

Brazilian test, plane strain, one quarter, elasticity

Brazilian test, elasticity

Deformation, vertical stress σ_yy and stress invariant J₂^σ

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Brazilian split test

Elasticity, mesh sensitivity of stresses

Stress σ_yy for coarse and fine meshes

Stress under the force goes to infinity (results depend on mesh density) - solution at odds with physics

Brazilian split test

Ideal Huber-Mises-Hencky plasticity

Final deformation and stress σ_yy

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Brazilian split test

Ideal Huber-Mises-Hencky plasticity

Final strain _yy and strain invariant J₂^

Brazilian split test

Ideal Huber-Mises-Hencky plasticity

0 0.2 0.4 0.6 0.8 1

Displacement 0

200 400 600 800

Force

0 0.2 0.4 0.6 0.8 1

Displacement 0

200 400 600 800

Force

This is correct!

For four-noded element load-displacement diagram exhibits artificial hardening due to so-called volumetric locking, since HMH flow theory contains kinematic constraint - isochoric plastic behaviour which cannot be 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 the number 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)

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 accuracy for coarse meshes.

Be careful with CST, Q4, T4 i H8

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In-plane shear locking (Cook)

In-plane shear locking

Only at the element centre γ_xy = 0 Incompatible quadrilateral Q6 u =P4

i =1N_iu_i + (1 − ξ²)g₁+ (1 − η²)g₂ v =P4

i =1N_iv_i + (1 − ξ²)g₃ + (1 − η²)g₄ γ_xy =P4

i =1

∂N_i

∂y u_i +P4 i =1

∂N_i

∂x v_i−

−^2y_b2g₂ − ^2x_a2g₃

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In-plane shear locking

Incompressibility locking

For plane strain or 3D when ν → 0.5

Pressure related to volumetric strain grows to infinity (isochoric deformation is impossible).

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Deviatoric-volumetric split

G = E

2(1 + ν), K = E 3(1 − 2ν) (G K_dev + K K_vol)ˇu = f

When ν → 0.5, K K_vol acts as a penalty constraint and locks the solution, unless K_vol is singular.

Mixed formulation

Linear elasticity

σ_ij = 2Gu_{i ,j} + λu_k,kδ_ij, λ = 2νG 1 − 2ν Incompressibility

u_k,k = 0 Modification of theory

σ_ij = 2Gu_{i ,j} − pδ_ij, p = 1

3σ_ii − extra unknown Incompressibility or compressibility

u_k,k + p λ = 0

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Mixed formulation

Strong form

L^Tσ + b = 0

∇^Tu + p λ = 0 Weak form

Z

V

(Lδu)^TσdV = Z

V

(δu)^TbdV + Z

S

(δu)^TtdS ∀δu Z

V

δp

∇^Tu + p λ



dV = 0 ∀δp Discretization of displacements and pressure

u = N_u ˇu, p = N_p ˇp

Mixed formulation

Two-field elements

 K G

G^T M

  ˇu ˇp



=

 f f_p



If M = 0 (incompressibility) then eliminate ˇu:

(1) → ˇu → (2) discrete Poisson equation → ˇp If M 6= 0 (compressibility) then eliminate ˇp:

(2) → ˇp → (1) standard → ˇu Constraint ratio

r = n_equ n_con

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

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 failure of stiff geomaterials based on element test results, University of Tokyo, 1994

From: P.B. Lourenco,

Computational strategies for masonry structures,

Delft University of Technology, 1996

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

Enhanced continuum description - no pathology

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Continuum vs discontinuum

Displacement and strain distribution in one dimension

displacement

strain

∞

displacement

strain

displacement

strain

Strong discontinuity Weak discontinuity Regularized discontinuity