Common pitfalls while using FEM J. Pamin Instytut Technologii Informatycznych w Inżynierii Lądowej Wydział Inżynierii Lądowej, Politechnika Krakowska e-mail: JPamin@L5.pk.edu.pl With thanks to: R. de Borst (Delft University of Technology) R.L. Taylor (University of California at Berkeley) M. Radwańska, Z. Waszczyszyn, A. Winnicki, A. Wosatko (Cracow Univ. of Technol.)
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 architektów 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
Understanding a structure tension compression Stress flow in panels
Numerical model Discretization (e.g. FEM) Simplest case: set of linear equations Kǔ = f K - stiffness matrix ǔ - vector of degrees of freedom f - loading vector
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
Where FE mesh should be finer (Felippa)
Variants of mesh refinement (Cook)
Adaptive mesh refinement Example from Altair Engineering http://www.comco.com
Mesh generation
Discretization error monitored Adaptive mesh refinement
Advanced problems solved using FEM Mechanics: Extreme load cases, e.g. impact Physical nonlinearities, e.g. damage, cracking, plasticity Geometrical nonlinearities, i.e. large displacements and/or strains, e.g. sponge Contact problems (unilateral constraints) Multiphysics: ANSYS simulations 1 2 3 4 ADINA simulations 1 2 3 4
Let s solve a simple problem Brazilian test, plane strain, one quarter, elasticity
Brazilian test, elasticity Deformation, vertical stress σ yy and stress invariant J σ 2
Brazilian test, elasticity Stress σ yy for coarse and fine mesh
Brazilian test, perfect Huber-Mises plasticity Final deformation and stress σ yy
Brazilian test, perfect Huber-Mises plasticity Load-displacement diagram with hardening???
Brazilian test, perfect Huber-Mises plasticity Eight-noded versus four-noded element
Limitations of finite elements Various kinds of locking (overstiff response) Zero-energy deformation modes Kinematic constraints (e.g. incompressibility) 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 (NDOF=4 2=8, NCON=4 3=12) Locking (overstiff response) Q4-RI (NDOF=4 2=8, NCON=1 3=3) Singularity (hourglass modes)
Remedies to locking Higher-order interpolation Special arrangement of elements (e.g. crossed-diagonal) Selective integration or B approach of Hughes Mixed formulations (e.g. pressure discretization) 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
In-plane shear locking (Cook)
In-plane shear locking Only at the element centre γ xy = 0 Incompatible quadrilateral Q6 u = 4 i=1 N iu i + (1 ξ 2 )g 1 + (1 η 2 )g 2 v = 4 i=1 N iv i + (1 ξ 2 )g 3 + (1 η 2 )g 4 γ xy = 4 N i i=1 y u i + 4 N i i=1 x v i 2y b 2 g 2 2x a 2 g 3
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).
Deviatoric-volumetric split G = E 2(1 + ν), K = E 3(1 2ν) (GK dev + KK vol )ǔ = f When ν 0.5, KK 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,k δ ij, λ = 2νG 1 2ν Incompressibility Modification of theory σ ij = 2Gu i,j pδ ij, u k,k = 0 p = 1 3 σ ii extra unknown Incompressibility or compressibility u k,k + p λ = 0
Mixed formulation Strong form L T σ + b = 0 T u + p λ = 0 Weak form V (Lδu) T σdv = V V (δu) T bdv + (δu) T tds S ( δp T u + p ) dv = 0 λ Discretization of displacements and pressure δp δu u = N u ǔ, p = N p ˇp
Mixed formulation Two-field elements [ K G G T M ] [ ǔ ˇp ] [ f = f p ] If M = 0 (incompressibility) then eliminate ǔ: (1) ǔ (2) discrete Poisson equation ˇp If M 0 (compressibility) then eliminate ˇp: (2) ˇp (1) standard ǔ Constraint ratio r = n equ n con Optimal r = 2, e.g. Q4p1 - constant pressure element ( B, SI)
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 Strain localization is a constitutive effect. It is a precursor to failure in majority of materials.
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
Pathological mesh sensitivity of numerical solution
Enhanced continuum description - no pathology
Continuum vs discontinuum Displacement and strain distribution in one dimension displacement displacement displacement strain strain strain Strong discontinuity Weak discontinuity Regularized discontinuity