Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem Introduction to Finite Element Method Georges CAILLETAUD & Saber EL AREM Centre des Mat´ eriaux, MINES ParisTech, UMR CNRS 7633 WEMESURF course, Paris 21-25 juin 1/79
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Introduction to Finite Element Method
Georges CAILLETAUD & Saber EL AREM
Centre des Materiaux, MINES ParisTech, UMR CNRS 7633
WEMESURF course, Paris 21-25 juin
1/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Contents
1 Introduction
2 Examples
3 Bibliography on finite element
4 Discrete versus continuous
5 ElementInterpolationElement list
6 Global problemFormulationMatrix formulationAlgorithm
2/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Contents
1 Introduction
2 Examples
3 Bibliography on finite element
4 Discrete versus continuous
5 ElementInterpolationElement list
6 Global problemFormulationMatrix formulationAlgorithm
Introduction 3/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Numerical methods for PDE solving
Many physical phenomena in engineering andscience can be described in terms of partialdifferential equations (PDE) .In general, solving these equations by classicalanalytical methods for arbitrary shapes is almostimpossible.The finite element method (FEM) is a numericalapproach by which these PDE can be solvedapproximately.The FEM is a function/basis-based approach tosolve PDE.FE are widely used in diverse fields to solve staticand dynamic problems − Solid or fluid mechanics,electromagnetics, biomechanics, etc.
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Google search results for FEM
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Google search results for FE+course
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
FE problem solving steps
Two key words: Discretization & Interpolation1 Definition of the physical problem: development of
the model.2 Formulation of the governing equations.
Systems of PDE, ODE, algebraic equations,define initial conditions and/or boundary conditions to get awell-posed problem.
3 Discretization of the equations.4 Solution of the discrete system of equations.5 Interpretation of the obtained results.6 Errors analysis.
Introduction 7/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Contents
1 Introduction
2 Examples
3 Bibliography on finite element
4 Discrete versus continuous
5 ElementInterpolationElement list
6 Global problemFormulationMatrix formulationAlgorithm
Examples 8/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
AERONAUTICS
Crash-test, Boeing 737 par la NASA* Static, dynamic, and coupled acoustic-structural analysis of aircraft frames *Simulations of large deployable space structures such as solar sails, space radars andreflector antennas * Simulating the performance of various aircraft components, such asbulkhead under pressurization, wing panel buckling, and crack propagation in the fuselage *Blade containment evaluations and bird strike simulations * Thermomechanical simulationof aircraft engines and rocket motors under different operating conditions * Verification ofturbine blade designs *Simulation of various aircraft mechanisms such as landing gears, wingflaps, and cargo doors
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Reference element
x
y
ξ
η
Actual geometryPhysical space (x , y)
η
ξ
1
−1
−1 1
Reference elementParent space (ξ, η)
∫Ω
f (x , y)dxdy =
∫ +1
−1
∫ +1
−1
f∗(ξ, η)Jdξdη
J is the determinant of the partial derivatives ∂x/∂ξ. . . matrixElement 25/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Remarks on geometrical mapping
The values on an edge depends only on the nodal values on the sameedge (linear interpolation equal to zero on each side for 2-node lines,parabolic interpolation equal to zero for 3 points for 3-node lines)
Continuity...
The mid node is used to allow non linear geometries
Limits in the admissible mapping for avoiding singularities
Element 26/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Jacobian and inverse jacobian matrix
(dxdy
)=
∂x
∂ξ
∂x
∂η∂y
∂ξ
∂y
∂η
(dξdη
)= [J]
(dξdη
)
(dξdη
)=
∂ξ
∂x
∂ξ
∂y∂η
∂x
∂η
∂y
(dxdy
)= [J]−1
(dxdy
)
Since (x , y) are known from Ni (ξ, η) and xi ,
[J]−1 is computed from the known quantities in [J], using also:
J = Det ([J]) =∂x
∂ξ
∂y
∂η− ∂y
∂ξ
∂x
∂η
Element 28/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Expression of the inverse jacobian matrix
[J]−1 =1
J
∂y
∂η−∂y
∂ξ
−∂x
∂η
∂x
∂ξ
For a rectangle [±a,±b] in the ”real world”, the mapping function isthe same for any point inside the rectangle. The jacobian is adiagonal matrix, with ∂x/∂ξ = a, ∂y/∂η = b, and the determinantvalue is abFor any other shape, the ”mapping” changes according to thelocation in the elementFor computing [B], one has to consider ∂Ni/∂x and ∂Ni/∂y :
∂Ni
∂x=
∂Ni
∂ξ
∂ξ
∂x+
∂Ni
∂η
∂η
∂x
∂Ni
∂y=
∂Ni
∂ξ
∂ξ
∂y+
∂Ni
∂η
∂η
∂y
then(
∂Ni/∂x∂Ni/∂y
)= [J]−1
(∂Ni/∂ξ∂Ni/∂η
)
Element 29/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Contents
1 Introduction
2 Examples
3 Bibliography on finite element
4 Discrete versus continuous
5 ElementInterpolationElement list
6 Global problemFormulationMatrix formulationAlgorithm
Element 30/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
2D solid elements
Type shape interpol # of polynomof disp nodes terms
C2D3 tri lin 3 1, ξ, ηC2D4 quad lin 4 1, ξ, η, ξηC2D6 tri quad 6 1, ξ, η, ξ2, ξη, η2
2 Compute ∆ε = [B].∆uiter+1 then ∆ε∼ for each Gauss point
3 Integrate the constitutive equation: ∆ε∼→ ∆σ∼, ∆αI ,∆σ∼∆ε∼
4 Compute int and ext forces: Fint(ut + ∆uiter+1) , Fe5 Compute the residual force: Riter+1 = Fint − Fe6 New displacement increment: δuiter+1 = −[K ]−1.Riter+1
Global problem 78/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Convergence
Value of the residual forces < Rε, e.g.
||R||n =
(∑i
Rni
)1/n
; ||R||∞ = maxi|Ri |
Relative values:
||Ri − Re ||||Re ||
< ε
Displacements ∣∣∣∣Uk+1 − Uk
∣∣∣∣n
< Uε
Energy [Uk+1 − Uk
]T. Rk < Wε
Global problem 79/79
Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Bathe, K. (1982).Finite element procedures in engineering analysis.Prentice Hall, Inc.
Batoz, J. and Dhatt, G. (1991).Modelisation des structures par elements finis, I—III.Hermes.
Belytschko, T., Liu, W., and Moran, B. (2000).Nonlinear Finite Elements for Continua and Structures.
Besson, J., Cailletaud, G., Chaboche, J.-L., and Forest, S. (2001).Mecanique non–lineaire des materiaux.Hermes.
Buchanan, G. (1995).Finite element analysis.Schaum’s outlines.
Ciarlet, P. and Lions, J. (1995).
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Handbook of Numerical Analysis : Finite Element Methods (P.1),Numerical Methods for Solids (P.2).North Holland.
Crisfield, M. (1991).Nonlinear Finite Element Analysis of Solids and Structures.Wiley.
Dhatt, G. and Touzot, G. (1981).Une presentation de la methode des elements finis.Maloine.
Hughes, T. (1987).The finite element method: Linear static and dynamic finite elementanalysis.Prentice–Hall Inc.
Kardestuncer, H., editor (1987).Finite Element Handbook.Mc Graw Hill.
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Introduction Examples Bibliography on finite element Discrete versus continuous Element Global problem
Mc Neal, R. (1993).Finite Element: their design and performance.Marcel Dekker.
Simo, J. and Hughes, T. (1997).Computational Inelasticity.Springer Verlag.
Zienkiewicz, O. and Taylor, R. (2000).The finite element method, Vol. I-III (Vol.1: The Basis, Vol.2: SolidMechanics, Vol. 3: Fluid dynamics).Butterworth–Heinemann.