Introduction to Finite Element computations

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1 Non Linear Computational Mechanics Athens MP06/2012 Introduction to Finite Element computations Vincent Chiaruttini, Georges Cailletaud

2 Outline Continuous to discrete problem Strong to weak formulation Galerkin method for approximate solution computation Isoparametric finite elements Finite element mesh Geometrical element Interpolation of displacements FE method for linear elastic problems Variational formulation with isoparametric elements Elemental computations Global problem Linear solution process 2

3 Continuous mechanical problem Solving PDE on a given time-space Studied domain Local r¾ + f = 0 1 T "(u) = ru +r u 2 ¾=K :" Boundary conditions Unable to get exact solution How to obtain an approximated solution? u = ud ¾ n = F

4 Continuous mechanical problem Local equations r¾ + f = 0 1 T "= ru + r u 2 ¾=K :" u = ud ¾ n = F Admissible spaces for displacements Uad = fuju continuous and regular on ; u = ud g 0 Uad = fuju continuous and regular on ; u = 0 g Variational formulation of the displacement problem 0 Finding u 2 Uad that verifies, for all u 2 Uad ¾ : " f :u d = F :u ds, a(u; u ) = L(u

5 Continuous mechanical problem Potential energy definition 1 P (v) = "(v) : K : "(v)d f :vd F :vds 1 P (v) = a(v; v) L(v) 2 Minimization problem Lax-Milgram theorm insures 0 u 2 Uad veri es 8u 2 Uad ; a(u; u ) = L(u ), u = argminv2uad P(v) Variational formulation = weak formulation on a stationary functional Galerkin based methods Approximate minimization problem

From Galerkin method to FE method Galerkin method Finding the solution of a variational formulation in an admissible subspace of Using polynomial base functions defined on the domain, for instance

6 From Galerkin method to FE method Galerkin method Finding the solution of a variational formulation in an admissible subspace of Using polynomial base functions defined on the domain, for instance v(x) = ud (x) + N X k Ák (x) k=1 ud (x) 2 Uad Uad 0 Ák (x) 2 Uad The variational formulation produces a NxN linear system 8h 2 f1; : : : ; N g; N X k=1 a(ák ; Áh ) k = L(Ák ) The approximate solution is optimal in the sense that it constitutes the projection of the solution on the given subspace (ie the error is orthogonal to the given subspace) Advantages/drawbacks Mesh and compact support base functions => FE method Easy to define a regular base (polynomials) Optimal approximation Non-sparse operators, not so easy for complex geometries

7 Some popular discrete methods for PDE Finite differences Strong formulation of PDE Differential approximations dti/dx=(ti+1-ti-1)/(2h) h Ti-1 Ti Ti+1 Regular grid Finite volume Integral approximations Flux Very popular for conservation laws Structured or unstructured mesh Finite elements Variational formulation Optimal solution for a given approximation space (Galerkin approach) Free meshes, efficient a priori and a posteriori error estimation

8 Outline Continuous to discrete problem Strong to weak formulation Galerkin method for approximate solution computation Isoparametric finite elements Finite element mesh Geometrical element Interpolation of displacements FE method for linear elastic problems Variational formulation with isoparametric elements Elemental computations Global problem Linear solution process 8

9 Isoparametric finite element Motivation Industrial need for a robust and efficient numerical tool to obtain accurate results in the simulation of mechanical problems with complex behaviour and geometries Meshing process Given a geometrical model (usually from CSG built using a CAD software) 235 parts 45 parts How to obtain a suitable discrete representation for FE computations?

10 Finite element mesh A 2D example triangular elements A continuous domain is approximated Ti using a conform partition based on rectilinear triangular regions => a triangular Mesh h To insure convergence of the solution process the elements size can be reduced If the studied domain boundary is polygonal, it is possible to get = h Interpolation functions Represent a scalar field Th (x) on the domain using nodal values Ti x 2 in element T Th (x) = N1 (x) T1 + N2 (x) T2 + N3 (x) T3 T3 3 T (x) T1 1 x T2 2 Can functions Ti be less dependant of the mesh?

11 Deformed and reference configurations Parametrization of an element positioned anywhere in the studied space (linear triangular example) Physical element»2 X3 x X1 X2 Reference element 3(0; 1)» 1(0; 0)»1 2(1; 0) x = N1 (») X 1 + N2 (») X 2 + N3 (») X 3 Ni (») are base or shape functions defined on the reference domain for the geometrical representation of the element, usually linear: 8 < N1 (»1 ;»2 ) = 1»1»2 N2 (»1 ;»2 ) =»1 : N3 (»1 ;»2 ) =»2

12 Deformed and reference configurations Parametrization of an element positioned anywhere in the studied space (quadratic triangular example) Physical element X5»2 X4 X6 X1 x X2 X3 Reference element 5 6 1» 4 2»1 3 Reference to physical transformation regularity (n nodes element) " n # n x= Nk (») X k J(») = = (»)Xkj J(») = j j k=1 k=1 The transformation between the reference and the physical configurations must be a bijection, the Jacobian must not reach the zero value. Misappropriate physical element geometry can produce such failure Element integration dv (x) = J(»)dV (»), if N is polynomial then J is also

13 Interpolation of displacements Isoparametric element The same base functions are used for geometry and displacement interpolation uh (x) = n X Nk (») U k where U k are the vectorial nodal displacements k=1

14 Operational computation of displacements Local vector of displacement unknowns for element e with n nodes A vector of elementary degrees of freedoms is usually built, for a 2D problem by Ue = (j) h it (1) (1) (2) (2) (j) U1 ; U2 ; U1 ; U2 ; : : : ; Ui ; : : : where Ui is the nodal displacement in direction i at node j Thus for any point x inside the specified element of coordinate system, the displacement is obtained by» in the reference uh (x) = N e (») U e N e (») is a matrix built using the base function, in 2D: N1 (») 0 N2 (») 0 : : : Nj (») N (») = 0 N1 (») 0 N2 (») : : : 0 where 0 Nj (») ::: ::: To insure correct convergence properties (ie representation of constant gradient and solid body displacement field), the following partition of unity property must be n verified anywhere in the element: X Nk (») = 1 k=1

15 Operational computation of displacement gradients Vectorial Voigt writing of second order tensors Symmetric gradients h it p p p "v = "11 "22 "33 2"12 2"13 2"23 h it p p p ¾ v = ¾11 ¾22 ¾33 2¾12 2¾13 2¾23 Linear behaviour relationship as a matrix-vector product ¾ v = [A] "v Gradient computation Physical system of coordinate gradient definition duh (x) = r uh (x) dx " n # To link with the reference coordinate (k) (») Ui duh (x) = H(») dx with H(») = [Hij ] k=0 using dx = J(») d» we get Symmetric gradient computation r uh (x) = H(») J 1 (») 1³ T ²h [u(»)] = H(») J 1 (») + H(») J 1 (») 2 v That can be expressed as ²h (») = B U e using the symmetric gradient operator

16 Outline Continuous to discrete problem Strong to weak formulation Galerkin method for approximate solution computation Isoparametric finite elements Finite element mesh Geometrical element Interpolation of displacements FE method for linear elastic problems Variational formulation with isoparametric elements Elemental computations Global problem Linear solution process 16

17 Finite element method for linear elasticity Global algorithm from the variational formulation in displacement Continuous problem Uad = fuju continuous and regular on ; u = ud 0 8u 2 Uad ; ¾ : " d = f:u d Finite element discrete problem Su Sf 8U j ; h On each element ek F :u uh (x) = N ke (») U e Prescribed displacements on Su nodes Usk k = U (x d s) h k Virtual displacement field uh (x) = N e (») U e ek Verifying null value on Su nodes Usk = 0 "vh [A] : "v d = f :u d + F :u h h h h h ds h, 8W; [W ]T [K][U ] = [W ]T [F ]

18 Computing elementary contributions Domain formulation based on element by element contributions X [W ]T [K][U ] = "vh [A] "v h dv fek g [W ]T [F ] = X fek g ek ek f :u h dv + Usually the two parts are separated in Internal force contributions External force contributions ef F :u h ds [Fint ] + [Fext ] = [0] [Fint ] = [Fext ] = X fek g X fek g ek ek "vh [A] "v h dv f :u h dv + ef F :u h ds

19 Computing elementary contributions Elementary rigidity matrices Using the FE gradient operators and the local vectors of unknowns we need to compute: ek T "vh [A] "v h dv = [We ] [B]T [A][B]dV [Ue ] = [We ][Ke ][Ue ] ek Even for polynomial base/shape function, the inverse of Jacobian used to compute the [B] symmetric gradient operator, make impossible to evaluate such elementary integral exactly for non-trivial meshes. Thus a Gauss integration process is used: ek f (»)dv (») ¼ G X wg f (» g ) g=1 Such process requires the knowledge of some predefined integration points» g (usually called Gauss points) and the associated weight wg for the reference configuration of the element. The parametric representation of both the geometry and the displacement field of the element constitutes one of most important aspect to insure the generic aspect of the FE method: only a few patterns of integration scheme are required corresponding to the associated finite element in reference configuration.

20 Computing elementary contributions Gauss integration 1D example

21 Computing elementary contributions Gauss integration 1D example

22 Computing elementary contributions Gauss integration 1D example

23 Computing elementary contributions Gauss integration 1D example

24 Computing elementary contributions Gauss integration In 3D

25 Computing elementary contributions External forces X [Fext ] = [W ]T [N ][f ]dv + fek g ek [W ][N ]T [F ]ds ef For the volume force a Gauss integration process is required For the nodal force nodal value must be correctly calculated

26 Computing elementary contributions External forces

27 Computing elementary contributions External forces X [Fext ] = fek g ek f :u h dv + ef F :u h ds For the volume force a Gauss integration process is required For the nodal force nodal value must be correctly calculated

28 Computing elementary contributions External forces X [Fext ] = fek g ek f :u h dv + ef F :u h ds For the volume force a Gauss integration process is required For the nodal force nodal value must be correctly calculated

29 Computing elementary contributions External forces X [Fext ] = fek g ek f :u h dv + ef F :u h ds For the volume force a Gauss integration process is required For the nodal force nodal value must be correctly calculated

30 Assembling the global problem Local to global DOF indices (thermal problem for shake of simplicity) 2 E1 1 E2 4 7 E9 Global DOF numbering E3 9 E4 E3 E E5 10 E8 E7 1 E6 Elementary DOF numbering Global matrix 1 Binary trace operators [Ue ] = [ e ][U ] [U ] = [ e ]T [Ue ] 2 3 Assembling global matrix 4 [K] = bandwidth X [ e ]T [Ke ][ e ] fek g Prescribed displacements When assembling is processed, prescribed displacement unknowns can be replaced and eliminated for the linear system

31 Solving efficiently a sparse linear system Large linear system to be solved [K][U ] = [F ] sparse with n equations (n from 104 to 109) K is a symmetric, definite positive for a linear-elastic problem Reduce bandwidth using an accurate numbering process (reduce computational cost) Optimization in memory requirement: sparse storage only keep non zero terms Direct solvers Factorization process [K] = [L][L]T with L a lower-triangular matrix Solving [L][Y ] = [F ] then [L]T [U ] = [Y ] (2 successive triangular systems solving) Interest: matrix storage, less memory consuming than required by inverse computation, triangular system solving is of the same complexity order than matrix vector product Optimized multifrontal, dissection, multithreated, solvers Iterative solvers Krylov type solvers (conjugate gradient, GMRes) need good preconditionners Domain decomposition solvers Split the structure in many subdomains (for supercomputers parallel computation) Efficient iterative strategy to achieve interface equilibrium condition (very good preconditionners)

32 Convergence proprieties Displacement convergence Polynomial approximations are used, on any element Ee, Taylor expansion gives: (x; x0 ) 2 Ee2 u(x) = u(x0 ) + ru(x0 ):(x x0 ) + : : : + O(kx x0 kp+1 ) Committed error on displacement using p order shape function and h characteristic size kx x0 k < h gives ku(x) uh (x)k = O(hp+1 ); x 2 Ee Strain and stress convergence FE error k"[u](x) " [u](x)k = O(hp ); x 2 Ee h k¾[u](x) ¾ h [u](x)k = O(hp ); x 2 Ee Energy norm 2 "[v] : A : "[v]d Defined by kvke = For order p 1 the FE method is convergent in energy: ku uh ke! 0 if h! 0 and if the problem is regular enough ku uh ke C hp kuke

33 Global FE solution process for a linear problem Approximate geometry build a mesh using a priori analysis Loop on elements Compute the local variational formulation contribution Loop on elements Gauss points Compute Jacobian, gradient matrices, get behaviour, multiply matrices to get local rigidity contribution Integrate the rigidity matrix and the local vector for external forces Assemble local rigidity matrix and local external forces Apply Dirichlet BC, MPC (linear relationship between unknowns) Compute global external forces Solve the linear system

34 References Bonnet M., Frangi A. (2006) Analyse des solides déformables par la méthode des éléments finis. Editions Ecole Polytechnique. 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 linéaire des matériaux. Hermes. Ciarlet, P. and Lions, J. (1995). Handbook of Numerical Analysis : Finite Element Methods (P.1), Numerical Methods for Solids (P.2). North Holland. Dhatt, G. and Touzot, G. (1981). Une présentation de la méthode des élements finis. Maloine. Hughes, T. (1987). The finite element method: Linear static and dynamic finite element analysis. Prentice Hall Inc. Simo, J. and Hughes, T. (1997). Computational Inelasticity. Springer Verlag. ienkiewicz, O. and Taylor, R. (2000). The finite element method, Vol. I-III (Vol.1: The Basis, Vol.2: Solid Mechanics, Vol. 3: Fluid dynamics). Butterworth Heinemann.

The Finite Element method for nonlinear structural problems

Non Linear Computational Mechanics Athens MP06/2013 The Finite Element method for nonlinear structural problems Vincent Chiaruttini vincent.chiaruttini@onera.fr Outline Generalities on computational strategies