The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems

The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 1-21 September, 2017 Institute of Structural Engineering Method of Finite Elements II 1

Learning Goals understanding the limits of static linear finite element analysis understanding the difference between linear and nonlinear differential operators introducing the Newton-Raphson method as a fundamental tool for solving nonlinear algebraic equations building a big picture of most common nonlinear problems encountered in engineering Institute of Structural Engineering Method of Finite Elements II 2

Linear Differential Operators Is a mapping acting on elements of a space to produce elements of the same space, e.g.: functions f : R R. Force balance: Differential operator: E d 2 u = L(u) = P dx 2 L( ) = E d 2 ( ) dx 2 The differential operator is linear: The superposition principle holds. L (au 1 + bu 2 ) = al (u 1 ) + bl (u 2 ) Institute of Structural Engineering Method of Finite Elements II 3

From Differential to Algebraic Equations Linear differential equation: L (u) = b, with L (u) = n i=0 a 2i d 2i u dx 2i { d j u dx j 0,L, j {0,..., n 1} essential BCs (primal quantities) d j u dx j 0,L, j {n,..., 2n 1} natural BCs (dual quantities) _ Linear algebraic equation: u K u (n n u) (n n u). d (n 1) u dx (n 1) = F u Institute of Structural Engineering Method of Finite Elements II 4

Nonlinear Differential Operator Is a mapping acting on elements of a space to produce elements of the same space, e.g.: functions f : R R. Force balance (in presence of large strains): Differential operator: L(u) = E d 2 u dx 2 + E du d 2 u dx dx 2 = P L( ) = E d 2 ( ) dx 2 The differential operator L is nonlinear: + E d( ) d 2 ( ) dx dx 2 L(au 1 + bu 2 ) = E d 2 (au 1 + bu 2 ) dx 2 + E d(au 1 + bu 2 ) dx al(u 1 ) + bl(u 2 ) d 2 (au 1 + bu 2 ) dx 2... the superposition principle does no longer apply. Institute of Structural Engineering Method of Finite Elements II 5

Nonlinear Static FEA The principle of virtual displacements facilitates the derivation of discretized equations: Ω R u (u) = F u {δɛ} T {σ}dω = Ω δu T P vol u dω + δu T P sur u dγ Γ σ : stress state generated by volume P vol u loads. and surface P sur u δu and δɛ : compatible variations of displacement u and strain ɛ fields. Institute of Structural Engineering Method of Finite Elements II 6

Nonlinear Static FEA Nonlinear differential equation: L (u) = b { d j u dx j 0,L, j {0,..., n 1} essential BCs (primal quantities) d j u dx j 0,L, j {n,..., 2n 1} natural BCs (dual quantities) _ Nonlinear algebraic equation: u R u. d (n 1) u dx (n 1) = F u Institute of Structural Engineering Method of Finite Elements II 7

Nonlinear Static FEA The solution of the governing BVs turns into a minimization problem that can be solved with standard optimization tools: Key idea: û n = argmin u n (F u (t n ) R u (u n )) R u (u + u) R u (u) + K u u where K u = u R u is the so called tangent stiffness matrix. Institute of Structural Engineering Method of Finite Elements II 8

The Newton-Raphson Algorithm Graphical representation of the Newton-Raphson algorithm for the monodimensional case: Institute of Structural Engineering Method of Finite Elements II 9

The Newton-Raphson Algorithm Graphical representation of the Newton-Raphson algorithm for the monodimensional case:... evaluations of tangent stiffness matrices may be reduced. Institute of Structural Engineering Method of Finite Elements II 9

The Modified Newton-Raphson Algorithm Graphical representation of the Newton-Raphson algorithm for the monodimensional case: Institute of Structural Engineering Method of Finite Elements II 10

Material Nonlinearities (a) elastic (brittle behavior materials); (b) increase of stiffness as load increases (pneumatic structures); (c) commonly observed behavior (concrete or steel) C.A. Felippa: A tour of Nonlinear Analysis. Institute of Structural Engineering Method of Finite Elements II 11

Material Nonlinearities Hysteresis loop Wikipedia ɛ = ɛ e + ɛ p ɛ e : elastic strain, ɛ p : plastic strain. Institute of Structural Engineering Method of Finite Elements II 12

Material Nonlinearities Von Mises s and Tresca s yielding surfaces Wikipedia if f (σ, σ y ) < 0 σ = E ɛ σ = E( ɛ ɛ p ), if f (σ, σ y (α)) = 0 ɛ p = if f (σ, σ y ) = 0 λ f α = λp f = 0 Institute of Structural Engineering Method of Finite Elements II 13

Large Strain Force balance: Constitutive model: dσ dx + P = 0 σ = Eɛ Strain measure: ɛ = du dx linear Institute of Structural Engineering Method of Finite Elements II 14

Large Strain Force balance: Constitutive model: dσ dx + P = 0 σ = Eɛ Strain measure: ɛ = du dx + 1 2 ( ) du 2 Nonlinear! dx Institute of Structural Engineering Method of Finite Elements II 14

Large displacements of a rigid beam Small displacements Large displacements Equilibrium equation: Fl = cϕ Equilibrium equation: Fl cos (ϕ) = cϕ Example taken from: Nonlinear Finite Element Methods by P. Wriggers, Springer, 2008 Institute of Structural Engineering Method of Finite Elements II 15

Large displacements of a rigid beam 10 8 6 Small displacements Large displacements Fl c 4 2 0 0 20 40 60 80 ϕ Force versus rotation Institute of Structural Engineering Method of Finite Elements II 16

Examples Example #1: Forming of a metal profile Example #2: Large deformation of a race-car tyre Example #3: Buckling of a truss structure Institute of Structural Engineering Method of Finite Elements II 17

Nonlinear Dynamic FEA Nonlinear differential equation: L (u, u, ü) = b { d j u dx j 0,L, j {0,..., n 1} essential BCs d j u dx j 0,L, j {n,..., 2n 1} natural BCs d j u dx j t=0, j {0,..., n 1} Initial Conditions (ICs) on essential BCs _ Nonlinear algebraic equation: ü u u M u (n n u) (n n u). d (n 1) ü dx (n 1) + R u. d (n 1) u dx (n 1),. d (n 1) u dx (n 1) = F u Institute of Structural Engineering Method of Finite Elements II 18

Nonlinear Dynamic FEA Also in this case, the solution of the governing BVs turns into a minimization problem that can be solved with standard optimization tools: {û n, û n, û n } = argmin u n, u n,ü n (F u (t n ) R u (u n, u n ) M u ü n ) In detail: A minimization problem must be solved at each load step. The previous time step solution is taken as starting point. Apparently the number of unknowns increased... Institute of Structural Engineering Method of Finite Elements II 19

Nonlinear Dynamic FEA: Examples Example #1: Shake table test of a concrete frame Example #2: Crash test of a car Institute of Structural Engineering Method of Finite Elements II 20

Thermal Stress Analysis Rail buckling driven by restrained thermal expansion Wikipedia Institute of Structural Engineering Method of Finite Elements II 21

Linear Thermoelasticity ɛ = ɛ e + ɛ T ɛ e : elastic strain, ɛ T : thermal strain. ɛ xx = 1 E (σ xx ν (σ yy + σ zz )) + α (T T 0 ) ɛ yy = 1 E (σ yy ν (σ zz + σ xx )) + α (T T 0 ) ɛ zz = 1 E (σ zz ν (σ xx + σ yy )) + α (T T 0 ) ɛ xy = σ xy 2G ɛ xz = σ xz 2G ɛ yz = σ yz 2G α : coef. of thermal expansion; T : abs. temperature; T 0 : ref. temperature. Institute of Structural Engineering Method of Finite Elements II 22

Thermal Stresses Analysis: Examples Example #1: Buckling of a railway Example #2: Spring forging Institute of Structural Engineering Method of Finite Elements II 23

Coupled Problems a) Impact on the containment vessel of a nuclear reactor casing; b) normal load applied at the point of impact; c) geometry of the structure. Combescure, A., Gravouil, A. (2002). A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis. Computer Methods in Applied Mechanics and Engineering, 191(1112), 11291157. Institute of Structural Engineering Method of Finite Elements II 24

Coupled Problems d) mesh of the structure; e) mesh of the interfaces. Combescure, A., Gravouil, A. (2002). A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis. Computer Methods in Applied Mechanics and Engineering, 191(1112), 11291157. Institute of Structural Engineering Method of Finite Elements II 24

Coupled Problems Task sequence of the partitioned algorithm. { M A u ü A n+1 + ( ) RA u u A n+1, u A n+1 = F A ( ) u (t n+1 ( ) + L ) AT Λ n+1 M B u ü B + R B n+ j u u B, u B = F B ss n+ j ss n+ j u t n+ j + L BT Λ ss ss n+ j ss L A u A + L B u B = 0 n+ j ss n+ j ss Combescure, A., Gravouil, A. (2002). A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis. Computer Methods in Applied Mechanics and Engineering, 191(1112), 11291157. Institute of Structural Engineering Method of Finite Elements II 24

Coupled Problems: Examples Example #1: Bullet impact Example #1: Fluid-structure interaction Institute of Structural Engineering Method of Finite Elements II 25