The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems. Prof. Dr. Eleni Chatzi, J.P. Escallo n Lecture December, 2013

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1 The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi, J.P. Escallo n Lecture December, 2013 Institute of Structural Engineering Method of Finite Elements II 1

2 NL FE Special Considerations - The Contact Problem What is Contact? Physically, contact stress is transmitted between two bodies when they touch. Numerically, contact is a severely discontinuous for of non-linearity. Difficulties Complex non linear behaviour = contact between two or more bodies. Relative sliding of the surfaces has to be evaluated iteratively. Deformable-to-deformable body contact generates non-linear time-dependent BC. Institute of Structural Engineering Method of Finite Elements II 2

3 Contact Discretization Node-to-Surface (Implicit only) Surface-to-Surface (Implicit only) Node-to-Face (Explicit only) Edge-to-Edge Institute of Structural Engineering Method of Finite Elements II 3

4 Node-to-Surface (strict master/slave formulation) Institute of Structural Engineering Method of Finite Elements II 4

5 Node-to-Surface Contact is enforced between a slave node and master surface facets local to the node: The opening/penetration distance is measured along the normal to the master surface A nodal area is assigned to each slave node to convert contact forces to contact stresses The more refined surface should act as the slave surface The stiffer body should be the master Institute of Structural Engineering Method of Finite Elements II 5

6 Surface-to-Surface Institute of Structural Engineering Method of Finite Elements II 6

7 Surface-to-Surface Contact is enforced between the slave node and a larger number of master surface facets around it: The opening/penetration distance is measured along the slave surface facet normal Sliding is measured perpendicular to the slave normal Institute of Structural Engineering Method of Finite Elements II 7

8 Contact Discretization comparison: Abaqus/Explicit Undetected penetrations of master nodes into the slave surface do not occur with surface-to-surface discretization: Source: Abaqus Analysis User s Manual Institute of Structural Engineering Method of Finite Elements II 8

9 Contact discretizations in Abaqus/Explicit Node-to-Face: No distinction is made between master/slave surfaces as in Node-to-Surface (i.e., contact is enforced everywhere). Edge-Edge: It is very effective in enforcing contact that cannot be detected as penetrations of nodes into faces. Source: Abaqus Analysis User s Manual Institute of Structural Engineering Method of Finite Elements II 9

10 Surface Description Discrete Surface: Discontinuities in the surface normal direction at surface facet boundaries can contribute to convergence difficulties. Smooth Surface: Surface smoothing is used to reduce the discretization error asociated with faceted representations of curved surfaces. Institute of Structural Engineering Method of Finite Elements II 10

11 Hard Contact Enforcement Methods Direct Enforcement Method: Strict enforcement of pressure-penetration relationship using Lagrange multiplier method. Penalty method: approximate enforcement using penalty stiffness. Institute of Structural Engineering Method of Finite Elements II 11

12 Direct Enforcement Variational formulation for a steady-state analysis without contact: Contact Constraint: Π = 1 2 U T KU U T F (1) U i = U i (2) Variational formulation for a steady-state analysis with contact enforcement using Lagrange multiplier method: Π = 1 2 U T KU U T F + λ(u i U i ) (3) Institute of Structural Engineering Method of Finite Elements II 12

13 Direct Enforcement Equilibrium Condition δπ = 0: δπ = δu T KU δu T F + λδu i + δλ(u i Ui ) = 0 (4) The above relationship can be written as: KU + λe i = F (5) e T i U = Ui (6) In matrix form: [ ] [ K ei U e T i 0 λ ] [ F = U i ] (7) λ is the vector of Lagrange multiplier degrees of freedom (constraint forces) *One per constraint. Institute of Structural Engineering Method of Finite Elements II 13

14 Direct Enforcement The contact virtual work contribution is: δπ c = λδu i + δλ(u i Ui ) (8) This expression is written in Abaqus Theory Manual as: δπ c = δph + pδh (9) where p is the Lagrangian multiplier, and h is the overclosure. Hard Contact p=0 for h < 0 contact is open h=0 for p = 0 contact is closed Institute of Structural Engineering Method of Finite Elements II 14

15 Direct Enforcement ADVANTAGES: Accuracy: The constraints are satisfied exactly DISADVANTAGES: Adds cost to the equation solver Potential convergence problems Institute of Structural Engineering Method of Finite Elements II 15

16 Penalty Method The right hand side of the potential function Π is amended in the following manner: Π = 1 2 U T KU U T F + α 2 (U i U i ) 2 (10) I use a large α in order to make U i = U i, i.e., α >> max(k ii) Equilibrium condition δπ = 0: δπ = δu T KU δu T F + α(u i U i )δu i = 0 (11) (K + αe i e T i )U = F + αu i e i (12) Institute of Structural Engineering Method of Finite Elements II 16

17 Penalty Method The Penalty method corresponds to having a spring to bring back the penetrating node to the surface. Institute of Structural Engineering Method of Finite Elements II 17

18 Penalty Method ADVANTAGES: Convergence rates significantly improve Better equation solver performance DISADVANTAGES: Small amount of penetration (typically insignificant) In some cases, the penalty stiffness needs to be adjusted Institute of Structural Engineering Method of Finite Elements II 18

19 Strain-free adjustments of initial overclosures Within the penetration tolerance, all initial overclosures are treated with strain-free adjustments. Initial overclosures can be due to pre-processing errors or discretization of curved surfaces. Source: Abaqus Analysis User s Manual Institute of Structural Engineering Method of Finite Elements II 19

20 Friction Models Coulomb friction model τ cr = µp (13) τ cr = min(µp, τ max ) (14) Source: Abaqus Analysis User s Manual Institute of Structural Engineering Method of Finite Elements II 20

21 Friction Models Additional features: Friction coefficient dependence on slip rate Friction coefficient dependence on contact pressure Anisotropic friction Source: Abaqus Analysis User s Manual Institute of Structural Engineering Method of Finite Elements II 21

22 Friction Enforcement Lagrangian multiplier method Penalty method Institute of Structural Engineering Method of Finite Elements II 22

23 Exact stick formulation Lagrange multipliers are used to enforce exact sticking conditions. The virtual work due to friction is evaluated as: δπ = (τ i δγ i + γ i δq i )ds (15) S q i are the Lagrangian multipliers used to enforce exact stick ( γ i = 0). If τ eq > τ crit the element passes from sticking to slipping. If γ i τ i (t) < 0 the element passes from slipping to sticking Institute of Structural Engineering Method of Finite Elements II 23

24 Penalty method It approximates stick with stiff elastic behaviour. In Abaqus/Explicit by default the same penalty stiffness used in hard contact is used for frictional constraints. On the contrary in Abaqus/Standard (Implicit) it depends on the elastic slip. The elastic slip γ crit is calculated as: γ crit = F f l i, where F f is the slip tolerance, and l i is the characteristic contact surface length. Institute of Structural Engineering Method of Finite Elements II 24

25 Elastic behaviour (τ eq < τ crit ): γ el i (t + t) = γ el i (t) + γ i (16) τ i = Gγ el i = τ crit γ crit γ el i = µp γ crit γ el i (17) dτ i = Gdγ i + τ i (µp + u p)dp (18) τ crit p The contributions from the contact pressure p are non-symmetric! Plastic behaviour (τ eq > τ crit ): γ i = γi el (t + t) γi el (t) + γi sl = γi el γ el i + γi sl (19) Institute of Structural Engineering Method of Finite Elements II 25

26 The shear stress at the end of the increment is evaluated with the elastic relationship: τ i (t + t) = τ i = Gγ el i = τ crit γ crit γ el i (20) Slip increment: γ sl i = τ i τ crit γ sl eq (21) Replacing γ el i and γ sl i in eq. 19 yields: γ i = τ i τ crit γ crit γ el i + τ i τ crit γ sl eq (22) τ i = γel i + γ i γ crit + γeq sl τ crit (23) Institute of Structural Engineering Method of Finite Elements II 26

27 The critical stress equality yields: τ crit = G(γ pr eq γ sl eq) = τ crit γ crit (γ pr eq γ sl eq) (24) where γ pr eq is the equivalent elastic predictor strain Institute of Structural Engineering Method of Finite Elements II 27

28 τ crit = G(γ pr eq γ sl eq) = τ crit γ crit (γ pr eq γ sl eq) (25) Replacing eq. 26 into eq. 23 yields: γ sl eq = γ pr eq γ crit (26) γ pr i τ i = γ crit + γeq sl τ crit = γpr i γeq pr τ crit = n i τ crit (27) where n i is the normalized slip direction. The iterative solution scheme for τ crit as a function of the slip rate ( γ eq sl = γeq/ t) sl yields: τ i = (δ ij n i n j ) τ crit γeq pr dγ j +n i (µ+p µ p )dp+n p in j t µ dγ j (28) γ eq The unsymmetric terms may have a strong effect on the speed of convergence of the Newton scheme! Institute of Structural Engineering Method of Finite Elements II 28

29 Contact Algorithm (Implicit calculations) Newton-Raphson iterative scheme (K T ) i i+1 n+1 Un+1 = (F int) i n+1 + (F ext) n+1 + (R c (u i+1 n+1 ))i+1 n+1 Un+1 i+1 = U n+1 i i+1 + Un+1 where K T is the tangent stiffness matrix, i refers to the ongoing iteration with the Newton-Raphson process and n refers to the loading increment. The contact forces vector R c depends on U which influences both the contact surface shape and the magnitude of the contact reaction. Institute of Structural Engineering Method of Finite Elements II 29

30 Inverting the tangent stiffness matrix yields: U i+1 n+1 = ((K T ) i n+1 ) 1 )((F int ) i n+1 + (F ext) n+1 )... + ((K T ) i n+1 ) 1 )(R c (u i+1 n+1 ))i+1 n+1 which may be written in a simpler way: U i+1 n+1 = ( U lib) i+1 n+1 + ( U c) i+1 n+1 with: ( U lib ) i+1 n+1 = ((K T ) i n+1 ) 1 )((F int ) i n+1 + (F ext) n+1 ) ( U c ) i+1 n+1 = ((K T ) i n+1 ) 1 )(R c (U i+1 n+1 ))i+1 n+1 The displacement is split into two parts, one independent from the contact problem (prediction), and a term depending exclusively on contact (correction). Institute of Structural Engineering Method of Finite Elements II 30

31 Source: A. Batailly et. al., 2013 Institute of Structural Engineering Method of Finite Elements II 31

32 Institute of Structural Engineering Method of Finite Elements II 32

33 Contact in Explicit Codes Calculation is advanced explicitly element-by-element (No need to assembly a global stiffness matrix and no Newton iterations are performed). No convergence problems related to faceted representation of curves (smoothing is not relevant). Contact forces do not depend on the displacements (no iterative process is carried out). Easier to deal with sliding friction because calculations are advanced explicitly element-by-element. Time step is very small and therefore is suitable to analyse short contact dynamic problems where friction plays an important role, i.e., impact. Institute of Structural Engineering Method of Finite Elements II 33