CHAP 5 Finite Element Analysis of Contact Problem Nam-Ho Kim 1 Introduction • Contact is boundary nonlinearity – The graph of contact force versus displacement becomes vertical – Both displacement and contact force are unknown in the interface • Objective of contact analysis 1. Whether two or more bodies are in contact 2. Where the location or region of contact is 3. How much contact force or pressure occurs in the interface 4. If there is a relative motion after contact in the interface • Finite element analysis procedure for contact problem 1. Find whether a material point in the boundary of a body is in contact with the other body 2. If it is in contact, the corresponding contact force must be calculated 2 Introduction • Equilibrium of elastic system: – finding a displacement field that minimizes the potential energy • Contact condition (constrained minimization) – the potential is minimized while satisfying the contact constraint • Convert to unconstrained optimization – Can be solved using either the penalty method or Lagrange multiplier method • Slave-master concept for contact implementation – the nodes on the slave boundary cannot penetrate the surface elements on the master boundary 3 Goals • Learn the computational difficulty in boundary nonlinearity • Understand the concept of variational inequality and its relation with the constrained optimization • Learn how to impose contact constraint and friction constraints using penalty method • Understand difference between Lagrange multiplier method and penalty method • Learn how to integrate contact constraint with the structural variational equation • Learn how to implement the contact constraints in finite element analysis • Understand collocational integration 4 5.2 1D CONTACT EXAMPLES 5 Contact Problem – Boundary Nonlinearity • Contact problem is categorized as boundary nonlinearity • Body 1 cannot penetrate Body 2 (impenetrability) Body 2 Body 1 Contact stress (compressive) • Why nonlinear? Contact boundary – Both contact boundary and contact stress are unknown!!! Contact force – Abrupt change in contact force (difficult in NR iteration) Penetration 6 Why are contact problems difficult? • Unknown boundary – Contact boundary is unknown a priori – It is a part of solution – Candidate boundary is often given Body 1 Body 2 • Abrupt change in force – Extremely discontinuous force profile – When contact occurs, contact force cannot be determined from displacement – Similar to incompressibility (Lagrange multiplier or penalty method) • Discrete boundary Contact force Penetration – In continuum, contact boundary varies smoothly – In numerical model, contact boundary varies node to node – Very sensitive to boundary discretization 7 Contact of a Cantilever Beam with a Rigid Block • q = 1 kN/m, L = 1 m, EI = 105 N∙m2, initial gap = 1 mm • Trial-and-error solution – First assume that the deflection is smaller than the gap qx2 vN (x) (x2 6L2 4Lx), 24EI qL4 vN (L) 0.00125m 8EI – Since vN(L) > , the assumption is wrong, the beam will be in contact 8 Cantilever Beam Contact with a Rigid Block cont. • Trial-and-error solution cont. – Now contact occurs. Contact in one-point (tip). – Contact force, l, to prevent penetration lx2 l vc (x) (3L x), vc (L) 6EI 3 105 – Determine the contact force from tip displacement = gap vtip vN (L) vc (L) 0.00125 l 5 3 10 0.001 l 75N v(x) vN (x) vc (x) 9 Cantilever Beam Contact with a Rigid Block cont. • Solution using contact constraint – Treat both contact force and gap as unknown and add constraint 2 qx2 l x v(x) (x2 6L2 4Lx) (3L x) 24EI 6EI – When l = 0, no contact. Contact occurs when l > 0. l < 0 impossible – Gap condition: g vtip 0 10 Cantilever Beam Contact with a Rigid Block cont. • Solution using contact constraint cont. – Contact condition No penetration: g≤0 Positive contact force: l ≥ 0 Consistency condition: lg = 0 – Lagrange multiplier method l lg l 0.00025 0 5 3 10 – When l = 0N g = 0.00025 > 0 violate contact condition – When l = 75N g = 0 satisfy contact condition Lagrange multiplier, l, is the contact force 11 Cantilever Beam Contact with a Rigid Block cont. • Penalty method – Small penetration is allowed, and contact force is proportional to it – Penetration function 1 N g g 2 N = 0 when g ≤ 0 N = g when g > 0 – Contact force l KN N – From KN: penalty parameter g vtip 0 KN 1 g 0.00025 g g 5 2 3 10 – Gap depends on penalty parameter 12 Cantilever Beam Contact with a Rigid Block cont. • Penalty method cont. – Large penalty allows small penetration Penalty parameter Penetration (m) Contact force (N) 3×105 1.25×10−4 37.50 3×106 2.27×10−5 68.18 3×107 2.48×10−6 74.26 3×108 2.50×10−7 74.92 3×109 2.50×10−8 75.00 13 Beam Contact with Friction • Sequence: q is applied first, followed by the axial load P • Assume no friction PL no-friction utip 1.0mm EA • Frictional constraint Stick condition: t l 0, utip 0 Slip condition: t l 0, utip 0 Consistency condition: u tip (t P=100N, =0.5 t: tangential friction force l) 0 14 Beam Contact with Friction cont. 1. Trial-and-error solution – First assume stick condition utip PL tL 0, EA EA Violate t t P 100N l 0 – Next, try slip condition t l 37.5N utip PL tL 0.625mm EA EA Satisfies utip > 0, therefore, valid 15 Beam Contact with Friction cont. 2. Solution using frictional constraint (Lagrange multiplier) – Use consistency condition utip (t l) 0 – Choose utip as a Lagrange multiplier and t-l as a constraint EA utip P utip l 0 L – When utip = 0, t = P, and t – l = 62.5 > 0, violate the stick condition – When utip (P l)L 0.625mm 0 EA t = l and the slip condition is satisfied, valid solution 16 Beam Contact with Friction cont. 3. Penalty method – Penalize when t – l > 0 1 T t l t l 2 – Slip displacement and frictional force utip KT T KT: penalty parameter for tangential slip – When t – l < 0 (stick), utip = 0 (no penalization) – When t – l > 0 (slip), penalize to stay close t = l – Friction force EA t P utip L 17 Beam Contact with Friction cont. • Penalty method cont. – Tip displacement – For large KT, utip utip KT L(P l) L KT EA (P l)L EA t P EA utip l L Penalty parameter Tip displacement (m) Frictional force (N) 1×10−4 5.68×10−4 43.18 1×10−3 6.19×10−4 38.12 1×10−2 6.24×10−4 37.56 1×10−1 6.25×10−4 37.50 1×100 6.25×10−4 37.50 18 Observations • Due to unknown contact boundary, contact point should be found using either direct search (trial-and-error) or nonlinear constraint equation – Both methods requires iterative process to find contact boundary and contact force – Contact function replace the abrupt change in contact condition with a smooth but highly nonlinear function • Friction force calculation depends on the sequence of load application (Path-dependent) • Friction function regularizes the discontinuous friction behavior to a smooth one 19 5.3 GENERAL FORMULATION OF CONTACT PROBLEMS 20 General Contact Formulation • Contact between a flexible body and a rigid body • Point x W contacts to xc rigid surface (param x) xc xc (x) • How to find xc(x) xc,x t et t xc,x (xc ) ( x xc (xc ))T et (xc ) 0 Unit tangent vector – Closest projection onto the rigid surface W Gc x en xc0 x gn e t xc gt Rigid Body 21 Contact Formulation cont. • Gap function gn ( x xc (xc ))T en (xc ) • Impenetrability condition gn 0, x Gc • Tangential slip function gt t 0 (xc xc0 ) boundary that has a possibility of contact Parameter at the previous contact point W Gc x en xc0 x gn e t xc Rigid Body gt 22 Ex) Project to a Parabola • Projection of x={3, 1} to y = • Let xc = {x, et xc,x xc,x x2 4 x2}T y = x2 3 1 2 2x 1 4x 1 2 xc en gn x 1 2x en et k 0 0 2 1 1 4x • Projection point 3 3 x 2 x (x) ( x xc (x))T et (x) 0 1 4x2 1 • Gap gn ( x xc )T en x2c 6xc 1 1 4x2c 1 2 3 4 xc = 1.29 xc = {1.29, 1.66} 1.83 23 Variational Inequality • Governing equation fb 0 u 0 n f s xW x Gg x Gs • Contact conditions (small deformation) uT en gn 0 n 0 x Gc n ( uT en gn ) 0 • Contact set (convex) K w [H1 (W)]N w G g 0 and wT en gn 0 on Gc satisfies all kinematic constraints (displacement conditions) 24 Variational Inequality cont. • Variational equation with ū = w – u (i.e., w is arbitrary) W ijij (w u) dW W ij,j (wi ui ) dW G G s c ijnj (wi ui ) dG ij,j fi b ( x W), W ijij (w u) dW G c G G G ijnj (wi ui ) dG ( w u) c c c G c ijnj fi s ( x Gs ) ijnj (wi ui ) dG n (wn un ) dG n (wn gn un gn ) dG Since it is arbitrary, we don’t know the value, but it is non-negative n (wn gn ) dG 0 W ijij (w u) dW ( w u), w K Variational inequality for contact problem 25 Illustration of Projection a( u, w u) ( w u), w K • If the solution u’ is out of set it is projected to u on , • Beam deflection example (rigid block with initial gap of 1mm) • v’(x): beam deflection without rigid block v – Contact condition is violated (penetration to the block) • v(x): projection of v’(x) onto convex set – by applying the contact force x 10-3 0 v(x) v'(x) 0.4 0.8 1.2 0 Rigid block 0.2 0.4 0.6 0.8 1 26 Variational Inequality cont. • For large deformation problem K w [H1 (W)]N w G g 0 and ( x xc (xc ))T en 0 on Gc . • Variational inequality is not easy to solve directly • We will show that V.I. is equivalent to constrained optimization of total potential energy • The constraint will be imposed using either penalty method or Lagrange multiplier method 27 Potential Energy and Directional Derivative • Potential energy ( u) 21 a( u, u) ( u) • Directional derivative d d ( u v ) 0 dd 21 a( u v, u v ) ( u v ) 21 a( u, v ) 21 a( v, u) ( u) 0 a( u, v ) ( u) • Directional derivative of potential energy D( u), v a( u, v ) ( v ) • For variational inequality a( u, w u) ( w u) D( u), w u 0, w K 28 Equivalence • V.I. is equivalent to constrained optimization – For arbitrary w ( w) ( u) 21 a( w, w) ( w) 21 a( u, u) ( u) a( u, w u) a( u, w u) a( u, w u) ( w u) 21 a( w, w) a( u, w) 21 a( u, u) D( u), w u 21 a( w u, w u) ( w) ( u) D( u), w u non-negative w K • Thus, (u) is the smallest potential energy in ( u) min ( w) min 21 a( w, w) ( w) wK wK – Unique solution if and only if (w) is a convex function and set closed convex is 29 Constrained Optimization • PMPE minimizes the potential energy in the kinematically admissible space • Contact problem minimizes the same potential energy in the contact constraint set K w [H1 (W)]N w G g 0 and ( x xc )T en 0 on Gc gn 0 on Gc • The constrained optimization problem can be converted into unconstrained optimization problem using the penalty method or Lagrange multiplier method – If gn < 0, penalize (u) using 1 1 2 P n gn dG t gt2 dG GC GC 2 2 penalty parameter 30 Constrained Optimization cont. • Penalized unconstrained optimization problem ( u) min ( w) min ( w) P( w) wK w unconstrained constrained Solution space w/o contact Contact force u uNoContact 31 Ex) Beam Deflection with Rigid Block • q = 1 kN/m, L = 1 m, EI = 105 N∙m2, initial gap = 1 mm • Assumed deflection: v(x) a2x2 a3x3 a4 x 4 • Penalty function: P 21 n gn2 gn vtip a2 a3 a4 • Penalized potential energy L 1 L 1 2 a P EI(v,xx ) dx qv dx n gn2 0 2 0 2 32 Ex) Beam Deflection with Rigid Block • Stationary condition a 0 ai 4EI n 6EI n 6EI n 12EI n 8EI 18EI n n 1 8EI n a2 3 q n 1 18EI n a3 4 q n 144 a 1 q EI n 4 n 5 5 • Penetration: a2 + a3 + a4 − • Contact force: −ngn Penalty parameter a1 a2 a3 Penetration (m) Contact force (N) 3×105 2.31×10−3 -1.60×10−3 4.17×10−4 1.25×10−4 37.50 3×106 2.16×10−3 -1.55×10−3 4.17×10−4 2.27×10−5 68.18 3×107 2.13×10−3 -1.54×10−3 4.17×10−4 2.48×10−6 74.26 3×108 2.13×10−3 -1.54×10−3 4.17×10−4 2.50×10−7 74.92 3×109 2.13×10−3 -1.54×10−3 4.17×10−4 2.50×10−8 75.00 True value 2.13×10−3 -1.54×10−3 4.17×10−4 0.0 75.00 33 Variational Equation • Structural Equilibrium a( u, u ) ( u ) P( u; u ) 0 ( u; u ) P( u; u ) 0 P( u; u ) n gn gn dG t gt gt dG b( u, u ) GC bN ( u, u ) GC bT ( u, u ), need to express in terms of u and ū • Variational equation a( u, u ) b( u, u ) ( u ), u 34 Frictionless Contact Formulation • Variation of the normal gap gn ( x xc (x))T en gn u T en • Normal contact form bN ( u, u ) gn u T en dG Gc 35 Linearization • It is clear that bN is independent of energy form – The same bN can be used for elastic or plastic problem • It is nonlinear with respect to u (need linearization) • Increment of gap function Dgn D ( x xc (x))T en DuT en ( x xc (x))T Den Dgn ( u; Du) DuT en Den et x xc en – We assume that the contact boundary is straight line Den = 0 – This is true for linear finite elements Rigid surface 36 Linearization cont. • Linearization of contact form bN ( u, u ) Gc g T u en dG bN* (u; Du, u ) u T en enT Du dG Gc • N-R iteration a* (u; Du, u ) bN* ( u; Du, u ) ( u ) a( u, u ) bN ( u, u ), u 37 Ex) Frictionless Contact of a Block • Calculate displacement, penetration and contact force at y q the contact interface. • EA = 105N, n = 0, q = 1.0kN/m, plane strain • Plane strain: u {ux , uy }T u {ux , uy }T • Contact boundary: en {0, 1}T x x, xc { xc , 0}T , Elastic body 0 x { xc ,uy }T Rigid body Contact boundary 1 x • Gap function: gn ( x xc )T en uy • Contact form: b ( u, u ) 1 u u N n 0 y y • Penalized potential energy 1 1 T P D dA ( q)uy 0 2 A y 0 dx 1 2 1 dx n gn y 1 0 2 y 1 dx 38 Ex) Frictionless Contact of a Block • From n = 0, D becomes a diagonal matrix, decoupled x & y • Since load is only y-direction, xx = gxy = 0 • Linear displacement in y-direction uy a0 a1 y uy a0 a1 y • Penalized potential P 1 1 A Ea1a1 dA 0 (q)(a0 a1 ) dx n 0 a0a0 dx 0 – Need to satisfy for arbitrary a0 , a1 q q a0 , a1 n EA q q uy y, 0 y 1 n EA n gn nuy y 0 q As n increases, penetration decreases but contact force remains constant 39 Frictional Contact Formulation • frictional contact depends on load history • Frictional interface law – regularization of Coulomb law • Friction form bT ( u, u ) t gt gt dG enT xc,xx GC gt t 0 xc nu T et bT ( u, u ) t ngt u T et dG Gc etT xc,xx g enT xc,xxx c t 2 gn n t t0 c -ngn t gt 40 Friction Force • During stick condition, fT = tgt (t: regularization param.) t gt n gn • When slip occurs, t gt n gn • Modified friction form t ngt u T et dG, if t gt n gn Gc bT ( u, u ) T sgn(g ) n g u et dG, otherwise . n t G n c 41 Linearization of Stick Form • Increment of slip Dgt ( u; Du) t 0 Dxc n etT Du • Increment of tangential vector Det e3 Den en ( etT Du) c • Incremental slip form for the stick condition bT* ( u; Du, u ) t n2 u T et etT Du dG Gc ngt T t u ( en etT et enT )Du dG Gc c ngt 2 T T t ( g t 2 )g t u e e n t t Du d G Gc c2 42 Linearization of Slip Form • Parameter for slip condition: t n sgn(gt ). • Friction form: bT ( u,u ) t n gn u T et dG Gc • Linearized slip form (not symmetric!!) bT* ( u; Du, u ) t nu T et enT Du dG Gc ngn T t u ( en etT et enT )Du dG Gc c ngn 2 T T t ( g t 2 )g t u e e n t t Du d G Gc c2 43 Ex) Frictional Slip of a Cantilever Beam • Distributed load q axial load P, t = 106, = 0.5 • Contact force: Fc = –ngn = 75N • Penalized potential energy (axial alone) 1 a EA(u,x ) dx Pu(L) t gt2 0 2 L a L 2 0 EAu,xu,x dx Pu(L) tgtgt x L x L 0, u P=100N 44 Ex) Frictional Slip of a Cantilever Beam • Linear axial displacement field: u(x) = a0 + a1x u(0) 0 u(x) a1x u,x a1 u,x a1 • Tangential slip in terms of displacement – Parametric coordinate x has an origin at x = L, and it has the same length as the x-coordinate xc,x t t 0 1 xc0 0 gt xc u(L) a1 • Assume the stick condition: t gt n gn a1 (EAa1 t a1 P) 0, a1 Px u(x) 9.09 10 5 x EA t t gt 90.9 37.5 n gn The assumption is violated!! 45 Ex) Frictional Slip of a Cantilever Beam • Assume the slip condition: a L 0 EAu,xu,x dx Pu(L) n sgn(gt )gngt x L 0, u – With contact force = 70N, n gn 37.5N a1 (EAa1 P 37.5) 0, a1 a1 62.5 10 5 utip 0.625mm 46 5.4 FINITE ELEMENT FORMULATION OF CONTACT PROBLEMS 47 Finite Element Formulation • Slave-Master contact – The rigid body has fixed or prescribed displacement – Point x is projected onto the piecewise linear segments of the rigid body with xc (x = xc) as the projected point – Unit normal and tangent vectors et x2 x1 , L en e3 et Flexible body x g x0 x1 xc0 x en xc L x2 et Rigid body 48 Finite Element Formulation cont. • Parameter at contact point xc 1 ( x x1 )T et L • Gap function gn ( x x1 )T en 0 Impenetrability condition • Penalty function NC 1 P g 2 I 1 2 I – Note: we don’t actually integrate the penalty function. We simply added the integrand at all contact node – This is called collocation (a kind of integration) – In collocation, the integrand is function value × weight 49 Finite Element Formulation cont. • Contact form (normal) bN ( u, u ) NC I 1 gn T d en I {d }T { fc } – (gn): contact force, proportional to the violation – Contact form is a virtual work done by contact force through normal virtual displacement • Linearization D gn Heaviside step function H(x) = 1 if x > 0 = 0 otherwise H( gn )enT Du H( gn )enT Dd bN* ( u; Du, u ) NC H(gn ) d I 1 { d }T [Kc ]{ Dd} T en enT Dd I {d } T NC H(gn ) en enT I { Dd} I 1 Contact stiffness 50 Finite Element Formulation cont. • Global finite element matrix equation for increment {d }T [KT Kc ]{ Dd} { d }T { fext fint fc } – Since the contact forms are independent of constitutive relation, the above equation can be applied for different materials • A similar approach can be used for flexible-flexible body contact – One body being selected as a slave and the other as a master • Computational challenge in finding contact points (for example, out of 10,000 possible master segments, how can we find the one that will actually in contact?) 51 Finite Element Formulation cont. • Frictional slip gt l0 ( xc xc0 ) • Friction force and tangent stiffness (stick condition) ftc t gt et ktc t et etT • Friction force and tangent stiffness (slip condition) ftc n sgn(gt )gn et , if t gt n gn ktc n sgn(gt )et enT 52 5.6 CONTACT ANALYSIS PROCEDURE AND MODELING ISSUES 53 Types of Contact Interface • Weld contact – – – – A slave node is bonded to the master segment (no relative motion) Conceptually same with rigid-link or MPC For contact purpose, it allows a slight elastic deformation Decompose forces in normal and tangential directions • Rough contact – Similar to weld, but the contact can be separated • Stick contact – The relative motion is within an elastic deformation – Tangent stiffness is symmetric, • Slip contact – The relative motion is governed by Coulomb friction model – Tangent stiffness become unsymmetric 54 Contact Search • Easiest case – User can specify which slave node will contact with which master segment – This is only possible when deformation is small and no relative motion exists in the contact surface – Slave and master nodes are often located at the same position and connected by a compression-only spring (node-to-node contact) – Works for very limited cases • General case – User does not know which slave node will contact with which master segment – But, user can specify candidates – Then, the contact algorithm searches for contacting master segment for each slave node – Time consuming process, because this needs to be done at every iteration 55 Contact Search cont. Node-to-surface contact search Surface-to-surface contact search 56 Slave-Master Contact • Theoretically, there is no need to distinguish Body 1 from Body 2 • However, the distinction is often made for numerical convenience • One body is called a slave body, while the other body is called a master body • Contact condition: the slave body cannot penetrate into the master body • The master body can penetrate into the slave body (physically not possible, but numerically it’s not checked) Master Slave 57 Slave-Master Contact cont. • Contact condition between a slave node and a master segment • In 2D, contact pair is often given in terms of {x, x1, x2} • Slave node x is projected onto the piecewise linear segments of the master segment with xc (x = xc) as the projected point • Gap: g ( x x1 ) en 0 • g > 0: no contact • g < 0: contact Slave x x0 x1 x xc g en 0 xc x2 et L Master 58 Contact Formulation (Two-Step Procedure) 1. Search nodes/segments that violate contact constraint Body 1 Violated nodes Contact candidates Body 2 2. Apply contact force for the violated nodes/segments (contact force) Contact force 59 Contact Tolerance and Load Increment • Contact tolerance – Minimum distance to search for contact (1% of element length) Out of contact Within tolerance Out of contact • Load increment and contact detection – Too large load increment may miss contact detection 60 Contact Force • For those contacting pairs, penetration needs to be corrected by applying a force (contact force) • More penetration needs more force • Penalty-based contact force (compression-only spring) FC Kn g • Penalty parameter (Kn): Contact stiffness – It allows a small penetration (g < 0) – It depends on material stiffness – The bigger Kn, the less allowed penetration x1 g<0 xc x x2 FC 61 Contact Stiffness • Contact stiffness depends on the material stiffness of contacting two bodies • Large contact stiffness reduces penetration, but can cause problem in convergence • Proper contact stiffness can be determined from allowed penetration (need experience) • Normally expressed as a scalar multiple of material’s elastic modulus Kn SF E SF 1.0 • Start with small initial SF and increase it gradually until reasonable penetration 62 Lagrange Multiplier Method • In penalty method, the contact force is calculated from penetration – Contact force is a function of deformation • Lagrange multiplier method can impose contact condition exactly – Contact force is a Lagrange multiplier to impose impenetrability condition – Contact force is an independent variable • Complimentary condition FC g 0 g 0 FC 0 contact g 0 FC 0 no contact • Stiffness matrix is positive semi-definite K T A A d F 0 FC 0 • Contact force is applied in the normal direction to the master segment 63 Observations • Contact force is an internal force at the interface – Newton’s 3rd law: equal and opposite forces act on interface F pC1 qC1 F pC2 qC2 Np Nq i=1 i=1 pci qci qC2 F • Due to discretization, force distribution can be different, but the resultants should be the same 64 Contact Formulation • Add contact force as an external force • Ex) Linear elastic materials F [K]{d} {F} {FC (d)} Internal force External force pC1 Contact force qC1 • Contact force depends on displacement (nonlinear, unknown) [K KC ]{ Dd} {F} {FC (d)} [K]{d} Tangent stiffness pC2 qC2 qC2 F Residual F [KC ] C Contact stiffness d 65 Friction Force • So far, contact force is applied to the normal direction – It is independent of load history (potential problem) • Friction force is produced by a relative motion in the interface – Friction force is applied to the parallel direction – It depends on load history (path dependent) • Coulomb friction model Ff FC FC Friction force Body 1 Stick Slip Friction force Relative motion Contact force 66 Friction Force cont. • Coulomb friction force is indeterminate when two bodies are stick (no unique determination of friction force) • In reality, there is a small elastic deformation before slip • Regularized friction model – Similar to elasto-perfectly-plastic model Ff Kt s FC Stick Kt: tangential stiffness Slip Friction force FC Kt Relative motion 67 Tangential Stiffness • Tangential stiffness determines the stick case Ff Kt s Stick FC Slip • It is related to shear strength of the material Kt SF E SF 0.5 • Contact surface with a large Kt behaves like a rigid body • Small Kt elongate elastic stick condition too much (inaccurate) Friction force FC Kt Relative motion 68 Selection of Master and Slave • Contact constraint – A slave node CANNOT penetrate the master segment – A master node CAN penetrate the slave segment – There is not much difference in a fine mesh, but the results can be quite different in a coarse mesh • How to choose master and slave – Rigid surface to a master – Convex surface to a slave – Fine mesh to a slave Master Slave Slave Master 69 Selection of Master and Slave • How to prevent penetration? – Can define master-slave pair twice by changing the role – Some surface-to-surface contact algorithms use this – Careful in defining master-slave pairs Master-slave pair 2 Master-slave pair 1 70 Flexible or Rigid Bodies? • Flexible-Flexible Contact – Body1 and Body2 have a similar stiffness and both can deform • Flexible-Rigid Contact – Stiffness of Body2 is significantly larger than that of Body1 – Body2 can be assumed to be a rigid body (no deformation but can move) 107 – Rubber contacting to steel 7 10 • Why not flexible-flexible? 1 – When two bodies have a large difference in 1 stiffness, the matrix becomes ill-conditioned – Enough to model contacting surface only for Body2 • Friction in the interface – Friction makes the contact analysis path-dependent – Careful in the order of load application 71 Effect of discretization • Contact stress (pressure) is high on the edge Uniform pressure Slave body • Contact stress is sensitive to discretization Non-uniform contact stress 72 Rigid-Body Motion • Contact constraint is different from displacement BC – Contact force is proportional to the penetration amount – A slave body between two rigid-bodies can either fly out or oscillate between two surfaces – Always better to remove rigid-body motion without contact Rigid master Rigid master Rigid master Rigid master 73 Rigid-Body Motion • When a body has rigid-body motion, an initial gap can cause singular matrix (infinite/very large displacements) • Same is true for initial overlap Rigid master Rigid master 74 Rigid-Body Motion • Removing rigid-body motion – A small, artificial bar elements can be used to remove rigid-body motion without affecting analysis results much Contact stress at bushing due to shaft bending 75 Convergence Difficulty at a Corner • Convergence iteration is stable when a variable (force) varies smoothly • The slope of finite elements are discontinuous along the curved surface • This can cause oscillation in residual force (not converging) • Need to make the corner smooth using either higherorder elements or many linear elements – About 10 elements in 90 degrees, or use higher-order elements 76 Curved Contact Surface • Curved surface – Contact pressure is VERY sensitive to the curvature – Linear elements often yield unsmooth contact pressure distribution – Less quadratic elements is better than many linear elements Linear elements Quadratic elements 77 Summary • Contact condition is a rough boundary nonlinearity due to discontinuous contact force and unknown contact region – Both force and displacement on the contact boundary are unknown – Contact search is necessary at each iteration • Penalty method or Lagrange multiplier method can be used to represent the contact constraint – Penalty method allows a small penetration, but easy to implement – Lagrange multiplier method can impose contact condition accurately, but requires additional variables and the matrix become positive semi-definite • Numerically, slave-master concept is used along with collocation integration (at slave nodes) • Friction makes the contact problem path-dependent • Discrete boundary and rigid-body motion makes the contact problem difficult to solve 78