Multi-Point Constraints

Multi-Point Constraints

Multi-Point Constraints Multi-Point Constraints Single point constraint examples Multi-Point constraint examples linear, homogeneous linear, non-homogeneous linear, homogeneous linear, non-homogeneous nonlinear, homogeneous 2

Multi-Point Constraints (Example) Modelling of joints Perfect connection ensured here turbine blade turbine disc nodes at interface (a) (b) 3

Multi-Point Constraints (Example) Modelling of joints 1 v u Mismatch between DOFs of beams and 2D solid beam is free to rotate (rotation not transmitted to 2D solid) v u 1 Perfect connection by artificially extending beam into 2D solid (Additional mass) 4

Multi-Point Constraints (Example) Modelling of joints Using MPC equations d d 1 5 blade model d d ad d 2 6 7 d 3 5 d d ad 4 6 7 d 1 d 2 d 7 d 6 d 5 d 4 d 3 ad 7 d 2 d 6 d 1 d 5 2a disc model 5

Multi-Point Constraints (Example) Creation of MPC equations for offsets Node of the beam element Node on neutralsurface of the plate Rigid body Neutral surface of the plate 6

Multi-Point Constraints (Example) Creation of MPC equations for offsets d 6 = d 1 + d 5 or d 1 + d 5 - d 6 = 0 d 7 = d 2 - d 4 or d 2 - d 4 - d 7 = 0 d 8 = d 3 or d 3 - d 8 = 0 d 9 = d 5 or d 5 - d 9 = 0 7

Multi-Point Constraints (Example) Modelling of joints Similar for plate connected to 3D solid Mesh for plate Mesh for Solid 8

Multi-Point Constraints (Example) Modelling of symmetric boundary conditions d n = 0 v n 1 1 90 0 u i cos + v i sin=0 or u i +v i tan =0 for i=1, 2, 3 y 2 3 2 3 u x Axis of symmetry Axis of symmetry 9

Multi-Point Constraints (Example) Enforcement of mesh compatibility Use lower order shape function to interpolate d x = 0.5(1-) d 1 + 0.5(1+) d 3 d y = 0.5(1-) d 4 + 0.5(1+) d 6 Substitute value of at node 2 0.5 d 1 - d 2 + 0.5 d 3 =0 Quad d 6 d 5 1 2 d 3 d 2 Linear 1 0 0.5 d 4 - d 5 + 0.5 d 6 =0 d 4 3 d 1-1 10

Multi-Point Constraints (Example) Enforcement of mesh compatibility Use shape function of longer element to interpolate d x = -0.5 (1-) d 1 + (1+)(1-) d 3 + 0.5 (1+) d 5 Substituting the values of for the two additional nodes d 2 = 0.251.5 d 1 + 1.50.5 d 3-0.250.5 d 5 d 4 = -0.250.5 d 1 + 0.51.5 d 3 + 0.251.5 d 5 Quad d 10 d 9 d 8 d 5 Quad d 4 1 0.5 0 d 7 d 6 d 3 d 2-0.5-1 d 1 11

Multi-Point Constraints (Example) Enforcement of mesh compatibility In x direction, 0.375 d 1 - d 2 + 0.75 d 3-0.125 d 5 = 0-0.125 d 1 + 0.75 d 3 - d 4 + 0.375 d 5 = 0 In y direction, Quad d 10 d 9 d 5 Quad 1 0.5 0.375 d 6 - d 7 +0.75 d 8-0.125 d 10 = 0 d 8 d 4 0-0.125 d 6 +0.75 d 8 - d 9 + 0.375 d 10 = 0 d 7 d 6 d 3 d 2-0.5-1 d 1 12

Multi-Point Constraints (Example) Modelling of constraints by rigid body attachment d 1 = q 1 d 2 = q 1 +q 2 l 1 l 1 l 2 l 3 d 3 =q 1 +q 2 l 2 d 1 d 2 d 3 d 4 Rigid slab d 4 =q 1 +q 2 l 3 q 2 q 1 Eliminate q 1 and q 2 (l 2 /l 1-1) d 1 - ( l 2 /l 1 ) d 2 + d 3 = 0 (l 3 /l 1-1) d 1 - ( l 3 /l 1 ) d 2 + d 4 = 0 (DOF in x direction not considered) 13

Multi-Point Constraints Sources of Multi-Point Constraints Skew displacement BCs Coupling nonmatched FEM meshes Global-local and multiscale analysis Incompressibility 14

Multi-Point Constraints MPC Application Methods Master-Slave Elimination Penalty Function Augmentation Lagrange Multiplier Adjunction 15

Multi-Point Constraints Example 1D Structure to Illustrate MPCs Multi-Point constraint: Linear homogeneous MPC 16

Multi-Point Constraints Example 1D Structure to Illustrate MPCs 17

Multi-Point Constraints- Master Slave Method Master Slave Method for Example Structure Recall: Taking u as master: 18

Multi-Point Constraints- Master Slave Method Forming the Modified Stiffness Equations Unconstrained master stiffness equations: Master-slave transformation: Congruential transformation: Modified stiffness equations: 19

Multi-Point Constraints- Master Slave Method Modified Stiffness Equations for Example Structure u 2 as master and u 6 as slave DOF. 20

Multi-Point Constraints- Master Slave Method Modified Stiffness Equations for Example Structure u 6 as master and u 2 as slave DOF. Although they are algebraically equivalent, the latter would be processed faster if a skyline solver is used for the modified equations. 21

Multi-Point Constraints- Master Slave Method Multiple MPCs Suppose take 3, 4 and 6 as slaves: and put in matrix form: 22

Multi-Point Constraints- Master Slave Method Nonhomogeneous MPCs In matrix form 23

Multi-Point Constraints- Master Slave Method Nonhomogeneous MPCs modified system: in which: For the example structure 24

Multi-Point Constraints- Master Slave Method The General Case of MFCs For implementation in general-purpose programs the master-slave method can be described as follows. The degrees of freedoms in u are classified into three types: independent or uncommitted, masters and slaves. The MFCs may be written in matrix form as Inserting into the partitioned stiffness matrix and symmetrizing 25

Multi-Point Constraints- Master Slave Method Assessment of Master-Slave Method ADVANTAGES exact if precautions taken easy to understand retains positive definiteness important applications to model reduction DISADVANTAGES requires user decisions messy implementation for general MPCs sensitive to constraint dependence restricted to linear constraints 26

Multi-Point Constraints- Penalty Function Method Penalty Function Method, Physical Interpretation Recall the example structure under the homogeneous MPC u 2 = u 6 27

Multi-Point Constraints- Penalty Function Method Penalty Function Method, Physical Interpretation "penalty element" of axial rigidity w 28

Multi-Point Constraints- Penalty Function Method Penalty Function Method, Physical Interpretation Upon merging the penalty element the modified stiffness equations are This modified system is submitted to the equation solver. Note that u retains the same arrangement of DOFs. 29

Multi-Point Constraints- Penalty Function Method Penalty Function Method - General MPCs Premultiply both sides by Scale by w and merge: 30

Theory of Penalty Function Method - General MPCs t CU Q (Constrain equations) Π p Multi-Point Constraints- Penalty Function Method 1 1 2 2 T T T U KU U F t t =[ 1 2... m ] is a diagonal matrix of penalty numbers stationary condition of the modified functional requires the derivatives of p with respect to the U i to vanish Π p T T du 0 KU F C CU + C Q 0 T T [ K C C] U F C Q Penalty matrix 31

Multi-Point Constraints- Penalty Function Method Theory of Penalty Function Method - General MPCs [Zienkiewicz et al., 2000] : = constant (1/h) p+1 characteristic size of element p is the order of element used 46 max (diagonal elements in the 1.0 10 stiffness matrix) or 58 1.010 Young s modulus 32

Multi-Point Constraints- Penalty Function Method Assessment of Penalty Function Method ADVANTAGES general application (inc' nonlinear MPCs) easy to implement using FE library and standard assembler no change in vector of unknowns retains positive definiteness insensitive to constraint dependence DISADVANTAGES selection of weight left to user accuracy limited by ill-conditioning the constraint equations can only be satisfied approximately. 33

Multi-Point Constraints- Lagrange Multiplier Method Lagrange Multiplier Method, Physical Interpretation 34

Multi-Point Constraints- Lagrange Multiplier Method Lagrange Multiplier Method Because λ is unknown, it is passed to the LHS and appended to the node-displacement vector: This is now a system of 7 equations and 8 unknowns. Needs an extra equation: the MPC. 35

Multi-Point Constraints- Lagrange Multiplier Method Lagrange Multiplier Method Append MPC as additional equation: This is the multiplier-augmented system. The new coefficient matrix is called the bordered stiffness. 36

Multi-Point Constraints- Lagrange Multiplier Method IMPLEMENTATION OF MPC EQUATIONS KU F (Global system equation) CU - Q 0 (Matrix form of MPC equations) Constant matrices Optimization problem for solution of nodal degrees of freedom: Find U to Minimize: Subject to: p 1 2 T U KU CU -Q 0 T U F 37

Multi-Point Constraints- Lagrange Multiplier Method Lagrange multiplier method T 1 2 m T { CU Q} 0 (Lagrange multipliers) Multiplied to MPC equations Find U and to 1 T T T Minimize: L U KU U F { CU Q} 2 U F C 0 λ Q T K C Added to functional The stationary condition requires the derivatives of L with respect to the U i and i to vanish. L T 0 KU F C 0 du L 0 CU Q 0 d Matrix equation is solved 38

Multi-Point Constraints- Lagrange Multiplier Method Lagrange Multiplier Method - Multiple MPCs Three MPCs: Recipe step #1: append the 3 constraints Recipe step #2: append multipliers, symmetrize and fill 39

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support E = 70 Gpa, A = 10-3 m 2, P = 20 kn. 40

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (1) 41

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (2) 42

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (3) 43

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (4) 44

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (4) 45

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support k (5) 46

Multi-Point Constraints- Lagrange Multiplier Method Example: Five bar truss with inclined support 47

Multi-Point Constraints- Lagrange Multiplier Method After adjusting for essential boundary conditions Multipoint constraint due to inclined support at node 1: The augmented global equations with the Lagrange multiplier are as follows. 48

Multi-Point Constraints- Lagrange Multiplier Method Solving the final system of global equations we get Global to local transformation matrix 49

Multi-Point Constraints- Lagrange Multiplier Method Element nodal displacements in global coordinates Element nodal displacements in local coordinates u l 50

Multi-Point Constraints- Lagrange Multiplier Method Assessment of Lagrange Multiplier Method ADVANTAGES General application Constraint equations are satisfied exactly DISADVANTAGES Difficult implementation Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal terms Efficiency of solving the system equations is lower sensitive to constraint dependence 51

Multi-Point Constraints- Lagrange Multiplier Method MPC Application Methods: Assessment Summary 52

References 1- Finite Element Method: A Practical Course by: S. S. Quek, G.R. Liu, 2003. 2- Introduction to Finite Element Methods, by: Carlos Felippa, University of Colorado at Boulder. http://www.colorado.edu/engineering/cas/courses.d/ifem.d/ 53