Introduction to FEM 9 MultiFreedom Constraints II IFEM Ch 9 – Slide 1
Introduction to FEM Penalty Function Method Physical Interpretation Recall the example structure u 1 , f 1 u 2 , f 2 u 3 , f 3 u 4 , f 4 u 5 , f 5 u 6 , f 6 u 7 , f 7 (1) (2) (3) (4) (5) (6) x 1 2 3 4 5 6 7 under the homogeneous MFC u 2 = u 6 IFEM Ch 9 – Slide 2
Introduction to FEM Penalty Function Method (cont'd) u 1 , f 1 u 2 , f 2 u 3 , f 3 u 4 , f 4 u 5 , f 5 u 6 , f 6 u 7 , f 7 (1) (2) (3) (4) (5) (6) x 2 6 1 3 4 5 7 (7) "penalty element" of axial rigidity w � � u 2 � f 2 (7) − 1 � � � 1 = w − 1 (7) 1 u 6 f 6 w = the "penalty weight" assigned to the constraint IFEM Ch 9 – Slide 3
Introduction to FEM Penalty Function Method (cont'd) Upon merging the penalty element the modified stiffness equations are K 11 K 12 0 0 0 0 0 u 1 f 1 K 12 K 22 + w K 23 0 0 − w 0 u 2 f 2 0 K 23 K 33 K 34 0 0 0 u 3 f 3 0 0 K 34 K 44 K 45 0 0 u 4 f 4 = 0 0 0 K 45 K 55 K 56 0 u 5 f 5 0 − w 0 0 K 56 K 66 + w K 67 u 6 f 6 0 0 0 0 0 K 67 K 77 u 7 f 7 This modified system is submitted to the equation solver. Note that u retains the same arrangement of DOFs. IFEM Ch 9 – Slide 4
Introduction to FEM But which penalty weight to use? From HW Log10(of solution error norm) Exercise 10.1 0 (Mathematica 4.1 on Mac G4) -2 -4 constraint solution error gap error best w -6 dominates dominates Log10(w) 4 6 8 10 12 14 Rough guideline: "square root rule"; see Notes. IFEM Ch 9 – Slide 5
Introduction to FEM Penalty Function Method - General MFCs 3 u 3 + u 5 − 4 u 6 = 1 � u 3 � [ 3 1 − 4 ] u 5 = 1 u 6 � � u 3 � 3 Premultiply both sides by � � � 9 3 − 12 "Penalty element" T [ 3 1 − 4 ] 3 1 − 4 u 5 1 = stiffness equations − 12 − 4 16 u 6 − 4 Scale by w and merge: K 11 K 12 0 0 0 0 0 u 1 f 1 K 12 K 22 K 23 0 0 0 0 u 2 f 2 0 K 23 K 33 + 9 w K 34 3 w − 12 w 0 u 3 f 3 + 3 w 0 0 K 34 K 44 K 45 0 0 u 4 f 4 = 0 0 3 w K 45 K 55 + w K 56 − 4 w 0 u 5 f 5 + w 0 0 − 12 w 0 K 56 − 4 w K 66 + 16 w K 67 u 6 f 6 − 4 w 0 0 0 0 0 K 67 K 77 u 7 f 7 IFEM Ch 9 – Slide 6
Introduction to FEM Assessment of Penalty Function Method ADVANTAGES general application (inc' nonlinear MFCs) 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 weights left to user - big burden accuracy limited by ill-conditioning IFEM Ch 9 – Slide 7
Introduction to FEM Lagrange Multiplier Method Physical Interpretation u 1 , f 1 u 2 , f 2 u 3 , f 3 u 4 , f 4 u 5 , f 5 u 6 , f 6 u 7 , f 7 (1) (2) (3) (4) (5) (6) x 2 1 3 4 5 6 7 −λ λ force-pair that enforces MFC K 11 K 12 0 0 0 0 0 u 1 f 1 K 12 K 22 K 23 0 0 0 0 u 2 f 2 − λ 0 K 23 K 33 K 34 0 0 0 u 3 f 3 0 0 K 34 K 44 K 45 0 0 u 4 f 4 = 0 0 0 K 45 K 55 K 56 0 u 5 f 5 0 0 0 0 K 56 K 66 K 67 u 6 f 6 + λ 0 0 0 0 0 K 67 K 77 u 7 f 7 IFEM Ch 9 – Slide 8
Introduction to FEM Lagrange Multiplier Method (cont'd) Because λ is unknown, it is passed to the LHS and appended to the node-displacement vector: u 1 K 11 K 12 0 0 0 0 0 0 f 1 u 2 0 0 0 0 1 K 12 K 22 K 23 f 2 u 3 0 K 23 K 33 K 34 0 0 0 0 f 3 u 4 0 0 K 34 K 44 K 45 0 0 0 f 4 = u 5 0 0 0 K 45 K 55 K 56 0 0 f 5 u 6 0 0 0 0 K 56 K 66 K 67 − 1 f 6 u 7 0 0 0 0 0 K 67 K 77 0 f 7 λ This is now a system of 7 equations and 8 unknowns. Need an extra equation: the MFC. IFEM Ch 9 – Slide 9
Introduction to FEM Lagrange Multiplier Method (cont'd) Append MFC as additional equation: K 11 K 12 0 0 0 0 0 0 u 1 f 1 K 12 K 22 K 23 0 0 0 0 1 u 2 f 2 0 K 23 K 33 K 34 0 0 0 0 u 3 f 3 0 0 K 34 K 44 K 45 0 0 0 u 4 f 4 = 0 0 0 K 45 K 55 K 56 0 0 u 5 f 5 0 0 0 0 K 56 K 66 K 67 − 1 u 6 f 6 0 0 0 0 0 K 67 K 77 0 u 7 f 7 0 1 0 0 0 − 1 0 0 λ 0 This is the multiplier-augmented system . The new coefficient matrix is called the bordered stiffness . IFEM Ch 9 – Slide 10
Introduction to FEM Lagrange Multiplier Method - Multiple MFCs u 2 − u 6 = 0 , 5 u 2 − 8 u 7 = 3 , 3 u 3 + u 5 − 4 u 6 = 1 Three MFCs: Recipe step #1: append the 3 constraints K 11 K 12 0 0 0 0 0 f 1 K 12 K 22 K 23 0 0 0 0 f 2 u 1 0 K 23 K 33 K 34 0 0 0 f 3 u 2 0 0 K 34 K 44 K 45 0 0 f 4 u 3 0 0 0 K 45 K 55 K 56 0 f 5 u 4 = 0 0 0 0 K 56 K 66 K 67 f 6 u 5 0 0 0 0 0 K 67 K 77 f 7 u 6 0 1 0 0 0 − 1 0 0 u 7 0 5 0 0 0 0 − 8 3 0 0 3 0 1 − 4 0 1 IFEM Ch 9 – Slide 11
Introduction to FEM Lagrange Multiplier Method - Multiple MFCs (cont'd) Recipe step #2: append multipliers, symmetrize & fill K 11 K 12 0 0 0 0 0 0 0 0 u 1 f 1 K 12 K 22 K 23 0 0 0 0 1 5 0 u 2 f 2 0 K 23 K 33 K 34 0 0 0 0 0 3 u 3 f 3 0 0 K 34 K 44 K 45 0 0 0 0 0 u 4 f 4 0 0 0 K 45 K 55 K 56 0 0 0 1 u 5 f 5 = 0 0 0 0 K 56 K 66 K 67 − 1 0 − 4 u 6 f 6 0 0 0 0 0 K 67 K 77 0 − 8 0 u 7 f 7 0 1 0 0 0 − 1 0 0 0 0 λ 1 0 0 5 0 0 0 0 − 8 0 0 0 3 λ 2 0 0 3 0 1 − 4 0 0 0 0 1 λ 3 IFEM Ch 9 – Slide 12
Introduction to FEM Assessment of Lagrange Multiplier Method ADVANTAGES general application exact no user decisions ("black box") DISADVANTAGES difficult implementation additional unknowns loses positive definiteness sensitive to constraint dependence IFEM Ch 9 – Slide 13
Introduction to FEM MFC Application Methods: Assessment Summary Master-Slave Penalty Lagrange Elimination Function Multiplier Generality fair excellent excellent Ease of implementation poor to fair good fair Sensitivity to user decisions high high small to none Accuracy variable mediocre excellent Sensitivity as regards high none high constraint dependence Retains positive definiteness yes yes no Modifies unknown vector yes no yes IFEM Ch 9 – Slide 14
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