Generalized Finite Element Methods

1. Generalized Finite Element Methods Constraints Suvranu De

2. Last lecture Local weak form Local weighted residual method (LUSWF I) Local Galerkin method (LSWF) Local H-1 method (LUSWF II)

3. This class • Imposition of essential boundary conditions • Penalty method • Lagrange multipliers • Physical interpretation of the Lagrange multipliers • Nitsche’s method • Coupling meshfree and finite element methods • Direct coupling • Indirect coupling • Internal constraint and volumetric locking

4. Essential boundary conditions Finite element shape functions satisfy “Kronecker delta” property which makes imposition of essential boundary conditions straightforward. In certain meshfree methods, the Kronecker delta property is absent and imposition of essential boundary conditions is tricky.

5. Example Paraboloid (J) Plane (x1 –x2= 0) x2 x1 Constrained minimization Find (x1, x2) such that is minimized subject to the constraint absolute minimum (-12, -15) constrained minimum (-12,-12)

6. Example Constrained minimization Direct substitution

7. Example Constrained minimization Penalty method

8. Example Constrained minimization Penalty method • Note: • The coefficient matrix is symmetric • The coefficient matrix is ill-conditioned for large a (numerical computations become error prone for large a)

9. n Gt W Gu The elasticity problem Constrained minimization Consider a solid occupying a domain with boundary

10. The elasticity problem Constrained minimization The constrained minimization problem may be stated as subject to the constraint where

11. The elasticity problem Constrained minimization the vectors and matrices in 2D are

12. The elasticity problem Constrained minimization the vectors and matrices in 2D are

13. The elasticity problem Constrained minimization Penalty method Construct a modified functional Solve the unconstrained minimization problem

14. Implementation Penalty method Approximate solution Xh is a finite dimensional subspace of H1 where the discretized modified functional Now, discretize where

15. Implementation Penalty method Part A Proof

16. Implementation Penalty method Proof (contd)

17. Implementation Penalty method Part B using

18. Implementation Penalty method Hence

19. Implementation Penalty method • Note • Ah is symmetric • Ah is illconditioned for large a • The formulation is not consistent for trial functions that do not vanish on the Dirichlet boundary ~ ~

20. Problem Penalty method The formulation is not consistent for trial functions that do not vanish on the Dirichlet boundary To see this, start with the modified functional

21. Problem Penalty Method Using Green’s Theorem

22. Problem Penalty method Hence Nothing to balance this unless =0 Neumann b.c =0 Dirichlet b.c =0 Equilibrium equation This lack of consistency is a major problem for methods where the trial function space does not vanish on the Dirichlet boundary. We will see that Nitsche’s method overcomes this problem. But before that we need to understand the physical interpretation of Lagrange multipliers

23. Example Paraboloid (J) Plane (x1 –x2= 0) x2 x1 Constrained minimization Find (x1, x2) such that is minimized subject to the constraint absolute minimum (-12, -15) constrained minimum (-12,-12)

24. Example Constrained minimization Lagrange Multipliers l is an unknown “Lagrange multiplier” (notice that in the penalty method, we chose the penalty parameter)

25. Example Constrained minimization Lagrange Multipliers • Notice • The matrix is symmetric but not positive definite • The number of unknowns has been increased (increasing the computational cost).

26. The elasticity problem Constrained minimization The constrained minimization problem may be stated as subject to the constraint where

27. The elasticity problem Constrained minimization Lagrange multipliers Construct a modified functional Solve the unconstrained minimization problem The problem posed on finite dimensional subspaces

28. Implementation Lagrange Multipliers Discretization The modified functional where

29. Implementation Lagrange Multipliers Minimization in matrix form Saddle point problem • Higher computational cost as number of unknowns increase • May not be positive definite (later) • Symmetric • Well conditioned

30. Physical interpretation Lagrange Multipliers Claim: The physical interpretation of the Lagrange multiplier is that it represents the traction at the Dirichlet boundary Proof: Start with the modified functional

31. Physical interpretation Lagrange Multipliers Lets first look at Part A

32. Physical interpretation Lagrange Multipliers Using Green’s Theorem

33. Physical interpretation Lagrange Multipliers Hence (Neumann b.c.) (Equilibrium equation) (Dirichlet b.c.) Hence, we identify

34. Physical interpretation Lagrange Multipliers Hence, we may now replace the Lagrange multiplier with its physical interpretation to define the modified functional Notice that , due to minor symmetry

35. Physical interpretation Lagrange Multipliers The advantage of using the modified functional is that the number of equations do not increase! In vector form with

36. Physical interpretation Lagrange Multipliers Minimizing • Number of unknowns does not increase • System matrix remains symmetric • Less accurate

37. Modified functional Nitsche’s Method • Overcomes the inaccuracy of the previous method by combining this with that of the Penalty method. The modified functional is Lagrange multiplier term with the Lagrange multiplier replaced by its physical interpretation Penalty term enforcing the same Dirichlet condition

38. =0 =0 Consistency Nitsche’s Method Hence With

39. =0 =0 =0 Consistency Nitsche’s Method Hence Hence, Nitsche’s method restores consistency in the formulation unlike the Penalty method. This is now becoming the standard for application of essential boundary conditions in meshfree methods.

40. Summary Imposition of constraints Penalty method: Assume a large penalty parameter Matrix problem is ill-conditioned Lagrange multipliers The Lagrange parameter is an unknown A “saddle point problem” results which is symmetric and well-conditioned. However, the problem is indefinite. Physical interpretation of the Lagrange multipliers Nitsche’s Method Restores consistency in the Penalty formulation

41. This class • Imposition of essential boundary conditions • Penalty method • Lagrange multipliers • Physical interpretation of the Lagrange multipliers • Nitsche’s method • Coupling meshfree and finite element methods • Direct coupling • Indirect coupling • Internal constraint and volumetric locking

42. Coupling finite element and meshfree methods • Why? • Meshfree methods are more accurate but also more costly than finite element methods. • Application of Dirichlet boundary conditions (however, a note of caution is that finite elements introduce errors at the boundary which might reduce overall convergence rates) • Techniques • Direct coupling • Ramp Function, Reproducing Conditions, Bridging Scale • Indirect coupling • Penalty method, Lagrange multipliers, Nitsche’s method

43. Ramp Function Coupling FE and MM Direct Coupling Coupling using Ramp Functions (Belytschko, 1995) where, the Ramp function

44. Ramp Function Coupling FE and MM Direct Coupling Coupling using Ramp Functions (Belytschko) The Ramp function is the sum of all the FE shape functions on the interface Hence and varies continuously between the two interfaces

45. Ramp Function Coupling FE and MM Direct Coupling Derivatives are discontinuous along the interfaces GMM and GFEM. This may reduce higher rate of convergence of MM.

46. n GMM nFEM GINT GFEM nMM Formulation Coupling FE and MM Indirect Coupling Total potential energy of the system Constraint conditions along GINT • Displacement continuity • Traction equilibrium Advantage: No transition region necessary

47. n GMM nFEM GINT GFEM nMM Lagrange multipliers Coupling FE and MM Indirect Coupling Define the modified functional Discretize in WFEM in WMM on GINT where Hl is the trace of the FEM shape functions on GINT

48. n GMM nFEM GINT GFEM nMM Lagrange multipliers Coupling FE and MM Indirect Coupling The resulting set of equations

49. n GMM nFEM GINT GFEM nMM Penalty Method Coupling FE and MM Indirect Coupling Define the modified functional The resulting set of equations

50. n GMM nFEM GINT GFEM nMM Physical interpretation Coupling FE and MM Indirect Coupling Realize that the Lagrange multipliers represent tractions on the interface where Possibilities: (1) (2) (3)