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

6. Constrained minimization
Example
Constrained minimization
Direct substitution

7. Constrained minimization
Example
Constrained minimization
Penalty method

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

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

20. Penalty method Problem
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. Nothing to balance this unless
Problem
Penalty method
Hence
Nothing to balance this unless =0
Neumann b.c =0
Equilibrium equation =0
Dirichlet b.c
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. Constrained minimization
Example
Constrained minimization
Find (x1, x2) such that
is minimized subject to the constraint x1 x2
Paraboloid (J)
Plane (x1 –x2= 0)
absolute minimum (-12, -15)
constrained minimum
(-12,-12)

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

25. Constrained minimization
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. Lagrange Multipliers Implementation Discretization
The modified functional
where

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

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

35. Lagrange Multipliers Physical interpretation
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. Nitsche’s Method Modified functional
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. Consistency
Nitsche’s Method
Hence
With =0 =0

39. Nitsche’s Method Consistency Hence =0 =0 =0
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. Physical interpretation of the Lagrange multipliers Nitsche’s Method
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. Coupling FE and MM Ramp Function Direct Coupling
Coupling using Ramp Functions (Belytschko, 1995)
where, the Ramp function

44. Coupling using Ramp Functions (Belytschko)
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. Coupling FE and MM Ramp Function Direct Coupling
Derivatives are discontinuous along the interfaces GMM and GFEM. This may reduce higher rate of convergence of MM.

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

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

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

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

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

51. Define the modified functional
Nitsche’s Method
Coupling FE and MM
Indirect Coupling n
GMM
GFEM
nFEM
nMM
GINT
Define the modified functional
where the Lagrange multipliers have been replaced by one of the previous three physical representations

52. 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

53. The elasticity problem
Constrained minimization
Volumetric Locking
Consider a solid occupying a domain with boundary W Gu Gt n
The minimization problem
where

54. The elasticity problem
Constrained minimization
Volumetric Locking
In a body composed of an almost incompressible material, we expect the volumetric strains (eV) to be small compared to the deviatoric strains (eij’) and therefore use the following form of the constitutive equation
(1)
where
(2)

55. Constrained minimization
The elasticity problem
Constrained minimization
Volumetric Locking
We will define a scalar known as the hydrostatic pressure within the body
(3)
the deviatoric strain
(4)
the deviatoric stress
(5)
Comparing with the constitutive equation (1)
(7)
(6)

56. The elasticity problem
Constrained minimization
Volumetric Locking
Going back to the expression of the functional
Notice
and

57. The elasticity problem
Constrained minimization
Volumetric Locking
Hence
Here the bulk modulus (k) acts exactly like a penalty parameter when the Poisson’s ratio tends towards 0.5. For the incompressible case, k becomes infinite and it enforces the condition that the volumetric strain is 0. However, the pressure is finite and of the order of the boundary tractions. Hence a pure displacement formulation cannot be used to compute the displacements and then compute the pressure from Eq (3). When the predictive capability of the finite element formulation degrades with increase in bulk modulus, we call this “volumetric locking”

58. The elasticity problem
Constrained minimization
u/p formulation
The key is to treat the pressure as an independent variable
Hence

59. The elasticity problem
Constrained minimization
u/p formulation
Notice
Implies
This formulation shows that the pressure (p) acts as a Lagrange multiplier enforcing the condition

60. The elasticity problem
Constrained minimization
u/p formulation
We will therefore use the following functional of w and p
Taking variations w.r.t the displacements and pressure
u/p formulation

61. The elasticity problem
Constrained minimization
u/p formulation
With the following discretizations:
Obtain the following discretized equations
with

62. The elasticity problem
Constrained minimization
Saddle point problem
Notice that in the limit of incompressible analysis
We obtain the saddle point problem
which is analogous to the one we obtained in our lecture on Lagrange multipliers. We will now look more deeply at this generic problem which arises when a constrained problem is solved using a Lagrange multiplier

63. Lagrange Multipliers Saddle point Problem Observations:
The matrix S is symmetric
The matrix S is usually not ill-conditioned
The matrix S is indefinite (i.e., can have eigenvalues that are +ve, -ve or zero)

64. Lagrange Multipliers Saddle point Problem Observations:
4. The system has a unique solution for wh if Ah is SPD.
5. The system has a unique solution for lh if ANY ONE of the following (equivalent) statements holds good
Statement 1: BT has a trivial null space
Statement 2: The inf-sup condition is satisfied
Statement 3: BA-1BT is SPD

65. Lagrange Multipliers Saddle point Problem Observations:
6. The system allows very attractive iterative solution procedures.
7. When a large number of constraints are imposed, solution cost increases considerably (since we have to solve a (N+M)x(N+M) system).

66. Uniqueness of solution
Lagrange Multipliers
Let and be two solutions of the above
equation (we will find the conditions for which it is not
possible to have two such solutions)
The difference solution
Satisfies the homogeneous equation

67. Uniqueness of solution
Lagrange Multipliers
Can be written as two sets of equations:
Premultiplying (1) by and (2) by and subtracting (2) from (1)

68. Uniqueness of solution
Lagrange Multipliers
If A is positive definite (on the set of vectors which lie in the null space of B), then we can say that the above implies
The next question is, what is the condition for l to be unique?
GO back to equation (1) with

69. Uniqueness of solution
Lagrange Multipliers
If BT has a trivial null space, then
The following statements are identical:
Statement 1: BT has a trivial null space
Statement 2: The inf-sup condition is satisfied

70. Lagrange Multipliers Inf-sup condition
Statement 2: The inf-sup condition is satisfied
What is this “inf-sup parameter”?
Using Cauchy-Schwartz inequality

71. Lagrange Multipliers Inf-sup condition Hence
Now consider the following eigenvalue problem
The matrix BBT is SPD if BT is full rank and therefore it will have M poisitive eigenvalues. If lmin is the minimum eigenvalue of BBT then

72. Lagrange Multipliers Inf-sup condition
If BT is full rank , then the inf sup condition is satisfied with
Where lmin is the minimum eigenvalue of BBT

73. Lagrange Multipliers Inf-sup condition Example: consider the matrix
With s1>s2> 0

74. Lagrange Multipliers Inf-sup condition
Using Cauchy-Schwartz inequality
Hence

75. Lagrange Multipliers Inf-sup condition Now, for arbitrary q=[q1, q2]T
The expression within the radical sign is a symmetric positive function of the variable ‘q’ whose minimum value is s2

76. Lagrange Multipliers Inf-sup condition Hence
Hence, the inf-sup condition provides a very practical means of evaluating whether BT has a null space…in this example, s2 needs to be positive