1. APPLICATION OF PERIODIC BOUNDARY
CONDITION USING IMPLICIT BOUNDARY
METHOD
Chair: Dr. Ashok V. Kumar
Committee member:
Dr. Bhavani Sankar
Aswath Manogaran
05/12/2016

2. 1. Introduction
2. Implicit Boundary Finite Element Method
3. Implicit boundary method for periodic boundary
condition
4. Effective property of composite materials
5. Results and Discussion
6. Conclusion
2
Outline

3. Introduction
• Motivation
– Periodic boundary conditions
are needed to model unit cells
and Representative Volume
Elements (RVE)
– Periodic condition are applied
using multi-point constraints
– Identical nodes are needed at
opposite faces of RVE to
impose multipoint constraints
– Generating conforming mesh
with identical nodes at opposite
ends is difficult
3

4. Introduction
• Motivation
– Use of structured grid ensures
the presence of identical
elements/nodes at opposite ends
of the RVE.
– Element and node numbers at
opposite ends are easily
identified in a structured mesh.
4

5. Introduction
• Goal
– To develop a formulation for applying periodic
boundary condition using implicit boundary
method
– To verify and validate the formulation by
comparing with known results in existing
literature.
5

6. 6
Implicit Boundary Finite Element Method
• The data from CAD file are used to represent the
analytical domain for FE calculation.
The geometry in the left is approximated with triangles using *.stl format from a
CAD software and imported into IBFEM for FE calculations.

7. 7
Implicit Boundary Finite Element Method
• A background structured mesh is used for interpolating
the field variable within the analytical domain
The background mesh is non conforming to the CAD geometry. Since the nodes
are not necessarily located on the geometry, special techniques are needed to
solve boundary value problem.

8. 8
Implicit Boundary Finite Element Method
• Elements in the FE model are classified as internal
elements and boundary elements
For internal elements traditional finite element formulation is used to compute
stiffness.
Heaviside step functions are used to construct solutions structures for boundary
elements

9. • Solution structure
– field variable.
– field variable associated with the grid
– essential boundary value.
– is a diagonal matrix
– n is the dimension of the problem.
Implicit Boundary Finite Element Method (cont.)
Step function configuration
9
g a s a
u Hu u u u   
g
u
a
u
   ,...,i nH diag H H
u

10. 10
Implicit Boundary Finite Element Method
• An approximate Heaviside step function is constructed
using the signed distance function for solving boundary
value problem.
Heaviside Step function
 
0 0
1 1 0
1
K
H
 

  
         
 
   

11. • Boundary value function :
– is defined such that its value is equal to the imposed essential
boundary condition.
– and are constructed using the same shape functions
– same shape functions ensure accurate representation of constant
strains.
Implicit Boundary Finite Element Method (cont.)
Step function configuration
11
.
a
u
a
u
a
u g
u
a a
i i
i
u N u 
Imposed essential boundary
value = 

12. • Material discontinuity:
– One of the application of periodic boundary condition is finding the
effective property of a composite
– Composite materials consist of fiber matrix interface, where material
discontinuity exist
– Special technique is required to treat material discontinuity when the
interface does not have any nodes on them
Implicit Boundary Finite Element Method (cont.)
Step function configuration
12
.

13. • Material discontinuity:
– The matrix and fiber of the composite has separate mesh associated
with them
– Near the interface the matrix and fiber elements overlap each other
– Heaviside step function associated with the inclusion is used to
impose interface continuity
Implicit Boundary Finite Element Method (cont.)
Step function configuration
13
.

14. • Solution structure:
– For solving problems with material discontinuity using structured grid
the following solution structure is used in determining field variables.
– Using the above solution structure, inside the inclusion region the field
variable (solution from inclusion region) and outside the
inclusion region (solution from matrix region).
– Within the narrow band, the filed variable is a blend of solution form
inclusion and matrix region.
Implicit Boundary Finite Element Method (cont.)
14
.
  1 2
1 inc g inc g
u H u H u  
2g
u u
1g
u u

15. • Heaviside step function
– The value of is one within the inclusion domain, varies
quadratic within the narrow band from the boundary, zero at the
boundary and outside the boundary
Implicit Boundary Finite Element Method (cont.)
15
.
inc
H


16. • Gradient of the solution structure:
– The gradient of the solution structure allows slope discontinuity at the
interface.
– The terms and in the above expression are continuous
at the interface while the other two terms and are
discontinuous, since they are zero outside the inclusion domain but
non-zero within the narrow band , facilitating discontinuity in normal
strain in inclusion and matrix near the interface.
Implicit Boundary Finite Element Method (cont.)
16
.
 
1 2
1 2
1
g ginc inc
inc g inc gi i i
i i
j j j j j
u u uH H
H u H u
x x x x x
   
    
    
 
1
1
g
inc i
j
u
H
x



2g
inc i
j
u
H
x


1
inc
g
i
j
H
u
x


2
inc
g
i
j
H
u
x



17. Solution structure for periodic boundary conditions:
displacement field within the structural domain periodic element displacement
displacement of the element in the opposite boundary corresponding to the periodic boundary
Heaviside step function applied strain
17
Periodic boundary condition
(1 ) (1 )h p
i ii i ii i ii iu H u H u H L    
𝑢𝑖 𝑢𝑖
ℎ
𝑢𝑖
𝑝
𝐻𝑖𝑖 𝐿𝑖
 
0 0
1 1 0
1
K
H
 

  
         
 
   

18. 18
Implicit boundary method for periodic
boundary condition
• Solution structure
– At the boundary H = 0, therefore
– For where H =1,
the solution structure reduces to
(1 ) (1 )h p
i ii i ii i ii iu H u H u H L    
p
i i iu u L 
  
h
i iu u
5
10
 ;

19. 19
Implicit boundary method for periodic
boundary condition
• Solution structure
– The above solution structure can be written in matrix form as follows,
(1 ) (1 )h p
i ii i ii i ii iu H u H u H L    
       
   
  
   
     
1 2
1 2
1
2
[ ]
[ ]
h
e L L
p
h
e
p
L
X
u N X X N N X
X
X
u N X N N
X
N H N
N H N
X H L
 
            
 
 
               
   
     
  
% % %
% % %
%
%

20. 20
Implicit boundary method for periodic
boundary condition
• Gradient of displacement
– For a 2D element the strain vector is given by,
(1 )
(1 ) ( )
h p
h pi i ii i ii ii
ii i ii i
j j j j j j
h p
h pi i ii
ii ii i i
j j j
u u H u H H
H u H u L
x x x x x x
u u H
H H u u L
x x x
     
     
     
  
     
  
       
   
    
      
    
1
1
1
2
2 1 2
2
12
1 2
2 1
1 2
1
2
[ ]
[ ]
h
e L L
p
h
e
p
L
u
x
Xu
B X B B
Xx
u u
x x
X
B X B B
X
B H N H B
B H B H N
H L
 
 
  
                                      
 
   
 
               
     
       
   
% % %
% % %
%
%

21. 21
Implicit boundary periodic boundary
condition
• Weak form
– The weak form for a elastostatic boundary value problem can be
expressed as,
– Assuming no body forces or tractions
– Modified weak form:
           
T T T
d u b d u t d   
  
     
     0
T
C d 

 
          
T TT T
e e e LX B C B X d X B C d  


              % % %

22. 22
Implicit boundary periodic boundary
condition
• Weak form
– The stiffness matrix and the internal force vector can be written as
follows,
 
   
   
 
  
  
1 1 1 2 1
2 1 2 2 2
Lh
h p h p
p
L
B C B B C B B CX
X X d X X d
XB C B B C B B C

   
 
                             
                         
 
% % % % %
% % % % %
        
 
   
   
   
   
 
 
 
  
  
1 1 1 21 2
3 4 2 1 2 2
11
2 2
T T
e e e
L
L
X K X X F
B C B B C BK K
K d
K K B C B B C B
B CF
F d
F B C
 





                     
                  
          
      


% % % %
% % % %
%
%

23. • Inhomogeneous composite can be assumed to be made of
infinite number of periodic elements
• Experimental analysis of composites to determine effective
properties is complicated
• Micromechanical theories have been proposed to
determine the effective properties of composite by
knowing the elastic properties of its constituents and their
arrangement within the composite
23
Effective property of composite materials

24. • Representative Volume Element:
– A composite material can be analyzed by considering smallest of
volume, which repeats infinitely to form the composite
– For periodic composite a unit cell is considered for analysis
– For non periodic composite RVE is considered for analysis
24
Effective property of composite materials
Unit Cell RVE

25. • Periodic boundary condition:
– The RVE tiles the composite domain
– To preserve continuity in displacement, stress and strain RVE
should be subjected to periodic boundary condition
– Periodic boundary condition ensures the adjacent boundaries has
the same shape of displacement and therefore displaces together.
25
Effective property of composite materials

26. • Periodic boundary condition imposed on 3D RVE
26
Effective property of composite materials
Periodic boundary conditions imposed on f-
front, bk-back, l-left, r-right, t-top and b-bottom
face.

27. • Macro stress and effective property:
– The elemental stress developed due to applied strain is averaged
over the RVE volume to determine the Macro stress.
– Knowing the Macro stress developed in the RVE and the strain
applied on the RVE, the stiffness matrix of the composite can be
determined using constitutive relationship between stress and
strain.
– The columns of the stiffness matrix can be determined from the
macro stress developed due to unit strains applied independently in
different orientation on the RVE.
27
Effective property of composite materials
1
V
dV
V
  
 K 

28. • Plane Stress – Aluminum Boron composite
• Element type: 4-node quadrilateral
28
Results and Discussion
379.3fE GPa
0.1f 
2L 
FE Model
Fiber
Matrix
68.3mE GPa
0.3m 
1.55fd 

29. • Implicit method vs Multi point constrain
29
Results and Discussion

30. • Implicit method vs Multi point constrain
30
Results and Discussion

31. • Implicit method vs Multi point constrain
31
Results and Discussion

32. • 3D– Aluminum Boron composite
• Element type: 8-node Hexa
32
Results and Discussion
379.3fE GPa
0.1f 
2L 
FE Model
Fiber
Matrix
68.3mE GPa
0.3m 
0.2t 
1.55fd 

33. • Implicit method vs Multi point constrain
33
Results and Discussion

34. • Implicit method vs Multi point constrain
34
Results and Discussion

35. • Implicit method vs Multi point constrain
35
Results and Discussion

36. • Implicit method vs Multi point constrain
36
Results and Discussion

37. • Implicit method vs Multi point constrain
37
Results and Discussion

38. • Implicit method vs Multi point constrain
38
Results and Discussion

39. • Effective Property Comparison
39
Results and Discussion
Effective
Properties
IBFEM 2D IBFEM 3D IBFEM 2D
(Multi point
constraint)
IBFEM 3D
(Multi point
constraint)
Ref: Sun and
Vaidya
(1996) [15]
Ref: Sun and
Chen (1990)
[16]
Ref: Chamis
(1984) [17]
Ref: Whitney
and Riley
(1966) [18]
E1(GPa) 216 214.83 214.85 215 214 214 215
E2(GPa) 146 147 144.07 143.85 144 135 156 123
E3(GPa) 146 147 144.07 143.85 144 135 156 123
G23(GPa) 45.6 46.1 46.07 45.97 45.9 43.6
G13(GPa) 55.3 54.44 54.35 57.2 51.1 62.6 53.9
G12(GPa) 56.3 54.44 54.35 57.2 51.1 62.6 53.9
ν23 0.29 0.24 0.25 0.25 0.29 0.31
ν13 0.19 0.198 0.195
ν12 0.19 0.198 0.195 0.19 0.19 0.20 0.19
ν32 0.29 0.24 0.25 0.25
ν31 0.13 0.13 0.13
ν21 0.13 0.13 0.13

40. • Weave composite
40
Results and Discussion

41. • Weave composite - Example
41
Results and Discussion
FE Model
Geometry
a b g h l
2 0.3 0.3 0.6 2

42. • Weave composite - Example
42
Results and Discussion
Assemble FE model
Fiber mesh
Matrix mesh

43. • Stress Plot
43
Results and Discussion
xx

44. • Stress Plot
44
Results and Discussion
yy

45. • Stress Plot
45
Results and Discussion
zz

46. • Stress Plot
46
Results and Discussion
zy

47. • Stress Plot
47
Results and Discussion
zx

48. • Stress Plot
48
Results and Discussion
xy

49. • Effective Property Comparison
49
Results and Discussion
Effective Properties IBFEM J. L. Kuhn and P. G.
Charalambides
0.76 0.83
0.29 0.32
0.2 0.23
ˆ
xE
ˆ
xyG
ˆxy

50. • Summary
– Successful extension of implicit boundary method to apply
periodic boundary condition
– The formulation is validated by comparing examples with known
results
• Future work
– Using the pixel density the material domain of the bone and voids
can be identified from the CT scan and can be used to construct the
representative volume element.
– Use of implicit boundary method for applying periodic boundary
condition to calculate effective property of composite material can
be extended to bones.
50
Summary and Scope of Future Work

51. Thank you
51

52. • Advantage:
– Generating automated conforming mesh for complicated geometry
results in poor quality mesh and need human intervention for mesh
correction. Use of structured mesh avoids mesh generation process.
– Use of structured mesh avoids numerical error associated with mesh
distortion
– The stiffness of internal elements are the same which reduces
computation time for FE calculations.
Implicit Boundary Finite Element Method (cont.)
Step function configuration
52
.

53. • Stiffness matrix comparison
Multi point constraint
Implicit boundary method
Implicit Boundary Finite Element Method (cont.)
Step function configuration
53
.
 
   
   
 
  
  
1 1 1 2 1
2 1 2 2 2
Lh
h p h p
p
L
B C B B C B B CX
X X d X X d
XB C B B C B B C

   
 
                             
                         
 
% % % % %
% % % % %