1. MANE 4240 & CIVL 4240 Introduction to Finite Elements
Prof. Suvranu De
Mapped element geometries and shape functions: the isoparametric formulation

2. Reading assignment:
Chapter , Lecture notes
Summary:
Concept of isoparametric mapping
1D isoparametric mapping
Element matrices and vectors in 1D
2D isoparametric mapping : rectangular parent elements
2D isoparametric mapping : triangular parent elements
Element matrices and vectors in 2D

3. For complex geometries
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General quadrilateral elements
Elements with curved sides

4. Displacement interpolation 2 1 1 1 1 s 1
Consider a special 4-noded rectangle in its local coordinate system (s,t) t
Displacement interpolation 2 1 1 1 1 s 1
Shape functions in local coord system 3 4

5. Recall that
Rigid body modes
Constant strain states

6. Global coordinate system Local coordinate system
Goal is to map this element from local coords to a general quadrilateral element in global coord system s y 1 2 3 4 x t
Global coordinate system s t 1 2 3 4
Local coordinate system

7. 1. Kronecker delta property
In the mapped coordinates, the shape functions need to satisfy
1. Kronecker delta property
Then
2. Polynomial completeness

8. The relationship
Provides the required mapping from the local coordinate system
To the global coordinate system and is known as isoparametric mapping
(s,t): isoparametric coordinates
(x,y): global coordinates

9. Examples t t 1 1 1 s 1 y s x t t 1 s s y 1 x

10. 1D isoparametric mapping
3 noded (quadratic) element x1 x3 x2 1 3 2 x s 1 3 2 1 1
Isoparametric mapping
Local (isoparametric) coordinates

11. NOTES
1. Given a point in the isoparametric coordinates, I can obtain the corresponding mapped point in the global coordinates using the isoparametric mapping equation
Question
x=? at s=0.5?

12. 2. The shape functions themselves get mapped
In the isoparametric coordinates (s) they are polynomials.
In the global coordinates (x) they are in general nonpolynomials
Lets consider the following numerical example 4 2 x 1 3 2
Isoparametric mapping x(s)
Simple polynomial
Inverse mapping s(x)
Complicated function

13. Now lets compute the shape functions in the global coordinates

14. Now lets compute the shape functions in the global coordinates
N2(x)
N2(s) 1 1 3 2 1 4 2 x s 1 3 2 1 1
N2(x) is a complicated function
N2(s) is a simple polynomial
However, thanks to isoparametric mapping, we always ensure
1. Knonecker delta property
2. Rigid body and constant strain states

15. Element matrices and vectors for a mapped 1D bar element3 2 x s 1 3 2 1 1
Displacement interpolation
Strain-displacement relation
Stress
The strain-displacement matrix

16. The only difference from before is that the shape functions are in the isoparametric coordinates
We know the isoparametric mapping
And we will not try to obtain explicitly the inverse map.
How to compute the B matrix?

17. Using chain rule
(*)
Do I know
Do I know
I know
Hence
From (*)

18. What does the Jacobian do?
Maps a differential element from the isoparametric coordinates to the global coordinates

19. The strain-displacement matrix
For the 3-noded element

20. The element stiffness matrix
NOTES
1. The integral on ANY element in the global coordinates in now an integral from -1 to 1 in the local coodinates
2. The jacobian is a function of ‘s’ in general and enters the integral. The specific form of ‘J’ is determined by the values of x1, x2 and x3. Gaussian quadrature is used to evaluate the stiffness matrix
3. In general B is a vector of rational functions in ‘s’

21. Isoparametric mapping in 2D: Rectangular parent elements
© 2002 Brooks/Cole Publishing / Thomson Learning™
Parent element
Mapped element in global coordinates
Isoparametric mapping

22. Shape functions of parent element in isoparametric coordinates
Isoparametric mapping

23. NOTES:
The isoparametric mapping provides the map (s,t) to (x,y) , i.e., if you are given a point (s,t) in isoparametric coordinates, then you can compute the coordinates of the point in the (x,y) coordinate system using the equations
2. The inverse map will never be explicitly computed.

24. 8-noded Serendipity element
© 2002 Brooks/Cole Publishing / Thomson Learning™ t 7 4 3 1 1 1 6 8 s 1 5 1 2

25. 8-noded Serendipity element: element shape functions in isoparametric coordinates

26. NOTES 1. Ni(s,t) is a simple polynomial in s and t
NOTES 1. Ni(s,t) is a simple polynomial in s and t. But Ni(x,y) is a complex function of x and y.
2. The element edges can be curved in the mapped coordinates
3. A “midside” node in the parent element may not remain as a midside node in the mapped element. An extreme example t 5 y 2 1 1 1 1 1 5 8
2,6,3 8 6 s x 1 7 3 4 7

27. 4. Care must be taken to ensure UNIQUENESS of mapping2 y t 2 1 1 1 1 3 1 x s 1 3 4 4

28. Isoparametric mapping in 2D: Triangular parent elements
Parent element: a right angled triangles with arms of unit length
Key is to link the isoparametric coordinates with the area coordinates 2 1
P(s,t) s t s 3 1 1

29. Isoparametric mapping Parent shape functions
Now replace L1, L2, L3 in the formulas for the shape functions of triangular elements to obtain the shape functions in terms of (s,t)
Example: 3-noded triangle s t
2 (x2,y2) t 2 1 1 y x
3 (x3,y3) s
1 (x1,y1) 3 1
Isoparametric mapping
Parent shape functions

30. Element matrices and vectors for a mapped 2D element
Recall: For each element
Displacement approximation
Strain approximation
Stress approximation
Element stiffness matrix
Element nodal load vector
STe e

31. In isoparametric formulation
Shape functions first expressed in (s,t) coordinate system
i.e., Ni(s,t)
2. The isoparamtric mapping relates the (s,t) coordinates with the global coordinates (x,y)
3. It is laborious to find the inverse map s(x,y) and t(x,y) and we do not do that. Instead we compute the integrals in the domain of the parent element.

32. NOTE
1. Ni(s,t) s are already available as simple polynomial functions
2. The first task is to find and
Use chain rule

33. In matrix form
This is known as the Jacobian matrix (J) for the mapping
(s,t) → (x,y)
We want to compute these for the B matrix
Can be computed

34. How to compute the Jacobian matrix?
Start from

35. Need to ensure that det(J) > 0 for one-to-one mapping

36. 3. Now we need to transform the integrals from (x,y) to (s,t)
Case 1. Volume integrals
h=thickness of element
This depends on the key result

37. Proof y s t ds dt db da j x i

38. ISOPARAMETRIC COORDINATES
Problem: Consider the following isoparamteric map y 1 2 3 4
(3,1)
(6,1)
(6,6)
(3,6) x s t 1 2 3 4 y
ISOPARAMETRIC COORDINATES
GLOBAL COORDINATES

39. Displacement interpolation
Shape functions in isoparametric coord system

40. In this case, we may compute the inverse map, but we will NOT do that!
The isoparamtric map
In this case, we may compute the inverse map, but we will NOT do that!

41. The Jacobian matrix since
NOTE: The diagonal terms are due to stretching of the sides along the x-and y-directions. The off-diagonal terms are zero because the element does not shear.

42. Hence, if I were to compute the first column of the B matrix along the positive x-direction
I would use
Hence

43. The element stiffness matrix

44. Case 2. Surface integrals
For dST we consider 2 cases
Case (a):
dST 2 t y ds 1 1 2 3 s
dST 4 x 3 4 ds

45. Case (b):
dST 2 t y 1 1 2 dt 3 s
dST dt 4 x 3 4

46. Summary of element matrices in 2D plane stress/strain
Quadrilateral element t 1 s 1
Suppose s=-1 gets mapped to ST 1 1

47. Summary of element matrices in 2D plane stress/strain
Triangular element t 1 t
s=1-t t s
Suppose t=0 gets mapped to ST 1