MANE 4240 & CIVL 4240 Introduction to Finite Elements

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

2. Reading assignment: Chapter 10.1-10.3, 10.6 + 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. © 2002 Brooks/Cole Publishing / Thomson Learning™ For complex geometries General quadrilateral elements Elements with curved sides

4. 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 4 3

5. Recall that Rigid body modes Constant strain states

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

7. 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 2 1 3 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 2 1 3 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 2 1 3 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 element 1 3 2 x s 2 1 3 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. © 2002 Brooks/Cole Publishing / Thomson Learning™ Isoparametric mapping in 2D: Rectangular parent elements 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. © 2002 Brooks/Cole Publishing / Thomson Learning™ 8-noded Serendipity element t 7 4 3 1 1 1 6 8 s 1 5 2 1

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

26. NOTES1. 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 1 2 1 1 1 5 1 2,6,3 8 8 6 s x 1 7 4 3 7

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

28. Isoparametric mapping in 2D: Triangular parent elements t 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. t s y x 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 2 (x2,y2) t 2 1 1 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. 1 2 (6,6) (3,6) t 2 1 1 1 1 y s 1 4 3 (3,1) (6,1) 4 3 x Problem: Consider the following isoparamteric map y ISOPARAMETRIC COORDINATES GLOBAL COORDINATES

38. Displacement interpolation Shape functions in isoparametric coord system

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

40. 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.

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

42. The element stiffness matrix