Finite Element Method

1. Finite Element Method Chapter 10 Isoparametric Formulation

2. Definition: • The term isoparametric (same parameters) is derived from the use of the same shape (interpolation) functions N to define the element’s geometric shape as are used to define the displacements within the element. Basic Principle of Isoparametric Elements Alternatively: The basic principle of isoparametric elements is that the interpolation functions for the displacements are also used to represent the geometry of the element.

3. In this formulation, displacements are expressed in terms of the natural (local) coordinates and then differentiated with respect to global coordinates. Accordingly, a transformation matrix [J], called Jacobian, is produced. • If the geometric interpolation functions are of lower order than the displacement shape functions, the element is called subparametric. If the reverse is true, the element is referred to as superparametric. • The isoparametric formulation is generally applicable to 1-, 2- and 3- dimensional stress analysis. The isoparametric family includes elements for plane, solid, plate, and shell problems. Also, it is applicable for nonstructural problems. Basic Principle of Isoparametric Elements

4. In this formulation, displacements are expressed in terms of the natural (local) coordinates and then differentiated with respect to global coordinates. Accordingly, a transformation matrix [J], called Jacobian, is produced. • If the geometric interpolation functions are of lower order than the displacement shape functions, the element is called subparametric. If the reverse is true, the element is referred to as superparametric. • The isoparametric formulation is generally applicable to 1-, 2- and 3- dimensional stress analysis. The isoparametric family includes elements for plane, solid, plate, and shell problems. Also, it is applicable for nonstructural problems. Basic Principle of Isoparametric Elements

5. The isoparametric formulation makes it possible to generate elements that are nonrectangular and have curved sides. So it can facilitate an accurate representation of irregular elements. • Numerous commercial computer programs have adopted this formulation for their various libraries of elements. Basic Principle of Isoparametric Elements

6. Step 1: Select the Element Type • Isoparametric Formulation ofthe Bar Element Stiffness Matrix We consider the bar element to have two degrees of freedom s. For the special case when the s and x axes are parallel to each other, the s and x coordinates can be related by where xc is the global coordinate of the element centroid. Using the global coordinates x1 and x2 in

7. Step 2: Select Displacement Functions We begin by relating the natural coordinate to the global coordinate by This linear shape functions map the s coordinate of any point in the element to the x coordinate

8. Step 2: Select Displacement Functions The displacement function within the bar is now defined by the same shape functions Step 3: Define the Strain/Displacement and Stress/Strain Relationships

10. Basic Principle of Isoparametric Elements As shown in the figure, the local (natural) coordinate system (,)for the two elements have their origins at the centroids of the elements, with (,)varying form –1 to 1. The natural coordinate system needs not to be orthogonal and neither has to be parallel to the x-y axes. The coordinate transformation will map the point (,)in the master element to x(,) and y(,)in the slave element.

12. Examples t t 1 1 1 s 1 y s x t t 1 s s y 1 x

13. Step 2: Select Displacement Functions In other words, we look for shape functions that map the regular shape element in isoparametric coordinates to the quadrilateral in the x-y coordinates whose size and shape are determined by the eight nodal coordinates x1, y1, x2, y2, ….., x4, y4.

14. Step 2: Select Displacement Functions

15. Step 2: Select Displacement Functions

16. Step 2: Select Displacement Functions Shape Function for 4-Nodes quadrilateral Elements These shape functions are seen to map the (,)coordinates of any point in the rectangular element in the above master element to x and y coordinates in the quadrilateral (slave) element. For example, consider the coordinates of node 1, where =-1,=-1 using the above equation, we get x=x1 , y=y1

18. Step 2: Select Displacement Functions Shape Function for 4-Nodes quadrilateral Elements

19. Step 2: Select Displacement Functions Shape Function for 4-Nodes quadrilateral Elements where n = the number of shape functions associated with number of nodes

20. Step 2: Select Displacement Functions where u and v are displacements parallel to the global x and y coordinates

21. Step 3: Define the Strain/Displacement and Stress/Strain Relationships Using Chain Rule

22. Step 3: Define the Strain/Displacement and Stress/Strain Relationships Using Chain Rule Can be computed We want to compute these for the B matrix This is known as the Jacobianmatrix (J) for the mapping (,) → (x,y)

23. Define the Strain/Displacement and Stress/Strain Relationships Basic Principle of Isoparametric Elements

24. Define the Strain/Displacement and Stress/Strain Relationships Since: Basic Principle of Isoparametric Elements

25. Define the Strain/Displacement and Stress/Strain Relationships Basic Principle of Isoparametric Elements

26. Define the Strain/Displacement and Stress/Strain Relationships Basic Principle of Isoparametric Elements

27. Derive the Element Stiffness Matrix and Equations Basic Principle of Isoparametric Elements

28. Derive the Element Stiffness Matrix and Equations The shape function are: Their derivatives:

29. Derive the Element Stiffness Matrix and Equations Basic Principle of Isoparametric Elements Explicit formulation for |J| for 4 node Element

30. Derive the Element Stiffness Matrix and Equations Basic Principle of Isoparametric Elements

31. The element body force matrix Basic Principle of Isoparametric Elements and Xb and Yb are the weight densities (body weight/unit volume) in the x and y directions, respectively.

32. The element surface force matrix Basic Principle of Isoparametric Elements We assumed surface loading at edge with overall length L (Since N1 = 0 and N2 = 0 along edge =1, and hence, no nodal forces exist at nodes 1 and 2, Note that for one-dimensional transformation | J | = L / 2. Also, p, p are the pressure distributions inand , respectively.

33. 3 4 (6,6) (3,6)  4 3 1 1 1 y  1 2 1 (3,1) (6,1) 2 1 x Problem:Consider the following isoparamteric map ISOPARAMETRIC COORDINATES GLOBAL COORDINATES

34. Displacement interpolation Shape functions in isoparametric coord system

35. The isoparamtric map

36. 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.

37. Hence, if I were to compute the columns of the B1matrix along the positive x-direction Hence

38. Numerical IntegrationGauss Quadrature • Gauss quadratureimplements a strategy of positioning any two points on a curve to define a straight line that would balance the positive and negative errors. • Hence, the area evaluated under this straight line provides an improved estimate of the integral.

41. Two points Gauss-Legendre Formula • Assume that the two Integration points are xo and x1 such that: • c0 and c1 are constants, the function arguments x0 and x1 are unknowns…….(4 unknowns)

42. Two points Gauss-Legendre Formula • Thus, four unknowns to be evaluated require four conditions. • If this integration is exact for a constant, 1st order, 2nd order, and 3rd order functions:

43. Two points Gauss-Legendre Formula • Solving these 4 equations, we can determine c0, c1, x0 and x1.

44. f(x) f(x) f(x1) f(xo) x xo x1 a b x -1 1 Two points Gauss-Legendre Formula • Since we used limits for the previous integration from –1 to 1 and the actual limits are usually from a to b, then we need first to transform both the function and the integration from the x-system to the xd-system

45. Integration point Weight Higher-Points Gauss-Legendre Formula

46. Multiple Points Gauss-Legendre Points Weighting factor Function argument Exact for 2 1.0 -0.577350269= up to 3rd 1.0 0.577350269= degree 3 0.5555556=5/9 0.774596669= up to 5th 0.8888889=8/9 0.0 degree 0.5555556=5/9 0.774596669= 4 0.3478548 -0.861136312 up to 7th 0.6521452 -0.339981044 degree 0.6521452 0.339981044 0.3478548 0.861136312 6 0.1713245 -0.932469514 up to 11th 0.3607616 -0.661209386 degree 0.4679139 -0.238619186 0.4679139 0.238619186 0.3607616 0.661209386 0.1713245 0.932469514

47. Double integral: Multiple Integration • In General transformation to natural coordinate:

48. Where Wij=WiWj

49. For n=2 Wij=WiWj=1  1 1 1  1

50. Shape function for Isoparametric 6-Nodes triangular Elements