For the fournode quadrilateral element a shown in the global

For the four-node quadrilateral element (a), shown in the global coordinate system, evaluate [J] (the Jacobian), and |J| (its determinant). For element (b), determine the x & y coordinates of the point located at s = t = 0.6. Use the node numbers as indicated. What is an acceptable error in a finite element analysis (in %)? Explain. need higher order elements in a pure heat-transfer analysis?

Solution

given the 4 node rectangle with coordinates (4,-3), (4,3), (-4,3),(-4,-3) we need to map to the quadrilateral

(4,3), (0,3),(-4,-3),(0,-3)

Writing the cooridnates for x and y as a matrix

[4 4 -4 -4] row 1 ( x\'s)

[-3 3 3 -3] row 2 (y\'s)

should transform to [4 0 -4 0]

                              [3 3 -3 -3]

The matrix that does this is the Jacobian, and by solving it is   {1/2 -2/3}

                                                                                            {3/4     0}

The determinant is (1/2)* 0 - (-2/3)*3/4 = 1/2

the area of the rectanglke is 48, that of the quadrilateral is 24, the mapping reduces the area by 1/2 which is what is shown by the determinant.

Mapping (.6,.6) thru the Jacobian, get [(1/2)*.6 -2/3*.6], [ 3/4*.6, 0] = -0.1,0

Acceptable error in FEM, depends on the mapping and the distortion measured by Jacobians.

Accpetable is 5-10%