Answer the following descriptive questions.

a. Explain what the "isoparametric" mapping means.

b. For a 1D 3node quadratic element, what is the condition to make the Jacobian constant?

c. When quadratic elements are used, will displacements be continutous across the element boundary with the adjacent element? Will strains be continuous?

d. The stress in a bar varies linearly. When three nodes are equally spaced in the bar, which result is more accurate, two linear elements or one quadratic element?

e. What will happen if the shape of an element is distorted in a such a way that there is a concave region within the element?

f. List possible problems when the Jacobian of an element is negative.
g. How many Gauss quadrature points are required to accurately integrate a polynomial of order 7 ?
h. How many zero eigenvalues are expected for the properly integrated stiffness matrix of a quadrilateral element?
i. When the stiffness matrix of a quadrilateral element has four zero eigenvalues, explain the possible cause of the problem.
j. Explain why it is not acceptable to put a 4node quadrilateral element adjacent to an 8node quadrilateral element in a finite element mesh.
k. Why is numerical quadrature needed for 2D and 3D elements?
l. What are the main sources of error in the solutions obtained using isoparametric elements?