AE420/ME471 – Introduction to the Finite Element Method Practice Problems Fall 2022
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AE420/ME471 – Introduction to the Finite Element Method
Practice Problems Fall 2022
Problem 1: (total 50 marks).
a. Suppose you want to use 3-noded one dimensional elements below, where the coordinates of nodes 1, 2, and 3 are x = 1, - 1 and 0.7. You have implemented this element in Abaqus. When you run the simulations, however, Abaqus terminates with an error. Show calculations to explain why. (10 marks)
b. Provide the expression in terms of the triangular coordinates Li (i = 1, 2, 3) of all the shape functions for the 6-node triangular element shown in the adjacent figure. (i) What convergence rate would you expect for this element compared to a 3 node triangular element? (ii) For a typical finite element problem, would it be more efficient to increase the no. of 3 node triangular elements such that the typical element dimension is h instead of 2h, or to use 6-node triangular elements of typical element dimension 2h, assuming that the computational cost is the same for both? Explain your answers with calculations. (10 marks)
c. Derive the expressions in terms of ξ,η for the Jacobian matrix [J], det[J] and for the following mapping: (10 marks)
d. Identify the problems associated with connecting four-node and eight-node elements in the pattern shown below. Consider all pairs of adjacent elements (1 and 2, 1 and 3, …). Indicate where compatibility problems would arise and why. (10 marks)
e. Finite element simulations are to be performed on the problem below, where a hemispherical stress-ball is indented by a vertical punch. Briefly describe how you would model this problem in finite element method in as computationally efficient a manner as possible. Indicate the appropriate boundary conditions and type of element you would choose. (10 marks)
2023-08-04