Finite Element Analysis

Q3. (20 points) Consider a generic 3-node triangle, numbered according to the figure in the class notes. Write a series of subroutines in a program language of your choice (Matlab, Python, C, ...) that does the following. Note: these routines may be very short. You should provide comments and also use names for variables that are suggestive, i.e. use good programming habits. (iv) Verify via hand calculations (i.e. the final formula for B in terms of x,y given in class) that your B matrix is correct.

Problem 1: Calculate the y-component of the displacement of node 4 and the axial force in each element for the three-bar truss shown in the figure. Let E = 2 (107)N/cm? and A = 5 cm? for each element.

A hot surface at 125°C is to be cooled by attaching 8 cm long, 0.8 cm in diameter aluminum pin fins(k = 237 W/m-°C and a = 97.1 X 106 m²/s) to it. The temperature of the surrounding medium is 17°C, and the heat transfer coefficient on the surfaces is 38 W/m²-°C. Initially, the fins are at a uniform temperature of 29°C, and at time t=0, the temperature of the hot surface is raised to 125°C.Assuming one-dimensional heat conduction along the fin and taking the nodal spacing to be Ax = 2.0 cm and a time step to be At = 1.5 s, determine the nodal temperatures after 5 minutes by using the explicit finite difference method. Also determine how long it will take for steady conditions to be reached.

[01] Given the language: L(M)=\left\{\omega: \omega \in\{\theta, 1\}^{*}\left|\omega_{n} \neq \omega_{n-1}, n=\right| \omega \mid>=0\right\} Do the following: Express L(M) in natural language. · Express L (M) as an RE. c. Express L(M) as an NFA State diagram. . Convert NFA to DFA e. Minimize the resulting DFA

The bar of length 2L (see the figure) is loaded along its axes with the load q(x)=q_{0} x / L The left end is fixed. At the right end,the spring of stiffness k connects the bar to the fixed support. It is fully relaxed when no external load exists. The elastic modulus E is uniform, and the cross-sectional area varies as A(x)=A_{0}(3-x / L) \text { The governing equation of the problem is } \frac{d}{d x}\left(E A(x) \frac{d u(x)}{d x}\right)+q(x)=0 \text {. } (a) (20%) Write the boundary conditions in terms of displacement and its derivative. Which one is the essential and which one is the natural boundary condition? [Hint: Watch the signs!] (b) (40%) Define the one condition that must be satisfied by the otherwise arbitrary test function w(x). Then derive the weak form of the problem. (c) (40%) Discretize the problem using the 2 finite elements (el and e2) and three nodes (seethe figure). The interpolation functions are piecewise linear and are illustrated in the figure. Derive the finite element equations for the two unknowns: \left[\begin{array}{ll} K_{22} & K_{23} \\ K_{32} & K_{33} \end{array}\right]\left\{\begin{array}{l} u_{2} \\ u_{3} \end{array}\right\}=\left\{\begin{array}{l} F_{2} \\ F_{3} \end{array}\right\} \text { Compute the coefficients } K_{22}, K_{23}, K_{33}, F_{2} \text {, and } F_{3} \text { in terms of known quantities } k, q_{0}, A_{0}, E \text { and } L

8. A simply supported steel beam is subjected to two point-loads as shown in the figure. Thegoverning differential equation for the deflection curve is E I \frac{d^{2} q}{d x^{2}}-M(x)=0 b) Redo part a using 8 elements (each element is 50 cm long). c) Briefly discuss the results in parts (a) and (b).

Q1. Using Hermite cubic shape functions in the strain-displacement matrix, show that for constant flexural rigidity EI, the element stiffness matrix is given as below. Hint: the functions are given on the next page, for an element spanning Ω = (0,l), so set h=l.

Consider a 2-D region shown below, where the boundary conditions are shown as symbols on the figure, and their corresponding values are shown in the table. Using the finite difference mesh shown: Write the finite-difference equations for each node. (22 nodes) b) Find the steady state temperatures of all the nodes. (22 nodes)

4. The problem is related to conversion of coordinate systems. b. Let node 3 be at x = 7 and node 4 be at x = 10 on the one-dimensional linear scale.Calculate f xxin(x) dx by converting it first to local, s coordinate system, and then to natural, l2 coordinate system.

Q1. (10 points) The element shown has 3 linear sides and one quadratic side. Determine the five shape functions N₁ (x, y). Dimensions are 2a by 2b. Hint: Use either method 1 or method 3 (demonstrated for the quadratic triangular element) described in the class notes, which are discussed in the attached pages similar to Ch 6 of the Cook textbook and in Handout_Ch04supp.pdf (Section 6.11).