Question 4 (20 marks) Consider a two-dimensional square which positions within the square are given by coordinates (x,y). This square has an external node 1 at coordinates (-1,-1), external node 2 at coordinates (1,-1), external node 3 at coordinates (1,1) and finally external node 4 at coordinates (-1,1). The interpolation function at node 1 is given by N1 = (1/4)(1-x)(1-y), the interpolation function at node 2 is given by N2 = (1/4)(1+x)(1-y), the interpolation function at node 3 is given by N3 = (1/4)(1+x)(1+y), and finally the interpolation function at node 4 is given by N4 = (1/4)(1- x)(y+1). The temperature at node 1 is 200K, the temperature at node 2 is 80K, the temperature at node 3 is 90K, and finally the temperature at node 4 is 110K. (a) Using the interpolation functions given above, calculate the temperature at positions - 0.5,0), (0,-0.5), (0.5,0) and (0,0.5) (10 marks) dT (b) The heat flux vector is given by q = (-ke dT dx' where T>is the temperature and k > 0 is the coefficient of thermal conductivity. Calculate the heat flux vector at coordinates (-0.5,0), (0,-0.5), (0.5,0) and (0,0.5) in terms of the thermal conductivity k. (10 marks)