Describe the role of the stiffness matrix for fluid flow problems in the finite element (FE) method.
The stiffness matrix, particularly, the global stiffness matrix allows us to assemble all of the local stiffness matrices of FE elements. For a 2D linear finite element, the triangularization of a uniform Cartesian is achieved by applying the Dirichlet boundary condition thus setting a sum of local stiffness matrices equal to a body integral. In fluid dynamics, finite elements using conservation laws gives us the final FE statement relating the summation of mass and stiffness matrices with boundary and body integrals. This also allows us to obtain the finite volume method statement which is the most commonly used method in modern day fluid dynamics algorithm
How many times does a 1 appear in the list of integers from 0 to 9,999,999?
We first determine how many total digits are used in the list. This problem is significantly simplified by the fact that we are searching the number of times 1 appears and we can ignore cases in which no 1’s appear. Thus we can rewrite numbers with 0’s preceding them such as: 1 → 0,000,001 and 1,000 → 0,001,000 Thus we use 0 digits for every number and there are a total of 10,000,000 numbers 10,000,000 ∗ 7 = 70,000,000 digits. Each digit has an equal probability of occurring because of the rewritten form, thus there is a 1 in 10 chance that any given digit is a 1. By multiplying the total number of digits by this probability, we conclude how many times the digit 1 appears: Number of occurrences = 170,000,000 = 7,000,000
For a control system with $$K_V = 2$$, what is the steady-state error for inputs $$2u(t), 2tu(t)$$, and $$2t^2u(t)$$
$$2u(t) = 0$$ $$2tu(t)=2/K_V=1$$ $$2t^2u(t)=\infty$$