Welcome to Computational Mechanics Module #4! In this module we will explore applied linear algebra for engineering problems and revisit the topic of linear regression with a new toolbox of linear algebra. Our main goal, is to transform large systems of equations into manageable engineering solutions.
- How to solve a linear algebra problem with
np.linalg.solve - Creating a linear system of equations
- Identify constants in a linear system
$\mathbf{A}$ and$\mathbf{b}$ - Identify unknown variables in a linear system
$\mathbf{x}$ - Identify a singular or ill-conditioned matrix
- Calculate the condition of a matrix
- Estimate the error in the solution based upon the condition of a matrix
- Graph 2D and 3D linear algebra problems to identify a solution (intersections
- of lines and planes)
- How to solve a linear algebra problem using Gaussian elimination (
GaussNaive) - Store a matrix with an efficient structure LU decomposition where
$\mathbf{A=LU}$ - Solve for
$\mathbf{x}$ using forward and backward substitution (solveLU) - Create the LU Decomposition using the Naive Gaussian elimination process (
LUNaive) - Why partial pivoting is necessary in solving linear algebra problems
- How to use the existing
scipy.linalg.luto create the PLU decomposition - How to use the PLU efficient structure to solve our linear algebra problem (
solveLU)
- How to use the general least squares regression method for almost any function
- How to calculate the coefficient of determination and correlation coefficient for a general least squares regression,
$r^2~ and~ r$ - Why we need to avoid overfitting
- How to construct general least squares regression using the dependent and independent data to form
$\mathbf{y}=\mathbf{Za}$ . - How to construct a piecewise linear regression
In this project we will perform a linear-elastic finite element analysis (FEA) on a support structure made of 11 beams that are riveted in 7 locations to create a truss as shown in the image below.
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The triangular truss shown above can be modeled using a direct stiffness method [1], that is detailed in the extra-FEA_material notebook. The end result of converting this structure to a FE model. Is that each joint, labeled $n1-7$, short for node 1-7 can move in the x- and y-directions, but causes a force modeled with Hooke's law. Each beam labeled $el1-11$, short for element 1-11, contributes to the stiffness of the structure. We have 14 equations where the sum of the components of forces = 0, represented by the equation