Computational Mechanics 4 - Linear Algebra

Computational Mechanics 4 - Linear Algebra#

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.

01_Linear-Algebra#

  • 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

02_Gauss_elimination#

  • 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.lu to create the PLU decomposition

  • How to use the PLU efficient structure to solve our linear algebra problem (solveLU)

03_Linear-regression-algebra#

  • 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

HW_04#

Project_04 - Practical Linear Algebra for Finite Element Analysis#

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.

Mesh image of truss

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 \(n~1-7\), 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 \(el~1-11\), 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