# Archives

### Reasons and solutions for the optimization algorithm convergence

Various reasons are responsible when the optimization algorithm’s convergence is not ensured. Some of the following steps are beneficial in fixing this issue:  Ensure that the objective function and the constraints are appropriately formulated. Ensure that the objective and constraint functions are continuous and differentiable at least up to the second order. If the objective […]

Gradient vector: If the partial derivative of a function f(x) (function having $n$ variables) with respect to the $n$ variables $x_1$, $x_2$…..$x_n$ at a point $x^{\star}$ is taken, then that partial derivative vector of f(x) represents the “gradient vector” which is represented by symbols like $c$ or $\triangledown {f}$, as: \$\mathbf{c}=\nabla f\left(\mathrm{x}^{*}\right)=\left[\begin{array}{c}\frac{\partial f\left(\mathrm{x}^{*}\right)}{\partial x_{1}} \\ \frac{\partial f\left(\mathrm{x}^{*}\right)}{\partial […]

### Norm and condition number of a matrix

In a numerical analysis with a vector involvement, norms are essential to predict the various errors involved in the numerical analysis. A norm is a function ||.|| in a vector space V. If A is a n*n matrix, then its norm is a real number and denoted by ||A||. A norm satisfies the following properties: […]

### IGA 2: Computational Geometries /Mathematical Preliminaries

This post will deal with the fundamental difference in mathematics behind both FEM and CAD.

### Variational Calculus: Deriving the strong and weak form

There are many ways in which we can obtain the weak form and the strong form. We focus on the Variational Principles to derive them because FEniCS works on the variational form

### A primer on shell structures

Through this post, I attempt to provide a starter pack for shell structures. This includes trying to understand their structural behaviour, a very brief overview of shell-kinematics and the basic methodology for shell analysis through FEM. The theory and implementation follows the widely popular Kirchhoff-Love theory described for thin shells (thickness/length < 20)

### Vector and Matrix Operations in PETSc (via petsc4py)

Through this post, you will learn how to do basic vector and matrix operations through petsc4py and how to solve a small linear problem using petsc4py. In addition, FEniCS PETSc formats and conversions are also briefly mentioned.

### Preconditioners

Today we are going to learn some important concepts related to matrix solvers. We will learn some of the preconditioners. They are often used as a black box in ABAQUS. We use its output ignoring the mechanism behind it. I hope with going through this blog post you will get a fair idea about the […]