# Blog

### 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, Hessian matrix and Quadratic forms

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 […]

### Uncertainities in Topology Optimization

Various types of uncertainties are considered in the topology optimization, like uncertainty in loading, geometry and material properties, etc. Two methods account for the uncertainties in the topology optimization framework. They are- 1. Probabilistic based methods (Statistical or reliability-based methods) 2. Non-probabilistic based methods The traditional approach to account for the uncertainties in the structure […]

### Extrude 2D geometry to 3D geometry in Comsol

Select 2D space dimension and select specific physics. Model some 2D geometry under Geometry 1 tab. Now, go to add component option under Home Tab and click on 3D. Right click on Geometry 2 and click on work plane. Work Plane 1 option now appears under Geometry 2 button and Plane Geometry option now appears […]

### Explicit Dynamic Analysis

Need for explicit dynamic analysis Implicit and explicit are the two main types of time integration methods that are used to perform dynamic simulations. The explicit time integration is more efficient for the following situations- 1. To simulate the buckling and complex contact problem. 2. To model material deformation and failure at a high strain […]

### Architected Materials: Metamaterials

Metamaterials are those materials that are not found in nature and are artificially manufactured materials. The properties of such materials are derived from their internal microstructure (geometrical configuration) and not from their chemical composition. The concept of metamaterials was initially coined in the context of optics and electromagnetism. The properties of the metamaterials on the […]

### Numerical Challenges occurs in SIMP based Topology Optimization

A) Checkerboard Pattern: a) Checkerboarding refers to the formation of alternate solid and void elements in the final topologically optimized structure and possesses artificial high stiffness to the numerical model. b) This issue is primarily due to the discretization error of the numerical FE model. c) It occurs mainly in those models which have been […]

### Mesh refinement in Topology Optimization

Mesh refinement or discretization is a fundamental step in finite element analysis. In finite element analysis, the refinement of mesh is done only in those areas that are of interest (eg: the region of high stress in the stress analysis) so that to obtain good accuracy with minimum computational efforts. However, in the context of […]