Exploration

Convex hull is the smallest envelope that contains the points set. It is used to construct the grain in grain-based model. There are several methods to generate the convex hull data, but it may need some efforts if you want to put it into ABAQUS model. Here, I post a simple method which is suitable for programming.

Generate convex hull data

Several methods can be used to generate the convex hull data. I am used to generate the convex hull data with my own code. But if you do not like write code, you can use Mathematica or Matlab to generate the data.

For example, in Mathematica, you can use following code to generate a 3D convex hull:

pts = RandomReal[{-1, 1}, {20, 3}];
ConvexHullMesh[pts]

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This is a homework of computational nanomechanics. The basic requirement is to use the Lagrangian function to describe the motion of particles. In homework, it requires 5 particles. As an enhancement, I rewrote the code with Qt and use GNU Scientific Library(GSL) to finish the task.

Concept

The Lagrangian equation of motion describe the particle motion with energy method, the equation is:

$$ \displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_k}-\frac{\partial L}{\partial x_k}=0$$

$ L$ is the Lagrangian function of the system, $ k$ denotes the degree of freedom.

$$ \displaystyle L = T – U$$

$ T$ is the kinetic energy and $ U$ is the potential energy.

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