Cfd with python: from modelling to simulation

This lesson clarifies the concept of finite element analysis and highlights the role of finite element simulations

In this lesson we start to give the prerequisites of functional analysis:

• test function

• distributions

• distributional derivatives

• functional spaces

The following lesson serves to clarify the concepts of:

• domain discretization

• 1°,2° order approximation functions

• lagrange basis

In this chapter we show the concept ofs:

• reference triangle

• map between generic and reference triangle

• finite element

Elastic membrane problem:

• model definition

• derivation of the weak formulation

• algebraic formulation via Galerkin method

  
  

This lesson is aimed at introducing the diffusion advection reaction equation:

• explain the model

• write weak formulation

• introduction to the Pèclet number

• Stabilization methods

This lesson explain the heat diffusion:

• explain the model

• write the weak problem in discrete form

• extract the mass and stifness matrices

• discretization via Crank-Nicholson method

The students will be guided through these steps, necessary to set up the simulation environment:

• WSL configuration

• ubuntu download

• FEniCs installation in ubuntu

• run a demo file

• solve recurring problems

The students will understand how subsequent lessons will be set up

• The student will learn how to generate a computational domain using the fenics library in python

• Generate a triangular mesh with variable resolution.

• The student will learn the basic workflow to create simple cad models in FreeCAD to use them in numerical simultation

  
  

• This lesson show interactively the mesh generation process using the open source software GMSH

• This lesson shows how to correctly export the mesh and convert the format into .xdmf which is the one for FEniCs.

• The student will learn how to implement the FEM  to solve the poisson equation proposed in section 2.

• The approximation error with the analytical solution, for different values of mesh size, will be estimated.

• The student will learn how to implement the FEM  to solve the diffusion-advection-reaction equation proposed in section 2.

• The presented problem is a simplified model in order to describe the dispersion of two pollutants that interact with each other giving rise to a reaction product.

• General introduction to the problem, the students will learn the equations of mass and momentum conservation for a generic body in the continuum mechanics framework. We then explain how to adapt them to recover the Navier-Stokes equation for an incompressible fluid.

• In this section we derive the variationl formulation of the Navier-Stokes system then the theory behind one of the best-known numerical methods, the Incremental Pressure Correction Scheme, for solving navier stokes equations  will be shown.

• In this lesson we exploit the implementation of the 2D numerical problem of turbolent flow against an obstacle in a channel

• Students will understand the Courant-Friedrichs-Lewy condition, which is really important in computational fluid dynamics

• We will introduce the Stokes problem and the student will learn how to solve it numerically.

• Students will use a geometry  generated with an external software.

• The student will learn how to do post processing to visualize and interpret simulation data.

• Through Paraview we will show you how to display streamlines, quivers, contours, sliced planes.

• The methods are applicable to all simulations seen