Project 1
AIM:
Derive FVM formulation for 2D Heat conduction equation and solve it in a distorted mesh stored in an unstructured format. Mesh will be supplied in a text file, build a data structure in python to read and store the mesh. Solve the heat diffusion problem for this unstructured mesh.
Finite Volume Method of 2D steady state heat conduction problem :
This method is used to convert the partial differential equation to a system of algebraic
equation that can be solved iteratively by the tri-diagonal method of linear algebra.
Grid generation:
The first step in the finite volume method is to divide the domain of the wall into discrete
control volumes. Each node is surrounded by a control volume or cell. The nodes to the west,
east, north, south are identified by W,E,N, and S respectively. Also, the side faces of the control volume is referred to by’‘w,’‘e ,’‘ n ,’‘s..
The distances between the nodes W and P, and between nodes P and E , are identified by
Discretization:
When the above equation is formally integrated over the control volume we obtain:
This is the derivation for the FVM formulation for 2D Heat conduction equation.
UNSTRUCTURED MESH PROCESSING:
Unstructured meshes are meshes with general connectivity (GCON) whose structure is arbitrary and therefore the connectivity of elements must be defined and stored.
DISORTED MESH:
Mesh distortion is controlled by adaptively remapping an internal mesh framework while maintaining its original topology. This technique has been used in studies (Donea et al., 2004) to simulate the cutting process of ductile materials, in which damage and fracture phenomena are not generally the primary concern.
DATA SRTRUCTURE:
A data structure is a specialized format for organizing, processing, retrieving and storing data. There are several basic and advanced types of data structures, all designed to arrange data to suit a specific purpose. Data structures make it easy for users to access and work with the data they need in appropriate ways.
Data Structure can be defined as the group of data elements which provides an efficient way of storing and organising data in the computer so that it can be used efficiently.
Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms.
CFD SOFTWARE ALGORITM:
Computational Fluid Dynamics (CFD) creates a digital simulation that visualises the flow of fluids and the way they are affected by objects. It is a very powerful tool that shows in advance how temperature, pressure and velocity are going to behave in a design.
Computational fluid dynamics (CFD) is the use of applied mathematics, physics and computational software to visualize how a gas or liquid flows -- as well as how the gas or liquid affects objects as it flows past. Computational fluid dynamics is based on the Navier-Stokes equations.
Exposure to real software development:
Nice software develpoment of CFD is mostly used to,
- analysis and evaluation of user comfort by using a 2D or 3D flow simulation
- evaluation of indoor air streams to prevent draughts
- analyses of glass roofs on top of atriums and malls and the implementation of smoke and heat exhaust ventilation systems
- planning and optimization of procedural air streams
- planning and fluid mechanical optimization of HVAC systems and components
- calculation and analysis of the spatial distribution of inside air, radiant and operative temperatures.
- this all will possible only because of the development of the CFD software development .
the codes for the 2d heat conduction and the unstructured mesh are submitted below,:
the reference diagram for the resuts are given in the question is :
the code results also match the diagram results.
the result diagram of project is ,
CONCLUSION:
The solution for the 2D Heat conduction equation by FVM formulation and solve it in a distorted mesh stored in an unstructured format are solved , and the results are expressed in detailed.