FIT (Finite Integration Technique)
Introduction
The Finite Integration Technique (FIT) is a numerical method used for solving partial differential equations (PDEs) in computational fluid dynamics (CFD). It was developed by a group of researchers at the University of Colorado in the early 1990s and has since become a popular method in the field of CFD due to its simplicity, accuracy, and efficiency.
The FIT method discretizes the governing equations of fluid flow over a computational mesh, and then integrates these equations over each cell of the mesh. This approach allows the solution to be obtained at the center of each cell, which is then used to update the solution at the next time step.
This article will provide an overview of the FIT method, including its basic principles, the discretization of the governing equations, and the solution procedure.
Basic principles of FIT
The FIT method is based on the principles of conservation of mass, momentum, and energy, which are expressed in the form of PDEs. These equations describe the behavior of fluid flow in terms of velocity, pressure, and temperature.
The FIT method discretizes these equations over a computational mesh, which is composed of a set of cells or control volumes. The control volumes are defined as the regions of the mesh where the conservation equations are integrated. The solution is obtained at the center of each control volume, which is then used to update the solution at the next time step.
Discretization of the governing equations
The governing equations of fluid flow can be expressed in the form of conservation laws. The conservation of mass, momentum, and energy for an incompressible fluid can be expressed as follows:
Conservation of mass:
∇ · u = 0
where u is the velocity vector and ∇ is the gradient operator.
Conservation of momentum:
∂u/∂t + (u · ∇)u = -1/ρ ∇p + ν∇^2u
where p is the pressure, ρ is the density, ν is the kinematic viscosity, and ∇^2 is the Laplacian operator.
Conservation of energy:
∂T/∂t + (u · ∇)T = α∇^2T
where T is the temperature and α is the thermal diffusivity.
The FIT method discretizes these equations over a computational mesh, which is composed of a set of control volumes. The control volumes are defined as the regions of the mesh where the conservation equations are integrated. The solution is obtained at the center of each control volume.
To discretize the conservation equations using the FIT method, the control volume is divided into smaller sub-volumes, where the solution is approximated using piecewise linear functions. The approximation is obtained using a least-squares fit, which ensures that the solution is consistent with the conservation equations.
The resulting system of equations can be expressed as:
[M]{U}n+1 = [A]{U}n + {F}
where [M] is the mass matrix, [A] is the advection matrix, {U}n+1 is the solution vector at the next time step, {U}n is the solution vector at the current time step, and {F} is the forcing vector.
Solution procedure
The solution procedure for the FIT method involves the following steps:
- Initialization: The initial conditions are specified at the start of the simulation.
- Time marching: The solution is updated at each time step using the discretized conservation equations.
- Boundary conditions: The boundary conditions are applied to the control volumes on the boundary of the domain.
- Convergence criteria: The solution is considered converged when a specified tolerance is reached.
- Output: The solution is outputted for visualization or further analysis.
Advantages and limitations
The FIT method has several advantages, including its simplicity, accuracy, and efficiency. The method is easy to implement and requires minimal computational resources, making it suitable for solving large-scale problems. FIT is also accurate and can produce reliable results for a wide range of fluid flow problems.
Another advantage of FIT is its ability to handle complex geometries and boundary conditions. The method is flexible and can handle irregular and non-uniform meshes, making it suitable for simulating real-world flow problems.
However, FIT also has some limitations. One limitation is that it is a first-order method, meaning that it may not be accurate enough for some problems that require higher-order methods. Additionally, the method can be prone to numerical diffusion, which can lead to a loss of accuracy.
Conclusion
The Finite Integration Technique is a powerful numerical method for solving partial differential equations in computational fluid dynamics. The method discretizes the governing equations over a computational mesh and integrates these equations over each cell of the mesh. The resulting system of equations is then solved to obtain the solution at the center of each cell.
FIT is a simple, accurate, and efficient method that can handle complex geometries and boundary conditions. While it has some limitations, the method is a valuable tool for simulating fluid flow problems in a wide range of applications.