ContactLensProject

Maintained by Roland Sanford ([email protected]).

##Table of Contents

1. Setup
2. Development Conditions
3. Compiling and Running
4. Implementation
5. Gauss-Seidel method
6. Finite-difference method
7. Computational Grid
8. General pseudocode

##1. Setup [top] ###1.1 Development Conditions The code for this project was written in C++ using the gcc (GCC) 4.9.2 (Red Hat 4.9.2-6) GNU compiler. This repository is under active development, so the current repo head may not compile.

###1.2 Compiling and Running The information below details how to compile and run the project's code for various operating systems. You can either clone or download the repository (and build, if you are running Linux).

####Linux Open a terminal and navigate to the main project directory. Type

$ g++ -o <executable name> c_eye.cpp helper.cpp

to manually compile the program. To run, enter

$ ./<executable name>

Alternatively, your can run build to produce an executable calld clp. This is done in the following way:

$ sh build 
$ ./clp

###Mac Open a terminal and navigate to the main project directory. This requires XCode. Type

$ clang++ -std=c++11 -stdlib=libc++ c_eye.cpp helper.cpp
$ ./a.out

to compile then run the program.

###Windows This requires Visual C++ .NET. Run Visual C++ 2010 Express Command Prompt. Type

> cl /EHsc c_eye.cpp helper.cpp
> c_eye.exe

to compile then run the program.

##2. Implementation [top] ###2.1 Gauss-Seidel method We use the Gauss-Seidel method (GSM) is our general technique for solving the cylindrical eye problem. In this section, we will provide a brief introduction to the method.

GSM is used to solve linear systems of equations in the form Ax=b one equation at a time. During each iteration, the new values are calculated based on the values from the previous iteration. An initial guess serves as the "old" values for the first run through. This process continues continues until a predefined convergence condition is satisfied. GSM convergence requires the matrix A to be strictly diagonally dominant, or symmetric and positive definite.

###2.2 Finite-difference method We discretized the governing partial differntial equations (PDEs) and boundary conditions using finite-difference method (FDM). FDM uses the values of a point and it's neighbors to approximate derivatives at that point. Combinations of forward, backward, and central finite difference equations were used thoughout the computational grid. Currently, a global second-order accuracy is held throughout the simulation.

###2.3 Computational grid We reprensent the eye as a radially-symmetric cylinder. Consequently, we only need sole the domain from the center of the eye to an edge to solve the system. The eye is mapped to an MxN grid. The variable i is used to represent rows and height, such that 0<=i<=M, and j to represent columns and width, such that 0<=j<=N. As a whole, the grid can be viewed as follows:

[M][0]  |---------------| [M][N]
 ^      |---------------|
 i      |---------------|
 v      |---------------|
[0][0]  |---------------| [0][N]
        <------ j ------>

Note: [] denotes array indexing in C (zero-based).

###2.4 General pseudocode Here, we present a high-level overview of the simulation.

Initialize the displacement functions R and W to be M+1xN+1 arrays.
Make the initial guess: R=0, W=0.
While the system has not converged to solution:
    Every k iterations update the suction pressure.
	Solve for R and W at each point on the computational grid.