Physical Formulation of the Direct Model
The first step in determining the parameter values of an incompletely defined system is to define a functional mathematical model of the physical system. This chapter develops the models that will be used in the analysis of the two dimensional, two body (2D, 2B) frictional system and its degenerates.
2D, 2B System
The most general system that will be considered is the 2D, 2B problem. This allows easier depictions of the system, as well as reduced computational time.
General Physical Schematic
depicts a schematic view of the unwrapped physical system of Figure 2.2. The top portion of the figure depicts the geometric arrangement with the dimensions and coordinate system shown. The double wavy lines at the left and right of the second body indicate that the boundary wraps around to the other side. This notation is used throughout the figures of this paper.
Formulation of the 2D, 2B Problem
Now that the heat addition through friction is represented as a heat flux distribution, the system can now be modeled as a standard heat transport process. The depiction of the problem setup can be found in Figure 3.3. Figure 3.3 depicts the flows of energy in and out of the control volumes in each body. One can see that the stationary Body 1 undergoes a purely conductive behavior, while the moving Body 2 exhibits advective behavior in addition to conduction.
Formulation of the 2D, 1B Problem
In order to arrive at the two dimensional, one body (2D, 1B) formulation, one simply must degenerate the 2D, 2B formulation results. In this case, in order to remove the second (non-moving) body, one needs to make the simplifying assumption that all of the heat evolved is transported into the moving body. The results in Figure 3.5, therefore, look very similar to the results achieved in Figure 3.4. The primary differences are that, obviously, the stationary body is removed, and that the coupling conditions have been reduced to a heat flux (as opposed to the coupled conditions).
Formulation of the 1D, 1B Problem
In order to arrive at the one dimensional, one body (1D, 1B) formulation, the static body must be removed from the 1D, 2B formulation. The reduction is much the same procedurally as the reduction to the 2D, 1B formulation from the 2D, 2B formulation. Figure 3.7 shows this formulation geometrically. As with the previous 1B case, all of the heat evolved at the interface enters the moving body
Solution of the Direct Problem via Modified Cellular Automata
This chapter deals with the development of the numerical solution to the direct problems formulated in Chapter 3. It will cover the modified cellular automata method and the rules developed for this particular set of physics.
The method of cellular automata is a method of breaking physics up into discrete, rule based, solutions that are applied to a given set of conditions in a serial manner. The formal method of cellular automata requires the entire system to be discrete, including the state variable. Temperature, by its nature, is not a naturally discrete state variable, and thus to use cellular automata, the state variable would have to be artificially discretized. This behavior is undesirable. However, the concept of breaking a difficult equation up (similar to operator splitting) into easier pieces and solving them in a serial manner is very attractive when computational times for the full system start to accumulate rapidly. Thus, we use a modified cellular automata method to arrive at the solution to the direct problem. The method of modified cellular automata is a technique of breaking up large, difficult to solve, multi-physics models into smaller, more easily digested mathematical pieces. This method requires feeding the result of each single (or reduced multi-) physics rule into the next, and solving the set of rules iteratively through small, appropriately chosen time steps. As a result of the breaking up of the physics into smaller rules, solved in a serial, rather than simultaneous, method, the order of the rule employment can have a small impact on the end result. However, it has been shown  that as the time step over which each rule is solved consecutively gets smaller, the differences in the results are reduced. Further, it was shown for the given examples that the order did not affect the results enough to make them physically unrealistic. The modified cellular automata method allows the user to draw from many well developed and understood solution methods. Finite difference methods, analytical solutions, and Runge-Kutta solvers are all available for use in this overall technique. After developing the mathematical model, one must then determine the breakdown of the mathematics into its simpler rules for the employment of this method. As an initial step, one must discretize the independent variable dimensions (space and time). Time is already discretized as a result of the cellular approach where the multi-physics are simulated for each major time step. Space also needs to be discretized based on any requirements regarding the spatial resolution. This discrete grid can visually be represented.
The advection rule models the bulk transport behavior of the system. It is represented by the bulk transport term of the governing equation. It appears in the equation governing the moving body. where is the initial temperature distribution and is the distribution at time equal to . The interpretation of this result is that the initial distribution, under no other driving behaviors, simply shifts along the ‘’ axis as time proceeds. This solution produces an extremely simple rule for a process that can be very difficult to deal with when mixed with other processes. This can be depicted as seen. It shows that, at each time step, that the shape does not change. The shape only translates as time progresses
Application to Real Area of Contact
As mentioned in Chapter 3, the estimation of the real area of contact is done by inference from the contact distribution. That requires the estimation method to determine what that distribution is. Since the model is discretized in space, only the mean value of the contact distribution over the length of each node needs to be determined. Thus the number of parameters that the model contains is equal to the number of nodes contained within the contact region. This discretization can be seen. Each node has its own value that can be estimated.
1.3 Previous Research
2 Overview of Research
2.1 The Physical System
2.2 Solution of the Direct Problem
2.3 Parameter Estimation
3 Physical Formulation of the Direct Model
3.1 2D, 2B System
3.2 Formulation of the 2D, 1B Problem
3.3 Formulation of the 1D, 2B Problem
3.4 Formulation of the 1D, 1B Problem
4 Solution of the Direct Problem via Modified Cellular Automata
4.1 Basic Concept
4.2 Advection Rule
4.3 Source Partition Rule
4.4 Convection plus Source Rule
4.5 Diffusion Rule
4.6 Rule Library
5 Estimation of Real Area of Contact
5.2 Application to Real Area of Contact
6 Results and Discussion
6.1 Non-Dimensionalization of the 2D, 2B Equations
6.2 High Peclet Simplification
6.3 One Body Problems
6.4 Two Body Problems
7 Recommendations and Conclusions
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