# Physical Formulation of the Direct Model

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## Overview of Research

This research requires the development of mathematical models of the physics pertaining to the temperature fields evolved in sliding solids undergoing frictional heating in order to infer the real area of contact. The system, both mathematically and physically, can be seen below in Figure 2.1.*
The model must start from a proposed physical system, and be evolved mathematically through a direct numerical model that is used in a parameter estimation scheme to approximate the contact distribution from which the real area of contact is inferred.

The Physical System

This paper deals with a two body system undergoing frictional heating. Figure 2.2 depicts a possible physical setup for this system, consisting of a light-colored rotating structure that is rubbing against a dark-colored static pad (1) over a localized area. Braking type systems inspired this construct. The presumed direction of rotation is depicted in the figure.
The inset to Figure 2.2 shows an exaggerated, schematic close-up of the contact region. One can see the rough, uneven contact. Frictional heat generation will occur at the points where the two regions are actually in contact, and be dependent on the pressure of the contact and the velocity of the moving material.

### Solution of the Direct Problem

Mathematically, the direct problem is the determination of the response of a system with known forcing functions and system parameters. This problem is represented in Figure 2.3. This is the classic type of problem found in most engineering analysis, where the system’s excitation and characteristics are known, but the response is not. Mathematically, the direct problem is well-posed. This statement implies that a solution exists, is stable, and is unique. This statement further means that for any given forcing function, there is one and only one solution, and that the result will not become unstable under small changes to input data [27].
Mathematically, the physics surrounding the direct problem in this case are complex enough to make it impossible to achieve an analytical solution. Thus, one must choose from any number of numerical methods to arrive at a computational solution. In order to achieve a faster simulation that would allow the multiple iterations necessary to perform parameter estimation, a modified cellular automata method was chosen to handle the multi-physics.

Modified Cellular Automata

This modified cellular automata technique is a means of breaking up large, difficult to solve, multi-physics models into smaller, more easily digested bits. It requires feeding the result of each single/reduced multi-physics system into the next, and solving iteratively through small, appropriately chosen time steps for the physics involved. A single advancement is represented pictorially in Figure 2.4. This rule based method of advancing each of the individual physics independently has the advantage of enabling each of the physics to be solved using the technique best suited to it. The mathematics of this method proceeds in a stepwise fashion and therefore lacks simultaneity, however. Thus, the time steps must be small to avoid the risk of nonphysical results.

Parameter Estimation

There are two major types of inverse problems. Both of the inverse problems are mathematically ill-posed. The solution is not necessarily unique, and very frequently is not stable for small changes in the input conditions. The two types of inverse problem are the estimation of the forcing functions, and the estimation of system parameters. Both require the estimation of something required to cause the observed response in the data. In the estimation of the forcing function case, the system is fully characterized. By knowing the response of the system, one can estimate what the excitation must have been. This problem is graphically represented in Figure 2.5.
The other type of inverse problem is system parameter estimation. Parameter estimation problems are the class of problem where the excitation is known and the response is measured. By knowing the response of the system to the excitation, one can estimate the system parameters that caused the response. This is the case that is discussed in this thesis. The distribution of contacts and the total real area of contact are physical parameters of the system. Parameter estimation frequently exhibits instability in its mathematics because of singular or near-singular matrices that appear in the math. The means by which these are handled are discussed in chapter 5. A schematic representation of the parameter estimation problem is shown in Figure 2.6.

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#### 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

Figure 3.1 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.
The bottom portion of the figure depicts a close up of the nominal contact region of the second body. One of the objectives of the research is to determine approximately what the real contact area must be to produce the observed temperatures. The figure depicts individual contacts of varying intensity and size. The real contact area is the sum of each of the individual contact areas.

General Contact Area Profile

The contacts can be thermally represented as a heat flux distribution (e.g. as uniform distributions for plastic contacts, or Hertzian distributions for elastic contacts). The only region where this distribution can be positive is over the nominal contact zone. Points of zero contact between the two bodies are, by definition, zero. Thus, the mathematical representation is The total power dissipated by friction.
The distributions show how bodies can come into contact over the nominal contact region. The ‘multiple contacts’ distribution is the viewpoint paradigm with which this paper approaches the problem. This paradigm allows for multiple, irregularly shaped contacts, with the minimum resolution being the size of the spatial discretization of the model.
Further, since the contact distribution was derived on the basis that it is zero in regions of zero contact, it also gives insight into the size of the real contact area. One can look at the distribution as being such that the regions with a large relative value of the contact distribution are contacts. The converse is also true, where low value regions of the contact distribution are where the two bodies are not in contact.

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.

Abstract
Acknowledgements
1 Introduction
1.1 Motivation
1.2 Objectives
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
4.3 Source Partition Rule
4.4 Convection plus Source Rule
4.5 Diffusion Rule
4.6 Rule Library
5 Estimation of Real Area of Contact
5.1 Algorithm
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
7.1 Summary
7.2 Conclusions
7.3 Recommendations
List of References
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