Abstract
Tribology is in every sense of the word multidisciplinary. A multitude of different physical processes can occur simultaneously, often over widely varying time and length scales. The contact region between sliding bodies is a particularly interesting area, where thermal, mechanical, chemical, and electrical affects all play a role and interact in complex ways. Often our understanding of these phenomena is limited by the lack of a convenient methodology to model and solve such complex problems. The objective of this paper is to describe a new modeling and solution method that is relatively simple yet powerful enough to handle complex, multi-physics problems.
Abstract
Mathematical modeling of tribological systems is important for numerous reasons, including the development of a fundamental understanding of the underlying physics, identification of critical intrinsic parameters (i.e., the inverse problem), and optimization of particular processes. However, simulations of complex tribological processes are extremely challenging, as they often involve adaptive multi-physics phenomena in complex heterogeneous systems. To address this challenge, this work offers new generalized approaches to modeling and analyzing simulations of multi-physical phenomena in complex tribological systems. Implementation of such complex models using traditional procedures (e.g., finite element and finite difference methods) is often not practically feasible or limited due to excessively long computation times and limited multi-physical and adaptive capabilities.
Abstract
The objective of this paper is to develop and implement general multi-physics models for the analyses of complex tribological processes. In order to attain this goal, a rule-based solution method is developed that combines fundamental ideas from traditional control volume finite differencing with those from cellular automata. The unique feature of this solution methodology is that rather than attempting to develop equations for entire complex processes, computationally explicit, yet numerically stable rules for elementary physical processes are developed individually. These elementary rules are then assembled in a cascading sequence at each time step to form a complete process. The cascading sequence of explicit rules is applied to the state variables to transform them from one state to the next over a typical time interval. These rules are applied repeatedly until the simulation has evolved for a sufficient total time.
INTRODUCTION
Mathematical modeling of tribological systems is important for numerous reasons, including the development of a fundamental understanding of the underlying physics, identification of critical intrinsic parameters (i.e., the inverse problem), and optimization of particular processes. However, simulations of complex tribological processes are extremely challenging, as they often involve adaptive multi-physics phenomena in complex heterogeneous systems. To address this challenge, this work offers new generalized approaches to modeling and analyzing simulations of multi-physical phenomena in complex tribological systems. Implementation of such complex models using traditional procedures (e.g., finite element and finite difference methods) is often not practically feasible or limited due to excessively long computation times and limited multi-physical and adaptive capabilities.
INTRODUCTION
The objective of this paper is to develop and implement general multi-physics models for the analyses of complex tribological processes. In order to attain this goal, a rule-based solution method is developed that combines fundamental ideas from traditional control volume finite differencing with those from cellular automata. The unique feature of this solution methodology is that rather than attempting to develop equations for entire complex processes, computationally explicit, yet numerically stable rules for elementary physical processes are developed individually. These elementary rules are then assembled in a cascading sequence at each time step to form a complete process. The cascading sequence of explicit rules is applied to the state variables to transform them from one state to the next over a typical time interval. These rules are applied repeatedly until the simulation has evolved for a sufficient total time.
BASIC TRANSPORT PROCESSES
A medium is considered that contains a collection of interacting species, denoted by the vector u. The individual species could be concentrations of chemical components, energy density, momentum, or any other state variable needed to describe the state of the system. The dynamics of these species is governed by transport due to the following fundamental mechanisms: 1) advection due to bulk motion of the medium, 2) diffusion, 3) chemical reactions, and 4) interactions with an external environment. Application of this principle to a differential volume produces a set of coupled partial differential equations for the rate of change of the state variables;
BASIC TRANSPORT PROCESSES
where
BASIC TRANSPORT PROCESSES
is the position vector, t is time,
BASIC TRANSPORT PROCESSES
is the velocity vector, D is the diffusivity (m 2 /s), R describes the chemical kinetics, and S is the effect of the external environment.
BASIC TRANSPORT PROCESSES
Transport processes are often modeled using variations of Eq.(1). Although such a formulation is quite general, analytical solutions are usually not possible, except for certain classes of linear models.
BASIC TRANSPORT PROCESSES
Direct numerical solutions using Finite Difference (FD) and Finite Element (FE) are always possible in principle, but transient, multidimensional problems involving a large number of state variables can be computationally prohibitive. As a result, a new modular, rule-based solution strategy is proposed.
THE ELEMENTARY PROCESSES
The current solution method is a unique combination of fundamental ideas from finite differencing (Patankar, 1980) and cellular automata (Wolfram, 2002) . The basic idea is to split complex processes into simpler, elementary processes, as first suggested by Strange (1968) . Each elementary process is advanced in a discrete manner in time using a computationally explicit yet numerically stable rule. The total process is then assembled by applying each elementary rule in a cascading, sequential manner.
THE ELEMENTARY PROCESSES
The general transport equation (1) is split into elementary processes, summarized in Table 1 . These processes are applied over a spatially discretized grid and are advanced in discrete time steps. The state variable vector at time step k is denoted as 
THE ELEMENTARY PROCESSES
where n Φ is the specific rule corresponding to process n that advances the state variable vector u k over one time step. The symbol '⊗' signifies an operator or rule. 
THE ELEMENTARY PROCESSES
Note that the environmental interaction has been linearized.
RULES FOR THE ELEMENTARY PROCESSES
In order to accurately and conveniently compute solutions to complex problems, appropriate rules must be available for the elementary physical processes. Each rule is required to have the following attributes:
RULES FOR THE ELEMENTARY PROCESSES
1. Physically realistic 2. Computationally explicit 3. Numerically stable Since the elementary processes are so much simpler than the total, complex process, rules satisfying the previous criteria are possible.
RULES FOR THE ELEMENTARY PROCESSES
The rules corresponding to the elementary transport processes listed in Table 1 are summarized in Table 2 for a onedimensional medium over time step ∆t. 
RULES FOR THE ELEMENTARY PROCESSES
Similar rules work naturally for multidimensional problems. The diffusion processes consists of a two step procedure where all the odd-even cells interact first and then the even-odd cells interact. This parallelizes the diffusion process as much as possible while retaining the required attributes.
CONSTRUCTION OF COMPLEX PROCESSES
With rules for elementary processes available, complex processes are constructed as a cascade of elementary processes in the form,
CONSTRUCTION OF COMPLEX PROCESSES
where N p is the number of elementary processes. This strategy can be applied to a model containing any combination of physical processes. The basic transport process, as described in the traditional form by Eq. (1), becomes,
CONSTRUCTION OF COMPLEX PROCESSES
The accuracy of this scheme depends on two major factors; (1) the accuracy of the elementary rules listed in Table 2 , and (2) the accuracy of the splitting and reconstruction process. Inaccuracies depend on both the time step and the order of reconstruction.
CONSTRUCTION OF COMPLEX PROCESSES
These issues will be demonstrated in subsequent papers (Vick and Furey, 2005) .
SUMMARY AND FUTURE WORK
A new methodology, based on a combination of cellular automata and control volume finite difference concepts, is described. It involves a cascading sequence of simple, explicit rules of evaluation, rather than complicated partial differential equations. The resulting scheme is computationally explicit yet numerically stable.
SUMMARY AND FUTURE WORK
In addition to significant modeling flexibility, the cellular automata environment lends itself to extremely efficient computational algorithms and hardware implementation due to its inherent use of local rules and potential for parallel computation.
SUMMARY AND FUTURE WORK
The power and flexibility of the proposed cellular automata methodology will be demonstrated in future work using several case studies of varying complexity relevant to phenomena occurring at sliding contacts (Vick and Furey, 2005) . The current methodology will be compared with traditional finite difference techniques.

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