During the past two decades, mathematical modeling has been gaining acceptance as a legitimate part of materials science and engineering. However, as common to all relatively new disciplines, we still lack a realistic perspective regarding the uses, limitations, and even the optimal methodologies of mathematical modeling techniques.

The term “mathematical modeling” covers a broad range of activities, including molecular dynamics, other atomistic scale systems, continuum fluid and solid mechanics, deformation processing, systems analysis, input-output models, and lifecycle analyses. The common point is that we use algebraic expressions or differential equations to represent physical systems to varying degrees of approximation and then manipulate these equations, using computers, to obtain graphical output.

While it is becoming an accepted fact that some kind of mathematical modeling will be needed to make most research programs complete, there is still considerable ambiguity as to what form this should take and what might be the actual usefulness of such an effort.

Among the more seasoned and successful practitioners of this art, clear guidelines have emerged regarding the uses and limitations of the mathematical modeling approach. We seek to illustrate these uses through the successful modeling examples presented by some leading practitioners. Some general principles may be worth repeating as an introduction to this interesting collection of articles.

Computational fluid dynamics was born principally in the aerospace field as a method for fluid flow and heat transfer research methods following experimental and analytical approaches. Along with progress in the cost performance of computers, computational fluid dynamics is now establishing itself as a tool to improve production processes and product quality in the steel, nonferrous metals, glass, plastics, and composite materials industries.

Materials manufacturers use computational fluid dynamics for diverse purposes:

• 1. Reduction in experimental conditions and costs;

• 2. Detailed analysis of mechanisms with multifaceted information unobtainable through experimentation;

• 3. Universal tool for scale-up; and

• 4. Evaluation of novel processes.

It can be readily imagined that accuracy, flexibility, and other requirements of computational fluid dynamics should vary with specific applications.

Fluids generally observed in materials manufacturing processes are molten materials such as metal, glass, and plastics, and gases for stirring and refining. In the flow of such fluids, materials quality and process characteristics are governed by the following:

1. 1. Transport phenomena in the bulk region (where fluid flow is normally turbulent);

2. 2. Chemical reaction at interfaces;

3. 3. Transport phenomena in boundary layers near the interfaces; and

4. 4. Complex coupled phenomena (heat transfer, diffusion, chemical reaction, phase transformation like solidification, free surface, electromagnetic force, and bubble flow).

Besides the numerical tools which have been developed for solving the continuity equations in materials processing, the prediction of microstructures and defects in such processes is becoming an important step in assessing the quality, and ultimately the mechanical properties, of the final products. Because the typical length scales associated with the process and with the microstructure differ widely (typically a factor of 104), special techniques have to be used when coupling the macroscopic and microscopic levels. This contribution will briefly present some of the modeling tools recently developed for solidification. The trend is now to replace deterministic models used over the last 20 years by stochastic approaches which directly generate computed micrographs. Among those, Monte Carlo techniques originally developed for the grain growth in solids were adapted to the solidification of alloys, and cellular automata models specifically take into account the dendrite growth mechanisms.

The production and use of aluminum alloys involves a wide range of processing and fabrication methods. For a number of reasons discussed in this article, process modeling has an increasingly important role in all areas of production. This article reviews some process modeling activities in aluminum processing. The aims of the review are to (1) describe the current state of research in two major processing areas, (2) establish the principal motivations in the aluminum industry for undertaking process modeling, and (3) to identify generic research issues most in need of attention. This review is not intended to be exhaustive, but considers only selected processes, based on work in Norwegian laboratories with which the authors are most familiar. There is considerable parallel modeling activity worldwide in industry and academia which will not be discussed here. The case studies selected are sufficient, however, to illustrate all the main points of the review.

A recent review of materials process modeling identifies a number of recurring themes which indicate the commercial reasons why process modeling is undertaken in various industries. They may be summarized as modeling to (1) develop new or scaled-up processes, (2) reduce the number of trials, (3) optimize an existing process, and (4) improve quality by reducing product variability.

So the overriding concerns in industry are improving productivity and quality, and reducing costs. Modeling is also important from the point of view of model based real-time control of processes, but developments in this area are at present limited to relatively slow processing operations. Models are also important tools for both industry and academia to improve scientific understanding of materials processing. Within a company, modeling serves as a means to assemble know-how about the company's processes, raising the level of training of the workforce and making it possible for people to learn a process quickly.

Welding is utilized in 50% of the industrial, commercial, and consumer products that make up the U.S. gross national product. In the construction of buildings, bridges, ships, and submarines, and in the aerospace, automotive, and electronic industries, welding is an essential activity. In the last few decades, welding has evolved from an empirical art to a more scientifically based activity requiring synthesis of knowledge from various disciplines. Defects in welds, or poor performance of welds, can lead to catastrophic failures with costly consequences, including loss of property and life.

Figure 1 is a schematic diagram of the welding process showing the interaction between the heat source and the base metal. During the interaction of the heat source with the material, several critical events occur: melting, vaporization, solidification, and solid-state transformations. The weldment is divided into three distinct regions: the fusion zone (FZ), which undergoes melting and solidification; the heat-affected zone (HAZ) adjacent to the FZ, that may experience solid-state phase changes but no melting; and the unaffected base metal (BM).

Creating the extensive experimental data base required to adequately characterize the highly complex fusion welding process is expensive and time consuming, if not impractical. One recourse is to simulate welding processes either mathematically or physically in order to develop a phenomenological understanding of the process. In mathematical modeling, a set of algebraic or differential equations are solved to obtain detailed insight of the process. In physical modeling, understanding of a component of the welding process is achieved through experiments designed to avoid complexities that are unrelated to the component investigated.

In recent years, process modeling has grown to be a powerful tool for understanding the welding process. Using computational modeling, significant progress has been made in evaluating how the physical processes in the weld pool influence the development of the weld pool and the macrostructures and microstructures of the weld.