Multiphysics in Porous Materials

1.1 “Multiphysics is there” and “Multiphysics is coming”

Multiphysics is there

A World of Multiphysics Nature

We live in a multiphysics world. Real-life processes are inherently multiphysics. From atoms to galaxies, from amino-acids to living organisms, naturally built systems involve various types of physical phenomena operating at different spatial and temporal scales. Loads and deformation on solids, complex flows, fluid-structure interactions, plasma and chemical processes, thermo-mechanical and electromagnetic systems are just a few representative examples in fundamental and applied sciences (Krzhizhanovskaya and Sun, 2007). Take a cell phone for example–the antenna receives electromagnetic waves; the touch screen or buttons handles for interaction between mechanical and electrical components; and the battery involves chemical reactions and the movement electrical current (IEEE, 2011). Therefore, multiphysics is neither a research concept far from daily life nor a theory or technique developed recently.

We have been dealing with this multiphysics world even before the civilization of human beings. For example, a nature fire from lightning is a typical electrostatic, thermal, and chemical phenomenon. Fire starting with a hand drill involves both mechanical and thermal processes. Flood protection and irrigation required to understand the interaction between hydrodynamics and groundwater movement. Construction and earthwork called for the understanding of a coupled mechanical and hydrologic process. The modern industrialization even stemmed from multiphysical processes such as steam engines, which is an art of thermal, mechanical and dynamics. What would be more surprising is not that the modern lives are full of multiphysics, but instead, we have been taking advantage of most of these multiphysical processes in such an adept way.

Surprisingly, while multiphysics is part of nature and history of lives, the study of multiphysics, is not new either. We can describe what happens in the world using sets of physical laws. Originally, due to the limited understanding of complex physical processes and computing resources, so physical effects were observed in isolation. For example, traditional courses in engineering disciplines were usually built upon of the understanding of one type of physical process. For example, various courses in civil engineering were established on the basis of mechanics while many in electrical engineering were on electrostatics. However, there are still many multiphysical phenomena, which are too fundamental and “multiphysics”, such as electromagnetics and elastodynamics. These multiphysics topics were studied and matured much earlier than other multiphysical phenomena, even before the advent and prevalence of the word “multiphysics”. Since the 1940s, we have been using computers to understand these physical phenomena. Thus, regardless of the trademark “multiphysics”, simulations that incorporate multiple physical phenomena are as old as simulations themselves (Keyes et al., 2012).

Despite the studies which are by nature multiphysics but were not traditionally counted, multiphysics research has been on the playground for a couple of decades. On the one hand, many physical problems of interest these days are complex in nature and consist of separate physical processes that each contributes to the problem as a whole. Also, some problems which were historically investigated as an isolated physical process are now needed to be considered as a holistic process. This fact requires that researchers capture multiple physical processes when constructing numerical models (Groen et al., 2013). On the other hand, the uprising computing power in virtue of the innovations in computer engineering and science has been lending us the power to deal with more complicated problems, which may be insurmountable in the old days due to complicated couplings between fields. The higher speed of computing unit, higher storage, and especially the popularization of computer cluster enable the consideration of the whole process in a holistic way, giving rise to various multiphysics techniques. Multiphysics studies under the name of multiphysics have been conducted and widespread at least for two decades.

In fact, multiphysics has been developed beyond a research concept. For example, in industrial design and product testing, engineers used to solve one physical phenomenon at a time, such as structural integrity, then import the results into another system to solve for the aerodynamic behavior. Multiphysics simulation tools now allow them to simulate and analyze both these behaviors, and many others, at the same time. Similar to those physical processes discussed in research papers, in reality, product performance usually depends on several physical phenomena interacting with each other: multiphysics. To acquire dependable and realistic simulation results that allow better design decisions, coupled simulations of the physical effects involved become necessary and essential. Also, in light of the advance and promotion of CAD tools such as COMSOL, multiphysics has become an accepted consensus, trend, and even standard practice in many industries. With a multiphysics capable simulation tool, designers and engineers now can better capture the important aspects of the behavior of the product.

Multiphysics is coming

Multiphysics has rapidly developed into a research and application area across many science and engineering disciplines. There is a clear trend that more and more challenging problems we are faced with do not involve physical processes covered by a single traditional discipline in science or engineering disciplines. This trend requires us to extend our analysis capacity to solve more complicated and more multidisciplinary problems, advance the front-edge of engineering and applied sciences by exploring these problems, contribute back to the software development and computing techniques, and establish and explore innovative but urgent application areas.

Modern academic communities are confronted with problems of rapidly increasing complexity, which straddle across the traditional disciplinary boundaries between physics, chemistry, material science and biology (ASME). To respond to this challenge, many disciplines have spontaneously initiated efforts at reaching out to other disciplines to target at emerging issues which appear as a traditional topic in those disciplines while also extend beyond the scope. At the same time, computational science, which is more focused on techniques rather than problems, has been receiving a steadfast development. Innovative modeling techniques have been investigated for the purpose of offering an optimal handling of the information transfer which connects different aspects, components, scales, and levels involved in the quantitative description of those multidisciplinary phenomena. This trend is entailing the seamless coupling between different mathematical representations of various physical phenomena at widely disparate scales, from continuum fields to probability distribution functions and atomistic trajectories. (ASME)

Multiphysics has also become a frontier in industrial applications. Simulation programs have been evolving into a tool familiar to many engineers in design, product development, and quality control. This change is reshaping traditional creation processes: from design blueprints to prototypes to testing to validation to production, which was once simple, to extremely complex ones involving engineers from several disciplines. During these creation processes, engineers now must think in areas outside of their training, even with the assistance of the simulation tools. As commented by E&T, it is more and more necessary for the modern engineers to know and grape the concept of what is known deep inside the engineering world as “multiphysics.”  Take auto industry for example; traditionally, there are different people focusing on the structure, fluids, electromagnets and the other individual aspect separately. Now, the intersection of aspects, which may represent two physics and once was a gray area, can be the essential link in the life cycle of the product. Such needs will enable multiphysics applications to attract a much wider engineering user base and allow more designers to come together (E&T). As commented by Nilsson (COMSOL), “Design engineers are running more and more multiphysics simulations every day because they need to add reality into their models.”

Multiphysics will continue to benefit from the progress in computer engineering and science. The advent of commodity computing has been leading to profound changes in numerical analysis of continuum mechanics problems. Teraflop-rate computing facilities, which were available only to chosen few until recently, is becoming commonplace in an industrial setting (Jasak, 2006). This just reflects the fact that the range and complexity of problems in numerical analysis are expanding, leading to increasing interest in coupled problems which was computationally prohibitive. Besides, spectacular advances in computer performance and emerging technologies of parallel distributed grid computing are offering tools to break the barriers and bring simulation to a higher level of detail and accuracy. On the other hand, the advances in hardware and computing techniques also call for new efficient numerical algorithms specific to the field where coupling different models or scales within one simulation is essential (Krzhizhanovskaya and Sun, 2007).

The developments in multiphysics have been in turn contributing to the software engineering. The need and purpose of multiphysics software are to bring different strands of engineering under one umbrella. However, this is not an easy and straightforward task as it sounds to be. This is because research on complicated multiphysical phenomena or computer-aided design can include issues as diverse as predictions of structural strength, electromagnetic behavior, and fluid flow. Fortunately, the needs of the academic and industrial communities have generated considerable momentum in the software market to transfer or extend from traditional software packages with separate physics modules to tools with multiphysics genes which can promote better communication and interaction during the project lifecycle. A general rule is that the more multiphysics phenomena, the more accurate the final model will be, though the cost of building and solving more complicated model usually discourage such effort, leading to a compromise between accuracy and effort. Despite the progress, multiphysics is still a field in need of integration (Sjodin COMSOL). An easy way it to “stitch” different packages together using file imports, which is still common today and continuous effort is being made to render it smoother. More recent techniques aim at building up all the applications on the same platform all the way from the bottom.

Multiphysics is not only reformatting traditional applications, but also giving rise to new applications in scientific prediction, engineering design to policy making. In the automotive industry, multiphysics can be employed for traditional testing such as forces of rotating wheels, environmental stresses, and shock and vibration, under the conditions of different geometries, materials and load scenarios. In the meanwhile, new application areas are under development in virtue of multiphysics. For example, multiphysics is a significant impetus for the research and application of cyber-physical systems, which combine electronic control with physical problems. It is challenging but essential to understand the multi-domain problems underlying these systems, particularly for those humanoid robots equipped with multiple systems with more complex electrical and mechanical elements. For example, engineer Joel Gibbard’s Open Hand project explored the idea of how an amputee can create their prosthetic hand using a 3D printer. This project involves the design using National Instruments LabView software and data acquisition hardware and the measurement of the electrical activity of muscles at rest and during contraction for the understanding of the way that the nerves can send electrical signals.

1.2 What is Multiphysics

As introduced, multiphysics has been studied since the beginning of numerical simulation and at least two decades under the name of “multiphysics”. However, there has been no consensus on the definition of multiphysics. First of all, this word might be a little misleading or controversial if used in areas such as physics or in a context when the concept is not well known. Second, even in numerical simulation, the word could still refer to significantly different things. In the following paragraphs, the possible meanings of multiphysics will be discussed and a clear definition will be reached to lay down the scope of this book. Based on this definition, more details will be shown about how to move from monolithic physics to multiphysics. The purpose is to bridge the gap as some readers who are familiar with monolithic physics simulations move towards multiphysics. Finally, common types of multiphysics will be discussed to outline a big picture for this topic. Hopefully, readers can get an idea regarding what is multiphysics.

Definitions and Scopes

Definitions of multiphysics can be found in some literature which can be categorized according to the broadness of the scope. The typical definitions are listed and explained in the sequence of scope broadness, i.e., broad to narrow, as follows.

Definition 1: Multiphysics treats simulations that involve multiple physical models or multiple simultaneous physical phenomena (Wikipedia). While the use of “physical phenomena” semantically rephrases “multiphysics”, the inclusion of “multiple physical models” makes this definition a very broad and general concept. For example, combining chemical kinetics and fluid mechanics is a multiphysical process according to “physical phenomena” in the definition. Likewise, combining finite elements with molecular dynamics is then also a type of multiphysics as it includes “multiple physical models”. This definition is little bit self-contradictory as the implication of physical models may include that of physical phenomena. In fact, the definition of physical models is possibly more appropriate for multimodel. The multimodel concept may include multiscale, multirate, multilevel, and multilevel problems (Keyes et al., 2012). In this definition, multiphysics typically involves the solution of coupled systems of partial differential equations.

Definition 2: In a relatively narrow sense, multiphysics includes 1. coupled physical phenomena in computer simulation and 2. the study of multiple interacting physical properties (COMSOL). This definition narrows the scope of Definition 1 to physical phenomena and the materials associated with them. Thus a coupling between either physical phenomena of different types or materials properties associated with these different physical phenomena is an essential characteristic. The use of physical phenomena implies that multiple physical processes, each of which should be a distinct physical field, such as thermal field,  and can be described by a partial differential equation(s) based on underlying physical mechanisms such as conduction. The use of “physical properties” in the definition makes it more closely related to applications because in many phenomena of practical interest, such as piezo-resistance, these physical properties instead of the coupled process are the major interest. However, it is worthwhile to mention that the physical properties should be more than two properties and the properties are associated with different fields. Otherwise, density changes due to variations in water content in porous materials can be accounted as a multiphysical phenomenon.

Definition 3: Semantically, a multiphysics system consists of more than one component governed by its own principle(s) for evolution or equilibrium, typically conservation or constitutive laws. (Krzhizhanovskaya and Sun, 2007; Groen et al., 2013). This definition is very close to Definition 3 except for that it does not emphasize physical properties.

Definition 4: In a more strict way, multiphysics can be defined as those processes including closely coupled interactions amongst separate continuum physics phenomena (Croft et al., 2007). In this definition, two-way exchange of information between physical fields, which could involve implicit convergence within a time step is the essential feature. Accordingly, those processes only involve using data generated by one code as input into another is not qualified as multiphysics, but instead, as multidisciplinary.

The book defines multiphysics in the following way. First, the above definitions all point out several features and leave out some others. Among these features, multiple physical processes which are governed by their own laws of evolution or equilibrium are necessary. It seems that this is a feature directly reflecting the name and characteristics of most multiphysical processes. Second, the multi-model definition is abandoned to differentiate this topic from multiscale and multilevel problems. Third, couplings are not adopted in the definition as those are more an implementation technique than a fundamental characteristic.

Besides, a few neglected details need to be further discussed. First, “physics” in “multiphysics” means a physical field, which is a physical state variable varying with respect to space and/or time according to physical laws for its evolution or equilibrium. Thus, a field is either a time- (and space-) dependent spatial distribution of a state variable in a transient problem or a spatial distribution of a state variable in an equilibrium problem. The word physical process/phenomenon is not that clear and closely related to mathematics. Second, the simultaneous occurrence of these physical fields seems to be more a rule than the exception. Though sequential occurrence is also possible, it is far less common and leads to the direct coupling of physical processes involved. Therefore, simultaneous occurrence will be emphasized to leave out this possibility.  Third, all the definitions implicitly imply a deterministic process where stochastic factors do not seem to be at presence. However, if “physical field” is used, then it will be unnecessary to emphasize this feature. Fourth, the above definitions were made from the perspective of processes, phenomena, or system. All of these make multiphysics equal to the problem under investigation. But in a more common context, we also frequently use multiphysics to refer to the studies of and/or knowledge pool for these problems. Finally, a consensus in the above definitions is that multiphysics is about computer simulations. However, the implementation of multiphysics is not necessary via computer simulations and is also relatively independent of the methods of discretization and solution. Finally, it is believed that a major pool of knowledge is about the establishment of the mathematical model based on physical laws, which will also be the focus of this book. Thus, computer simulations will not be emphasized in the definition.

Based on the above discussion, in this book, multiphysics is defined as the coupled processes or systems involving more than one simultaneously occurring physical fields and also the studies of and knowledge about these processes and systems. For example, objects moving due to Newton’s second law will be viewed as an example of multiphysics as long as velocity will be calculated at every point of the domain, that is, to obtain a field. However, a model of a star cluster that resolves Newtonian gravitational interactions or molecular dynamics, which is built upon Newton’s second law, is not multiphysics according to this definition. From a mathematical perspective, systems consisting of partial differential equations (PDEs) of different types (e.g., elliptic-parabolic, elliptic-hyperbolic, or parabolic-hyperbolic) may be thought of as multiphysics because each of the classical PDE archetypes possibly represents a different physical phenomenon (Keyes et al., 2012).

Multiphysics problems vary widely in their nature, both in the physics involved and in the manner of coupling. To further clarify the scope, we will first classify multiphysics into two major categories based on the way in which different physical processes are coupled. In the first category, the coupling occurs in the bulk through source terms or constitutive relations that are active in the overlapping domains of the individual components. In the second category, the coupling occurs over an idealized interface or a narrow buffer zone through boundary conditions that transmit fluxes, pressures, or displacements (Higham, 2015). Typical examples of bulk-coupled multiphysics systems are radiation with hydrodynamics in astrophysics (radiation hydrodynamics), electricity and magnetism with hydrodynamics in plasma physics (magnetohydrodynamics), and chemical reaction with transport in combustion or subsurface flows (reactive transport). Typical examples of interface-coupled multiphysics systems are ocean-atmosphere dynamics in geophysics, fluid-structure dynamics in aeroelasticity, and core-edge coupling in tokamaks.

Multiscale is frequently mentioned with multiphysics. This is possibly because multiscale and multiphysics are different sides of a physical process. Multiphysics, which lies on fundamental physical laws, comes into being as a result of interactions originating from atomic scales and below and upscaling level by level to the scale of interest. Here we use the term multiscale modeling to refer to both multiscale modeling and multiscale simulation of physical problems, and the term multiscale application to refer to the program used to do the modeling. Groen et al. (2013) differentiated multiphysics and multiscale using the concept of submodel. Multiscale and multiphysics are therefore two distinct concepts, but they are common in that both of them consist of a number of submodels which are combined or coupled. Therefore, both multiphysics and multiscale face the same challenge of coupling these submodels such that the overall model is both accurate enough to be scientifically relevant and reproducible, and efficient enough to be executed conveniently by modern computational resources (Groen et al., 2013).

From Monolithic Physics to Multiphysics: Couplings

Though multiphysics is not necessarily obtained from monolithic physics, an intuitive and widely accepted idea is to reach multiphysics by means of monolithic physics. For such an idea, the first step in a general multi-physics platform is to provide the required models in a “single-physics” mode (Jasak 2006). This is possibly related to the historical development of numerical simulation. Monolithic physics simulations are the traditional commonplace and still predominant, and accordingly, numerous commercial software programs exist. It is thus intuitive and possibly more reasonable to attempt to establish multiphysics frameworks based on the current knowledge and tools for monolithic physics.

The relationship between monolithic physics and multiphysics can be treated in two ways. A common and straightforward way is to approach multiphysics bottom-up as the assembly of individual physical fields. The other way is to view the multiphysical process from a complementary perspective: that problems are intrinsically coupled while a monolithic physics application is merely an idealization made in asymptotic limits (Keyes et al., 2012). The strength of couplings between the physical fields can be used to determine which way to take. The strength of a coupling here is measured by its influence on the overall process. For example, in most cases, a flow of pore water within a porous material could change the porous skeleton, which in turn changes the flow regime. If this flow occurs in stiff rocks, the effect of this coupling to the flow or the fluid flow-rock deformation process is likely to be small. When this small effect is not of concern, a feasible way to consider this problem is to establish a model based on pore water flow and deformation of solids, for which monolithic models are available. The major effort to be made is to find a way to formulate this deformation and, if needed, changes in permeability, both of which are caused by the flow. However, if this process occurs in a soft clay, then there will be a process called consolidation, which can take place over years. The couplings between water flow and solid deformation are very strong and a bottom-up coupling may not be able to provide the accuracy. This is the reason why Terzaghi’s theory of consolidation and Biot’s theory of poroelasticity came into play. In this book, the introduction to multiphysics will be made in a bottom-up way. That is, monolithic physics will be introduced first to prepare the ingredients for cooking multiphysics. However, whenever possible, multiphysical processes will be described from a complementary perspective.

Couplings between different physical fields can be classified as one-way and two-way couplings according to the way in which two fields interact, and as explicit and implicit according to solution techniques. The definitions of one-way and two-way coupling are pretty straightforward. A one-way coupling indicates that, for a specific type of interaction, one field affects the other field but not in the opposite way. While for two-way coupling, two fields will influence each other via the same physical mechanism (s). Further examinations of coupled-field analysis reveal that two types of coupling generally exist for multiphysics problems including implicit (direct) coupling and explicit (sequential) coupling [15]. In an implicit (direct) coupled system, a single matrix system of equations based on all of the relevant physics is assembled and then solved. One drawback of direct-coupled systems is that finding a solution can be costly in terms of required processing power and computer memory requirements. Alternatively, an explicitly (sequentially) coupled system adopts a segregated solver, where the solution to the first set of field equations is passed to the second set of field equations, which is solved and then passed to a third set of equations, etc. This segregated process is repeated until a final solution is obtained. Many commercial multiphysics finite element analysis packages now are able to automatically choose an appropriate default coupling depending on the physics involved or let the user customize these settings, as needed, based on solution time, available computing power, and numerical convergence of the problem at hand (Dede et al., 2014).

The two classification methods can then be combined to categorize couplings based on the level of coupling (Jasak, 2007). The first category is the one-way explicit coupling, which is usually shorted as one-way interaction. This is the simplest way to couple two fields. One-way implicit coupling has rarely been mentioned possibly because the incorporation of a one-way coupling implicitly into the stiffness matrix may not make a significant difference. The next level is the explicit two-way coupling, which is typical when multiple software or discretization methods are involved. In this coupling, simulations run side-by-side and exchange coupling information during the run. This approach formally operates in Picard iterations in computational fluid mechanics and fails even in modestly interacting problems. The level with even stronger coupling is implicit two-way coupling. This level provides closer model-to-model coupling involves matrix-level interaction, where two physical models and model-to-model coupling terms are discretized separately and combined into a single linear system before the solution. This is an implicit variant of the above, with additional stability provided by the linear solver. Here, we can see the benefit of shared matrix and solver modules between multiple discretization schemes. The highest level of coupling is the Equation-level coupling [5], which is however only available for special cases. This type of coupling can be achieved based on the fact that continuum mechanics models originate from mass, momentum and energy balance models. Thus, a fluid-structure system may be considered as a single continuum governed by a single PDE, with different constitutive relations in various regions. Representing the strongest coupling, this level of interaction, in fact, represents a coupled physical model and may require special numerical techniques. Such techniques are particularly suitable for strong coupling or cases involving phase change and transition regions, e.g. melting and solidification.

Some facts have not been explicitly discussed in the literature but may deserve our attention in the future. The first fact is that most existing couplings were understood in the context of binary field systems. It is thus hypothesized that physical fields interact in a pair-wise way. Therefore, the possibility that three or more physical fields work together with multi-channel entangled couplings is totally ruled out. It is believed necessary to mention this possibility here, but also, to point out that the couplings introduced in this book are pair-wise couplings if not mentioned otherwise. The second fact is that it is usually unclear regarding how to count couplings. One way is to define the interaction(s) between two different physical fields as “a” coupling. The other way is to define a specific interaction as a coupling, which is identified according to nature of this interaction including its direction and physical mechanism. This book adopts the latter way. Again, take the interactions between water flow and contaminant transport for example–advection and dispersion will be treated as two different couplings because their underlying mechanisms are different. Similarly, the water flow due to particle movement in electrophoresis and particle movement due to water flow are believed to be two couplings because they have different directions. However, when coupling is used as an uncountable noun, coupling also means any or the overall interaction between two fields.

Establishing multiphysics-based on monolithic physics is not a “one plus one equals two” game (Keyes et al., 2011). The job of coupling individual simulations may introduce limitations on stability, accuracy, or robustness that are severer than the limitations imposed by the individual components. Furthermore, the data motion and data structure conversions require to iterate between independent simulations for each component may be more costly in latency and electrical power than those of the individually tuned components. As a result, ‘‘one plus one’’ may cost significantly more than ‘‘two’’ and may be less amenable to scalable execution (Keyes et al., 2011). Therefore, though the involved single-physics, e.g., fluid flow, structural mechanics, electromagnetic fields, and acoustics, are well-understood, coupling them into one multiphysics simulation is not trivial.

The ’plus’ in this one-plus-one procedure is the main difficulty due to several reasons (Mehl). Firstly, discretizing the whole set of equations for a multiphysics problem is prone to ill-conditioned system matrices that are hard to solve with sufficient accuracy. Also, the implementation of a new code for every possible (and required) combination of single-physics phenomena would be an immense effort. Thirdly, reuse of existing single-physics codes by just gluing them together requires a lot of numerics such as data mapping between non-matching grids and numerical iteration schemes to regain the solution of the fully coupled system and technical solutions for code-to-code communication. Fourth, the high accuracy of multiphysics models can only be exploited with a very high resolution of the underlying computational grids, necessitating the use of massively parallel supercomputers.

Perhaps one of the greatest challenges for multiphysics simulation is to understand all of the relevant physics involved and to set up a geometrically accurate model with appropriate loads and boundary conditions (Dede et al., 2014). The purpose of this book is to provide such a framework to multiphysics especially those in porous materials. There is currently no such a comprehensive framework. Therefore, whenever a designer would like to conduct a multiphysics analysis for which he or she does not have such background, it is intended that this book can be either a comprehensive introduction or a quick reference. However, we do not consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. In fact, many multiphysics applications can degenerate into an algebraic paradigm via linearization in which individual components, i.e., monolithic physics, is represented by diagonal blocks and the multiphysics coupling between them, as off-diagonal blocks (Keyes et al., 2012).

Physical Fields and Types of Multiphysics

The part “physics” in “multiphysics” denotes “physical field”. There, multiphysics means the coexistence of multiple physical fields in a process or a system. In physics, a field is a physical quantity that has a value for each point in space and time. For example, on a weather map, a vector at each point of the map is assigned to represent the surface wind velocity, including both speed and direction of the movement of air at that point. In another classic example, an electric field can be viewed as a “condition in space” emanating from an electric charge and extending throughout the whole of space. If a test electric charge is placed in this electric field, the particle will accelerate due to the force associated with the “condition in space”. Possibly attributed to the latter example, physicists tend to think of the notion of a field to be the cause of force.

The origin of “field” is also related to force. In the eighteenth century, a new concept was proposed to simplify the all these gravitational forces. This quantity, namely the gravitational field, was believed to exist at each point in space, whereby the total gravitational force which can be felt by an object with a unit mass at that point. This concept just provided another way to interpret the force rather than change the physics. With the field concept, two treatments will be equivalent: one calculates all the gravitational forces on an object individually and then adds them together, or one first adds all the contributions together as a gravitational field and then applies it to an object. [9] Then in the 19th century, the concept of a field received substantive developments due to the development of the theory of electromagnetism. In the early stages, Newton-style laws were employed to express the forces between pairs of electric charges or electric currents. Later the field concept became much more predominant for expressing these laws in terms of electric and magnetic fields. It is believed that Michael Faraday was the first to coin the term “field” in 1849. [9]

The scope of field may be wider than what is needed in multiphysics. In this book, the fields we are dealing with are constrained to classical fields, which means the quantum field will not be considered. Actually, the discussion of the quantum field in the context of multiphysics is not only interesting but also necessary. First, the quantum field has seldom been discussed in the multiphysics literature. This has been at least a convention, though it is not necessarily be in this way forever as the scope of multiphysics is expanding. The quantum field will not be counted or discussed in the book. However, it is necessary to point out that the physical fields we are dealing with occur in virtue of quantum mechanics. This is because common physical processes occur due to the basic forces. These basic forces except gravity, i.e., strong interaction, electromagnetic force, weak interaction, can find their origin in the quantum field. As a result, multiphysics will be constrained above the scale of atoms while it is not necessarily a solely macroscopic theory. For example, hydrodynamics can span over scales from geologic scale down to Kolmogorov microscale and chemical field can reach down to the molecular scale. Another presumption is that the fields in multiphysics are non-relativistic. Descriptions of physical fields were given before the advent of relativity theory and then revised in light of this theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic. Relativistic field theory can explain gravity. But we do not need to worry too much about the origins of the basic forces because we just deal with the non-relativistic fields and have physical laws above the quantum level.

The discussion in the above paragraph essentially determines the mathematical languages to be used in multiphysics. A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to the type of physical quantity they represent: a scalar, a vector, a spinor or a tensor. A field has a unique tensorial character at every point where it is defined. That is, a field can only exclusively be a scalar field or a vector field. Take the Newtonian gravitational field for example–it requires three numbers to specify the components along the three spatial dimensions at any point, leading to a vector field. However, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively. As we do not deal with the quantum mechanics here, so we just to use the normal operators in calculus analysis.

The use of physical fields extends all over most science and engineering disciplines. However, a discipline or sub-discipline only primarily deals with one a few physical fields. This fact makes multiphysics an extremely interdisciplinary topic and also poses challenges in enumerating the fields. However, the following fields can be identified by overviewing the core courses in the disciplines and publications on multiphysics: heat transfer (thermo-), pore water movement (hydro-), concentration field (concentro or diffuso/convecto/advecto), stress and strain (mechano-), dynamics (dyno-), chemical reactions (chemo- or chemico-), electrostatics (electro-), and magnetostatics (magneto-). In multiphysics, a multiphysical process is usually titled using compound words such as “thermo-hydro-mechanical”. These words now appear as the birth birthmarks or tattoos of multiphysics. The prefixes, roots or suffixes in the parentheses after each physical field are the ingredients for cooking multiphysics. Each physical field is called a monolithic physical field, uniphysics, or single physics.

Combinations of the above monolithic physics can lead to 247 possible types of multiphysics. However, the world of multiphysics is not only a game of combination in statistics. But instead, we coin a name and investigate a type of multiphysical phenomenon under this name based on the observations in nature and sciences and the needs from practice. Based on a review of multiphysics with emphasis on porous materials, the most representative multiphysical processes are believed to the follows: thermomechanics, hydromechanics, thermohydromechanics, electrokinetics, electromagnetics, elastodynamics, fluid dynamics, hydrodynomechanics, thermoelectricity, thermoelectromagnetics, magnetomechanics, electromechanics, and electromagnetomechanics. Another point deserving attention is that the orders of the roots in the name. Generally, it would be desirable to put the major process, cause or process of primary interest in the front. However, there is no widely accepted rule. For a process such as poroelasticity, it may even be a difficult job to judge which process is dominant, i.e., water movement and solid skeleton deformation.

There is more than one way to categorize and name multiphysics. For example, the multiphysics software COMSOL categorizes multiphysical phenomena based on applications. The most common multiphysics applications were divided into four major categories: electrical, mechanical, fluid, and chemical. Electrical applications include Joule heating, induction heat, microwave heating, piezoelectric effect, piezoresistive effect and electromechanical effects. The mechanical section consists of thermal expansion, thermal stress, and acoustic-structure interaction. The fluid section contains Navier-Stokes equation, Boussinesq approximation, non-isothermal flow, and fluid-structure interaction, poroelasticity, squeezed and sliding films. The chemical section comprises convection and diffusion. The advantage of this classification is that it is conducted based on applications, so it uses terms closer to applications. But the disadvantages are also very clear. The adopted terms do not effectively reveal the underlying mechanisms and cannot really reflect the real hierarchy and evolution of multiphysics and their relationships to monolithic physics. On the contrary, a classification purely based on the combinations of monolithic physics is not able to reflect the applications in the real word. For example, many combinations of monolithic physics may not have any existing applications at all. The classification system used in this book is a compromise between theories and applications. So you can see it still keeps the birthmarks or tattoos, but at the same time, adopts terms that are widely accepted but actually do not reflect the involved monolithic physics, such as hydrodynamics and poroelasticity.