Multiphysics simulation

Multiphysics is defined as the coupled processes or systems involving more than one simultaneously occurring physical fields and the studies of and knowledge about these processes and systems . As an interdisciplinary study area, multiphysics spans over many science and engineering disciplines. Multiphysics is a practice built on mathematics, physics, application, and numerical analysis. The mathematics involved usually contains partial differential equations and tensor analysis. The physics refers to common types of physical processes, e.g., 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-). The applications are underlain by physics processes involving one or more of the these monolithic physical processes. Typical applications are soil consolidation theories, fluid flow simulation, electrokinetic applications, computational electromagnetics, sensor deisign with piezoelectric materials, fluid-structure interaction analysis, and energy and climate change considerations in porous materials.

Definition

There are multiple definitions for multiphysics. In a broad sense, multiphysics refers to simulations that involve multiple physical models or multiple simultaneous physical phenomena. The inclusion of “multiple physical models” makes this definition a very broad and general concept, but this definition is a little bit self-contradictory as the implication of physical models may include that of physical phenomena . COMSOL defines multiphysics in a relatively narrow sense: multiphysics includes 1. coupled physical phenomena in computer simulation and 2. the study of multiple interacting physical properties. In another definition, a multiphysics system consists of more than one component governed by its own principle(s) for evolution or equilibrium, typically conservation or constitutive laws . This definition is very close to the previous one except for that it does not emphasize physical properties. In a more strict way, multiphysics can be defined as those processes including closely coupled interactions among separate continuum physics phenomena . In this definition, two-way exchange of information between physical fields, which could involve implicit convergence within a time step is the essential feature. Based on the above definitions, 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 .

History and Future

Multiphysics is neither a research concept far from daily life nor a recently-developed theory or technique. In fact, we live in a multiphysics world. Natural and artificial systems are running with various types of physical phenomena at different spatial and temporal scales: from atoms to galaxies and from pico-seconds to centuries. A few representative examples in fundamental and applied sciences are loads and deformations on solids, complex flows, fluid-structure interactions, plasma and chemical processes, thermo-mechanical and electromagnetic systems . Even the tablet that you use to view this Wikipedia page is a "multiphysics" example – the WIFI antenna receives electromagnetic waves; the touchscreen is designed for the interaction between mechanical and electrical components; and the battery involves chemical reactions and electrical currents [1].

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 involve physical processes that cannot be covered by a single traditional discipline. This trend requires us to extend our analysis capacity to solve more complicated and more multidisciplinary problems. 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. Multiphysics has also become a frontier in industrial practice. Simulation programs have been evolving into a tool in design, product development, and quality control. During these creation processes, engineers are now required to think in areas outside of their training, even with the assistance of the simulation tools. 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.” The auto industry gives out a good example. Traditionally, different groups of people focus on the structure, fluids, electromagnets and the other individual aspect separately. By constrast, the intersection of aspects, which may represent two physics topics and once was a gray area, can be the essential link in the life cycle of the product. As commented by , “Design engineers are running more and more multiphysics simulations every day because they need to add reality into their models.” Therefore, multiphysics represents the trend of industrial collaboration.

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 can used to represent the surface wind velocity with both speed and direction for the movement of air at that point.

A review of multiphysics with an emphasis on porous materials yields the following most representative multiphysical processes:

• thermomechanics,
• hydromechanics,
• thermohydromechanics,
• electrokinetics,
• electromagnetics,
• elastodynamics,
• fluid dynamics,
• hydrodynomechanics,
• thermoelectromagnetics,
• electromagnetomechanics.

It is noted that the orders of the roots in the name. A general rule is to put the major process, cause or process of primary interest in the front, though exceptions can be found in the literature. The naming of many multiphysics application, such as poroelasticity, gained names in other ways. For such multiphysics applications, it may even be difficult to judge the dominant mechanisms, i.e., water movement and solid skeleton deformation.

How to do Multiphysics?

The implementation of multiphysics usually follows the following procedure: identifying a multiphysical process/system, developing a mathematical description of this process/system, discretizing this mathematical model into an algebraic system, and solving this algebraic equation system and postprocessing the data. The abstraction of a multiphysical problem from a complex phenomenon and the description of such a problem are usually not emphasized but very critical to the success of the multiphysics analysis. This requires to identify the system to be analyzed, including geometry, materials and dominant mechanisms. The identified system will be interpreted using mathematics languages (function, tensor, differential equation) as computational domain, boundary conditions, auxiliary equations and governing equations. Discretization, solution and postprocessing are carried out using computers. Therefore, the above procedure is not much different from those in general numerical simulation based on the discretization of partial differential equations.

Mathematics Model

A mathematical model is essentially a set of equations. The equations can be divided into three categories according to the nature and intended role. The first category is governing equations. A governing equation describes the major physical mechanisms and process without further revealing the change and nonlinearity of the material properties. For example, in a heat transfer problem, the governing equation could describe a process in which the thermal energy (represented using temperature or enthalpy) at an infinitesimal point or a representative element volume is changed due to energy transferred from surrounding points via conduction, advection, radiation, and internal heat sources or any combinations of these four heat transfer mechanisms as the following equation :

${\underbrace {\frac {\partial u}{\partial t}} _{\rm {Accumulation}}\underbrace {+\nabla \cdot \left(uv\right)} _{\rm {Advection}}\underbrace {-\nabla \cdot \left(K\nabla u\right)} _{{\rm {Diffusion\;}}\left({\rm {Conduction}}\right)}\underbrace {-\nabla \cdot \left(D\nabla u\right)} _{\rm {Dispersion}}=\underbrace {Q} _{\rm {Source}}}$.

Couplings between fields can be achieved in each category.

Discretization

Discretization is the step right following the establishment of a mathematical model. This is because the mathematical model consisting of continuous equations usually cannot be solved directly using computers, though it is possible to solve some mathematical models by hand or by symbolic computation programs. The discretization of the mathematical model leads to an algebraic equations system as

$\mathrm {K} \cdot \mathrm {u} =\mathrm {F}$,                                                                                       where $\mathrm {K}$ is the stiffness matrix, $\mathrm {u}$ is the matrix of unknowns and $\mathrm {F}$ is the force matrix.

Many different methods are available for discretization. In multiphysics, the most popular discretization methods are the finite difference method, finite volume method, and finite element method . These numerical methods employ significantly different ways for discretization.

Solution and Postprocessing

Solution to multiphysics problems can be obtained by solving the above algebraic equation system using direct or iterative methods. Compared to meshing, selecting and setting the solvers and obtaining a solution to the equations constituting the numerical model within a reasonable computational time is possibly an even more difficult task. After solution we usually conduct postprocessing, in which we visualize simulation results in terms of 2D/3D plots. Error estimation is frequently adopted in post processing. To compare errors, we employ different mesh sizes to estimate the convergence of the numerical solution. Sensitivity analysis is also utilized in some multiphysics studies. Different input, such as material properties, initial conditions, and boundary conditions are used to test the influence of some parameters on the overall behavior of the numerical model. Sensitivity analysis is useful because it helps understand the structure of the problem, hidden bugs in the code, and errors in the model.

Discretization Method

Multiphysics is usually numerical implemented with discrectization methods such Finite Element Method, Finite Difference Method, and Finite Volume Method. Many software packages mainly rely on the finite element method or similar commonplace numerical methods for simulating coupled physics: thermal stress, electro- and acousto- magnetomechanical interaction, fluid structure interaction (FSI), fluid flow with heat transport and chemical reactions, electromagnetic fluids (magnetohydrodynamics or plasma), and electromagnetically induced heating. In many cases, to get accurate results, it is important to include mutual dependencies where the material properties significant for one field (such as the electric field) vary with the value of another field (such as temperature) and vice versa.

Multiple Discretization Methods

There are cases where each subset of partial differential equations has different mathematical behavior, for example when compressible fluid flow is coupled with structural analysis or heat transfer. To perform an optimal simulation in those cases, a different discretization procedure must be applied to each subset. For example, the compressible flow is discretized with a finite volume method and the conjugate heat transfer with a finite element analysis. Another example is the use of electromagnetic or electrostatic Particle-in-cell (PIC, EMPIC, ESPIC) methods combined with Direct simulation Monte Carlo, where the particles may interact with an electromagnetic (EM) field or other fields, with each other, and with fluids evolved by finite volume or other methods. The particles interact with the EM fields through the charges and currents they create and by being accelerated by the EM field. Particles collide with each other, and they collide with fluids.

References

• Susan L. Graham, Marc Snir, and Cynthia A. Patterson (Editors), Getting Up to Speed: The Future of Supercomputing, Appendix D. The National Academies Press, Washington DC, 2004. ISBN 0-309-09502-6.
• Paul Lethbridge, Multiphysics Analysis, p26, The Industrial Physicist, Dec 2004/Jan 2005, [2], Archived at: [3]