## Components of Multiphysics Models

### Theory

The implementation of multiphysics includes several essential sequential components: identifying a multiphysical process/system, developing a mathematical description of this process/system, discretizing this mathematical model into an algebraic system, and finally, solving this algebraic equation system and postprocessing the data to obtain information of interest. A mathematical model boils down to a bunch of equations. According to the nature and intended role, the equations can be divided into three categories. Couplings between fields can be achieved in each category. As shown in Figure below, the mathematical model which is defined in a continuous domain will be discretized into an algebraic model defined on a meshed domain for solution.

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: $\underbrace{\frac{\partial u}{\partial t} }_{{\rm Accumulation}}\underbrace{+\nabla \cdot \left(uv\right)}_{{\rm Advection}}\underbrace{-\nabla \cdot \left(K\nabla u\right)}_{{\rm Diffusion\; }\left({\rm Conduction}\right)}\underbrace{-\nabla \cdot \left(D\nabla u\right)}_{{\rm Dispersion}}=\underbrace{Q}_{{\rm Source}}$ According to the definition of multiphysics, one physical field of one component in one region only has one governing equation. Therefore, the mathematical model for any multiphysical process should contain at least two equations. Likewise, a mathematical model for a physical process containing two or more governing equations indicates multiphysics according to this criterion. Couplings can be embedded into in governing equations. For example, in the thermal field, heat advection as a separate term describes the coupling from the fluid movement to the thermal field.

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