Conversion Between Material and Spatial Frames
This section explains how the user inputs are converted between material and spatial frames. The conversion depends on the dimension of the variables (scalars, vectors, or tensors) and on the density order.
As described in the previous paragraph Material and Spatial Frames, lowercase letters are used to denote the spatial frame coordinates while uppercase letters denote the material frame coordinates. In the followings, a physical quantity A will be referred to as A(x, y, z) in the spatial frame and to as A(X, Y, Z) in the material frame.
The equations solved by the heat transfer interfaces are written in the spatial frame. When an input is specified in the material frame, conversion is necessary to deduce A(x, y, z) from A(X, Y, Z).
Density, Heat Source, Heat Flux
In heat transfer, the following variables are relative scalars of weight one (also called scalar densities):
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Mass density, ρ (SI unit: kg/m3),
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Heat source, Q0 (SI unit: W/m3),
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Production/absorption coefficient, qs (SI unit: W/(m3·K)),
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Heat flux, q0 (SI unit: W/m2),
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Heat transfer coefficient, h (SI unit: W/(m2·K)).
For all these variables, the conversion between material and spatial frame follows the relation:
This way, the integral of volumetric quantities over the domain, such as the mass density, is invariant between frames:
In these equalities, Ω0 and Ω denote the same domain but represented in material or in spatial frame, respectively. As expected, the same mass is found by integrating ρ(X, Y, Z) over the domain in the material frame or by integrating ρ(x, y, z) over the domain in the spatial frame. The same invariance principle applies to quantities per unit area, in particular heat flux and heat transfer coefficient:
Here, ∂Ω0 and ∂Ω are the boundaries of the same domain in material and spatial frames, respectively.
Velocity
The relationship between the velocity vectors in material and spatial frames, u(X, Y, Z) and u(x, y, z), is
This is directly deduced from the differential relation of Equation 4-178.
Thermal Conductivity
Thermal conductivity, k, is a tensor density. The relationship between the value on the spatial frame and the material frame is:
With this relation, and recalling that
the total conductive heat flux through a boundary, computed in both frames according to the integrals below, gives the same result:
Here, ∂Ω0 and ∂Ω are the boundaries of the same domain in material and spatial frames, respectively.
Thermal Conductivity of a Layer
The same transformations are applied to thermal conductivity but with different transformation matrices. The deformation gradient tensor depends on the layer type:
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When the layer is resistive, the deformation gradient tensor Fxdim is equal to the deformation gradient tensor F defined in Equation 4-177.
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When the layer is conductive, the deformation gradient tensor Ft is defined using tangential derivatives as follows:
where xTX corresponds to the tangential derivative x with respect to X, and so on.
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When the layer is an extra dimension, the deformation gradient tensor Fxdim is defined as follows:
where xTX corresponds to the tangential derivative x with respect to X, and so on. The (nx, ny, nz) vector corresponds to the normal vector in the spatial frame, and the (nX, nY, nZ) vector corresponds to the normal vector in the material frame.
Time Derivative
Partial differential equations often involve time derivative of a physical quantity such as temperature or internal energy in heat transfer. The variations of such state variables during an elementary time step are studied for a same elementary volume that could be subjected to spatial transformations. The material derivative, denoted d ⁄ dt, is the derivation operator used in such cases. The following relation defines the material derivative in the spatial frame.
The right-hand side of this relation shows a new term u ⋅ ∇ corresponding to convection in the case of fluids, or convected quantity by translational motion of a solid.
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About Frames in the COMSOL Multiphysics Reference Manual.
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Handling Frames in Heat Transfer