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+<!-- source-page: 231 -->
+
+# Procedures
+
+repetition about an axis in three dimensions. Combinations of the different types of symmetry are supported.
+
+To illustrate the handling of symmetries during viewfactor calculation, consider the case of a simple reflection symmetry in two-dimensional space (Figure 2.11.5-4). Radiation between facet i (with its centroid at point $A )$ and facet $j$ (with its centroid at point B) has two contributions: one arising from the ray between points A and B and the other coming from the ray between points A and $B ^ { \prime }$ , where $B ^ { \prime }$ is the mirror image of B. The length of ray AB is defined directly in the model. The definition of the length of ray $A B ^ { \prime }$ requires that point $C$ on the reflection symmetry line be located such that $A C$ and $B C$ make equal angles to it; ray $A B ^ { \prime }$ then has length $A C + B C$ . Similar logic can be extended to the three-dimensional case.
+
+In axisymmetric cases symmetry about the axis of symmetry of the model is always implied, and the only other symmetries allowed are simple reflection through a plane normal to the axis of symmetry or periodic repetition in the direction of the axis of symmetry.
+
+Figure 2.11.5-4 Reflection symmetry example.   
+![](images/page-231_4a1f3899abf4fcaff33ccb5e30f4cbc01802e80f419e856ebc1f4f450f0dab78.jpg)
+
+<details>
+<summary>text_image</summary>
+
+i
+A
+B
+j
+C
+B'
+B
+</details>
+
+# Viewfactor checking
+
+The viewfactor is a purely geometrical quantity, and it has some special properties. One property that allows us to check the accuracy of the calculation is that for a completely enclosed cavity:
+
+Equation 2.11.5-2
+
+$$
+\frac {1}{A _ {i}} \sum_ {j} F _ {i j} = 1,
+$$
+
+indicating that any ray from surface i in whatever direction it leaves the surface will reach another surface in the enclosed cavity.
+
+The quantity in Equation 2.11.5-2 is calculated for every facet of each cavity, and its value is used to provide a check to control the accuracy of viewfactor calculation.
+
+<!-- source-page: 232 -->
+
+# Radiation to ambient
+
+The quantity calculated in Equation 2.11.5-2 can deviate from unity so long as the cavity is not fully enclosed. The user can define such an open cavity by giving the value of the ambient temperature in the cavity definition. In this case the difference between one and the quantity calculated in Equation 2.11.5-2 for each facet of the open cavity is considered to be the fraction radiating from that facet to the surrounding medium.
+
+# 2.12 Coupled thermal-electrical analysis
+
+# 2.12.1 Coupled thermal-electrical analysis
+
+Joule heating arises when the energy dissipated by an electrical current flowing through a conductor is converted into thermal energy. ABAQUS/Standard provides a fully coupled thermal-electrical procedure for analyzing this type of problem. Coupling arises from two sources: the conductivity in the electrical problem is temperature dependent, and the internal heat generated in the thermal problem is a function of electrical current. The thermal part of the problem includes all the heat conduction and heat storage (specific and latent heat) features described in \`\`Uncoupled heat transfer analysis,'' Section 2.11.1. (Forced heat convection caused by fluid flowing through the mesh is not considered.)
+
+The thermal-electrical elements have both temperature and electrical potential as nodal variables.
+
+This section describes the governing equilibrium equations, the constitutive model, boundary conditions, the surface interaction model, finite element discretization, and the components of the Jacobian used.
+
+# Governing equations
+
+The electric field in a conducting material is governed by Maxwell's equation of conservation of charge. Assuming steady-state direct current, the equation reduces to
+
+Equation 2.12.1-1
+
+$$
+\int_ {S} \mathbf {J} \cdot \mathbf {n} d S = \int_ {V} r _ {c} d V,
+$$
+
+where V is any control volume whose surface is S, n is the outward normal to S, J is the electrical current density (current per unit area), and $r _ { c }$ is the internal volumetric current source per unit volume.
+
+The divergence theorem is used to convert the surface integral into a volume integral:
+
+$$
+\int_ {V} \left[ \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {J} - r _ {c} \right] d V = 0,
+$$
+
+and since the volume is arbitrary, this provides the pointwise differential equation
+
+$$
+\frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {J} - r _ {c} = 0.
+$$
+
+<!-- source-page: 233 -->
+
+# Procedures
+
+The equivalent weak form is obtained by introducing an arbitrary, variational, electrical potential field, $\delta \varphi$ , and integrating over the volume:
+
+$$
+\int_ {V} \delta \varphi \left[ \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {J} - r _ {c} \right] d V = 0.
+$$
+
+Using first the chain rule and then the divergence theorem, this statement can be rewritten as
+
+$$
+- \int_ {V} \frac {\partial \delta \varphi}{\partial \mathbf {x}} \cdot \mathbf {J} d V = \int_ {S} \delta \varphi J d S + \int_ {V} \delta \varphi r _ {c} d V,
+$$
+
+where ${ \boldsymbol { J } } { \stackrel { \mathrm { d e f } } { = } } - \mathbf { J } \cdot \mathbf { n }$ is the current density entering the control volume across $S .$
+
+# Constitutive behavior
+
+The flow of electrical current is described by Ohm's law:
+
+$$
+\mathbf {J} = \boldsymbol {\sigma} ^ {E} \cdot \mathbf {E},
+$$
+
+where ${ \pmb \sigma } ^ { E } ( \theta , f ^ { \alpha } )$ is the electrical conductivity matrix; µ is the temperature; and $f ^ { \alpha } , \alpha = 1 , 2 .$ :: are any predefined field variables. The conductivity can be isotropic, orthotropic, or fully anisotropic. $\mathbf { E } ( \mathbf { x } )$ is the electrical field intensity defined as
+
+Equation 2.12.1-2
+
+$$
+\mathbf {E} = - \frac {\partial \varphi}{\partial \mathbf {x}}.
+$$
+
+Since a potential rise occurs when a charged particle moves against the electrical field, the direction of the gradient is opposite to that of the electrical field. Using this definition of the electrical field, Ohm's law is rewritten as
+
+Equation 2.12.1-3
+
+$$
+\mathbf {J} = - \boldsymbol {\sigma} ^ {E} \cdot \frac {\partial \varphi}{\partial \mathbf {x}}.
+$$
+
+The constitutive relation is linear; that is, it assumes that the electrical conductivity is independent of the electrical field.
+
+Introducing Ohm's law, the governing conservation of charge equation becomes
+
+Equation 2.12.1-4
+
+$$
+\int_ {V} \frac {\partial \delta \varphi}{\partial \mathbf {x}} \cdot \pmb {\sigma} ^ {E} \cdot \frac {\partial \varphi}{\partial \mathbf {x}} d V = \int_ {V} \delta \varphi r _ {c} d V + \int_ {S} \delta \varphi J d S.
+$$
+
+# Thermal energy balance
+
+The heat conduction behavior is described by the basic energy balance relation
+
+<!-- source-page: 234 -->
+
+Equation 2.12.1-5
+
+$$
+\int_ {V} \rho \dot {U} \delta \theta d V + \int_ {V} \frac {\partial \delta \theta}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}} d V = \int_ {V} \delta \theta r d V + \int_ {S} \delta \theta q d S,
+$$
+
+where V is a volume of solid material, with surface area $S ; \rho$ is the density of the material; U is the internal energy; k is the thermal conductivity matrix; q is the heat flux per unit area of the body, flowing into the body; and r is the heat generated within the body. The thermal problem is discussed in detail in \`\`Uncoupled heat transfer analysis,'' Section 2.11.1.
+
+Equation 2.12.1-4 and Equation 2.12.1-5 describe the electrical and thermal problems, respectively. Coupling arises from two sources: the conductivity in the electrical problem is temperature dependent, $\pmb { \sigma } ^ { E } = \pmb { \sigma } ^ { E } ( \theta )$ , and the internal heat generation in the thermal problem is a function of electrical current, $r = r _ { e c } ( \mathbf { J } )$ , as described below.
+
+# Thermal energy due to electrical current
+
+Joule's law describes the rate of electrical energy, $P _ { e c }$ , dissipated by current flowing through a conductor as
+
+$$
+P _ {e c} = \mathbf {E} \cdot \mathbf {J}.
+$$
+
+Using Equation 2.12.1-2 and Equation 2.12.1-3, Joule's law is rewritten as
+
+$$
+P _ {e c} = \mathbf {E} \cdot \pmb {\sigma} ^ {E} \cdot \mathbf {E}.
+$$
+
+In a steady-state analysis $P _ { e c }$ is evaluated at time $t + \Delta t$ . In a transient analysis an averaged value of $P _ { e c }$ is obtained over the increment
+
+$$
+\begin{array}{l} P _ {e c} = \frac {1}{\Delta t} \int_ {\Delta t} P _ {e c} d t \\ = \mathbf {E} \cdot \pmb {\sigma} ^ {E} \cdot \mathbf {E} - \mathbf {E} \cdot \pmb {\sigma} ^ {E} \cdot \Delta \mathbf {E} + \frac {1}{3} \Delta \mathbf {E} \cdot \pmb {\sigma} ^ {E} \cdot \Delta \mathbf {E}, \\ \end{array}
+$$
+
+where E and ${ \pmb { \sigma } } ^ { E }$ are values at time $t + \Delta t$ . The amount of this energy released as internal heat is
+
+$$
+r = \eta_ {v} P _ {e c},
+$$
+
+where $\eta _ { v }$ is an energy conversion factor.
+
+# Surface conditions
+
+The surface--S--of the body consists of parts on which boundary conditions can be prescribed-- $S _ { p } { \mathrm { - } } { \mathrm { a n d } }$ parts that can interact with nearby surfaces of other bodies $\ -- S _ { i }$ . Prescribed boundary conditions include the electrical potential, $\varphi = \varphi ( \mathbf { x } , t )$ ; temperature, $\theta = \theta ( \mathbf { x } , t )$ ; electrical current density, $J = J ( \mathbf { x } , t )$ ; heat flux, $q = q ( \mathbf { x } , t )$ ; and surface convection and radiation conditions. The surface interaction model includes heat conduction and radiation effects between the interface surfaces and electrical current flowing across the interface. Heat conduction and radiation are modeled
+
+<!-- source-page: 235 -->
+
+# Procedures
+
+by
+
+$$
+q _ {c} = k _ {g} (\theta_ {B} - \theta),
+$$
+
+and
+
+$$
+q _ {r} = F _ {B} (\theta_ {B} - \theta^ {z}) ^ {4} - F (\theta - \theta^ {z}) ^ {4},
+$$
+
+respectively, where $\theta$ is the temperature on the surface of the body under consideration, $\theta _ { B }$ is the temperature on the surface of the other body, $\theta ^ { z }$ is the value of absolute zero temperature on the temperature scale being used, $k _ { g } ( \bar { \theta } , \bar { f } ^ { \alpha } )$ is the gap thermal conductance, $\begin{array} { r } { \bar { \theta } = \frac { 1 } { 2 } ( \theta + \theta _ { B } ) } \end{array}$ is the average interface temperature, $\begin{array} { r } { \bar { f } ^ { \alpha } = \frac { 1 } { 2 } ( f _ { A } ^ { \alpha } + f _ { B } ^ { \alpha } ) } \end{array}$ is the average of any predefined field variables at A and B, and $F$ and $F _ { B }$ are constants.
+
+The electrical current flowing between the interface surfaces is modeled as
+
+$$
+J = \sigma_ {g} (\varphi_ {B} - \varphi),
+$$
+
+where $\varphi$ is the electrical potential on the surface of the body under consideration, $\varphi _ { B }$ is the electrical potential on the surface of the other body, and $\sigma _ { g } ( \bar { \theta } , \bar { f } ^ { \alpha } )$ is the gap electrical conductance. The electrical energy dissipated by the current flowing across the interface,
+
+$$
+P _ {e c} = J (\varphi_ {B} - \varphi) = \sigma_ {g} (\varphi_ {B} - \varphi) ^ {2},
+$$
+
+is released as heat on the surfaces of the bodies:
+
+$$
+q _ {e c} = f \eta_ {g} P _ {e c}, \qquad \mathrm{and} \qquad q _ {e c} ^ {B} = (1 - f) \eta_ {g} P _ {e c},
+$$
+
+where $\eta _ { g }$ is an energy conversion factor and f specifies how the total heat is distributed between the interface surfaces. $P _ { e c }$ is evaluated at the end of the time increment in a steady-state analysis, and an averaged value over the time increment is used in a transient analysis. This is described in detail in \`\`Heat generation caused by electrical current,'' Section 5.2.6.
+
+Introducing the surface interaction effects and electrical energy released as thermal energy, the governing electric and thermal equations become
+
+Equation 2.12.1-6
+
+$$
+\int_ {V} \frac {\partial \delta \varphi}{\partial \mathbf {x}} \cdot \pmb {\sigma} ^ {E} \cdot \frac {\partial \varphi}{\partial \mathbf {x}} d V = \int_ {V} \delta \varphi r _ {c} d V + \int_ {S _ {p}} \delta \varphi J d S + \int_ {S _ {i}} \delta \varphi \sigma_ {g} (\varphi_ {B} - \varphi) d S,
+$$
+
+and
+
+Equation 2.12.1-7
+
+<!-- source-page: 236 -->
+
+$$
+\begin{array}{l} \int_ {V} \rho \dot {U} \delta \theta d V + \int_ {V} \frac {\partial \delta \theta}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}} d V = \int_ {V} \delta \theta r d V + \int_ {V} \delta \theta \eta_ {v} P _ {e c} d V \\ + \int_ {S _ {p}} \delta \theta q d S + \int_ {S _ {i}} \delta \theta (q _ {c} + q _ {r} + q _ {e c}) d S. \\ \end{array}
+$$
+
+# Spatial discretization
+
+In a finite element model equilibrium is approximated as a finite set of equations by introducing interpolation functions. Discretized quantities are indicated by uppercase superscripts (for example, $\varphi ^ { N } )$ . The summation convention is adopted for the superscripts. The discretized quantities represent nodal variables, with nodes shared between adjacent elements and appropriate interpolation chosen to provide adequate continuity of the assumed variation.
+
+The virtual electrical potential field is interpolated by
+
+$$
+\delta \varphi = \mathrm{N} ^ {N} \delta \varphi^ {N},
+$$
+
+where $\mathrm { N } ^ { N }$ are the interpolation functions. The discretized electrical equation is then written as
+
+$$
+\delta \varphi^ {N} \left\{\int_ {V} \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \pmb {\sigma} ^ {E} \cdot \frac {\partial \varphi}{\partial \mathbf {x}} d V = \int_ {V} N ^ {N} r _ {c} d V + \int_ {S _ {p}} N ^ {N} J d S + \int_ {S _ {i}} N ^ {N} \sigma_ {g} (\varphi_ {B} - \varphi) d S \right\}.
+$$
+
+Since $\delta \varphi$ is arbitrary,
+
+Equation 2.12.1-8
+
+$$
+I _ {\varphi} ^ {N} = \int_ {V} \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \pmb {\sigma} ^ {E} \cdot \frac {\partial \varphi}{\partial \mathbf {x}} d V - \int_ {V} N ^ {N} r _ {c} d V - \int_ {S _ {p}} N ^ {N} J d S - \int_ {S _ {i}} N ^ {N} \sigma_ {g} (\varphi_ {B} - \varphi) d S = 0.
+$$
+
+The temperature field in the thermal problem is approximated by the same set of interpolation functions:
+
+$$
+\delta \theta = \mathrm{N} ^ {P} \delta \theta^ {P}.
+$$
+
+Using these interpolation functions and a backward difference operator to integrate the internal energy rate, $\dot { U } _ { ; }$ , the thermal energy balance relation is obtained:
+
+Equation 2.12.1-9
+
+$$
+\begin{array}{l} I _ {\theta} ^ {P} = \frac {1}{\Delta t} \int_ {V} N ^ {P} \rho (U _ {t + \Delta t} - U _ {t}) d V + \int_ {V} \frac {\partial N ^ {P}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}} d V - \int_ {V} N ^ {P} r d V \\ - \int_ {V} N ^ {P} \eta_ {v} P _ {e c} d V - \int_ {S _ {p}} N ^ {P} q d S - \int_ {S _ {i}} N ^ {P} (q _ {c} + q _ {r} + q _ {e c}) d S = 0. \\ \end{array}
+$$
+
+# Jacobian contributions
+
+<!-- source-page: 237 -->
+
+The Jacobian contributions are obtained by taking variations of Equation 2.12.1-8 and Equation 2.12.1-9 with respect to the electrical potential, ', and the temperature, µ, at time $t + \Delta t$ . This yields
+
+$$
+\mathbf {K} _ {\varphi \varphi} ^ {N M} = \frac {\partial I _ {\varphi} ^ {N}}{\partial \varphi^ {M}} = \int_ {V} \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \pmb {\sigma} ^ {E} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V + \int_ {S _ {i}} N ^ {N} \sigma_ {g} N ^ {M} d S,
+$$
+
+$$
+\mathbf {K} _ {\varphi \theta} ^ {N Q} = \frac {\partial I _ {\varphi} ^ {N}}{\partial \theta^ {Q}} = - \int_ {V} \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \frac {\partial \pmb {\sigma} ^ {E}}{\partial \theta} \cdot \mathbf {E} N ^ {Q} d V - \frac {1}{2} \int_ {S _ {i}} N ^ {N} \frac {\partial \sigma_ {g}}{\partial \bar {\theta}} (\varphi_ {B} - \varphi) N ^ {Q} d S,
+$$
+
+$$
+\mathbf {K} _ {\theta \varphi} ^ {P M} = \frac {\partial I _ {\theta} ^ {P}}{\partial \varphi^ {M}} = \int_ {V} N ^ {P} \eta_ {v} \left(\mathbf {J} - \frac {1}{3} \Delta \mathbf {J}\right) \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V + 2 \int_ {S _ {i}} N ^ {P} f \eta_ {g} \sigma_ {g} (\varphi_ {B} - \varphi) N ^ {M} d S,
+$$
+
+$$
+\begin{array}{l} \mathbf {K} _ {\theta \theta} ^ {P Q} = \frac {\partial I _ {\theta} ^ {P}}{\partial \theta^ {Q}} = \frac {1}{\Delta t} \int_ {V} N ^ {P} \rho \frac {d U}{d \theta} N ^ {Q} d V + \int_ {V} \frac {\partial N ^ {P}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial N ^ {Q}}{\partial \mathbf {x}} d V + \int_ {V} \frac {\partial N ^ {P}}{\partial \mathbf {x}} \cdot \frac {\partial \mathbf {k}}{\partial \theta} \cdot \frac {\partial \theta}{\partial \mathbf {x}} N ^ {Q} d V \\ - \int_ {V} N ^ {P} \eta_ {v} \left(\mathbf {E} \cdot \frac {\partial \pmb {\sigma} ^ {E}}{\partial \theta} \cdot \mathbf {E} - \mathbf {E} \cdot \frac {\partial \pmb {\sigma} ^ {E}}{\partial \theta} \cdot \Delta \mathbf {E} + \frac {1}{3} \Delta \mathbf {E} \cdot \frac {\partial \pmb {\sigma} ^ {E}}{\partial \theta} \cdot \Delta \mathbf {E}\right) N ^ {Q} d V \\ - \int_ {S _ {p}} N ^ {P} \frac {\partial q}{+ \partial \theta} N ^ {Q} d S - \int_ {S _ {i}} N ^ {P} \left(\frac {\partial q _ {c}}{\partial \theta} + \frac {\partial q _ {r}}{\partial \theta} + \frac {\partial q _ {e c}}{\partial \theta}\right) N ^ {Q} d S. \\ \end{array}
+$$
+
+The term $\partial q / \partial \theta$ in the $\mathbf { K } _ { \theta \theta } ^ { P Q }$ component includes prescribed surface convection and radiation conditions. The surface interaction terms $\partial q _ { c } / \partial \theta \ , \partial q _ { r } / \partial \theta$ , and $\partial { { q } _ { e c } } / \partial { { \theta } }$ are evaluated in \`\`Heat generation caused by electrical current,'' Section 5.2.6.
+
+The Jacobian contributions give rise to an unsymmetric system of equations, requiring the use of the nonsymmetric matrix storage and solution scheme.
+
+# 2.13 Mass diffusion
+
+# 2.13.1 Mass diffusion analysis
+
+ABAQUS/Standard provides for the modeling of the transient or steady-state diffusion of one material through another, such as the diffusion of hydrogen through a metal (Crank (1956), deGroot and Mazur (1962)). The governing equations are an extension of Fick's equations, to allow for nonuniform solubility of the diffusing substance in the base material.
+
+The basic solution variable (used as the degree of freedom at the nodes of the mesh) is the "normalized concentration" (often referred to as the "activity" of the diffusing material), $\phi \ { \stackrel { \mathrm { d e f } } { = } } \ c / s _ { \mathrm { i } }$ , where c is the mass concentration of the diffusing material and s is its solubility in the base material. This means that when the mesh includes dissimilar materials that share nodes, the normalized concentration is continuous across the interface between the different materials. Since Á is the square root of the partial pressure of the diffusing phase, the partial pressure is the same on both sides of the interface; Sievert's law is assumed to hold at the interface.
+
+# Governing equations
+
+<!-- source-page: 238 -->
+
+# Procedures
+
+The diffusion problem is defined from the requirement of mass conservation for the diffusing phase:
+
+Equation 2.13.1-1
+
+$$
+\int_ {V} \frac {d c}{d t} d V + \int_ {S} \mathbf {n} \cdot \mathbf {J} d S = 0,
+$$
+
+where V is any volume whose surface is S, n is the outward normal to S, J is the flux of concentration of the diffusing phase, and n ¢ J is the flux of concentration leaving S.
+
+Using the divergence theorem,
+
+$$
+\int_ {V} \left(\frac {d c}{d t} + \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {J}\right) d V = 0.
+$$
+
+Because the volume is arbitrary, this provides the pointwise equation
+
+$$
+{\frac {d c}{d t}} + {\frac {\partial}{\partial \mathbf {x}}} \cdot \mathbf {J} = 0.
+$$
+
+The equivalent weak form is
+
+$$
+\int_ {V} \delta \phi \left(\frac {d c}{d t} + \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {J}\right) d V = 0,
+$$
+
+where ±Á is an arbitrary, suitably continuous, scalar field.
+
+This statement can be rewritten as
+
+$$
+\int_ {V} \left[ \delta \phi \left(\frac {d c}{d t}\right) + \frac {\partial}{\partial \mathbf {x}} \cdot (\delta \phi \mathbf {J}) - \mathbf {J} \cdot \frac {\partial \delta \phi}{\partial \mathbf {x}} \right] d V = 0.
+$$
+
+Using the divergence theorem again yields
+
+Equation 2.13.1-2
+
+$$
+\int_ {V} \left[ \delta \phi \left(\frac {d c}{d t}\right) - \frac {\partial \delta \phi}{\partial \mathbf {x}} \cdot \mathbf {J} \right] d V + \int_ {S} \delta \phi \mathbf {n} \cdot \mathbf {J} d S = 0.
+$$
+
+# Constitutive behavior
+
+The diffusion is assumed to be driven by the gradient of a chemical potential, which gives the general behavior
+
+Equation 2.13.1-3
+
+$$
+\mathbf {J} = - s \mathbf {D} \cdot \left[ \frac {\partial \phi}{\partial \mathbf {x}} + \kappa_ {s} \frac {\partial}{\partial \mathbf {x}} \bigg (\ln (\theta - \theta^ {Z}) \bigg) + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}} \right],
+$$
+
+<!-- source-page: 239 -->
+
+# Procedures
+
+where $\mathbf { D } ( c , \theta , \mathbf { f } )$ is the diffusivity; $s ( \theta , \mathbf { f } )$ is the solubility; $\kappa _ { s } \left( c , \theta , \mathbf { f } \right)$ is the "Soret effect" factor, providing diffusion because of the temperature gradient; µ is the temperature; $\theta ^ { Z }$ is the absolute zero on the temperature scale used; $\kappa _ { p } ( c , \theta , \mathbf { f } )$ is the pressure stress factor, providing diffusion driven by the gradient of the equivalent pressure stress, $p \stackrel { \mathrm { d e f } } { = } - \mathrm { t r a c e } ( { \pmb \sigma } ) / 3 ;$ and f are any predefined field variables.
+
+An example of a particular form of this constitutive model is the assumption made for hydrogen diffusion in a metal:
+
+$$
+\mathbf {J} = - \frac {\mathbf {D} c}{R (\theta - \theta^ {Z})} \cdot \frac {\partial \mu}{\partial \mathbf {x}},
+$$
+
+with the chemical potential, $\mu ,$ defined as
+
+$$
+\mu = \mu^ {0} + R (\theta - \theta^ {Z}) \ln \phi + p \overline {{V}} _ {H},
+$$
+
+where $\mu ^ { 0 }$ is a fixed datum, R is the universal gas constant, and $\overline { { V } } _ { H }$ is the partial molar volume of hydrogen in the solid solution. This form is similar to that used by Sofronis and McMeeking (1989) and results in a constitutive expression of the form
+
+$$
+\mathbf {J} = - s \mathbf {D} \cdot \left[ \frac {\partial \phi}{\partial \mathbf {x}} + \phi \ln \phi \frac {\partial}{\partial \mathbf {x}} \left(\ln (\theta - \theta^ {Z})\right) + \phi \frac {\overline {{{V}}} _ {H}}{R (\theta - \theta^ {Z})} \frac {\partial p}{\partial \mathbf {x}} \right].
+$$
+
+To implement this particular form, data for $\kappa _ { s }$ and $\kappa _ { p }$ must be calculated from the equations
+
+$$
+\kappa_ {s} = \phi \ln \phi \quad \mathrm{and} \quad \kappa_ {p} = \phi \frac {\overline {{V}} _ {H}}{R (\theta - \theta^ {Z})}.
+$$
+
+Changing variables $( c = \phi s )$ and introducing the constitutive assumption of Equation 2.13.1-3 into Equation 2.13.1-2 yields
+
+Equation 2.13.1-4
+
+$$
+\int_ {V} \left[ \delta \phi \left(s \frac {d \phi}{d t} + \phi \frac {d s}{d \theta} \frac {d \theta}{d t}\right) + \frac {\partial \delta \phi}{\partial \mathbf {x}} \cdot s \mathbf {D} \cdot \left(\frac {\partial \phi}{\partial \mathbf {x}} + \frac {\kappa_ {s}}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right) \right] d V = \int_ {S} \delta \phi q d S,
+$$
+
+where
+
+$$
+q \stackrel {\mathrm{def}} {=} - \mathbf {n} \cdot \mathbf {J}
+$$
+
+is the concentration flux entering the body across S.
+
+# Discretization and time integration
+
+Equilibrium in a finite element model is approximated by a finite set of equations through the introduction of appropriate interpolation functions. Discretized quantities are indicated by uppercase superscripts (for example, $\phi ^ { N } )$ . The summation convention is adopted for the superscripts. These
+
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+
+# Procedures
+
+represent nodal variables, with nodes shared between adjacent elements and appropriate interpolation chosen to provide adequate continuity of the assumed variation. The interpolation is based on material coordinates $S _ { i , i } = 1 , 2 , 3$ .
+
+The virtual normalized concentration field is interpolated by
+
+$$
+\delta \phi = \mathrm{N} ^ {N} \delta \phi^ {N},
+$$
+
+where $\mathrm { N } ^ { N } ( S _ { i } )$ are interpolation functions. Then, the discretized equations are written as
+
+Equation 2.13.1-5
+
+$$
+\int_ {V} \left[ \mathrm{N} ^ {N} \left(s \frac {d \phi}{d t} + \phi \frac {d s}{d \theta} \frac {d \theta}{d t}\right) + \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot s \mathbf {D} \cdot \left(\frac {\partial \phi}{\partial \mathbf {x}} + \frac {\kappa_ {s}}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right) \right] d V = \int_ {S} \mathrm{N} ^ {N} q d S.
+$$
+
+Time integration in transient problems utilizes the backward Euler method (the modified
+
+Crank-Nicholson operator). Adopting the convention that any quantity not explicitly associated with a point in time is taken at $t { + \Delta } t$ , we can drop the subscript $t { + \Delta } t$ and write the integrated equations as
+
+Equation 2.13.1-6
+
+$$
+\int_ {V} \left[ \mathrm{N} ^ {N} \left(s \frac {(\phi - \phi_ {t})}{\Delta t} + \phi \frac {d s}{d \theta} \frac {d \theta}{d t}\right) + \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot s \mathbf {D} \cdot \left(\frac {\partial \phi}{\partial \mathbf {x}} + \frac {\kappa_ {s}}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right) \right] d V = \int_ {S} \mathrm{N} ^ {N} q d S.
+$$
+
+# Jacobian contribution
+
+The Jacobian contribution from the conservation equation is obtained from the variation of Equation 2.13.1-6 with respect to $\phi$ at time $t + \Delta t$ . This yields
+
+$$
+\begin{array}{l} \int_ {V} \left[ \mathrm{N} ^ {N} \left(\frac {s}{\Delta t} d \phi + \frac {d s}{d \theta} \frac {d \theta}{d t} d \phi\right) + \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot s \frac {\partial \mathbf {D}}{\partial \phi} \cdot \left(\frac {\partial \phi}{\partial \mathbf {x}} + \frac {\kappa_ {s}}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right) d \phi \right. \\ \left. \right.\left. + \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot s \mathbf {D} \cdot \left(\frac {\partial d \phi}{\partial \mathbf {x}} + \frac {1}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} \frac {\partial \kappa_ {s}}{\partial \phi} d \phi + \frac {\partial p}{\partial \mathbf {x}} \frac {\partial \kappa_ {p}}{\partial \phi} d \phi\right)\right] d V. \\ \end{array}
+$$
+
+Rearranging and using the interpolation $d \phi = \mathrm { N } ^ { N } d \phi ^ { N }$ , we obtain
+
+$$
+\begin{array}{l} \int_ {V} \left[ \left(\frac {s}{\Delta t} + \frac {d s}{d \theta} \frac {d \theta}{d t}\right) \mathrm{N} ^ {N} \mathrm{N} ^ {M} + \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot s \mathbf {D} \cdot \frac {\partial \mathrm{N} ^ {M}}{\partial \mathbf {x}} \right. \\ + s \frac {\partial \mathrm{N} ^ {N}}{\partial \mathbf {x}} \cdot \left\{\frac {\partial \mathbf {D}}{\partial \phi} \cdot \left(\frac {\partial \phi}{\partial \mathbf {x}} + \frac {\kappa_ {s}}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right) + \mathbf {D} \cdot \left(\frac {1}{(\theta - \theta^ {Z})} \frac {\partial \theta}{\partial \mathbf {x}} \frac {\partial \kappa_ {s}}{\partial \phi} + \frac {\partial p}{\partial \mathbf {x}} \frac {\partial \kappa_ {p}}{\partial \phi}\right)\right\} \mathrm{N} ^ {M} \left. \right] d V. \\ \end{array}
+$$
+
+Inspecting the above equation, we observe that the Jacobian becomes unsymmetric whenever the diffusivity, $\mathbf { D } ;$ the temperature-driven diffusion coefficient, $\kappa _ { s } ;$ or the pressure-driven diffusion