BCs

book/_toc.yml

@@ -21,6 +21,14 @@ parts:
     - file: content/material/material_htm
     - file: content/material/material_advanced
 
+  - caption: Boundary Conditions
+    chapters: 
+    - file: content/boundary_conditions/bc_intro
+    - file: content/boundary_conditions/h_transport_basic
+    - file: content/boundary_conditions/h_transport_advanced
+    - file: content/boundary_conditions/examples
+    - file: content/boundary_conditions/heat_transfer
+
   - caption: Species & Reactions
     chapters:
     - file: content/species_reactions/species

book/content/boundary_conditions/bc_intro.md

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+---
+jupytext:
+  formats: ipynb,md:myst
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.19.1
+---
+
+# Fixed value and flux boundary condition mathematics
+
+This section discusses the math behind fixed temperature/concentration and flux boundary conditions (BCs).
+
++++
+
+## Fixed concentration/temperature boundary conditions
+
+A fixed concentration (Dirichlet) boundary condition prescribes the value of the mobile hydrogen isotope concentration at a boundary. This enforces the concentration to remain constant in time and space on the specified boundary, independent of the solution in the bulk. This also applies to a fixed temperature boundary condition.
+
+For hydrogen transport, this boundary condition is typically used to represent surfaces in equilibrium with a gas, imposed implantation conditions, or experimentally controlled concentrations. Fixed temperatures are commonly used to model infinite heat reservoirs or sinks.
+
+### Mathematical formulation
+
+On a boundary $\Gamma_D$, the mobile concentration satisfies
+
+$$
+c(\mathbf{x}, t) = c_0 \quad \text{for } \mathbf{x} \in \Gamma_D,
+$$
+
+where $c_0$ is the prescribed concentration value.
+
+This condition is enforced by directly constraining the degrees of freedom associated with the boundary.
+
+## Flux boundary conditions
+
+A particle flux (Neumann) boundary condition prescribes the normal flux of mobile hydrogen isotopes across a boundary. Unlike fixed concentration conditions, flux boundary conditions do not directly constrain the concentration value at the surface. Instead, they control the rate at which particles enter or leave the domain.
+
+Flux boundary conditions are commonly used to represent implantation from a plasma, outgassing to vacuum, permeation through a surface, or symmetry boundaries where no net transport occurs.
+
+This also applies for heat flux boundary conditions. 
+
+### Mathematical formulation
+
+The particle flux $\mathbf{J}$ is typically given by Fick’s law,
+
+$$
+\mathbf{J} = -D \nabla c,
+$$
+
+where $D$ is the diffusion coefficient and $c$ is the mobile concentration.
+
+On a boundary $\Gamma_N$, a prescribed normal flux is imposed as
+
+$$
+\mathbf{J} \cdot \mathbf{n} = g(\mathbf{x}, t) \quad \text{for } \mathbf{x} \in \Gamma_N,
+$$
+
+where:
+- $\mathbf{n}$ is the outward unit normal vector,
+- $g(\mathbf{x}, t)$ is the imposed particle flux (positive for outward flux, negative for inward flux).
+
+A special case is the **zero-flux (symmetry) boundary condition**, given by
+
+$$
+\mathbf{J} \cdot \mathbf{n} = 0 \quad \text{on } \Gamma,
+$$
+
+which implies no net particle transport across the boundary.
+
+#### Weak formulation
+
+In the weak form, flux boundary conditions appear naturally as surface integrals after integration by parts. For a test function $v$, the boundary contribution is
+
+$$
+\int_{\Gamma_N} g(\mathbf{x}, t)\, v \, \mathrm{d}\Gamma.
+$$
+
+Because of this, flux boundary conditions are often referred to as *natural boundary conditions* and do not require explicit modification of the solution space, unlike Dirichlet conditions.