from IPython.display import HTML
+from IPython.display import HTML
HTML('''<script>
code_show=true;
@@ -23,7 +22,7 @@
<form action="javascript:code_toggle()"><input type="submit" value="Code Toggle"></form>''')
-from IPython.display import HTML
+from IPython.display import HTML
HTML('''
<a href="{{ site.links.github }}/raw/nist-pages/benchmarks/benchmark1.ipynb"
@@ -76,7 +73,7 @@
''')
-from IPython.display import HTML
+from IPython.display import HTML
HTML('''{% include jupyter_benchmark_table.html num="[1]" revision=1 %}''')
-See the journal publication entitled "Benchmark problems for numerical implementations of phase field models" for more details about the benchmark problems. Furthermore, read the extended essay for a discussion about the need for benchmark problems.
+See the journal publication entitled "Benchmark problems for numerical implementations of phase field models" for more details about the benchmark problems. Furthermore, read the extended essay for a discussion about the need for benchmark problems.
Spinodal decomposition is one of the oldest problems in the phase field canon, and its formulation in terms of continuum fields goes back to the seminal works by Cahn and Hilliard [1]. The Cahn-Hilliard equation thus predates the name "phase field" in this context, but the term has subsequently been adopted by the community. While spinodal decomposition may be one of the simplest problems to model, it is highly relevant, as a large number of phase field models include the diffusion of a solute within a matrix. Furthermore, precipitation and growth may also be modeled with the same formulation if the appropriate initial conditions are chosen. For the benchmark problem, we select a simple formulation that is numerically tractable so that results may be obtained quickly and interpreted easily, testing the essential physics while minimizing model complexity and the chance to introduce coding errors.
+Spinodal decomposition is one of the oldest problems in the phase field canon, and its formulation in terms of continuum fields goes back to the seminal works by Cahn and Hilliard [1]. The Cahn-Hilliard equation thus predates the name "phase field" in this context, but the term has subsequently been adopted by the community. While spinodal decomposition may be one of the simplest problems to model, it is highly relevant, as a large number of phase field models include the diffusion of a solute within a matrix. Furthermore, precipitation and growth may also be modeled with the same formulation if the appropriate initial conditions are chosen. For the benchmark problem, we select a simple formulation that is numerically tractable so that results may be obtained quickly and interpreted easily, testing the essential physics while minimizing model complexity and the chance to introduce coding errors.
Because $c$ must obey a continuity equation -- the flux of $c$ is conserved -- the evolution of $c$ is given by the Cahn-Hilliard equation [1], which is derived from an Onsager force-flux relationship [[2][balluffi2005kinetics]]: +
Because $c$ must obey a continuity equation -- the flux of $c$ is conserved -- the evolution of $c$ is given by the Cahn-Hilliard equation [1], which is derived from an Onsager force-flux relationship [2]: $$ \frac{\partial c}{\partial t}=\nabla\cdot\Bigg\{M\nabla\left(\frac{\partial f_{chem}}{\partial c}-\kappa\nabla^{2}c\right)\Bigg\} $$ @@ -692,16 +676,14 @@