Merge pull request #154 from autotwin/resmooth

finish Laplace on bracket wo hierarchical control

book/examples/smoothing/README.md

@@ -2,7 +2,7 @@
 
 ## Double X
 
-We examine the most basic type of smoothing, Laplace smoothing
+We examine the most basic type of smoothing, Laplace smoothing, $\lambda = 0.3$,
 without hierarchical control, with the [Double X](../unit_tests/README.md#double-x) example.
 
 ![../unit_tests/double_x.png](../unit_tests/double_x.png)
@@ -110,7 +110,65 @@ To begin to examine hiearchical control, we consider the [Bracket](../unit_tests
 
 Figure: The **Bracket** example.
 
-The goal of this example is to demonstrate `PRESCRIBED` hierarchical smoothing.
+### Laplace Smoothing without Hierarchical Control
+
+As a baseline, let's examine what Laplace smoothing, $\lambda = 0.3$, *without* hierarchical control performs.
+
+![bracket_laplace_iter_100.gif](bracket_laplace_iter_100.gif)
+
+Figure: The `Bracket` test problem (left) original configuration, (right) subject to `[1, 2, 3, 4, 5, 10, 20, 30, 100`] iterations of Laplace smoothing. Animation created with [Ezgif](https://ezgif.com/).
+
+As an example, the nodal positions after 10 interations are as follows:
+
+node | `x` | `y` | `z`
+:---: | :--- | :--- | :---
+1 | 0.6603416706977089 | 0.6603416706977089 | 0.42058348557613
+2 | 1.164014406316456 | 0.5922705223353653 | 0.4003570849733875
+3 | 1.9979372129260855 | 0.5706936094999626 | 0.39548539946279243
+4 | 2.8325693635137097 | 0.5703120664922405 | 0.40180333889841546
+5 | 3.332396179530681 | 0.6196854057408008 | 0.4228468310236131
+6 | 0.5922705223353653 | 1.164014406316456 | 0.4003570849733875
+7 | 1.129330412354556 | 1.129330412354556 | 0.3779268501553354
+8 | 1.986117815900869 | 1.100245269915641 | 0.3744217105825115
+9 | 2.8536168286772536 | 1.0284532492877596 | 0.3839611664938703
+10 | 3.3805688588919414 | 1.007196857251266 | 0.40846995582593837
+11 | 0.5706936094999626 | 1.9979372129260853 | 0.39548539946279243
+12 | 1.100245269915641 | 1.986117815900869 | 0.37442171058251145
+13 | 1.9089262792820898 | 1.90892627928209 | 0.3766933485101331
+14 | 2.816962753463538 | 1.5457873563122884 | 0.3970154773256839
+15 | 3.3296020281899956 | 1.409074280806729 | 0.42165070606234384
+16 | 0.5703120664922405 | 2.8325693635137097 | 0.40180333889841546
+17 | 1.0284532492877596 | 2.8536168286772536 | 0.3839611664938703
+18 | 1.5457873563122884 | 2.816962753463538 | 0.3970154773256839
+19 | 0.6196854057408008 | 3.332396179530681 | 0.4228468310236131
+20 | 1.007196857251266 | 3.3805688588919414 | 0.40846995582593837
+21 | 1.409074280806729 | 3.3296020281899956 | 0.42165070606234384
+22 | 0.6603416706977089 | 0.6603416706977089 | 0.5794165144238701
+23 | 1.164014406316456 | 0.5922705223353653 | 0.5996429150266126
+24 | 1.9979372129260853 | 0.5706936094999626 | 0.6045146005372077
+25 | 2.8325693635137097 | 0.5703120664922404 | 0.5981966611015848
+26 | 3.332396179530681 | 0.6196854057408007 | 0.5771531689763871
+27 | 0.5922705223353654 | 1.164014406316456 | 0.5996429150266126
+28 | 1.129330412354556 | 1.129330412354556 | 0.6220731498446648
+29 | 1.986117815900869 | 1.100245269915641 | 0.6255782894174887
+30 | 2.8536168286772536 | 1.0284532492877596 | 0.6160388335061299
+31 | 3.3805688588919414 | 1.0071968572512657 | 0.5915300441740619
+32 | 0.5706936094999626 | 1.9979372129260853 | 0.6045146005372076
+33 | 1.100245269915641 | 1.986117815900869 | 0.6255782894174885
+34 | 1.90892627928209 | 1.9089262792820898 | 0.623306651489867
+35 | 2.816962753463538 | 1.5457873563122881 | 0.6029845226743162
+36 | 3.3296020281899956 | 1.409074280806729 | 0.5783492939376563
+37 | 0.5703120664922404 | 2.8325693635137097 | 0.5981966611015848
+38 | 1.0284532492877596 | 2.8536168286772536 | 0.6160388335061298
+39 | 1.5457873563122884 | 2.816962753463538 | 0.6029845226743162
+40 | 0.6196854057408007 | 3.332396179530681 | 0.5771531689763871
+41 | 1.0071968572512657 | 3.3805688588919414 | 0.5915300441740617
+42 | 1.409074280806729 | 3.3296020281899956 | 0.5783492939376562
+
+### Hierarchical Control
+
+Next, we illustrate the how hierarchical control changes the smoothing. 
+Conside the `PRESCRIBED` hierarchical smoothing below:
 
 ```python
 node_smoothing_categories: Hierarchy = (