Merge branch 'feature/connectivity'

README.md

@@ -39,6 +39,13 @@ The following aspects are implemented (✓) and planned (⌛):
 - ⌛ Degree elevation / reduction
 - ⌛ Construction of common geometries
 
+##### Connectivity
+- ✓ Determine patch connectivity
+ - identify interfaces between patches
+ - introduce per patch local numbering for vertices and edges
+- ✓ Virtual Bezier mesh connectivty (for FEM)
+ - introduce on each patch a virtual Bezier mesh
+ - determine adjacency information of mesh cells
 
 
 ## Citation

docs/make.jl

@@ -10,16 +10,26 @@ makedocs(;
  ),
  pages=[
  "Introduction" => "index.md",
- "Definitions" => Any["Bases" => "basis_def.md", "Curves" => "curves_def.md", "Surfaces" => "surfaces_def.md"],
- "Fundamental Operations" => Any[
- "Knot Insertion" => "knotInsertion.md",
- "Knot Removal" => "knotRemoval.md",
- "Degree Elevation" => "degreeElevation.md",
- "Degree Reduction" => "degreeReduction.md",
+ "Definitions & Terminology" => Any[
+ "B-Spline Bases" => "basis_def.md",
+ "Manifolds" => Any["Curves" => "curves_def.md", "Surfaces" => "surfaces_def.md"],
+ "Fundamental Operations" => Any[
+ "Knot Insertion" => "knotInsertion.md",
+ "Knot Removal" => "knotRemoval.md",
+ "Degree Elevation" => "degreeElevation.md",
+ "Degree Reduction" => "degreeReduction.md",
+ ],
+ "Connectivity" => Any["Patch Interfaces" => "interfaces.md", "Bezier Mesh" => "bezier.md"],
+ ],
+ "Manual" => Any[
+ "Bases Evaluation" => "basis.md",
+ "Curves" => Any["Defining & Evaluating" => "curves.md", "Manipulating" => "curves_man.md"],
+ "Surfaces" => Any["Defining & Evaluating" => "surfaces.md", "Manipulating" => "surfaces_man.md"],
+ "Connectivity" => "meshes.md",
+ "Transformations" => "trafos.md",
+ "File I/O" => "fileio.md",
+ "Utils" => "utils.md",
  ],
- "Manual" => Any["Bases" => "basis.md", "Curves" => "curves.md", "Surfaces" => "surfaces.md", "Transformations" => "trafos.md"],
- "File I/O" => "fileio.md",
- "Utils" => "utils.md",
  "Contributing" => "contributing.md",
  "API Reference" => "apiref.md",
  ],

docs/src/basis.md

@@ -8,7 +8,7 @@ The considered
 basis functions are defined by initializing corresponding structures. A naive evaluation method is available for all, an efficient evaluation only for B-splines and Curry-Schoenberg splines.
 
 ---
-## Define Structures
+## Defining Structures
 
 B-spline, Curry-Schoenberg, and NURBS bases are defined by initializing a [`Bspline`](@ref Bspline), a [`CurrySchoenberg`](@ref CurrySchoenberg), or a [`NURB`](@ref NURB) structure, respectively.
 
@@ -67,7 +67,7 @@ Plots.plot(evalpoints, bspline, w=2,
 Plots.plot!(evalpoints, nurb, w=2, label="NURB")
 Plots.plot!(evalpoints, cspl, w=2, label="Curry-Schoenberg")
 xlims!(0, 1) # hide
-ylims!(0, 1) # hide
+ylims!(0, 3) # hide
 savefig("plotBspl.html"); nothing # hide
 ```
 
@@ -112,7 +112,7 @@ Plots.plot(evalpoints, bspline, w=2,
  xlabel="𝑢", 
  ylabel="𝑁ᵢ,₂(𝑢)")
 xlims!(0, 1) # hide
-ylims!(0, 1) # hide
+ylims!(0, 12) # hide
 savefig("plotBspleff.html"); nothing # hide
 ```
 

docs/src/basis_def.md

@@ -1,5 +1,5 @@
 
-# Bases
+# B-Spline Bases
 
 The considered 
 - B-spline,
@@ -29,68 +29,68 @@ where ``u_i \leq u_{i+1}``, i.e., the entries are sorted in ascending order, ``u
 ---
 ## Definitions 
 
-### [B-Splines](@id bspl)
+#### [B-Splines](@id bspl)
 
-The B-spline basis functions of degree ``p`` are defined as ``N_{i,p}`` on the knot vector ``U`` recursively, starting with the piecewise constant (``p=0``) [[1, p. 50]](@ref refs)
+The B-spline basis functions of degree ``p`` are defined as ``B_{i,p}`` on the knot vector ``U`` recursively, starting with the piecewise constant (``p=0``) [[1, p. 50]](@ref refs)
 ```math
-N_{i,0}(u) = \begin{cases} 1 & u_i \leq u < u_{i+1} \\ 0 & \text{otherwise}\end{cases}
+B_{i,0}(u) = \begin{cases} 1 & u_i \leq u < u_{i+1} \\ 0 & \text{otherwise}\end{cases}
 ```
 as well as for ``p>0``
 ```math
-N_{i,p}(u) = \cfrac{u - u_i}{u_{i+p} - u_i} \, N_{i, p-1}(u) + \cfrac{u_{i+p+1} - u}{u_{i+p+1} - u_{i+1}} \, N_{i+1, p-1}(u) \,.
+B_{i,p}(u) = \cfrac{u - u_i}{u_{i+p} - u_i} \, B_{i, p-1}(u) + \cfrac{u_{i+p+1} - u}{u_{i+p+1} - u_{i+1}} \, B_{i+1, p-1}(u) \,.
 ```
 Whenever one of the contained quotients exhibits a division by 0, the quotient is defined to be 0 itself.
 
 !!! note
- The number of basis functions ``B`` is related to the length of the knot vector ``M`` and the polynomial degree ``p`` as
+ The number of basis functions ``N`` is related to the length of the knot vector ``M`` and the polynomial degree ``p`` as
  ```math
- B = M - p - 1 \,.
+ N = M - p - 1 \,.
  ```
 
-### [Curry-Schoenberg Splines](@id csspl)
+#### [Curry-Schoenberg Splines](@id csspl)
 
 The Curry-Schoenberg spline basis functions [[3, p. 88]](@ref refs)
 ```math
-n_{i,p}(u) = \cfrac{p+1}{u_{i+p+1} - u_i} N_{i,p}(u)
+b_{i,p}(u) = \cfrac{p+1}{u_{i+p+1} - u_i} B_{i,p}(u)
 ```
 are defined as a normalized version of the B-splines. 
 
 
 
-### [NURBS](@id nurbs)
+#### [NURBS](@id nurbs)
 
-Introducing ``B`` weights ``w_i \in \mathbb{R}_+`` the rational B-spline basis functions [[1, p. 118]](@ref refs)
+Introducing ``N`` weights ``w_i \in \mathbb{R}_+`` the rational B-spline basis functions [[1, p. 118]](@ref refs)
 ```math
-R_{i, p}(u) = \cfrac{N_{i, p}(u) w_i}{\sum_{j=1}^B N_{j, p}(u) w_i}
+R_{i, p}(u) = \cfrac{B_{i, p}(u) w_i}{\sum_{j=1}^N B_{j, p}(u) w_i}
 ```
-are defined based on the B-spline basis ``N_{i, p}`` on the knot vector ``U``.
+are defined based on the B-spline basis ``B_{i, p}`` on the knot vector ``U``.
 
 
 ---
 ## [Derivatives](@id derB)
 
-### B-Splines
+#### B-Splines
 
-The ``k``-th derivative of the B-splines ``N_{i, p}`` can be computed as [[1, p. 61]](@ref refs)
+The ``k``-th derivative of the B-splines ``B_{i, p}`` can be computed as [[1, p. 61]](@ref refs)
 ```math
-N_{i,p}^{(k)}(u) = p \left( \cfrac{N_{i,p-1}^{(k-1)}}{u_{i+p} - u_i} - \cfrac{N_{i+1,p-1}^{(k-1)}}{u_{i+p+1} - u_{i+1}} \right) \,.
+B_{i,p}^{(k)}(u) = p \left( \cfrac{B_{i,p-1}^{(k-1)}}{u_{i+p} - u_i} - \cfrac{B_{i+1,p-1}^{(k-1)}}{u_{i+p+1} - u_{i+1}} \right) \,.
 ```
 
 !!! note
  The [Curry-Schoenberg](@ref csspl) splines are related to the derivatives of the B-splines as
  ```math
- N_{i,p}^{(1)}(u) = n_{i,p-1}(u) - n_{i+1,p-1}(u) \,.
+ B_{i,p}^{(1)}(u) = b_{i,p-1}(u) - b_{i+1,p-1}(u) \,.
  ``` 
  As this is their common use, derivatives of the Curry-Schoenberg splines are not used/implemented in this package.
 
 
-### NURBS
+#### NURBS
 
 The ``k``-th derivative of the NURBS basis functions ``R_{i, p}`` can be computed as [[2]](@ref refs)
 ```math
-R_{i,p}^{(k)}(u) = \cfrac{w_i \, N_{i,p}^{(k)}(u) - \sum_{j=1}^k \begin{pmatrix} k \\j\end{pmatrix} W^{(j)}(u) R_{i,p}^{k-j}(u)}{W^{(0)}(u)}
+R_{i,p}^{(k)}(u) = \cfrac{w_i \, B_{i,p}^{(k)}(u) - \sum_{j=1}^k \begin{pmatrix} k \\j\end{pmatrix} W^{(j)}(u) R_{i,p}^{k-j}(u)}{W^{(0)}(u)}
 ```
 with
 ```math
-W^{(k)}(u) = \sum_{i=1}^B N_{i, p}^{(k)}(u) w_i \,.
+W^{(k)}(u) = \sum_{i=1}^N B_{i, p}^{(k)}(u) w_i \,.
 ```

docs/src/bezier.md

@@ -0,0 +1,33 @@
+
+# Bezier Mesh
+
+For finite-element (FEM) applications of NURBS and B-splines it is often required to introduce a cell-structure on the NURBS surfaces that is largely independent of the surface description.
+Especially, in isogeometric approaches B-splines are used to describe quantities such as currents or charges on the surface.
+To this end, it is customary to introduce on each NURBS patch a virtual mesh induced by the knot vectors of the B-splines: the Bezier mesh.
+
+!!! note
+ The Bezier mesh is independent of the knot vectors of the B-splines used to describe the surface.
+
+
+---
+## Per Patch Conventions
+
+The local edges and corner points numbering as defined in the [`patch conventions`](@ref localNum) is passed on to the Bezier cells.
+Moreover, the numbering of the cells follows
+- starting at patch one, there is a consecutive number over all patches
+- on each patch the numbering starts at ``u=v=0`` and is increased first along the ``v`` axes
+
+```@raw html
+<div align="center">
+<img src="../assets/BezierCells.svg" width="400"/>
+</div>
+<br/>
+```
+
+
+---
+## Adjacency Information
+
+Most importantly, adjacency information about the Bezier cells is extraced: for a given Bezier cell, what other Bezier cells:
+- have a common edge at which local edge numbers
+- have a common corner point at which local corner numbers

docs/src/curves.md

@@ -8,7 +8,7 @@ Two type of curves are available:
 Both are defined by initializing corresponding structures.
 
 ---
-## Define Structures
+## Defining Structures
 
 NURBS and B-spline curves are defined by initializing a [`BsplineCurve`](@ref BsplineCurve) or a [`NURBScurve`](@ref NURBScurve) structure, respectively.
 
@@ -96,123 +96,4 @@ savefig(t, "cruve3Dder.html"); nothing # hide
 
 ```@raw html
 <object data="../cruve3Dder.html" type="text/html" style="width:100%;height:50vh;"> </object>
-```
-
----
-## Refining a Curve
-
-Based on the principles of [knot insertion](@ref knotInsert), a single knot can be inserted multiple times into a curve (without changing the points the curve describes) by the [`insertKnot`](@ref insertKnot) function.
-
-```@example curves
-NCurve2 = insertKnot(NCurve, 0.75, 2) # insert knot at 0.75 twice
-C2 = NCurve2(evalpoints)
-
-plotCurve3D(C2, controlPoints=NCurve2.controlPoints)
-t = plotCurve3D(C2, controlPoints=NCurve2.controlPoints) # hide
-savefig(t, "cruve3DInserted.html"); nothing # hide
-```
-
-```@raw html
-<object data="../cruve3DInserted.html" type="text/html" style="width:100%;height:50vh;"> </object>
-```
-
-Several knots can be inserted by the [`refine`](@ref refine) function.
-
-```@example curves
-NCurve2 = refine(NCurve, [0.1, 0.2, 0.3, 0.8221]) # insert the array of knots
-C2 = NCurve2(evalpoints)
-
-plotCurve3D(C2, controlPoints=NCurve2.controlPoints)
-t = plotCurve3D(C2, controlPoints=NCurve2.controlPoints) # hide
-savefig(t, "cruve3Drefined.html"); nothing # hide
-```
-
-```@raw html
-<object data="../cruve3Drefined.html" type="text/html" style="width:100%;height:50vh;"> </object>
-```
-
-!!! note
- Refining a curve does not change the points in space described by the curve. Effectively, in the plots it can be seen that the number of control points is increased. 
- However, also underlying properties such as the differentiability are changed.
-
-
----
-## Splitting a Curve
-
-To split a curve into multiple separate curves the function [`split`](@ref NURBS.split) is provided which returns an array of curves.
-
-```@example curves
-cVec = split(NCurve, [0.2, 0.5]) # split at 0.2 and 0.5
-
-# plot all three curves
-data = PlotlyJS.GenericTrace[]
-for (i, spC) in enumerate(cVec)
- push!(data, plotCurve3D(spC(evalpoints), returnTrace=true, controlPoints=spC.controlPoints)...)
-end
-PlotlyJS.plot(data)
-t = PlotlyJS.plot(data) # hide
-savefig(t, "cruve3Dsplit.html"); nothing # hide
-```
-
-```@raw html
-<object data="../cruve3Dsplit.html" type="text/html" style="width:100%;height:50vh;"> </object>
-```
-
-To equally split a curve into ``n`` curves as a second argument an integer can be provided:
-
-```@example curves
-cVec = split(NCurve, 4) # split into 4 curves
-```
-
-
----
-## Removing Knots from a Curve
-
-Based on the principles of [knot removal](@ref knotRemoval), an interior knot can potentially be removed multiple times from a curve (without changing the points the curve describes) by the [`removeKnot`](@ref removeKnot) function.
-
-```@example curves
-p = 3
-
-P1 = SVector(0.0, 0.0, 1.0)
-P2 = SVector(0.0, 2.0, 0.0)
-P3 = SVector(1.5, 3.0, 0.0)
-P4 = SVector(3.0, 3.0, 0.0)
-P5 = SVector(4.5, 3.0, 0.0)
-P6 = SVector(6.0, 2.0, 0.0)
-P7 = SVector(6.0, 0.0, 1.0)
-
-cP = [P1, P2, P3, P4, P5, P6, P7]
-
-kVec = Float64[0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2]
-
-BCurve1 = BsplineCurve(Bspline(p, kVec), cP)
-C1 = BCurve1(evalpoints)
-
-BCurve2 = removeKnot(BCurve1, 0.5, 2) # remove knot at 0.5 twice
-C2 = BCurve2(evalpoints)
-
-# --- plot both curves 
-t1 = plotCurve3D(C1, controlPoints=BCurve1.controlPoints, returnTrace=true)
-t2 = plotCurve3D(C2, controlPoints=BCurve2.controlPoints, returnTrace=true)
-
-fig = make_subplots(
- rows=1, cols=2,
- specs=fill(Spec(kind="scene"), 1, 2)
-)
-
-add_trace!(fig, t1[1], row=1, col=1)
-add_trace!(fig, t1[2], row=1, col=1)
-add_trace!(fig, t2[1], row=1, col=2)
-add_trace!(fig, t2[2], row=1, col=2)
-fig
-savefig(fig, "cruve3DRemoved.html"); nothing # hide
-```
-
-```@raw html
-<object data="../cruve3DRemoved.html" type="text/html" style="width:100%;height:50vh;"> </object>
-```
-
-!!! note
- Removing a knot from a curve is only possible when the continuity of the curve is sufficient at the knot.
- A central part of the [`removeKnot`](@ref removeKnot) function is to verify if the knot can actually be removed.
- If not, warnings are generated, indicating the encountered limitations.
+```