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1558
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1 """
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2 ParameterSpace{D}
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3
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4 A space of parameters of dimension `D`. Used with `Chart` to indicate which
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5 parameters are valid for that chart.
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6
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7 Common parameter spaces are created using the functions unit sized spaces
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8 * `unitinterval`
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9 * `unitrectangle`
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10 * `unitbox`
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11 * `unittriangle`
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12 * `unittetrahedron`
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13 * `unithyperbox`
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14 * `unitsimplex`
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15
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16 See also: [`Interval`](@ref), [`Rectangle`](@ref), [`Box`](@ref),
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17 [`Triangle`](@ref), [`Tetrahedron`](@ref), [`HyperBox`](@ref),
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18 [`Simplex`](@ref),
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19 """
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20 abstract type ParameterSpace{D} end
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21
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22 struct HyperBox{T,D} <: ParameterSpace{D}
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23 a::SVector{D,T}
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24 b::SVector{D,T}
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25 end
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26
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27 function HyperBox(a,b)
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28 T = SVector{length(a)}
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29 HyperBox(convert(T,a), convert(T,b))
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30 end
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31
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32 Interval{T} = HyperBox{T,1}
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33 Rectangle{T} = HyperBox{T,2}
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34 Box{T} = HyperBox{T,3}
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35
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36 limits(box::HyperBox, d) = (box.a[d], box.b[d])
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37 limits(box::HyperBox) = (box.a, box.b)
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38
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39 unitinterval(T=Float64) = unithyperbox(T,1)
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40 unitsquare(T=Float64) = unithyperbox(T,2)
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41 unitcube(T=Float64) = unithyperbox(T,3)
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42 unithyperbox(T, D) = HyperBox((@SVector zeros(T,D)), (@SVector ones(T,D)))
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43 unithyperbox(D) = unithyperbox(Float64,D)
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44
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45
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46 struct Simplex{T,D} <: ParameterSpace{D}
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47 verticies::NTuple{D,SVector{D,T}}
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48 end
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49
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50 Triangle{T} = Simplex{T,2}
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51 Tetrahedron{T} = Simplex{T,3}
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52
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53 unittriangle(T) = unitsimplex(T,2)
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54 unittetrahedron(T) = unitsimplex(T,3)
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55 function unitsimplex(T,D)
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56 z = @SVector zeros(T,D)
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57 unitelement = one(eltype(z))
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58 verticies = ntuple(i->setindex(z, unitelement, i), 4)
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59 return Simplex(verticies)
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60 end
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61
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62
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63 """
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64
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65 A parametrized description of a manifold or part of a manifold.
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66
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67 Should implement a methods for
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68 * `parameterspace`
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69 * `(::Chart)(ξs...)`
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70
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71 There is a default implementation for `(::Chart{D})(::SVector{D})`
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72 """
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73 abstract type Chart{D} end
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74 # abstract type Chart{D,R} end
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75
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76 domain_dim(::Chart{D}) where D = D
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77 # range_dim(::Chart{D,R}) where {D,R} = R
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78
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79 """
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80 The parameterspace of a chart
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81 """
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82 function parameterspace end
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83
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84 (c::Chart{D})(x̄::SVector{D}) where D = c(x̄...)
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85
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86
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87 struct ConcereteChart{PST<:ParameterSpace, MT}
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88 parameterspace::PST
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89 mapping::MT
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90 end
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91
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92 (c::Chart)(x̄) = c.mapping(x̄)
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93
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94
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95 """
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96 Atlas
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97
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98 A collection of charts and their connections.
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99 Should implement methods for `charts` and
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100 """
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101 abstract type Atlas end
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102
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103 """
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104 charts(::Atlas)
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105
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106 The colloction of charts in the atlas.
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107 """
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108 function charts end
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109
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110 """
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111 connections
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112
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113 TBD: What exactly should this return?
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114
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115 """
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116
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117 struct CartesianAtlas <: Atlas
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118 charts::Matrix{Chart}
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119 end
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120
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121 charts(a::CartesianAtlas) = a.charts
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122
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123 struct UnstructuredAtlas <: Atlas
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124 charts::Vector{Chart}
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125 connections
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126 end
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127
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128 charts(a::UnstructuredAtlas) = a.charts
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129
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130
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131 ###
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132 # Geometry
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133 ###
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134
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135 abstract type Curve end
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136 abstract type Surface end
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137
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138
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139 struct Line{PT} <: Curve
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140 p::PT
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141 tangent::PT
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142 end
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143
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144 (c::Line)(s) = c.p + s*c.tangent
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145
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146
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147 struct LineSegment{PT} <: Curve
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148 a::PT
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149 b::PT
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150 end
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151
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152 (c::LineSegment)(s) = (1-s)*c.a + s*c.b
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153
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154
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155 struct Circle{T,PT} <: Curve
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156 c::PT
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157 r::T
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158 end
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159
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160 (c::Circle)(θ) = c.c + r*@SVector[cos(Θ), sin(Θ)]
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161
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162 struct TransfiniteInterpolationSurface{T1,T2,T3,T4} <: Surface
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163 c₁::T1
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164 c₂::T2
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165 c₃::T3
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166 c₄::T4
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167 end
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168
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169 function (s::TransfiniteInterpolationSurface)(u,v)
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170 c₁, c₂, c₃, c₄ = s.c₁, s.c₂, s.c₃, s.c₄
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171 P₀₀ = c₁(0)
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172 P₁₀ = c₂(0)
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173 P₁₁ = c₃(0)
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174 P₀₁ = c₄(0)
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175 return (1-v)*c₁(u) + u*c₂(v) + v*c₃(1-u) + (1-u)*c₄(1-v) - (
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176 (1-u)*(1-v)*P₀₀ + u*(1-v)*P₁₀ + u*v*P₁₁ + (1-u)*v*P₀₁
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177 )
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178 end
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179
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180 function (s::TransfiniteInterpolationSurface)(ξ̄::AbstractArray)
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181 s(ξ̄...)
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182 end
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183
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184
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185 function polygon_sides(Ps...)
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186 n = length(Ps)
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187 return [t->line(t,Ps[i],Ps[mod1(i+1,n)]) for i ∈ eachindex(Ps)]
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188 end
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