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annotate src/SbpOperators/volumeops/laplace/laplace.jl @ 1751:f3d7e2d7a43f feature/sbp_operators/laplace_curvilinear

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Merge feature/grids/manifolds
author Jonatan Werpers <jonatan@werpers.com>
date Wed, 11 Sep 2024 16:26:19 +0200
parents 29b96fc75bee b5690ab5f0b8
children 1f42944d4a72
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1 """
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2 Laplace{T, Dim, TM} <: LazyTensor{T, Dim, Dim}
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3
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4 The Laplace operator, approximating ∑d²/xᵢ² , i = 1,...,`Dim` as a
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5 `LazyTensor`.
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6 """
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7 struct Laplace{T, Dim, TM<:LazyTensor{T, Dim, Dim}} <: LazyTensor{T, Dim, Dim}
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8 D::TM # Difference operator
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9 stencil_set::StencilSet # Stencil set of the operator
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10 end
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11
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12 """
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13 Laplace(g::Grid, stencil_set::StencilSet)
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14
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15 Creates the `Laplace` operator `Δ` on `g` given `stencil_set`.
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16
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17 See also [`laplace`](@ref).
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18 """
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19 function Laplace(g::Grid, stencil_set::StencilSet)
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20 Δ = laplace(g, stencil_set)
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21 return Laplace(Δ, stencil_set)
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22 end
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23
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24 LazyTensors.range_size(L::Laplace) = LazyTensors.range_size(L.D)
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25 LazyTensors.domain_size(L::Laplace) = LazyTensors.domain_size(L.D)
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26 LazyTensors.apply(L::Laplace, v::AbstractArray, I...) = LazyTensors.apply(L.D,v,I...)
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27
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28 # TODO: Implement pretty printing of Laplace once pretty printing of LazyTensors is implemented.
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29 # Base.show(io::IO, L::Laplace) = ...
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31 """
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32 laplace(g::Grid, stencil_set)
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33
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34 Creates the Laplace operator operator `Δ` as a `LazyTensor` on `g`.
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35
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36 `Δ` approximates the Laplace operator ∑d²/xᵢ² , i = 1,...,`Dim` on `g`. The
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37 approximation depends on the type of grid and the stencil set.
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38
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39 See also: [`second_derivative`](@ref).
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40 """
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41 function laplace end
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42 function laplace(g::TensorGrid, stencil_set)
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43 # return mapreduce(+, enumerate(g.grids)) do (i, gᵢ)
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44 # Δᵢ = laplace(gᵢ, stencil_set)
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45 # LazyTensors.inflate(Δᵢ, size(g), i)
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46 # end
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47
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48 Δ = LazyTensors.inflate(laplace(g.grids[1], stencil_set), size(g), 1)
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49 for d = 2:ndims(g)
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50 Δ += LazyTensors.inflate(laplace(g.grids[d], stencil_set), size(g), d)
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51 end
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52 return Δ
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53 end
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54
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55 laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set)
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56
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57 function laplace(grid::MappedGrid, stencil_set)
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58 J = jacobian_determinant(grid)
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59 J⁻¹ = DiagonalTensor(map(inv, J))
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60
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61 Jg = map(*, J, metric_tensor_inverse(grid))
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62 lg = logicalgrid(grid)
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63
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64 return mapreduce(+, CartesianIndices(first(Jg))) do I
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65 i, j = I[1], I[2]
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66 Jgⁱʲ = componentview(Jg, i, j)
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67
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68 if i == j
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69 J⁻¹∘second_derivative_variable(lg, Jgⁱʲ, stencil_set, i)
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70 else
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71 Dᵢ = first_derivative(lg, stencil_set, i)
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72 Dⱼ = first_derivative(lg, stencil_set, j)
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73 J⁻¹∘Dᵢ∘DiagonalTensor(Jgⁱʲ)∘Dⱼ
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74 end
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75 end
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76 end
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77
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78
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79 """
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80 sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
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81
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82 The operators required to construct the SAT for imposing a Dirichlet
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83 condition. `H_tuning` and `R_tuning` are used to specify the strength of the
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84 penalty.
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85
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86 See also: [`sat`](@ref), [`DirichletCondition`](@ref), [`positivity_decomposition`](@ref).
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87 """
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88 function sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning = 1., R_tuning = 1.)
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89 id = boundary(bc)
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90 set = Δ.stencil_set
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91 H⁻¹ = inverse_inner_product(g,set)
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92 Hᵧ = inner_product(boundary_grid(g, id), set)
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93 e = boundary_restriction(g, set, id)
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94 d = normal_derivative(g, set, id)
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95 B = positivity_decomposition(Δ, g, boundary(bc); H_tuning, R_tuning)
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96 penalty_tensor = H⁻¹∘(d' - B*e')∘Hᵧ
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97 return penalty_tensor, e
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98 end
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99
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100 """
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101 sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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102
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103 The operators required to construct the SAT for imposing a Neumann condition.
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104
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105 See also: [`sat`](@ref), [`NeumannCondition`](@ref).
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106 """
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107 function sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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108 id = boundary(bc)
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109 set = Δ.stencil_set
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110 H⁻¹ = inverse_inner_product(g,set)
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111 Hᵧ = inner_product(boundary_grid(g, id), set)
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112 e = boundary_restriction(g, set, id)
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113 d = normal_derivative(g, set, id)
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114
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115 penalty_tensor = -H⁻¹∘e'∘Hᵧ
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116 return penalty_tensor, d
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117 end
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118
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119 """
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120 positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
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121
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122 Constructs the scalar `B` such that `d' - 1/2*B*e'` is symmetric positive
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123 definite with respect to the boundary quadrature. Here `d` is the normal
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124 derivative and `e` is the boundary restriction operator. `B` can then be used
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125 to form a symmetric and energy stable penalty for a Dirichlet condition. The
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126 parameters `H_tuning` and `R_tuning` are used to specify the strength of the
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127 penalty and must be greater than 1. For details we refer to
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128 <https://doi.org/10.1016/j.jcp.2020.109294>
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129 """
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130 function positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
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131 @assert(H_tuning ≥ 1.)
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132 @assert(R_tuning ≥ 1.)
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133 Nτ_H, τ_R = positivity_limits(Δ,g,b)
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134 return H_tuning*Nτ_H + R_tuning*τ_R
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135 end
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136
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137 function positivity_limits(Δ::Laplace, g::EquidistantGrid, b::BoundaryIdentifier)
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138 h = spacing(g)
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139 θ_H = parse_scalar(Δ.stencil_set["H"]["closure"][1])
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140 θ_R = parse_scalar(Δ.stencil_set["D2"]["positivity"]["theta_R"])
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141
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142 τ_H = one(eltype(Δ))/(h*θ_H)
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143 τ_R = one(eltype(Δ))/(h*θ_R)
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144 return τ_H, τ_R
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145 end
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146
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147 function positivity_limits(Δ::Laplace, g::TensorGrid, b::BoundaryIdentifier)
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148 τ_H, τ_R = positivity_limits(Δ, g.grids[grid_id(b)], b)
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149 return τ_H*ndims(g), τ_R
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150 end