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annotate src/SbpOperators/volumeops/laplace/laplace.jl @ 1607:7216448d0c5a feature/boundary_conditions

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REVIEW: Suggest deduplication of positivity decompostion code
author Jonatan Werpers <jonatan@werpers.com>
date Sun, 09 Jun 2024 00:02:40 +0200
parents 93b86625fcfd
children 8315c456e3b4
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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 laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set)
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55
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56
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57 """
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58 sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; tuning)
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59
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60 The operators required to construct the SAT for imposing a Dirichlet condition.
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61 `tuning` specifies the strength of the penalty. See
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62
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63 See also: [`sat`,`DirichletCondition`, `positivity_decomposition`](@ref).
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64 """
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65 function sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning = 1., R_tuning = 1.)
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66 id = boundary(bc)
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67 set = Δ.stencil_set
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68 H⁻¹ = inverse_inner_product(g,set)
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69 Hᵧ = inner_product(boundary_grid(g, id), set)
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70 e = boundary_restriction(g, set, id)
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71 d = normal_derivative(g, set, id)
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72 B = positivity_decomposition(Δ, g, bc; H_tuning, R_tuning)
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73 penalty_tensor = H⁻¹∘(d' - B*e')∘Hᵧ
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74 return penalty_tensor, e
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75 end
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76
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77 """
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78 sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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79
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80 The operators required to construct the SAT for imposing a Neumann condition
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81
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82
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83 See also: [`sat`,`NeumannCondition`](@ref).
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84 """
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85 function sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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86 id = boundary(bc)
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87 set = Δ.stencil_set
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88 H⁻¹ = inverse_inner_product(g,set)
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89 Hᵧ = inner_product(boundary_grid(g, id), set)
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90 e = boundary_restriction(g, set, id)
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91 d = normal_derivative(g, set, id)
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92
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93 penalty_tensor = -H⁻¹∘e'∘Hᵧ
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94 return penalty_tensor, d
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95 end
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96
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97
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98 function positivity_decomposition(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
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99 Nτ_H, τ_R = positivity_limits(Δ,g,bc)
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100 return H_tuning*Nτ_H + R_tuning*τ_R
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101 end
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102
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103
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104 # TODO: We should consider implementing a proper BoundaryIdentifier for EquidistantGrid and then
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105 # change bc::BoundaryCondition to id::BoundaryIdentifier
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106 function positivity_limits(Δ::Laplace, g::EquidistantGrid, bc::DirichletCondition)
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107 pos_prop = positivity_properties(Δ)
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108 h = spacing(g)
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109 θ_H = pos_prop.theta_H
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110 τ_H = 1/(h*θ_H)
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111 θ_R = pos_prop.theta_R
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112 τ_R = 1/(h*θ_R)
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113 return τ_H, τ_R
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114 end
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115
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116 function positivity_limits(Δ::Laplace, g::TensorGrid, bc::DirichletCondition)
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117 τ_H, τ_R = positivity_limits(Δ, g.grids[grid_id(boundary(bc))], bc)
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118 return τ_H*ndims(g), τ_R
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119 end
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Vidar Stiernström <vidar.stiernstrom@gmail.com>
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120
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121
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122 function positivity_properties(Δ::Laplace)
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123 D2_pos_prop = parse_named_tuple(Δ.stencil_set["D2"]["positivity"])
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124 H_closure = parse_tuple(Δ.stencil_set["H"]["closure"])
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125 return merge(D2_pos_prop, (theta_H = H_closure[1],))
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126 end