From ebf7fdbaa0c9dd11d80abfaba762eb66b6ab0866 Mon Sep 17 00:00:00 2001 From: gregoirepourtier Date: Fri, 13 Sep 2024 10:29:10 +0200 Subject: [PATCH 1/3] add first draft for solving heat equation (one step at time) --- docs/src/examples/transient/Manifest.toml | 1983 ++++++++++++++++++ docs/src/examples/transient/Project.toml | 9 + docs/src/examples/transient/heat_equation.jl | 334 +++ 3 files changed, 2326 insertions(+) create mode 100644 docs/src/examples/transient/Manifest.toml create mode 100644 docs/src/examples/transient/Project.toml create mode 100644 docs/src/examples/transient/heat_equation.jl diff --git a/docs/src/examples/transient/Manifest.toml b/docs/src/examples/transient/Manifest.toml new file mode 100644 index 00000000..e2652e04 --- /dev/null +++ b/docs/src/examples/transient/Manifest.toml @@ -0,0 +1,1983 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = 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+[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+2" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "35976a1216d6c066ea32cba2150c4fa682b276fc" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "10164.0.0+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "dcc541bb19ed5b0ede95581fb2e41ecf179527d2" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.6.0+0" diff --git a/docs/src/examples/transient/Project.toml b/docs/src/examples/transient/Project.toml new file mode 100644 index 00000000..59eec073 --- /dev/null +++ b/docs/src/examples/transient/Project.toml @@ -0,0 +1,9 @@ +[deps] +CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" +Gmsh = "705231aa-382f-11e9-3f0c-b7cb4346fdeb" +HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" +Inti = "fb74042b-437e-4c5b-88cf-d4e2beb394d5" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Meshes = "eacbb407-ea5a-433e-ab97-5258b1ca43fa" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +RecursiveArrayTools = "731186ca-8d62-57ce-b412-fbd966d074cd" diff --git a/docs/src/examples/transient/heat_equation.jl b/docs/src/examples/transient/heat_equation.jl new file mode 100644 index 00000000..7c6846c3 --- /dev/null +++ b/docs/src/examples/transient/heat_equation.jl @@ -0,0 +1,334 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═╡ f4a0859a-6b96-11ef-0c3b-7b7ba5c458ce +begin + using Pkg, Revise + + Pkg.activate(joinpath(@__DIR__, "Project.toml")) + + using Inti # dev version as dependency + using Meshes, Gmsh, CairoMakie, PlutoUI, LinearAlgebra, RecursiveArrayTools +end + +# ╔═╡ 95e65f76-f1ec-434d-8c1c-c314b2eca7c3 +using HypertextLiteral + +# ╔═╡ c0df4cc3-55cf-4f0c-a5a9-7e912724bb11 +md""" +# Solving an (interior/exterior) heat equation with Inti.jl +""" + +# ╔═╡ 9db5ce89-9a3b-41b7-bf46-2c7327dc0f2a +md""" +## 1. Model: Mathematical formulation +""" + +# ╔═╡ 64fe3119-aa5a-4b82-8562-872de48d11b6 +md"""$\begin{equation} \begin{cases} +\partial_t u(t,x) - \Delta u(t,x) &= 0 \quad &\text{in } \Omega\\[0.3cm] +\hspace{3.6cm} u &= g &\text{on } \partial\Omega\\[0.3cm] +\hspace{2.7cm} u(0,x) &= 0 \quad & x \in \Omega\\ +\end{cases}\tag{1}\end{equation}$ +""" + +# ╔═╡ 9730531f-8b7b-4bdb-af50-3a76ff781e69 +md""" +Following [a], the system $(1)$ can be reformulated as an elliptic BVP which can be solve __one step at time__, we have for $n= 1, \dots, N$ + +$\begin{equation} \begin{cases} +u^n - \tau\Delta u^n &= u^{n-1} \quad &\text{in } \Omega\\[0.3cm] +\hspace{1.6cm} u^n &= g^n &\text{on } \partial\Omega +\end{cases}\tag{2}\end{equation}$ +""" + +# ╔═╡ fc0e5d68-309e-471d-a99c-3966f6789803 +md""" +We can rewrite $(2)$ to resemble the Helmholtz equation and obtain + +$\begin{equation} \begin{cases} +\Delta u + k^2u &= f \quad &\text{in } \Omega\\[0.3cm] +\hspace{1.6cm} u &= g &\text{on } \partial\Omega +\end{cases}\tag{3}\end{equation}$ +where $k = \frac{i}{\sqrt\tau}$, $u = u^n$, $f = -\frac{1}{\tau}u^{n-1}$. +""" + +# ╔═╡ 7344c3a8-52b3-4a4d-baf8-9d1c58d3e3a2 +md""" +To deal with the inhomogeneous PDE $(3)$, we follow the same process as in the Poisson problem tutorial. We split the solution into a particular solution $u_p$ and a homogeneous solution $u_h$: +```math +u = u_p + u_h \;\tag{4}. +``` +The function ``u_p`` is given as a volume potential: +```math +u_p(x) = \int_{\Omega} G(x, y) f(y) \;\mathrm{d}y \; \tag{5}. +``` +""" + +# ╔═╡ a20158e0-ad00-4af0-826f-0dc64073b993 +md""" +The function ``u_h`` satisfies the homogeneous problem + +```math +\begin{align} + \Delta u_h + k^2u_h &= 0, \quad &&\text{in } \quad \Omega, \\ + u_h &= g - u_p, \quad &&\text{on } \; \partial\Omega, +\end{align}\tag{6} +``` +which can be solved using BIE method. +""" + +# ╔═╡ deffe780-72ed-4c84-b8e5-6f59316a16d3 +md""" +One possible formulation is the following: + +```math +\left(\frac{I}{2} + D - ikS\right)[\sigma] = g - u_p \;\; \tag{7}, +``` +where $S$ and $D$ are the single- and double-layer operators and $\sigma$ the density function. + +Or look at one of the formulations from Costabel [a]. +""" + +# ╔═╡ 89d4ca3b-e3c7-4519-917e-706a0fd6aea0 +md""" +## 2. Discretization +""" + +# ╔═╡ 6029fb91-1263-4c7c-8d13-fd220a00a90c +md""" +### 2.1 Geometry and mesh +""" + +# ╔═╡ 9e57c112-6328-4179-ab0b-5953978f4bd4 +begin + meshsize = 0.1 + gmsh.initialize() + circle = Inti.gmsh_curve(0, 2π; meshsize) do s + return Inti.Point2D(cos(s), sin(s)) + end + cl = gmsh.model.occ.addCurveLoop([circle]) + surf = gmsh.model.occ.addPlaneSurface([cl]) + gmsh.model.occ.synchronize() + gmsh.option.setNumber("Mesh.MeshSizeMax", meshsize) + gmsh.model.mesh.generate(2) + gmsh.model.mesh.setOrder(2) + msh = Inti.import_mesh(; dim = 2) + gmsh.finalize() +end + +# ╔═╡ 6e8f8c46-6902-4a88-8423-ae2046467ea6 +begin + Ω = Inti.Domain(e -> Inti.geometric_dimension(e) == 2, msh) + Γ = Inti.boundary(Ω) + Ω_msh = view(msh, Ω) + Γ_msh = view(msh, Γ) +end + +# ╔═╡ 9ab4251b-a061-44f3-b8d1-1ec9133832ee +begin + fig = viz(Ω_msh ; segmentsize=1, showsegments=true, axis=(aspect=DataAspect(),), figure=(;size=(500,400))) + viz!(Γ_msh; color=:red) + fig +end + +# ╔═╡ c5811230-dcbf-403b-bb94-9f056cdb92bb +md""" +### 2.2 Integral Operators +""" + +# ╔═╡ baefc33b-9f0c-4caf-b77d-a0dc118e911e +begin + Ω_quad = Inti.Quadrature(Ω_msh; qorder=4) + Γ_quad = Inti.Quadrature(Γ_msh; qorder=4) +end + +# ╔═╡ 1dd4afd5-4383-438b-8a4a-5b2b5cbbea75 + + +# ╔═╡ 218e2978-5388-4aec-a48f-73bc450145a3 +begin + # Initial Condition + u₀(x) = 1.0 + f = map(q -> u₀(q.coords), Ω_quad) + # Boundary Conditions + bc(x) = 0.0 # consider making it time dependent for future tests + g = map(q -> bc(q.coords), Γ_quad) +end + +# ╔═╡ 5529e3aa-cabb-4714-93e1-138b0e7431f9 +function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=1.0, τ=0.05) + + k = im/τ + pde = Inti.Helmholtz(; dim = 2, k=k) + + ## Initialize Volume Potential, single and double layer operators for solving the problem and then for solution evaluation + + # Integral operator to build the LSE + V_d2b = Inti.volume_potential(; + pde, + target = Γ_quad, + source = Ω_quad, + compression = (method = :none,), + correction = ( method = :none,), + ) + S, D = Inti.single_double_layer(; + pde, + target = Γ_quad, + source = Γ_quad, + compression = (method = :none,), + correction = (method = :dim,), + ) + # For the solution evaluation + V_d2d = Inti.volume_potential(; + pde, + target = Ω_quad, + source = Ω_quad, + compression = (method = :none,), + correction = (method = :dim,), + ) + S_b2d, D_b2d = Inti.single_double_layer(; + pde, + target = Ω_quad, + source = Γ_quad, + compression = (method = :none,), + correction = (method = :none,), + ) + + times, sol_u = [t_start], [f] + + for i in 1:floor(Int64,(t_end - t_start)/τ) + + L = I / 2 + D - im * k * S + rhs = g - V_d2b*f + + σ = L \ rhs + + # Solution Evaluation (to improve with compression and correction methods) + f = D_b2d*σ - im*k*S_b2d*σ + V_d2d*f # f = u^{n} -> used for the next iteration as source term of the PDE + + push!(sol_u, real(f)) # Taking the real part of the solution (im(f)≈0) + push!(times, τ*i) + end + + DiffEqArray(sol_u, times) +end + +# ╔═╡ 0f5436a7-4eac-4bd9-aca4-15b1f9d811b8 +md""" + __Run next cell__: $(@bind allow_run PlutoUI.CheckBox(default=false)) +""" + +# ╔═╡ 79527886-824d-4cb7-bee7-00041a6151b6 +if allow_run + tsol = main(Ω_quad, Γ_quad, f, g) +end + +# ╔═╡ 825a4196-8cfe-4a71-bb4a-34d9f445afd7 +md""" +## 3. Plots and results +""" + +# ╔═╡ 7f2aaacb-e159-4e24-8ed3-a9237287dd6b +md""" +Timestep: $(if allow_run @bind t_plot PlutoUI.Slider(1:length(tsol.t),default=1,show_value=true) else "tick the checkbox to run the solver :)" end) +""" + +# ╔═╡ 5ae1b44d-98cd-497b-b187-638ed42bad26 +if allow_run + nodes = Inti.nodes(Ω_msh) + u_quad = tsol[t_plot] + u_nodes = Inti.quadrature_to_node_vals(Ω_quad, u_quad) + + if t_plot == 1 + colorrange = (0.0, f[1]) + fig_sol = Figure(; size = (800, 300)) + ax = Axis(fig_sol[1, 1]; aspect = DataAspect()) + viz!(Ω_msh; colorrange, color=u_nodes, interpolate = true) + cb = Colorbar(fig_sol[1, 2]; label = "u", colorrange) + fig_sol + else + colorrange = extrema(u_nodes) + fig_sol = Figure(; size = (800, 300)) + ax = Axis(fig_sol[1, 1]; aspect = DataAspect()) + viz!(Ω_msh; colorrange, color=u_nodes, interpolate = true) + cb = Colorbar(fig_sol[1, 2]; label = "u", colorrange) + fig_sol + end +end + +# ╔═╡ c39c9560-b050-4fe4-9cb0-376fe1ffc532 +md""" +## 4. References +""" + +# ╔═╡ b6ec9ab6-ab4f-4d26-b2b0-9256621e472d +md""" +[a] _Time-dependent problems with the boundary integral equation method_, Martin Costabel, see [here](https://perso.univ-rennes1.fr/martin.costabel/publis/Co_ECM.pdf). +""" + +# ╔═╡ 8182287b-f56b-40bd-a24d-3fb69409d9a7 +begin + highlight(mdstring,color)= htl"""
$(mdstring)
""" + + macro important_str(s) :(highlight(Markdown.parse($s),"#ffcccc")) end + macro definition_str(s) :(highlight(Markdown.parse($s),"#ccccff")) end + macro statement_str(s) :(highlight(Markdown.parse($s),"#ccffcc")) end + + + html""" + +""" +end + +# ╔═╡ cec3125d-5b90-42b5-8485-7926dbbac91c +TableOfContents() + +# ╔═╡ Cell order: +# ╠═f4a0859a-6b96-11ef-0c3b-7b7ba5c458ce +# ╟─c0df4cc3-55cf-4f0c-a5a9-7e912724bb11 +# ╟─9db5ce89-9a3b-41b7-bf46-2c7327dc0f2a +# ╟─64fe3119-aa5a-4b82-8562-872de48d11b6 +# ╟─9730531f-8b7b-4bdb-af50-3a76ff781e69 +# ╟─fc0e5d68-309e-471d-a99c-3966f6789803 +# ╟─7344c3a8-52b3-4a4d-baf8-9d1c58d3e3a2 +# ╟─a20158e0-ad00-4af0-826f-0dc64073b993 +# ╟─deffe780-72ed-4c84-b8e5-6f59316a16d3 +# ╟─89d4ca3b-e3c7-4519-917e-706a0fd6aea0 +# ╟─6029fb91-1263-4c7c-8d13-fd220a00a90c +# ╟─9e57c112-6328-4179-ab0b-5953978f4bd4 +# ╟─6e8f8c46-6902-4a88-8423-ae2046467ea6 +# ╠═9ab4251b-a061-44f3-b8d1-1ec9133832ee +# ╟─c5811230-dcbf-403b-bb94-9f056cdb92bb +# ╠═baefc33b-9f0c-4caf-b77d-a0dc118e911e +# ╟─1dd4afd5-4383-438b-8a4a-5b2b5cbbea75 +# ╠═218e2978-5388-4aec-a48f-73bc450145a3 +# ╟─5529e3aa-cabb-4714-93e1-138b0e7431f9 +# ╟─0f5436a7-4eac-4bd9-aca4-15b1f9d811b8 +# ╠═79527886-824d-4cb7-bee7-00041a6151b6 +# ╟─825a4196-8cfe-4a71-bb4a-34d9f445afd7 +# ╟─7f2aaacb-e159-4e24-8ed3-a9237287dd6b +# ╟─5ae1b44d-98cd-497b-b187-638ed42bad26 +# ╟─c39c9560-b050-4fe4-9cb0-376fe1ffc532 +# ╟─b6ec9ab6-ab4f-4d26-b2b0-9256621e472d +# ╟─95e65f76-f1ec-434d-8c1c-c314b2eca7c3 +# ╟─8182287b-f56b-40bd-a24d-3fb69409d9a7 +# ╟─cec3125d-5b90-42b5-8485-7926dbbac91c From c631d9a5c10c59fa77126eae006995d87994e597 Mon Sep 17 00:00:00 2001 From: gregoirepourtier Date: Sun, 29 Sep 2024 19:02:15 +0200 Subject: [PATCH 2/3] update notebook --- docs/src/examples/transient/Manifest.toml | 1983 ----------------- docs/src/examples/transient/Project.toml | 9 - docs/src/examples/transient/heat_equation.jl | 2038 +++++++++++++++++- 3 files changed, 1996 insertions(+), 2034 deletions(-) delete mode 100644 docs/src/examples/transient/Manifest.toml delete mode 100644 docs/src/examples/transient/Project.toml diff --git a/docs/src/examples/transient/Manifest.toml b/docs/src/examples/transient/Manifest.toml deleted file mode 100644 index e2652e04..00000000 --- a/docs/src/examples/transient/Manifest.toml +++ /dev/null @@ -1,1983 +0,0 @@ -# This file is machine-generated - editing it directly is not advised - -julia_version = "1.10.5" -manifest_format = "2.0" -project_hash = "ed7ca7f1ef5cb1b46faaf41c044e390981d8bf4c" - -[[deps.AbstractFFTs]] -deps = ["LinearAlgebra"] -git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef" -uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" -version = "1.5.0" -weakdeps = ["ChainRulesCore", "Test"] - - [deps.AbstractFFTs.extensions] - AbstractFFTsChainRulesCoreExt = "ChainRulesCore" - AbstractFFTsTestExt = "Test" - -[[deps.AbstractPlutoDingetjes]] -deps = ["Pkg"] -git-tree-sha1 = "6e1d2a35f2f90a4bc7c2ed98079b2ba09c35b83a" -uuid = "6e696c72-6542-2067-7265-42206c756150" -version = "1.3.2" - 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-[[deps.x265_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl"] -git-tree-sha1 = "dcc541bb19ed5b0ede95581fb2e41ecf179527d2" -uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" -version = "3.6.0+0" diff --git a/docs/src/examples/transient/Project.toml b/docs/src/examples/transient/Project.toml deleted file mode 100644 index 59eec073..00000000 --- a/docs/src/examples/transient/Project.toml +++ /dev/null @@ -1,9 +0,0 @@ -[deps] -CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" -Gmsh = "705231aa-382f-11e9-3f0c-b7cb4346fdeb" -HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" -Inti = "fb74042b-437e-4c5b-88cf-d4e2beb394d5" -LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" -Meshes = "eacbb407-ea5a-433e-ab97-5258b1ca43fa" -PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" -RecursiveArrayTools = "731186ca-8d62-57ce-b412-fbd966d074cd" diff --git a/docs/src/examples/transient/heat_equation.jl b/docs/src/examples/transient/heat_equation.jl index 7c6846c3..35049ba4 100644 --- a/docs/src/examples/transient/heat_equation.jl +++ b/docs/src/examples/transient/heat_equation.jl @@ -16,12 +16,11 @@ end # ╔═╡ f4a0859a-6b96-11ef-0c3b-7b7ba5c458ce begin - using Pkg, Revise - - Pkg.activate(joinpath(@__DIR__, "Project.toml")) - - using Inti # dev version as dependency - using Meshes, Gmsh, CairoMakie, PlutoUI, LinearAlgebra, RecursiveArrayTools + using Inti + using Meshes, Gmsh + using CairoMakie + using PlutoUI + using LinearAlgebra, RecursiveArrayTools end # ╔═╡ 95e65f76-f1ec-434d-8c1c-c314b2eca7c3 @@ -100,7 +99,11 @@ One possible formulation is the following: ``` where $S$ and $D$ are the single- and double-layer operators and $\sigma$ the density function. -Or look at one of the formulations from Costabel [a]. +Or look at one of the formulations from Costabel [a]: + +```math +\left(\frac12 + K \right) [\sigma] = g - u_p \;\; \tag{D4}, +``` """ # ╔═╡ 89d4ca3b-e3c7-4519-917e-706a0fd6aea0 @@ -159,21 +162,27 @@ end # ╔═╡ 1dd4afd5-4383-438b-8a4a-5b2b5cbbea75 +# ╔═╡ 8fffe2b3-437b-49f7-812d-ad66fae7b542 +inival2d(x,y) = exp(-5(x^2+y^2)) + # ╔═╡ 218e2978-5388-4aec-a48f-73bc450145a3 begin # Initial Condition - u₀(x) = 1.0 - f = map(q -> u₀(q.coords), Ω_quad) + u₀(x,y) = 1.0 # 2*sin(x)*sin(2*y) + 3*sin(4*x)*sin(5*y) + f = map(Ω_quad) do q + x, y = q.coords + return u₀(x,y) # return x^2 + y^2 < 0.2 ? u₀(x, y) : 0.0 + end # Boundary Conditions - bc(x) = 0.0 # consider making it time dependent for future tests + bc(x) = 0.0 # time dependent for future cases g = map(q -> bc(q.coords), Γ_quad) end # ╔═╡ 5529e3aa-cabb-4714-93e1-138b0e7431f9 -function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=1.0, τ=0.05) +function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=0.2, τ=0.01) - k = im/τ - pde = Inti.Helmholtz(; dim = 2, k=k) + k = 1/√(τ) # im/√(τ) + pde = Inti.Yukawa(; dim = 2, λ=k) ## Initialize Volume Potential, single and double layer operators for solving the problem and then for solution evaluation @@ -190,7 +199,7 @@ function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=1.0, τ=0.05) target = Γ_quad, source = Γ_quad, compression = (method = :none,), - correction = (method = :dim,), + correction = (method = :none,), ) # For the solution evaluation V_d2d = Inti.volume_potential(; @@ -198,7 +207,7 @@ function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=1.0, τ=0.05) target = Ω_quad, source = Ω_quad, compression = (method = :none,), - correction = (method = :dim,), + correction = (method = :none,), ) S_b2d, D_b2d = Inti.single_double_layer(; pde, @@ -210,18 +219,22 @@ function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=1.0, τ=0.05) times, sol_u = [t_start], [f] + f *= -1/√(τ) + + L = I / 2 + D # S # I / 2 + D - im * k * S + for i in 1:floor(Int64,(t_end - t_start)/τ) - L = I / 2 + D - im * k * S rhs = g - V_d2b*f - σ = L \ rhs # Solution Evaluation (to improve with compression and correction methods) - f = D_b2d*σ - im*k*S_b2d*σ + V_d2d*f # f = u^{n} -> used for the next iteration as source term of the PDE + f = D_b2d*σ + V_d2d*f # S_b2d*σ - V_d2d*f # D_b2d*σ - im*k*S_b2d*σ + V_d2d*f # f = -u^n -> used for the next iteration as source term of the PDE - push!(sol_u, real(f)) # Taking the real part of the solution (im(f)≈0) + push!(sol_u, f) push!(times, τ*i) + + f *= -1/√(τ) end DiffEqArray(sol_u, times) @@ -253,8 +266,15 @@ if allow_run u_quad = tsol[t_plot] u_nodes = Inti.quadrature_to_node_vals(Ω_quad, u_quad) - if t_plot == 1 - colorrange = (0.0, f[1]) + colorrange = (0.0, 1.0) # extrema(u_nodes) + fig_sol = Figure(; size = (800, 300)) + ax = Axis(fig_sol[1, 1]; aspect = DataAspect()) + viz!(Ω_msh; colorrange, color=u_nodes, interpolate = true, colormap = :jet) + cb = Colorbar(fig_sol[1, 2]; label = "u", colorrange, colormap = :jet) + fig_sol + + #= if t_plot == 1 + colorrange = (0.0, extrema(u_nodes)[2]) fig_sol = Figure(; size = (800, 300)) ax = Axis(fig_sol[1, 1]; aspect = DataAspect()) viz!(Ω_msh; colorrange, color=u_nodes, interpolate = true) @@ -267,7 +287,7 @@ if allow_run viz!(Ω_msh; colorrange, color=u_nodes, interpolate = true) cb = Colorbar(fig_sol[1, 2]; label = "u", colorrange) fig_sol - end + end =# end # ╔═╡ c39c9560-b050-4fe4-9cb0-376fe1ffc532 @@ -280,27 +300,1958 @@ md""" [a] _Time-dependent problems with the boundary integral equation method_, Martin Costabel, see [here](https://perso.univ-rennes1.fr/martin.costabel/publis/Co_ECM.pdf). """ +# ╔═╡ cec3125d-5b90-42b5-8485-7926dbbac91c +TableOfContents() + # ╔═╡ 8182287b-f56b-40bd-a24d-3fb69409d9a7 -begin - highlight(mdstring,color)= htl"""
$(mdstring)
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+git-tree-sha1 = "dcc541bb19ed5b0ede95581fb2e41ecf179527d2" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.6.0+0" +""" # ╔═╡ Cell order: # ╠═f4a0859a-6b96-11ef-0c3b-7b7ba5c458ce @@ -320,8 +2271,9 @@ TableOfContents() # ╟─c5811230-dcbf-403b-bb94-9f056cdb92bb # ╠═baefc33b-9f0c-4caf-b77d-a0dc118e911e # ╟─1dd4afd5-4383-438b-8a4a-5b2b5cbbea75 +# ╠═8fffe2b3-437b-49f7-812d-ad66fae7b542 # ╠═218e2978-5388-4aec-a48f-73bc450145a3 -# ╟─5529e3aa-cabb-4714-93e1-138b0e7431f9 +# ╠═5529e3aa-cabb-4714-93e1-138b0e7431f9 # ╟─0f5436a7-4eac-4bd9-aca4-15b1f9d811b8 # ╠═79527886-824d-4cb7-bee7-00041a6151b6 # ╟─825a4196-8cfe-4a71-bb4a-34d9f445afd7 @@ -329,6 +2281,8 @@ TableOfContents() # ╟─5ae1b44d-98cd-497b-b187-638ed42bad26 # ╟─c39c9560-b050-4fe4-9cb0-376fe1ffc532 # ╟─b6ec9ab6-ab4f-4d26-b2b0-9256621e472d +# ╟─cec3125d-5b90-42b5-8485-7926dbbac91c # ╟─95e65f76-f1ec-434d-8c1c-c314b2eca7c3 # ╟─8182287b-f56b-40bd-a24d-3fb69409d9a7 -# ╟─cec3125d-5b90-42b5-8485-7926dbbac91c +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 From 49bba0972d45fbf0a40e1459d0f7ca8f566af1d8 Mon Sep 17 00:00:00 2001 From: gregoirepourtier Date: Fri, 4 Oct 2024 10:43:11 +0200 Subject: [PATCH 3/3] push corrections --- docs/src/examples/transient/heat_equation.jl | 22 ++++++++++---------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/docs/src/examples/transient/heat_equation.jl b/docs/src/examples/transient/heat_equation.jl index 35049ba4..5a9063c5 100644 --- a/docs/src/examples/transient/heat_equation.jl +++ b/docs/src/examples/transient/heat_equation.jl @@ -56,13 +56,13 @@ u^n - \tau\Delta u^n &= u^{n-1} \quad &\text{in } \Omega\\[0.3cm] # ╔═╡ fc0e5d68-309e-471d-a99c-3966f6789803 md""" -We can rewrite $(2)$ to resemble the Helmholtz equation and obtain +We can rewrite $(2)$ as the modified Helmholtz equation (or Yukawa equation) and obtain $\begin{equation} \begin{cases} -\Delta u + k^2u &= f \quad &\text{in } \Omega\\[0.3cm] +\Delta u - k^2u &= f \quad &\text{in } \Omega\\[0.3cm] \hspace{1.6cm} u &= g &\text{on } \partial\Omega \end{cases}\tag{3}\end{equation}$ -where $k = \frac{i}{\sqrt\tau}$, $u = u^n$, $f = -\frac{1}{\tau}u^{n-1}$. +where $k = \frac{1}{\sqrt\tau} \;$ ; $\;u = u^n \;$ ; $\;f = -\frac{1}{\tau}u^{n-1}$ . """ # ╔═╡ 7344c3a8-52b3-4a4d-baf8-9d1c58d3e3a2 @@ -83,7 +83,7 @@ The function ``u_h`` satisfies the homogeneous problem ```math \begin{align} - \Delta u_h + k^2u_h &= 0, \quad &&\text{in } \quad \Omega, \\ + \Delta u_h - k^2u_h &= 0, \quad &&\text{in } \quad \Omega, \\ u_h &= g - u_p, \quad &&\text{on } \; \partial\Omega, \end{align}\tag{6} ``` @@ -179,10 +179,10 @@ begin end # ╔═╡ 5529e3aa-cabb-4714-93e1-138b0e7431f9 -function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=0.2, τ=0.01) +function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=0.2, τ=0.005) - k = 1/√(τ) # im/√(τ) - pde = Inti.Yukawa(; dim = 2, λ=k) + λ = 1/√(τ) + pde = Inti.Yukawa(; dim = 2, λ) ## Initialize Volume Potential, single and double layer operators for solving the problem and then for solution evaluation @@ -219,7 +219,7 @@ function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=0.2, τ=0.01) times, sol_u = [t_start], [f] - f *= -1/√(τ) + f *= -1/τ L = I / 2 + D # S # I / 2 + D - im * k * S @@ -229,12 +229,12 @@ function main(Ω_quad, Γ_quad, f, g ; t_start=0.0, t_end=0.2, τ=0.01) σ = L \ rhs # Solution Evaluation (to improve with compression and correction methods) - f = D_b2d*σ + V_d2d*f # S_b2d*σ - V_d2d*f # D_b2d*σ - im*k*S_b2d*σ + V_d2d*f # f = -u^n -> used for the next iteration as source term of the PDE + f = D_b2d*σ - V_d2d*f # S_b2d*σ - V_d2d*f # D_b2d*σ - im*k*S_b2d*σ + V_d2d*f # f = -u^n -> used for the next iteration as source term of the PDE push!(sol_u, f) push!(times, τ*i) - f *= -1/√(τ) + f *= -1/τ end DiffEqArray(sol_u, times) @@ -2267,7 +2267,7 @@ version = "3.6.0+0" # ╟─6029fb91-1263-4c7c-8d13-fd220a00a90c # ╟─9e57c112-6328-4179-ab0b-5953978f4bd4 # ╟─6e8f8c46-6902-4a88-8423-ae2046467ea6 -# ╠═9ab4251b-a061-44f3-b8d1-1ec9133832ee +# ╟─9ab4251b-a061-44f3-b8d1-1ec9133832ee # ╟─c5811230-dcbf-403b-bb94-9f056cdb92bb # ╠═baefc33b-9f0c-4caf-b77d-a0dc118e911e # ╟─1dd4afd5-4383-438b-8a4a-5b2b5cbbea75