example on LVPP method example for obstacle problem (#83)

Author: chmerdon

examples/Example227_ObstacleProblemLVPP.jl

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+#=
+
+# 227 : Obstacle Problem LVPP
+([source code](@__SOURCE_URL__))
+
+This example computes the solution ``u`` of the nonlinear obstacle problem that seeks
+the minimiser of the energy functional
+```math
+\begin{aligned}
+	E(u) = \frac{1}{2} \int_\Omega \lvert \nabla u \rvert^2 dx - \int_\Omega f u dx
+\end{aligned}
+```
+with some right-hand side ``f`` within the set of admissible functions that lie above an obstacle ``\chi``
+```math
+\begin{aligned}
+	\mathcal{K} := \lbrace u \in H^1_0(\Omega) : u \geq \chi \rbrace.
+\end{aligned}
+```
+
+Opposite to Example225 the solution is computed by the latent variable proximal point (LVPP) method
+that solves the problem via a series of nonlinear mixed problems that guarantee the decay of the
+energy. Given ``\alpha_k`` and initial guess ``u_0`` and ``\psi_0``, the subproblem for ``k \geq 1``
+seeks a solution ``u_{k} \in V := H^1_0(\Omega)`` and ``\psi_{k} \in W := L^\infty(\Omega)`` such that
+```math
+\begin{aligned}
+\alpha_k (\nabla u_k, \nabla v) + (\psi_k, v) = (\alpha_k f + \psi_{k-1},v)
+&& \text{for all } v \in V\\
+(u_k, w) - (\varchi + \exp(\psi_k), w) & = 0 &&
+\text{for all } w \in W
+\end{aligned}
+```
+The parameter ``\alpha_k`` is initialized with ``\alpha_0 = 1`` and updated according to
+``\alpha_k = min(max(r^(q^k) - α), 10^3)`` with ``r = q = 1.5``. The problem for each ``k``
+is solved by the Newton method. This implements Algorithm 3 in the reference below.
+
+
+!!! reference
+
+	''Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints''
+	Brendan Keith, Thomas M. Surowiec, Found Comput Math (2024)
+	[>Link<](https://doi.org/10.1007/s10208-024-09681-8)
+
+
+![](example227.png)
+=#
+
+module Example227_ObstacleProblemLVPP
+
+using ExtendableFEM
+using ExtendableFEMBase
+using ExtendableGrids
+using LinearAlgebra
+using Metis
+using Test #hide
+
+## define obstacle
+const b = 9 // 20
+const d = sqrt(1 // 4 - b^2)
+function χ(x)
+    r = sqrt(x[1]^2 + x[2]^2)
+    if r <= b
+        return sqrt(1 // 4 - r^2)
+    else
+        return d + b^2 / d - b * r / d
+    end
+end
+
+## transformation of latent variable ψ to constrained variable u
+function ∇R!(result, input, qpinfo)
+    return result[1] = χ(qpinfo.x) + exp(input[1])
+end
+
+## boundary data for latent variable ψ (such that ̃u := χ + ∇R(ψ) satisfies Dirichlet boundary conditions)
+function bnd_ψ!(result, qpinfo)
+    return result[1] = log(-χ(qpinfo.x))
+end
+
+function main(;
+        nrefs = 5,
+        α0 = 1.0,
+        order = 1,
+        parallel = false,
+        npart = 8,
+        tol = 1.0e-12,
+        Plotter = nothing,
+        kwargs...
+    )
+
+    ## choose initial mesh
+    xgrid = uniform_refine(grid_unitsquare(Triangle2D; scale = (2, 2), shift = (-0.5, -0.5)), nrefs)
+    if parallel
+        xgrid = partition(xgrid, RecursiveMetisPartitioning(npart = npart))
+    end
+
+    ## create finite element space
+    FES = [FESpace{H1Pk{1, 2, order}}(xgrid), FESpace{H1Pk{1, 2, order}}(xgrid)]
+
+    ## init proximal parameter
+    α = α0
+
+    ## prepare Laplacian and mass matrix
+    L = FEMatrix(FES[1], FES[1])
+    assemble!(L, BilinearOperator([grad(1)], [grad(1)]; parallel = parallel))
+    function scaled_laplacian!(A, b, args; assemble_matrix = true, assemble_rhs = true, kwargs...)
+        return if assemble_matrix
+            ## add Laplacian scaled by α
+            ExtendableFEMBase.add!(A, L.entries; factor = α)
+        end
+    end
+    M = FEMatrix(FES[1], FES[2])
+    b = FEVector(FES[1])
+    assemble!(M, BilinearOperator([id(1)], [id(1)]; parallel = parallel))
+
+    ## problem description
+    PD = ProblemDescription()
+    u = Unknown("u"; name = "solution")
+    ψ = Unknown("ψ"; name = "latent variable")
+    assign_unknown!(PD, u)
+    assign_unknown!(PD, ψ)
+    assign_operator!(PD, CallbackOperator(scaled_laplacian!, [u]; kwargs...))
+    assign_operator!(PD, BilinearOperator([id(u)], [(id(ψ))]; transposed_copy = 1, store = true, parallel = parallel, kwargs...))
+    assign_operator!(PD, NonlinearOperator(∇R!, [id(ψ)], [id(ψ)]; parallel = parallel, factor = -1, kwargs...))
+    assign_operator!(PD, HomogeneousBoundaryData(u; regions = 1:4, kwargs...))
+    assign_operator!(PD, InterpolateBoundaryData(ψ, bnd_ψ!; regions = 1:4, kwargs...))
+    assign_operator!(PD, LinearOperator(b, [u]; kwargs...))
+
+    ## solve
+    sol = FEVector(FES; tags = PD.unknowns)
+    sol_prev = FEVector(FES; tags = PD.unknowns)
+    SC = nothing
+    r, q = 3 // 2, 3 // 2
+    converged = false
+    k = 0
+    while !converged
+        k += 1
+        @info "Step $k: α = $(α)"
+
+        ## save previous solution and update right-hand side
+        b.entries .= M.entries * view(sol[ψ])
+        sol_prev.entries .= sol.entries
+
+        ## solve nonlinear problem
+        sol, SC = solve(PD, FES, SC; init = sol, maxiterations = 20, verbosity = -1, timeroutputs = :hide, return_config = true, kwargs...)
+        niterations = length(ExtendableFEM.residuals(SC))
+
+        ## compute distance
+        dist = norm(view(sol[u]) .- view(sol_prev[u]))
+        @info "dist = $dist, niterations = $(niterations - 1)"
+        if dist < tol
+            converged = true
+        else
+            # increase proximal parameter
+            α = min(max(r^(q^k) - α), 10^3)
+        end
+    end
+
+    ## postprocess latent variable v = ϕ + exp ψ = ∇R!(ψ)
+    u2 = Unknown("̃u"; name = "solution 2")
+    append!(sol, FES[1]; tag = u2)
+    lazy_interpolate!(sol[u2], sol, [id(ψ)]; postprocess = ∇R!)
+
+    ## plot
+    plt = plot([id(u), id(ψ), id(u2)], sol; Plotter = Plotter, ncols = 3)
+
+    return sol, plt
+end
+
+generateplots = ExtendableFEM.default_generateplots(Example227_ObstacleProblemLVPP, "example227.png") #hide
+end # module