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Stokes2D_vep_reg_vc.jl
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Stokes2D_vep_reg_vc.jl
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# Initialisation
using Plots, Printf, Statistics, LinearAlgebra
Dat = Float64 # Precision (double=Float64 or single=Float32)
# Macros
@views av(A) = 0.25*(A[1:end-1,1:end-1].+A[2:end,1:end-1].+A[1:end-1,2:end].+A[2:end,2:end])
@views av_xa(A) = 0.5*(A[1:end-1,:].+A[2:end,:])
@views av_ya(A) = 0.5*(A[:,1:end-1].+A[:,2:end])
# 2D Stokes routine
@views function Stokes2D_vep()
do_DP = true # do_DP=false: Von Mises, do_DP=true: Drucker-Prager (friction angle)
η_reg = 1.2e-2 # regularisation "viscosity"
# Physics
Lx, Ly = 1.0, 1.0 # domain size
radi = 0.01 # inclusion radius
τ_y = 1.6 # yield stress. If do_DP=true, τ_y stand for the cohesion: c*cos(ϕ)
sinϕ = sind(30)*do_DP # sinus of the friction angle
μ0 = 1.0 # viscous viscosity
G0 = 1.0 # elastic shear modulus
Gi = G0/(6.0-4.0*do_DP) # elastic shear modulus perturbation
εbg = 1.0 # background strain-rate
# Numerics
nt = 10 # number of time steps
nx, ny = 63, 63 # numerical grid resolution
Vdmp = 4.0 # convergence acceleration (damping)
Vsc = 2.0 # iterative time step limiter
Ptsc = 6.0 # iterative time step limiter
ε = 1e-6 # nonlinear tolerence
iterMax = 3e4 # max number of iters
nout = 200 # check frequency
# Preprocessing
dx, dy = Lx/nx, Ly/ny
dt = μ0/G0/4.0 # assumes Maxwell time of 4
# Array initialisation
Pt = zeros(Dat, nx ,ny )
∇V = zeros(Dat, nx ,ny )
Vx = zeros(Dat, nx+1,ny )
Vy = zeros(Dat, nx ,ny+1)
Exx = zeros(Dat, nx ,ny )
Eyy = zeros(Dat, nx ,ny )
Exyv = zeros(Dat, nx+1,ny+1)
Exx1 = zeros(Dat, nx ,ny )
Eyy1 = zeros(Dat, nx ,ny )
Exy1 = zeros(Dat, nx ,ny )
Exyv1 = zeros(Dat, nx+1,ny+1)
Txx = zeros(Dat, nx ,ny )
Tyy = zeros(Dat, nx ,ny )
Txy = zeros(Dat, nx ,ny )
Txyv = zeros(Dat, nx+1,ny+1)
Txx_o = zeros(Dat, nx ,ny )
Tyy_o = zeros(Dat, nx ,ny )
Txy_o = zeros(Dat, nx ,ny )
Txyv_o = zeros(Dat, nx+1,ny+1)
# for vertices implementation
Ptv = zeros(Dat, nx+1,ny+1)
Exxv = zeros(Dat, nx+1,ny+1)
Eyyv = zeros(Dat, nx+1,ny+1)
Exxv1 = zeros(Dat, nx+1,ny+1)
Eyyv1 = zeros(Dat, nx+1,ny+1)
Txxv = zeros(Dat, nx+1,ny+1)
Tyyv = zeros(Dat, nx+1,ny+1)
Txxv_o = zeros(Dat, nx+1,ny+1)
Tyyv_o = zeros(Dat, nx+1,ny+1)
Fchkv = zeros(Dat, nx+1,ny+1)
Fv = zeros(Dat, nx+1,ny+1)
Plav = zeros(Dat, nx+1,ny+1)
λv = zeros(Dat, nx+1,ny+1)
dQdTxxv = zeros(Dat, nx+1,ny+1)
dQdTyyv = zeros(Dat, nx+1,ny+1)
dQdTxyv = zeros(Dat, nx+1,ny+1)
Tiiv = zeros(Dat, nx+1,ny+1)
Eiiv = zeros(Dat, nx+1,ny+1)
# for vertices implementation
Tii = zeros(Dat, nx ,ny )
Eii = zeros(Dat, nx ,ny )
F = zeros(Dat, nx ,ny )
Fchk = zeros(Dat, nx ,ny )
Pla = zeros(Dat, nx ,ny )
λ = zeros(Dat, nx ,ny )
dQdTxx = zeros(Dat, nx ,ny )
dQdTyy = zeros(Dat, nx ,ny )
dQdTxy = zeros(Dat, nx ,ny )
Rx = zeros(Dat, nx-1,ny )
Ry = zeros(Dat, nx ,ny-1)
dVxdt = zeros(Dat, nx-1,ny )
dVydt = zeros(Dat, nx ,ny-1)
dtPt = zeros(Dat, nx ,ny )
dtVx = zeros(Dat, nx-1,ny )
dtVy = zeros(Dat, nx ,ny-1)
Rog = zeros(Dat, nx ,ny )
η_v = μ0*ones(Dat, nx ,ny )
η_e = dt*G0*ones(Dat, nx ,ny )
η_ev = dt*G0*ones(Dat, nx+1,ny+1)
η_ve = ones(Dat, nx ,ny )
η_vep = ones(Dat, nx ,ny )
η_vepv = ones(Dat, nx+1,ny+1)
η_vev = ones(Dat, nx+1,ny+1) # for vertices implementation
η_vv = μ0*ones(Dat, nx+1,ny+1) # for vertices implementation
# Initial condition
xc, yc = LinRange(dx/2, Lx-dx/2, nx), LinRange(dy/2, Ly-dy/2, ny)
xc, yc = LinRange(dx/2, Lx-dx/2, nx), LinRange(dy/2, Ly-dy/2, ny)
xv, yv = LinRange(0.0, Lx, nx+1), LinRange(0.0, Ly, ny+1)
(Xvx,Yvx) = ([x for x=xv,y=yc], [y for x=xv,y=yc])
(Xvy,Yvy) = ([x for x=xc,y=yv], [y for x=xc,y=yv])
radc = (xc.-Lx./2).^2 .+ (yc'.-Ly./2).^2
radv = (xv.-Lx./2).^2 .+ (yv'.-Ly./2).^2
η_e[radc.<radi] .= dt*Gi
η_ev[radv.<radi].= dt*Gi
η_ve .= (1.0./η_e + 1.0./η_v).^-1
η_vev .= (1.0./η_ev + 1.0./η_vv).^-1
Vx .= εbg.*Xvx
Vy .= .-εbg.*Yvy
# Time loop
t=0.0; evo_t=[]; evo_Txx=[]
for it = 1:nt
iter=1; err=2*ε; err_evo1=[]; err_evo2=[]
Txx_o.=Txx; Tyy_o.=Tyy; Txy_o.=Txy; λ.=0.0
Txxv_o.=Txxv; Tyyv_o.=Tyyv; Txyv_o.=Txyv; λv.=0.0
local itg
while (err>ε && iter<=iterMax)
# divergence - pressure
∇V .= diff(Vx, dims=1)./dx .+ diff(Vy, dims=2)./dy
Pt .= Pt .- dtPt.*∇V
# strain rates
Exx .= diff(Vx, dims=1)./dx .- 1.0/3.0*∇V
Eyy .= diff(Vy, dims=2)./dy .- 1.0/3.0*∇V
Exyv[2:end-1,2:end-1] .= 0.5.*(diff(Vx[2:end-1,:], dims=2)./dy .+ diff(Vy[:,2:end-1], dims=1)./dx)
Exxv[2:end-1,2:end-1] .= av(Exx); Exxv[1,:].=Exxv[2,:]; Exxv[end,:].=Exxv[end-1,:]; Exxv[:,1].=Exxv[:,2]; Exxv[:,end].=Exxv[:,end-1]
Eyyv[2:end-1,2:end-1] .= av(Eyy); Eyyv[1,:].=Eyyv[2,:]; Eyyv[end,:].=Eyyv[end-1,:]; Eyyv[:,1].=Eyyv[:,2]; Eyyv[:,end].=Eyyv[:,end-1]
Ptv[2:end-1,2:end-1] .= av(Pt); Ptv[1,:].= Ptv[2,:]; Ptv[end,:].= Ptv[end-1,:]; Ptv[:,1].= Ptv[:,2]; Ptv[:,end].= Ptv[:,end-1]
# visco-elastic strain rates
Exx1 .= Exx .+ Txx_o ./2.0./η_e
Eyy1 .= Eyy .+ Tyy_o ./2.0./η_e
Exy1 .= av(Exyv) .+ Txy_o ./2.0./η_e
Eii .= sqrt.(0.5*(Exx1.^2 .+ Eyy1.^2) .+ Exy1.^2)
# visco-elastic strain rates vertices
Exxv1 .= Exxv .+ Txxv_o./2.0./η_ev
Eyyv1 .= Eyyv .+ Tyyv_o./2.0./η_ev
Exyv1 .= Exyv .+ Txyv_o./2.0./η_ev
Eiiv .= sqrt.(0.5*(Exxv1.^2 .+ Eyyv1.^2) .+ Exyv1.^2)
# trial stress
Txx .= 2.0.*η_ve.*Exx1
Tyy .= 2.0.*η_ve.*Eyy1
Txy .= 2.0.*η_ve.*Exy1
Tii .= sqrt.(0.5*(Txx.^2 .+ Tyy.^2) .+ Txy.^2)
# trial stress vertices
Txxv .= 2.0.*η_vev.*Exxv1
Tyyv .= 2.0.*η_vev.*Eyyv1
Txyv .= 2.0.*η_vev.*Exyv1
Tiiv .= sqrt.(0.5*(Txxv.^2 .+ Tyyv.^2) .+ Txyv.^2)
# yield function
F .= Tii .- τ_y .- Pt.*sinϕ
Pla .= 0.0
Pla .= F .> 0.0
λ .= Pla.*F./(η_ve .+ η_reg)
dQdTxx .= 0.5.*Txx./Tii
dQdTyy .= 0.5.*Tyy./Tii
dQdTxy .= Txy./Tii
# yield function vertices
Fv .= Tiiv .- τ_y .- Ptv.*sinϕ
Plav .= 0.0
Plav .= Fv .> 0.0
λv .= Plav.*Fv./(η_vev .+ η_reg)
dQdTxxv.= 0.5.*Txxv./Tiiv
dQdTyyv.= 0.5.*Tyyv./Tiiv
dQdTxyv.= Txyv./Tiiv
# plastic corrections
Txx .= 2.0.*η_ve.*(Exx1 .- λ.*dQdTxx)
Tyy .= 2.0.*η_ve.*(Eyy1 .- λ.*dQdTyy)
Txy .= 2.0.*η_ve.*(Exy1 .- 0.5.*λ.*dQdTxy)
Tii .= sqrt.(0.5*(Txx.^2 .+ Tyy.^2) .+ Txy.^2)
Fchk .= Tii .- τ_y .- Pt.*sinϕ .- λ.*η_reg
η_vep .= Tii./2.0./Eii
# plastic corrections vertices
Txxv .= 2.0.*η_vev.*(Exxv1 .- λv.*dQdTxxv)
Tyyv .= 2.0.*η_vev.*(Eyyv1 .- λv.*dQdTyyv)
Txyv .= 2.0.*η_vev.*(Exyv1 .- 0.5.*λv.*dQdTxyv)
Tiiv .= sqrt.(0.5*(Txxv.^2 .+ Tyyv.^2) .+ Txyv.^2)
Fchkv .= Tiiv .- τ_y .- Ptv.*sinϕ .- λv.*η_reg
η_vepv .= Tiiv./2.0./Eiiv
# η_vepv[2:end-1,2:end-1] .= av(η_vep); η_vepv[1,:].=η_vepv[2,:]; η_vepv[end,:].=η_vepv[end-1,:]; η_vepv[:,1].=η_vepv[:,2]; η_vepv[:,end].=η_vepv[:,end-1]
# Txyv .= 2.0.*η_vepv.*Exyv1
# PT timestep
dtVx .= min(dx,dy)^2.0./av_xa(η_vep)./4.1./Vsc
dtVy .= min(dx,dy)^2.0./av_ya(η_vep)./4.1./Vsc
dtPt .= 4.1.*η_vep./max(nx,ny)./Ptsc
# velocities
Rx .= .-diff(Pt, dims=1)./dx .+ diff(Txx, dims=1)./dx .+ diff(Txyv[2:end-1,:], dims=2)./dy
Ry .= .-diff(Pt, dims=2)./dy .+ diff(Tyy, dims=2)./dy .+ diff(Txyv[:,2:end-1], dims=1)./dx .+ av_ya(Rog)
dVxdt .= dVxdt.*(1-Vdmp/nx) .+ Rx
dVydt .= dVydt.*(1-Vdmp/ny) .+ Ry
Vx[2:end-1,:] .= Vx[2:end-1,:] .+ dVxdt.*dtVx
Vy[:,2:end-1] .= Vy[:,2:end-1] .+ dVydt.*dtVy
# convergence check
if mod(iter, nout)==0
norm_Rx = norm(Rx)/length(Rx); norm_Ry = norm(Ry)/length(Ry); norm_∇V = norm(∇V)/length(∇V)
err = maximum([norm_Rx, norm_Ry, norm_∇V])
push!(err_evo1, err); push!(err_evo2, itg)
@printf("it = %d, iter = %d, err = %1.2e norm[Rx=%1.2e, Ry=%1.2e, ∇V=%1.2e] (Fchk=%1.2e - Fchkv=%1.2e) \n", it, itg, err, norm_Rx, norm_Ry, norm_∇V, maximum(Fchk), maximum(Fchkv))
end
iter+=1; itg=iter
end
t = t + dt
push!(evo_t, t); push!(evo_Txx, maximum(Txx))
# Plotting
p1 = heatmap(xv, yc, Vx' , aspect_ratio=1, xlims=(0, Lx), ylims=(dy/2, Ly-dy/2), c=:inferno, title="Vx")
# p2 = heatmap(xc, yv, Vy' , aspect_ratio=1, xlims=(dx/2, Lx-dx/2), ylims=(0, Ly), c=:inferno, title="Vy")
p2 = heatmap(xc, yc, η_vep' , aspect_ratio=1, xlims=(dx/2, Lx-dx/2), ylims=(0, Ly), c=:inferno, title="η_vep")
p3 = heatmap(xc, yc, Tii' , aspect_ratio=1, xlims=(dx/2, Lx-dx/2), ylims=(0, Ly), c=:inferno, title="τii")
p4 = plot(evo_t, evo_Txx , legend=false, xlabel="time", ylabel="max(τxx)", linewidth=0, markershape=:circle, framestyle=:box, markersize=3)
plot!(evo_t, 2.0.*εbg.*μ0.*(1.0.-exp.(.-evo_t.*G0./μ0)), linewidth=2.0) # analytical solution for VE loading
plot!(evo_t, 2.0.*εbg.*μ0.*ones(size(evo_t)), linewidth=2.0) # viscous flow stress
if !do_DP plot!(evo_t, τ_y*ones(size(evo_t)), linewidth=2.0) end # von Mises yield stress
display(plot(p1, p2, p3, p4))
end
return
end
Stokes2D_vep()