diff --git a/docs/src/BeamElement.jl b/docs/src/BeamElement.jl new file mode 100644 index 0000000..1b93e7c --- /dev/null +++ b/docs/src/BeamElement.jl @@ -0,0 +1,427 @@ +# # Euler beam element + +using StaticArrays, LinearAlgebra, Muscade +@functor with() g̃_default(t) = SVector(0.,0.,1.); + +""" + BeamCrossSection + +Data structure containing the cross section material properties, for example to a [`EulerBeam3D`](@ref) +# Arguments to the constructor +- `EA :: 𝕣` is the axial stiffness [N] +- `EI₂ :: 𝕣` is the bending stiffness [Nm/(1/m)] about second axis +- `EI₃ :: 𝕣` is the bending stiffness [Nm/(1/m)] about third axis +- `GJ :: 𝕣` is the torsional stiffness [Nm/(rad/m)] about longitudinal axis +- `μ :: 𝕣` is the mass per unit length [kg/m] +- `ι₁ :: 𝕣` is the (mass) moment of inertia about longitudial axis per unit length [kgm²/m] + +# Optional argument to the constructor (all set to zero by default) +- `w :: 𝕣` is the weight per unit length [N/m] +- `g̃ :: Functor` describes the gravity field divided by acceleration of gravity [-], function of time, set to SVector(0.,0.,1.) by default +- `Ca₁ :: 𝕣` is the tangential added mass per unit length [kg/m] +- `Cl₁ :: 𝕣` is the tangential linear damping coefficient per unit length [N/m/(m/s)] +- `Cq₁ :: 𝕣` is the tangential quadratic damping coefficient per unit length [N/m/(m/s)^2], for example from drag +- `Ca₂ :: 𝕣` is the transverse added mass per unit length [kg/m] for motions along second axis +- `Cl₂ :: 𝕣` is the transverse linear damping coefficient per unit length [N/m/(m/s)] for motions along second axis +- `Cq₂ :: 𝕣` is the transverse quadratic damping coefficient per unit length [N/m/(m/s)^2], for motions along second axis +- `Ca₃ :: 𝕣` is the transverse added mass per unit length [kg/am] for motions along third axis +- `Cl₃ :: 𝕣` is the transverse linear damping coefficient per unit length [N/m/(m/s)] for motions along third axis +- `Cq₃ :: 𝕣` is the transverse quadratic damping coefficient per unit length [N/m/(m/s)^2], for motions along third axis + +# Example +``` +EA = 10. +EI₂ = 1. +EI₃ = 1. +GJ = 1. +μ = 1. +ι₁ = 0.1 +mat = BeamCrossSection(EA=EA,EI₂=EI₂,EI₃=EI₃,GJ=GJ,μ=μ,ι₁=ι₁) +``` + +See also: [`EulerBeam3D`](@ref), [`AxisymmetricBarCrossSection`](@ref) +""" +struct BeamCrossSection{Tg̃} + EA :: 𝕣 + EI₂ :: 𝕣 + EI₃ :: 𝕣 + GJ :: 𝕣 + μ :: 𝕣 + ι₁ :: 𝕣 + w :: 𝕣 + g̃ :: Tg̃ # a Functor + Ca₁ :: 𝕣 + Cl₁ :: 𝕣 + Cq₁ :: 𝕣 + Ca₂ :: 𝕣 + Cl₂ :: 𝕣 + Cq₂ :: 𝕣 + Ca₃ :: 𝕣 + Cl₃ :: 𝕣 + Cq₃ :: 𝕣 +end +BeamCrossSection(;EA,EI₂=EI₂,EI₃=EI₃,GJ=GJ,μ=μ,ι₁=ι₁,w=0.,g̃=g̃_default,Ca₁=0.,Cl₁=0.,Cq₁=0.,Ca₂=0.,Cl₂=0.,Cq₂=0.,Ca₃=0.,Cl₃=0.,Cq₃=0.) = BeamCrossSection(EA,EI₂,EI₃,GJ,μ,ι₁,w,g̃,Ca₁,Cl₁,Cq₁,Ca₂,Cl₂,Cq₂,Ca₃,Cl₃,Cq₃); + +# Resultant function that computes the internal loads from the strains and curvatures, and external loads on the element. +@espy function resultants(o::BeamCrossSection,ε,κ,xᵧ,rₛₘ,vᵢ,t) + r₀ = ∂0(rₛₘ) # orientation of the element's local refsys + vᵢ₁ = ∂1(vᵢ) # intrinsic rotation rate of the element's local refsys + vᵢ₂ = ∂2(vᵢ) # intrinsic rotation acceleration of the element's local refsys + xᵧ₀,xᵧ₁,xᵧ₂ = ∂0(xᵧ),∂1(xᵧ),∂2(xᵧ) + xₗ₁ = xᵧ₁ ∘₁ r₀ + xₗ₂ = xᵧ₂ ∘₁ r₀ + ## Weight + fw = - o.w * o.g̃(t) + ## Compute translational inertia force + fi = o.μ * xᵧ₂ + ## Compute added mass force + faₗ = SVector(o.Ca₁ * xₗ₂[1], o.Ca₂ * xₗ₂[2], o.Ca₃ * xₗ₂[3]) + faᵧ = r₀ ∘₁ faₗ + ## Compute linear damping force + flₗ = SVector( o.Cl₁ * xₗ₁[1], o.Cl₂ * xₗ₁[2], o.Cl₃ * xₗ₁[3]) + flᵧ = r₀ ∘₁ flₗ + ## Compute quadratic damping force + fq1ₗ = o.Cq₁ * xₗ₁[1]^2; if xₗ₁[1] < 0; fq1ₗ = -fq1ₗ end + fq2ₗ = o.Cq₂ * xₗ₁[2]^2; if xₗ₁[2] < 0; fq2ₗ = -fq2ₗ end + fq3ₗ = o.Cq₃ * xₗ₁[3]^2; if xₗ₁[3] < 0; fq3ₗ = -fq3ₗ end + fqₗ = SVector(fq1ₗ,fq2ₗ,fq3ₗ) + fqᵧ = r₀ ∘₁ fqₗ + # Summing up forces + ☼fₑ = fi + faᵧ + flᵧ + fqᵧ + fw # external forces at Gauss point. + ## Compute roll inertia moment + m₁ₗ = o.ι₁*vᵢ₂[1] #local + mᵧ = ∂0(rₛₘ)[:,1] * m₁ₗ #global + ☼mₑ = mᵧ # external couples at Gauss point. + ## Compute internal loads + ☼fᵢ = o.EA*∂0(ε) + ## WARNING: curvatures are defined as rate of rotation along the element, not second derivatives of deflection. + ## Hence κ[3]>0 implies +2 direction is inside curve, + ## κ[2]>0 implies -3 direction is inside curve. + ☼mᵢ = SVector(o.GJ*∂0(κ)[1],o.EI₃*∂0(κ)[2],o.EI₂*∂0(κ)[3]) + return fᵢ,mᵢ,fₑ,mₑ +end; + +## Static Euler beam element, with two nodes, four Gauss points and 12 degrees of freedom. +const ngp = 4 +const ndim = 3 +const nXdof = 12 +const nUdof = 3 +const nXnod = 2; + +# Shape functions for a beam element with support ζ∈[-1/2,1/2]. Though the shape function matrices are sparse, do not "unroll" them. That would be applyer but considerably clutter the code +yₐ(ζ) = 2ζ # differential axial displacement or roll field +yᵤ(ζ) = -4ζ^3 +3ζ # deflection due to differential nodal transverse translation +yᵥ(ζ) = ζ^2 - 1/4 # deflection due to differenttial rotation (bending, not torsion) +κₐ(ζ) = 2 # torsion . κₐ = yₐ′ . Divide by L . +κᵤ(ζ) = -24ζ # curvature. κᵤ = yᵤ′′. Divide by L². +κᵥ(ζ) = 2; # curvature. κᵥ = yᵥ′′. Divide by L . + +# Data structure describing an EulerBeam3D element as meshed +""" + EulerBeam3D <: AbstractElement + +A three-dimensional Euler beam element, with two nodes, twelve X-dofs and three U-dofs. +# Arguments to the constructor +- `nod :: Vector{Node}` contains the element's nodes +- `mat :: Mat` contains the material properties ([`BeamCrossSection`](@ref), for example) + +# Optional argument to the constructor +- `orient2 :: SVector{3,𝕣}` defines the direction of the first bending axis in the global coordinate system. +Default is `SVector(0.,1.,0.)`. + +# Example +``` +EA = 10. +EI₂ = 3. +EI₃ = 3. +GJ = 4. +μ = 1. +ι₁ = 1.0 +L = 5 +model = Model(:TestModel) +node1 = addnode!(model,𝕣[0,0,0]) +node2 = addnode!(model,𝕣[L,0,0]) +mat = BeamCrossSection(EA=EA,EI₂=EI₂,EI₃=EI₃,GJ=GJ,μ=μ,ι₁=ι₁) +addelement!(model,EulerBeam3D,[node1 node2];mat) +``` + +See also: [`BeamCrossSection`](@ref), [`Bar3D`](@ref) +""" +struct EulerBeam3D{Mat,Uforce} <: AbstractElement + cₘ :: SVector{3,𝕣} # Position of the middle of the element, as meshed + rₘ :: Mat33{𝕣} # Orientation of the element, as meshed, represented by a rotation matrix + ζgp :: SVector{ngp,𝕣} # Location of the Gauss points for the normalized element with length 1 + ζnod :: SVector{nXnod,𝕣} # Location of the nodes for the normalized element with length 1 + tgₘ :: SVector{ndim,𝕣} # Vector connecting the nodes of the element in the global coordinate system + tgₑ :: SVector{ndim,𝕣} # Vector connecting the nodes of the element in the local coordinate system + yₐ :: SVector{ngp,𝕣} # Value at gp of shape function for differential axial displacement or roll field + yᵤ :: SVector{ngp,𝕣} # Value at gp of shape function for deflection due to differential nodal transverse translation + yᵥ :: SVector{ngp,𝕣} # Value at gp of shape function for deflection due to differenttial rotation (bending, not torsion). Multiplied by L . + κₐ :: SVector{ngp,𝕣} # Value at gp of shape function for torsion . κₐ = yₐ′ . Divided by L . + κᵤ :: SVector{ngp,𝕣} # Value at gp of shape function for curvature. κᵤ = yᵤ′′. Divided by L². + κᵥ :: SVector{ngp,𝕣} # Value at gp of shape function for curvature. κᵥ = yᵥ′′. Divided by L . + L :: 𝕣 # as meshed length of the element + dL :: SVector{ngp,𝕣} # weight associated to each Gauss point + mat :: Mat # used to store material properties (BeamCrossSection, for example) +end; + +# For performance, `residual` will only accept differentiation to first order +Muscade.no_second_order(::Type{<:EulerBeam3D}) = Val(true) + +# Define nodes, classes, and field names of dofs +Muscade.doflist( ::Type{EulerBeam3D{Mat,false}}) where{Mat} = + (inod = (1,1,1,1,1,1, 2,2,2,2,2,2), + #class= ntuple(i->:X,nXdof), # not typestable + class= (:X,:X,:X,:X,:X,:X,:X,:X,:X,:X,:X,:X), + field= (:t1,:t2,:t3,:r1,:r2,:r3, :t1,:t2,:t3,:r1,:r2,:r3) ) +Muscade.doflist( ::Type{EulerBeam3D{Mat,true}}) where{Mat} = + (inod = (1,1,1,1,1,1, 2,2,2,2,2,2, 3,3,3), + #class= (ntuple(i->:X,nXdof)...,ntuple(i->:U,nUdof)...), # not typestable + class= (:X,:X,:X,:X,:X,:X,:X,:X,:X,:X,:X,:X,:U,:U,:U), + field= (:t1,:t2,:t3,:r1,:r2,:r3, :t1,:t2,:t3,:r1,:r2,:r3, :t1,:t2,:t3) ) + +# ElementType for the EulerBeam3D element. Arguments: node list, material, and direction of the first bending axis in the global coordinate system. +EulerBeam3D(nod;kwargs...) = EulerBeam3D{false}(nod;kwargs...) # by default, EulerBeam3D does not have Udof. +function EulerBeam3D{Udof}(nod::Vector{Node};mat,orient2::SVector{ndim,𝕣}=SVector(0.,1.,0.)) where {Udof} + c = coord(nod) + ## Position of the middle of the element in the global coordinate system (as-meshed) + cₘ = SVector{ndim}((c[1]+c[2])/2) + ## Length and tangential vector to the element in the global coordinate system + tgₘ = SVector{ndim}( c[2]-c[1] ) + L = norm(tgₘ) + t = tgₘ/L + ## Create t, n, b which are the longitudinal and two transverse unit vectors to the element (as-meshed). + ## NB: orient2, provided by the user, will define the first bending axis. + orient2/= norm(orient2) + n = orient2 - t*dot(orient2,t) + nn = norm(n) + nn>1e-3 || muscadeerror("Provide a 'orient' input that is not nearly parallel to the element") + n /= nn + b = cross(t,n) + rₘ = SMatrix{ndim,ndim}(t...,n...,b...) + ## Tangential vector and node coordinates in the local coordinate system + tgₑ = SVector{ndim}(L,0,0) + ## Weight associated to each Gauss point + dL = SVector{ngp}(L/2*(18-sqrt(30))/36,L/2*(18+sqrt(30))/36 ,L/2*(18+sqrt(30))/36,L/2*(18-sqrt(30))/36 ) + ## Location ζgp of the Gauss points for a unit-length beam element, with nodes at ζnod=±1/2. + ζgp = SVector{ngp }(-1/2*sqrt(3/7+2/7*sqrt(6/5)),-1/2*sqrt(3/7-2/7*sqrt(6/5)), +1/2*sqrt(3/7-2/7*sqrt(6/5)),+1/2*sqrt(3/7+2/7*sqrt(6/5))) + ζnod = SVector{nXnod}(-1/2 ,1/2 ) + shapes = (yₐ.(ζgp), yᵤ.(ζgp), yᵥ.(ζgp)*L, κₐ.(ζgp)/L, κᵤ.(ζgp)/L^2, κᵥ.(ζgp)/L) + return EulerBeam3D{typeof(mat),Udof}(cₘ,rₘ,ζgp,ζnod,tgₘ,tgₑ,shapes...,L,dL,mat) +end; + +# Define now the residual function for the EulerBeam3D element. +@espy function Muscade.residual(o::EulerBeam3D{Mat,Udof}, X,U,A,t,SP,dbg) where{Mat,Udof} + ## Compute all quantities at Gauss point, their time derivatives, including intrinsic roll rate and acceleration + P,ND,X_ = motion(X) + gp_,ε_,vₛₘ_,rₛₘ_,vₗ₂_,_,_ = kinematics{:direct}(o,X_) + gpval,☼ε,☼rₛₘ = motion⁻¹{P,ND}(gp_,ε_,rₛₘ_) + vᵢ = intrinsicrotationrates(rₛₘ) + ## compute all Jacobians of the above quantities with respect to X₀ + X₀ = ∂0(X) + TX₀ = revariate{P+1}(X₀) + Tgp,Tε,Tvₛₘ,_,_,_,_ = kinematics{:direct}(o,TX₀) # the crux + gp∂X₀,ε∂X₀,vₛₘ∂X₀ = chainrule_Jacobian((Tgp,Tε,Tvₛₘ),X₀) + ## Quadrature loop: compute resultants + gp = ntuple(ngp) do igp + ☼x,☼κgp = gpval[igp].x, gpval[igp].κ + x∂X₀,κ∂X₀ = gp∂X₀[igp].x, gp∂X₀[igp].κ + fᵢ,mᵢ,fₑ,mₑ = ☼resultants(o.mat,ε,κgp,x,rₛₘ,vᵢ,t) # call the "resultant" function to compute loads (local coordinates) from strains/curvatures/etc. using material properties. Note that output is dual of input. + fₑ = Udof ? fₑ-∂0(U) : fₑ # U is per unit length + R_ = (fᵢ ∘₀ ε∂X₀ + mᵢ ∘₁ κ∂X₀ + fₑ ∘₁ x∂X₀ + mₑ ∘₁ vₛₘ∂X₀) * o.dL[igp] # Contribution to the local nodal load of this Gauss point [nXdof] = scalar*[nXdof] + [ndim]⋅[ndim,nXdof] + [ndim]⋅[ndim,nXdof] + @named(R_) + end + R = sum(gpᵢ.R_ for gpᵢ∈gp) + ♢κ = motion⁻¹{P,ND}(SVector(vₗ₂_[1],vₗ₂_[3],-vₗ₂_[2])).*(2/o.L) + return R,noFB +end; + +struct kinematics{Mode} end # Mode: +function kinematics{Mode}(o::EulerBeam3D,X₀) where{Mode} + cₘ,rₘ,tgₘ,tgₑ,ζnod,ζgp,L = o.cₘ,o.rₘ,o.tgₘ,o.tgₑ,o.ζnod,o.ζgp,o.L # As-meshed element coordinates and describing tangential vector + vₛₘ,rₛₘ,uₗ₂,vₗ₂,cₛₘ = corotated{Mode}(o,X₀) + ε = √((uₗ₂[1]+L/2)^2+uₗ₂[2]^2+uₗ₂[3]^2)*2/L - 1. + gp = ntuple(ngp) do igp # gp[igp].κ, gp[igp].x + yₐ,yᵤ,yᵥ,κₐ,κᵤ,κᵥ = o.yₐ[igp],o.yᵤ[igp],o.yᵥ[igp],o.κₐ[igp],o.κᵤ[igp],o.κᵥ[igp] + κ = SVector( κₐ*vₗ₂[1], κᵤ*uₗ₂[2]+κᵥ*vₗ₂[3], κᵤ*uₗ₂[3]-κᵥ*vₗ₂[2]) + y = SVector(yₐ*uₗ₂[1] , yᵤ*uₗ₂[2]+yᵥ*vₗ₂[3], yᵤ*uₗ₂[3]-yᵥ*vₗ₂[2]) + x = rₛₘ∘₁(tgₑ*ζgp[igp]+y)+cₛₘ + (κ=κ,x=x) + end + return gp,ε,vₛₘ,rₛₘ,vₗ₂,uₗ₂,cₛₘ +end + +struct corotated{Mode} end +function corotated{Mode}(o::EulerBeam3D,X₀) where{Mode} + cₘ,rₘ,tgₘ,tgₑ,ζnod,ζgp = o.cₘ,o.rₘ,o.tgₘ,o.tgₑ,o.ζnod,o.ζgp # As-meshed element coordinates and describing tangential vector + uᵧ₁,vᵧ₁,uᵧ₂,vᵧ₂ = vec3(X₀,1:3), vec3(X₀,4:6), vec3(X₀,7:9), vec3(X₀,10:12) + Δvᵧ,rₛₘ,vₛₘ = apply{Mode}((vᵧ₁=vᵧ₁,vᵧ₂=vᵧ₂)) do v + rₛ₁ = apply{Mode}(Rodrigues,v.vᵧ₁) + rₛ₂ = apply{Mode}(Rodrigues,v.vᵧ₂) + Δvᵧ_ = 0.5*Rodrigues⁻¹(rₛ₂ ∘₁ rₛ₁') + rₛₘ_ = apply{Mode}(Rodrigues,Δvᵧ_) ∘₁ rₛ₁ ∘₁ o.rₘ + vₛₘ_ = Rodrigues⁻¹(rₛₘ_) + return Δvᵧ_,rₛₘ_,vₛₘ_ + end + cₛ = 0.5*(uᵧ₁+uᵧ₂) + uₗ₂ = rₛₘ' ∘₁ (uᵧ₂+tgₘ*ζnod[2]-cₛ)-tgₑ*ζnod[2] #Local displacement of node 2 + vₗ₂ = rₛₘ' ∘₁ Δvᵧ + return vₛₘ,rₛₘ,uₗ₂,vₗ₂,cₛ+cₘ +end; + +# The following functions explain how the beam element should be drawn +using GLMakie +""" + +Drawing a `EulerBeam3D`. + + draw!(axis,state) + + draw!(axis,state;EulerBeam3D=(;style=:shape)) + + α = 2π*(0:19)/20 + circle = 0.1*[cos.(α) sin.(α)]' + draw!(axis,state;EulerBeam3D=(;style=:solid,section = circle)) + +`style=:shape` shows the deformed neutral axis of the element. It has optional arguments `frame=true` +(draws the element's corotated frame of reference) +and `nseg=10` (number of points to show the deflected shape of each element). + +`style=:solid` shows the deformed shape of the element. It requires the input `section=...` to be given +a matrix of size `(2,nsec)` describing `nsec` points around the cross section of the element (no need to close +the circumference by repeating the first point at the end). It has optional arguments `nseg=10` as above, `marking=true` +to draw a longitudinal marking and `solid_color=:yellow`. + +Other optional arguments (and their default values) are +- `Udof` (`true` iff element has Udofs) wether to draw U-forces. +- `draw_frame = false` wether to draw the local reference frame of each element +- `draw_marking = true` wether to draw "longitudinal marking" along the element. Will only draw if style=:solid. +- `nseg = 1` number of segments to display the shape of a deformed element +- `solid_color = :yellow` color of the surface if `style=:solid` +- `line_color = :black` color of the line if `style=:sshape` +- `Uscale = 1.` How many meter is a Newton per meter? +""" +function Muscade.allocate_drawing(axis,o::AbstractVector{EulerBeam3D{Tmat,Udof}};kwargs...) where{Tmat,Udof} + args = default{:EulerBeam3D }(kwargs,(;) ) + section = default{:section }(args,zeros(2,0)) + nsec = size(section,2) + opt = (default(args,(style=:shape,draw_frame=false,draw_marking=true,nseg=1, + solid_color=:yellow,line_color=:black,Uscale=1.,Udof=Udof))..., + nel = length(o) , + nsec = nsec , + section = section , + markrad = nsec==0 ? 0. : 1.01*maximum(section[1,:]) + ) + opt.style==:solid && nsec<2 && muscadeerror("An section description must be provided for 'solid' plot") + nel_shape = opt.style==:shape ? opt.nel : 0 + nel_shape_frame = opt.draw_frame ? nel_shape : 0 + nel_solid = opt.style==:solid ? opt.nel : 0 + nel_solid_marking = opt.draw_marking ? nel_solid : 0 + nel_udof = opt.Udof ? opt.nel : 0 + + mut=( + node = 𝕣2(undef,3,3*opt.nel) , + shape_x = 𝕣2(undef,3,(opt.nseg+2)*nel_shape) , + shape_frame = 𝕣2(undef,3,3*3*nel_shape_frame) , # idim, point-point-lift, ivec, iel + solid_vertex = 𝕣2(undef,3,opt.nsec*(opt.nseg+1)*nel_solid) , + solid_face = 𝕫2(undef,2*opt.nsec* opt.nseg *nel_solid,3), + solid_mark = 𝕣2(undef,3,(opt.nseg+2)*nel_solid_marking) , + ucrest = 𝕣2(undef,3,5*nel_udof) , # idim, 6point-lift,iel + ) + return mut,opt +end + +function Muscade.update_drawing(axis,o::AbstractVector{EulerBeam3D{Tmat,Udof}},oldmut,opt, Λ,X,U,A,t,SP,dbg) where{Tmat,Udof} + mut = oldmut + X₀ = ∂0(X) + U₀ = ∂0(U) + it1,ir1,it2,ir2 = SVector{3}(1:3),SVector{3}(4:6),SVector{3}(7:9),SVector{3}(10:12) + nsec = size(opt.section,2) + node = reshape(mut.node,(3,3,opt.nel)) + for (iel,oᵢ) = enumerate(o) + node[:,1,iel] = oᵢ.cₘ - oᵢ.tgₘ/2 + X₀[it1,iel] + node[:,2,iel] = oᵢ.cₘ + oᵢ.tgₘ/2 + X₀[it2,iel] + node[:,3,iel].= NaN + end + + if opt.style==:shape + ζ = range(-1/2,1/2,opt.nseg+1) + if opt.draw_frame shape_frame = reshape(mut.shape_frame ,(3,3,3 ,opt.nel)) end + if opt.Udof ucrest = reshape(mut.ucrest, (3,5 ,opt.nel)) end + shape_x = reshape(mut.shape_x ,(3,opt.nseg+2,opt.nel)) + for (iel,oᵢ) = enumerate(o) + cₘ,rₘ,tgₘ,tgₑ,ζnod,ζgp,L = oᵢ.cₘ,oᵢ.rₘ,oᵢ.tgₘ,oᵢ.tgₑ,oᵢ.ζnod,oᵢ.ζgp,oᵢ.L + X₀ₑ = view(X₀,:,iel) + vₛₘ,rₛₘ,uₗ₂,vₗ₂,cₛₘ = corotated{:direct}(oᵢ,X₀ₑ) + if opt.draw_frame + for ivec = 1:3 + shape_frame[:,1,ivec,iel] = cₛₘ + shape_frame[:,2,ivec,iel] = cₛₘ + oᵢ.L/3*rₛₘ[:,ivec] + shape_frame[:,3,ivec,iel].= NaN + end + end + if opt.Udof + ucrest[:,1,iel] = node[:,1,iel] + ucrest[:,2,iel] = node[:,1,iel] + rₛₘ ∘₁ view(U₀,:,iel) * opt.Uscale + ucrest[:,3,iel] = node[:,2,iel] + rₛₘ ∘₁ view(U₀,:,iel) * opt.Uscale + ucrest[:,4,iel] = node[:,2,iel] + ucrest[:,5,iel].= NaN + end + for (i,ζᵢ) ∈ enumerate(ζ) + y = SVector(yₐ(ζᵢ)*uₗ₂[1] , yᵤ(ζᵢ)*uₗ₂[2]+L*yᵥ(ζᵢ)*vₗ₂[3], yᵤ(ζᵢ)*uₗ₂[3]-L*yᵥ(ζᵢ)*vₗ₂[2]) + shape_x[:,i ,iel] = rₛₘ∘₁(tgₑ*ζᵢ+y)+cₛₘ + shape_x[:,opt.nseg+2,iel].= NaN + end + end + elseif opt.style==:solid + ζ = range(-1/2,1/2,opt.nseg+1) + idx(iel,iseg,isec) = Muscade.mod_onebased(isec,opt.nsec)+opt.nsec*(iseg-1+(opt.nseg+1)*(iel-1)) # 1st index into rvertex + if opt.Udof ucrest = reshape(mut.ucrest ,(3,5 ,opt.nel)) end + if opt.draw_marking solid_mark = reshape(mut.solid_mark ,(3,opt.nseg+2 ,opt.nel)) end + solid_face = reshape(mut.solid_face ,(2,opt.nsec, opt.nseg ,opt.nel,3)) + solid_vertex = reshape(mut.solid_vertex,(3,opt.nsec, opt.nseg+1 ,opt.nel)) + for (iel,oᵢ) = enumerate(o) + cₘ,rₘ,tgₘ,tgₑ,ζnod,ζgp,L = oᵢ.cₘ,oᵢ.rₘ,oᵢ.tgₘ,oᵢ.tgₑ,oᵢ.ζnod,oᵢ.ζgp,oᵢ.L + X₀ₑ = view(X₀,:,iel) + vₛₘ,rₛₘ,uₗ₂,vₗ₂,cₛₘ = corotated{:direct}(oᵢ,X₀ₑ) + vᵧ₁,vᵧ₂ = vec3(X₀ₑ,4:6), vec3(X₀ₑ,10:12) + rₛ₁ = Rodrigues(vᵧ₁) + rₛ₂ = Rodrigues(vᵧ₂) + if opt.Udof + ucrest[:,1,iel] = node[:,1,iel] + ucrest[:,2,iel] = node[:,1,iel] + rₛₘ ∘₁ view(U₀,:,iel) * opt.Uscale + ucrest[:,3,iel] = node[:,2,iel] + rₛₘ ∘₁ view(U₀,:,iel) * opt.Uscale + ucrest[:,4,iel] = node[:,2,iel] + ucrest[:,5,iel].= NaN + end + Δv = Rodrigues⁻¹(rₛ₂ ∘₁ rₛ₁')/opt.nseg + for (iseg,ζᵢ) ∈ enumerate(ζ) # actualy iterating over nseg+1 segment boundaries + y = SVector(yₐ(ζᵢ)*uₗ₂[1] , yᵤ(ζᵢ)*uₗ₂[2]+L*yᵥ(ζᵢ)*vₗ₂[3], yᵤ(ζᵢ)*uₗ₂[3]-L*yᵥ(ζᵢ)*vₗ₂[2]) # interpolate + xn = rₛₘ∘₁(tgₑ*ζᵢ+y)+cₛₘ # point on neutral axis + r = Rodrigues((iseg-1)*Δv) ∘₁ rₛ₁ ∘₁ rₘ + if opt.draw_marking + solid_mark[:, iseg ,iel] = xn .+ r[:,2]*opt.markrad + solid_mark[:,opt.nseg+2,iel].= NaN + end + for isec = 1:opt.nsec + solid_vertex[:,isec,iseg,iel] = xn .+ r[:,2]*opt.section[1,isec] + r[:,3]*opt.section[2,isec] + if iseg≤opt.nseg + i1,i2,i3,i4 = idx(iel,iseg,isec),idx(iel,iseg ,isec+1),idx(iel,iseg+1,isec ),idx(iel,iseg+1,isec+1) + solid_face[1,isec,iseg,iel,:] = SVector(i1,i2,i4) + solid_face[2,isec,iseg,iel,:] = SVector(i1,i4,i3) + end + end + end + end + end + return mut +end + +function Muscade.display_drawing!(axis,::Type{EulerBeam3D{Tmat,Udof}},obs,opt) where{Tmat,Udof} + scatter!( axis, obs.node ,color = opt.line_color , marker=:circle,markersize=3) + opt.style==:shape && lines!( axis, obs.shape_x ,color = opt.line_color ,linewidth=.5 ) + opt.style==:shape && opt.draw_frame && lines!( axis, obs.shape_frame ,color = :grey ,linewidth=.5 ) + opt.style==:solid && mesh!( axis, obs.solid_vertex, obs.solid_face ,color = opt.solid_color ) + opt.style==:solid && opt.draw_marking && lines!( axis, obs.solid_mark ,color = opt.line_color ) + opt.Udof && lines!( axis, obs.ucrest ,color = :red ,linewidth=.5 ) +end \ No newline at end of file diff --git a/docs/src/DecayAnalysis.md b/docs/src/DecayAnalysis.md new file mode 100644 index 0000000..b07854b --- /dev/null +++ b/docs/src/DecayAnalysis.md @@ -0,0 +1,281 @@ +```@meta +EditURL = "../../examples/DecayAnalysis.jl" +``` + +# Estimating model parameters + +We estimate the mass and damping matrices of a coupled linear oscillator (floater moving in the surge, sway and yaw) based on a decay tests +In the following, we define the necessary element and residual function describing the dynamic behaviour of the floater. + +NB: In two places in this script, `solve` is called with optional `verbose=false`, because this script is part of the generation +of `Muscade`'s online documentation. Setting `verbose=true` would be more relevant in other contexts. + +````@example DecayAnalysis +using Muscade,StaticArrays,Interpolations,GLMakie + +fold(x::SVector{6}) = SMatrix{3,3}( x[1],x[2],x[3], + x[2],x[4],x[5], + x[3],x[5],x[6]) + +const floatermotion = (:surge,:sway,:yaw) +const idx = (:11,:12,:16,:22,:26,:66) + +struct FloaterOnCalmWater <: AbstractElement + K :: SMatrix{3,3,𝕣} + C :: SMatrix{3,3,𝕣} + M :: SMatrix{3,3,𝕣} +end +FloaterOnCalmWater(nod::Vector{Node};K,C,M ) = FloaterOnCalmWater(K,C,M) + +Muscade.no_second_order(::Type{<:FloaterOnCalmWater}) = Val(true) + +Muscade.doflist(::Type{<:FloaterOnCalmWater}) = (inod = (ntuple(i-> 1,3)...,ntuple(i-> 1,3)...,ntuple(i-> 1,6)..., ntuple(i-> 1,6)... ), + class = (ntuple(i->:X,3)...,ntuple(i->:U,3)...,ntuple(i->:A,6)..., ntuple(i->:A,6)... ), + field = (floatermotion... ,floatermotion... ,ntuple(i->Symbol(:M,idx[i]),6)..., ntuple(i->Symbol(:C,idx[i]),6)...)) + +@espy function Muscade.residual(o::FloaterOnCalmWater, X,U,A,t,SP,dbg) + x,x′,x″ = ∂0(X),∂1(X),∂2(X) + ☼u = ∂0(U) + a = exp10.(A) + ☼r₂ = (o.M.*fold(a[@SVector [i for i∈1:6 ]]))∘₁x″ + ☼r₁ = (o.C.*fold(a[@SVector [i for i∈7:12]]))∘₁x′ + ☼r₀ = o.K∘₁x + return r₀+r₁+r₂-u, noFB +end +```` + +This is a tailor-made cost element where the cost is made dependent on the iteration number. In practice, this is used to first solve an XU problem (costs on A are prohibitive) before solving the actual XUA problem. + +````@example DecayAnalysis +struct SingleDecayAcost{Field,Tcost,Tcostargs} <: AbstractElement + cost :: Tcost + costargs :: Tcostargs + fac :: 𝕣1 +end + +SingleDecayAcost(nod::Vector{Node};field::Symbol,fac,cost::Functor ,costargs=()) = SingleDecayAcost{field,typeof(cost),typeof(costargs)}(cost,costargs,fac) +Muscade.doflist(::Type{<:SingleDecayAcost{Field,Tcost,Tcostargs}}) where{Field,Tcost,Tcostargs} = (inod=(1,),class=(:A,),field=(Field,)) +@espy function Muscade.lagrangian(o::SingleDecayAcost,Λ,X,U,A,t,SP,dbg) + iter = min(length(o.fac),default{:iter}(SP,length(o.fac))) + ☼cost = o.cost( A[1] ,o.costargs...) + return cost*o.fac[iter],noFB +end +```` + +Define stiffness, damping and mass matrix for the true system + +````@example DecayAnalysis +K = fold(SVector{6}([1.0, 0.0, 0.0, 1.0, 0.0, 1.0])) +C = fold(SVector{6}([0.25, -0.2, 0.1, 0.15, -0.15, 0.03])) +M = fold(SVector{6}([1.0, 0.1, 0.2, 0.5, 0.1, 0.1])); +nothing #hide +```` + +Solve direct problem + +````@example DecayAnalysis +model = Model(:MooredFloater) +n1 = addnode!(model,𝕣[0,0,0]) +e1 = addelement!(model,FloaterOnCalmWater,[n1]; K,C,M) +initialstate = initialize!(model;time=0.) +initialstate = setdof!(initialstate,[2.0]; field=:surge, nodID=[n1], order=0) +initialstate = setdof!(initialstate,[1.0]; field=:sway, nodID=[n1], order=0) +initialstate = setdof!(initialstate,[-5.0]; field=:yaw, nodID=[n1], order=0) +T = 0.1 *(1:250) +state = solve(SweepX{2}; initialstate,time= T,verbose=false); +surge = [s.X[1][1] for s∈state] +sway = [s.X[1][2] for s∈state] +yaw = [s.X[1][3] for s∈state]; +nothing #hide +```` + +Create fake measurements + +````@example DecayAnalysis +surgeMeas = surge + .05 * randn(length(T)) +swayMeas = sway + .05 * randn(length(T)) +yawMeas = yaw + .1 * randn(length(T)); +nothing #hide +```` + +Create intial guesses for M and C + +````@example DecayAnalysis +Cguess = fold(SVector{6}([0.1, -0.1, 0.1, 0.1, -0.1, 0.1])) +Mguess = fold(SVector{6}([1.0, 1.0, 1.0, 1.0, 1.0, 1.0])); +nothing #hide +```` + +Create XUA model + +````@example DecayAnalysis +modelXUA = Model(:MooredFloater) +n1 = addnode!(modelXUA,𝕣[0,0,0]) +e1 = addelement!(modelXUA,FloaterOnCalmWater,[n1]; K,C=Cguess,M=Mguess); +nothing #hide +```` + +Assign costs to unknown forces + +````@example DecayAnalysis +Quu = @SVector [0.05 ^-2 for i=1:3 ] +@functor with(Quu) cost1(u,t,i) = 0.5*Quu[i]*u^2 +e2 = [addelement!(modelXUA,SingleDofCost ,[n1]; class=:U,field=f , cost=cost1,costargs=(i,)) for (i,f)∈enumerate(floatermotion)]; +nothing #hide +```` + +Assign costs to variations of model parameters (wrt guess). + +````@example DecayAnalysis +fac = [256,128,64,32,16,8,4,2,1] +QCaa = @SVector [.1 ^-2 for i=1:6 ] +@functor with(QCaa,T) cost2(a,i) = 0.5*QCaa[i]/length(T)*a^2 +e3 = [addelement!(modelXUA,SingleDecayAcost ,[n1]; field=f,fac, cost=cost2,costargs=(i,)) for (i,f)∈enumerate((:C11,:C12,:C16,:C22,:C26,:C66))] +QMaa = @SVector [.1 ^-2 for i=1:6 ] +@functor with(QMaa,T) cost3(a,i) = 0.5*QMaa[i]/length(T)*a^2 +e4 = [addelement!(modelXUA,SingleDecayAcost ,[n1]; field=f,fac, cost=cost3,costargs=(i,)) for (i,f)∈enumerate((:M11,:M12,:M16,:M22,:M26,:M66))]; +nothing #hide +```` + +Assign costs to measurement errors + +````@example DecayAnalysis +surgeInt = linear_interpolation(T, surgeMeas) +swayInt = linear_interpolation(T, swayMeas) +yawInt = linear_interpolation(T, yawMeas) +@functor with(surgeInt) devSurge(surge,t) = 1e-1 ^-2 * (surge-surgeInt(t))^2 +@functor with(swayInt ) devSway(sway,t) = 1e-1 ^-2 * (sway-swayInt(t))^2 +@functor with(yawInt ) devYaw(yaw,t) = 1e-1 ^-2 * (yaw-yawInt(t))^2 +e5 = addelement!(modelXUA,SingleDofCost,[n1];class=:X,field=:surge, cost=devSurge) +e6 = addelement!(modelXUA,SingleDofCost,[n1];class=:X,field=:sway, cost=devSway) +e7 = addelement!(modelXUA,SingleDofCost,[n1];class=:X,field=:yaw, cost=devYaw); + + + + + +#Solve inverse problem +initialstateXUA = initialize!(modelXUA;time=0.) +stateXUA = solve(DirectXUA{2,0,1};initialstate=[initialstateXUA],time=[T], + maxiter=100,saveiter=true,verbose=false, + maxΔx=1e-5,maxΔλ=Inf,maxΔu=1e-5,maxΔa=1e-5); +nothing #hide +```` + +Fetch and display estimated model parameters + +````@example DecayAnalysis +lastIter = findlastassigned(stateXUA); niter = lastIter; iexp=1; +Mest = Mguess .* fold(exp10.(SVector{6}(stateXUA[niter][iexp][1].A[1:6 ]))) +Cest = Cguess .* fold(exp10.(SVector{6}(stateXUA[niter][iexp][1].A[7:12]))); +nothing #hide +```` + +Fetch response and loads + +````@example DecayAnalysis +surgeRec = [s.X[1][1] for s∈stateXUA[niter][iexp]] +swayRec = [s.X[1][2] for s∈stateXUA[niter][iexp]] +yawRec = [s.X[1][3] for s∈stateXUA[niter][iexp]] +surgeExtF = [s.U[1][1] for s∈stateXUA[niter][iexp]] +swayExtF = [s.U[1][2] for s∈stateXUA[niter][iexp]] +yawExtF = [s.U[1][3] for s∈stateXUA[niter][iexp]] +req = @request r₂,r₁,r₀ +loads = getresult(stateXUA[niter][iexp],req,[e1]) +inertiaLoads = [loads[i][:r₂] for i∈1:length(T)] +dampingLoads = [loads[i][:r₁] for i∈1:length(T)] +stiffnessLoads = [loads[i][:r₀] for i∈1:length(T)]; +nothing #hide +```` + +Create a figure with the results + +````@example DecayAnalysis +fig = Figure(size = (2000,1000)); +nothing #hide +```` + +Display response + +````@example DecayAnalysis +ax=Axis(fig[1,1], ylabel="Surge", yminorgridvisible = true,xminorgridvisible = true) +scatter!(fig[1,1],T,surgeMeas, color=RGBf(.8, .8, .8), label=L"\text{Measurements}") +lines!(fig[1,1],T,surge, color=:black, linestyle=:dash, label=L"\text{Exact direct solution } (M,C,K)") +lines!(fig[1,1],T,surgeRec, color=:black, label=L"\text{Inverse solution } (\hat{M},\hat{C},K)") +ylims!(ax,minimum(surgeMeas),maximum(surgeMeas)) +ax.title="Displacements [m,deg]" +axislegend() + +ax=Axis(fig[2,1], ylabel="Sway", yminorgridvisible = true,xminorgridvisible = true) +scatter!(fig[2,1],T,swayMeas, color=RGBf(.8, .8, .8)) +lines!(fig[2,1],T,sway, color=:black, linestyle=:dash) +ylims!(ax,-1,2) +lines!(fig[2,1],T,swayRec, color=:black) +ylims!(ax,minimum(swayMeas),maximum(swayMeas)) + +ax=Axis(fig[3,1], ylabel="Yaw", yminorgridvisible = true,xminorgridvisible = true,xlabel="Time [s]") +scatter!(fig[3,1],T,yawMeas, color=RGBf(.8, .8, .8)) +lines!(fig[3,1],T,yaw, color=:black, linestyle=:dash) +lines!(fig[3,1],T,yawRec, color=:black) +ylims!(ax,minimum(yawMeas),maximum(yawMeas)) +```` + +Display loads + +````@example DecayAnalysis +ax=Axis(fig[1,2], yminorgridvisible = true,xminorgridvisible = true) +lines!(fig[1,2],T,-[inertiaLoads[i][1] for i∈1:length(T)], color=:red, label=L"\text{Inertia } (\hat{M})") +lines!(fig[1,2],T,-[dampingLoads[i][1] for i∈1:length(T)], color=:blue, label=L"\text{Damping } (\hat{C})") +lines!(fig[1,2],T,-[stiffnessLoads[i][1] for i∈1:length(T)],color=:green, label=L"\text{Stiffness } (K)") +lines!(fig[1,2],T,surgeExtF, color=:black, label=L"\text{Unknown}") +ax.title="Loads [N, Nm]" +ylims!(ax,-2,2) +axislegend() + +ax=Axis(fig[2,2], yminorgridvisible = true,xminorgridvisible = true) +lines!(fig[2,2],T,-[inertiaLoads[i][2] for i∈1:length(T)], color=:red ) +lines!(fig[2,2],T,-[dampingLoads[i][2] for i∈1:length(T)], color=:blue ) +lines!(fig[2,2],T,-[stiffnessLoads[i][2] for i∈1:length(T)], color=:green) +lines!(fig[2,2],T,swayExtF, color=:black) +ylims!(ax,-2,3) + +ax=Axis(fig[3,2], yminorgridvisible = true,xminorgridvisible = true,xlabel="Time [s]") +lines!(fig[3,2],T,-[inertiaLoads[i][3] for i∈1:length(T)], color=:red ) +lines!(fig[3,2],T,-[dampingLoads[i][3] for i∈1:length(T)], color=:blue) +lines!(fig[3,2],T,-[stiffnessLoads[i][3] for i∈1:length(T)], color=:green) +lines!(fig[3,2],T,yawExtF, color=:black) +ylims!(ax,-4,4) + +ax=Axis(fig[1,3], limits=(nothing,nothing,0,2), + xticks = (1:6, [L"\hat{M}_{11}/M_{11}", L"\hat{M}_{12}/M_{12}", L"\hat{M}_{16}/M_{16}",L"\hat{C}_{11}/C_{11}", L"\hat{C}_{12}/C_{12}", L"\hat{C}_{16}/C_{16}"]), + yminorgridvisible = true,xminorgridvisible = true) +barplot!(ax,[1,2,3,4,5,6], [Mest[1,1]/M[1,1],Mest[1,2]/M[1,2],Mest[1,3]/M[1,3], Cest[1,1]/C[1,1],Cest[1,2]/C[1,2],Cest[1,3]/C[1,3]], + bar_labels = :y,color=:white,strokewidth=1,strokecolor=[:red,:red,:red,:blue,:blue,:blue]) +ax.title="Estimated model parameters compared to exact solution (iteration " * string(niter) * "/" * string(lastIter) * ")" + +ax=Axis(fig[2,3], limits=(nothing,nothing,0,2), + xticks = (1:6, [L"\hat{M}_{21}/M_{21}", L"\hat{M}_{22}/M_{22}", L"\hat{M}_{26}/M_{26}",L"\hat{C}_{21}/C_{21}", L"\hat{C}_{22}/C_{22}", L"\hat{C}_{26}/C_{26}"]), + yminorgridvisible = true,xminorgridvisible = true) +barplot!(ax,[1,2,3,4,5,6], [Mest[2,1]/M[2,1],Mest[2,2]/M[2,2],Mest[2,3]/M[2,3], Cest[2,1]/C[2,1],Cest[2,2]/C[2,2],Cest[2,3]/C[2,3]], + bar_labels = :y,color=:white,strokewidth=1,strokecolor=[:red,:red,:red,:blue,:blue,:blue]) + +ax=Axis(fig[3,3], limits=(nothing,nothing,0,2), + xticks = (1:6, [L"\hat{M}_{61}/M_{61}", L"\hat{M}_{62}/M_{62}", L"\hat{M}_{66}/M_{66}",L"\hat{C}_{61}/C_{61}", L"\hat{C}_{62}/C_{62}", L"\hat{C}_{66}/C_{66}"]), + yminorgridvisible = true,xminorgridvisible = true) +barplot!(ax,[1,2,3,4,5,6], [Mest[3,1]/M[3,1],Mest[3,2]/M[3,2],Mest[3,3]/M[3,3], Cest[3,1]/C[3,1],Cest[3,2]/C[3,2],Cest[3,3]/C[3,3]], + bar_labels = :y,color=:white,strokewidth=1,strokecolor=[:red,:red,:red,:blue,:blue,:blue]) + +currentDir = @__DIR__ +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","decay.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"decay.png")),fig) +end +```` + +![Result](assets/decay.png) + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/DryFriction.md b/docs/src/DryFriction.md new file mode 100644 index 0000000..d49df55 --- /dev/null +++ b/docs/src/DryFriction.md @@ -0,0 +1,89 @@ +```@meta +EditURL = "../../examples/DryFriction.jl" +``` + +````@example DryFriction +using Muscade,StaticArrays +```` + +# DryFriction + +Besides providing a general example of how to implement an element in `Muscade`, this element illustrates how to implement hysteretic behaviour +(here: dry friction, but this would also apply to plasticity) without internal variables, since these make problems with `U`dofs intractable. +This is handled by making what would otherwise have been an internal variable into an additional `X`dof. In this case, the element's second degree +of freedom is the friction force. +## Type +The struct contains the values provided (indirectly) by the user. Note `Fx` and `Ff` which are type parameters: these will be `Symbol`s that +represent the field of the `X`dof on which to apply the friction, and a `X`dof to represent the friction force + +````@example DryFriction +struct DryFriction{Fx,Ff} <: AbstractElement + fric :: 𝕣 + x′scale :: 𝕣 + k⁻¹ :: 𝕣 # ∈ [0,∞[, so k ∈ ]0,∞] +end +```` + +## Constructor +We provide a constructor, which will be called by `AddElement!`. The keyword arguments can, or must be given by the user when calling `AddElement!`, and are passed on +to the constructor. Note that the constructor is type unstable: it gets `fields` and `fieldf` as values and uses them as type parameters. This is not deemed to be a problem for +the constructor (type instability in `residual` would be another matter) + +````@example DryFriction +DryFriction(nod::Vector{Node};fieldx::Symbol,fieldf::Symbol=:f, + friction::𝕣,Δx::𝕣=0.,x′scale::𝕣=1.) = + DryFriction{fieldx,fieldf}(friction,x′scale,Δx/friction) +```` + +## `residual` +The `residual` function is prepended by `@espy` to facilitate the extraction of element-results . +Variables `old` and `new` are prepended by `☼` (`\sun`), to tell `@espy` the values of these variables can +be requested using [`getresult`](@ref). + +The full name `Muscade.residual` must be used, because we are adding a method to a function defined in the `module` `Muscade`. + +````@example DryFriction +@espy function Muscade.residual(o::DryFriction, X,U,A, t,SP,dbg) + x,x′,f,f′ = ∂0(X)[1],∂1(X)[1], ∂0(X)[2], ∂1(X)[2] + stick = (x′-o.k⁻¹*f′)/o.x′scale + slip = abs(f)/o.fric -1 + ☼old = abs(slip)o.fric ☼new = :slip + elseif old==:slip && f*x′<0 ☼new = :stick + else ☼new = old + end + return (new==:slip ? SVector(f,slip) : SVector(f,stick)), noFB +end; +nothing #hide +```` + +In the above `f` (a force) uses the "nod-on-el" convention (force exterted by the element's node on the element), so the sign is unusual. + +If the element was in stick the previous iteration, the variable `stick = (x′-o.k⁻¹*f′)/o.x′scale` will be very close to zero. +Similarly, if the element was in slip the previous iteration, the variable `slip = abs(f)/o.fric -1` will be very close to zero. +`old` is then either `:stick` or `:slip` depending on which of the two above is smallest. + +The `if` construct can be read as follows: +- If we were in stick at previous iteration but `|f|` from previous iteration exceeds `o.fric`, we slip in this iteration. +- If we were in slip at previous iteration but force from previous iteration is in the wrong direction, we stick in this iteration. +- Otherwise, no change. + +The function returns a 2-vector of residuals (corresponding to the two `X`dofs). The first residual `f` is the friction force applied to the dof `fieldx`. + +The second residual corresponds to dof `fieldf`. Importantly, this later dof must not be shared with any other element, so the that solver will set +the second residual to zero in this iteration: depending on `new`, this will enforce `slip==0` or `stick==0`. The relevant condition will +be enforced exactly (to rounding errors) because both conditions are linear in `X`. + +## `doflist` +Another function that must be overloaded, in order to tell `Muscade` what dofs the element provides. Note that this is a function of the element *type*, not +of the element *variable*: elements of the same concrete type must have the same dofs. + +````@example DryFriction +Muscade.doflist( ::Type{DryFriction{Fx,Ff}}) where{Fx,Ff} = + (inod =(1 ,1 ), class=(:X,:X), field=(Fx,Ff)) +```` + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/DynamicBeamAnalysis.md b/docs/src/DynamicBeamAnalysis.md new file mode 100644 index 0000000..3214fe6 --- /dev/null +++ b/docs/src/DynamicBeamAnalysis.md @@ -0,0 +1,246 @@ +```@meta +EditURL = "../../examples/DynamicBeamAnalysis.jl" +``` + +# Dynamic analysis of a beam + +We perform the dynamic analysis of a steel catenary riser (SCR) subject to forced top motions. We compare the results against the corresponding SIMA/RIFLEX example case. The riser consists of three segments with two different cross-sections. + +NB: In two places in this script, `solve` is called with optional `verbose=false`, because this script is part of the generation +of `Muscade`'s online documentation. Setting `verbose=true` would be more relevant in other contexts. + +````@example DynamicBeamAnalysis +using Muscade, StaticArrays, GLMakie, Muscade.Toolbox, Interpolations, CSV, DataFrames +```` + +Read RIFLEX motions, tensions, bending moment and curvatures time series (verification data) + +````@example DynamicBeamAnalysis +file_path = "SCR.csv" +df = CSV.read(file_path, DataFrame; delim=' ') +dynMotionX = vcat(0,df[:,"Displacement in x - direction"].-df[1,"Displacement in x - direction"],df[end,"Displacement in x - direction"]-df[1,"Displacement in x - direction"]) +dynMotionZ = vcat(0,df[:,"Displacement in z - direction"].-df[1,"Displacement in z - direction"],df[end,"Displacement in z - direction"]-df[1,"Displacement in z - direction"]) +dynMotionT = vcat(-10.,df[:,"x"],df[end,"x"]+100) +xMotion = linear_interpolation(dynMotionT,-dynMotionX) +zMotion = linear_interpolation(dynMotionT,dynMotionZ); +nothing #hide +```` + +Some physical constants + +````@example DynamicBeamAnalysis +const g=9.81 +const ρ=1025.; +nothing #hide +```` + +Normalized gravity field, ramping up during static analysis + +````@example DynamicBeamAnalysis +@functor with() ramping_gravityField(t)= clutch(t,-9.,-5.,0.,1.,1) * SVector(0.,0.,-1.); +nothing #hide +```` + +Define parameters for cross-section 1 + +````@example DynamicBeamAnalysis +x1_D = 0.429 # Outer diameter [m] +x1_t = 0.022 # Thickness [m] +x1_steelArea = (x1_D-x1_t)*π*x1_t # Pipe wall area [m^2] +x1_innerArea = (x1_D-2*x1_t)^2*π/4 # Inner area [m^2] +x1_ρₛ = 7850. # Steel denstiy [kg/m^3] +x1_ρᵢ = 200. # Internal fluid density [kg/m^3] +x1_EA = 5.823e9 # Axial stiffness [N] +x1_EI₂ = 1.209e8 # Bending stiffness [Nm/(1/m)] +x1_EI₃ = 1.209e8 # Bending stiffness [Nm/(1/m)] +x1_GJ = 9.347e7 # Torsional stiffness [Nm/(rad/m)] +x1_μ = x1_steelArea*x1_ρₛ + x1_innerArea*x1_ρᵢ # Mass per unit length [m] +x1_ι₁ = 0.2053^2*x1_steelArea*x1_ρₛ # Moment of inertia about x-axis per unit length [kgm²/m] +x1_w = x1_μ*g - π*x1_D^2/4*ρ*g # Weight per unit length [N/m] +x1_Dh = 0.459 # Hydrodynamic outer diameter [m] +x1_Ca₂ = 1.0 * ρ * π*x1_Dh^2/4 # Transverse added mass coefficients [N/m/(m/s^2)] +x1_Ca₃ = 1.0 * ρ * π*x1_Dh^2/4 +x1_Cq₂ = 1.0 * 0.5 * ρ * x1_Dh # Transverse drag coefficients [N/m/(m/s)^2] +x1_Cq₃ = 1.0 * 0.5 * ρ * x1_Dh +x1_mat = BeamCrossSection(EA=x1_EA, EI₂=x1_EI₂, EI₃=x1_EI₃, GJ=x1_GJ, μ=x1_μ, ι₁=x1_ι₁, w=x1_w, g̃=ramping_gravityField, Ca₂=x1_Ca₂, Cq₂=x1_Cq₂, Ca₃=x1_Ca₃, Cq₃=x1_Cq₃) +```` + +Define parameters for cross-section 2 + +````@example DynamicBeamAnalysis +x2_D = 0.441 # Outer diameter [m] +x2_t = 0.028 # Thickness [m] +x2_steelArea = (x2_D-x2_t)*π*x2_t # Pipe wall area [m^2] +x2_innerArea = (x2_D-2*x2_t)^2*π/4 # Inner area [m^2] +x2_ρₛ = 7850. # Steel denstiy [kg/m^3] +x2_ρᵢ = 200. # Internal fluid density [kg/m^3] +x2_EA = 7.520e9 # Axial stiffness [N] +x2_EI₂ = 1.611e8 # Bending stiffness [Nm/(1/m)] +x2_EI₃ = 1.611e8 # Bending stiffness [Nm/(1/m)] +x2_GJ = 1.245e8 # Torsional stiffness [Nm/(rad/m)] +x2_μ = x2_steelArea*x2_ρₛ + x2_innerArea*x2_ρᵢ # Mass per unit length [m] +x2_ι₁ = 0.2084^2*x2_steelArea*x2_ρₛ # Moment of inertia about x-axis per unit length [kgm²/m] +x2_w = x2_μ*g - π*x2_D^2/4*ρ*g # Weight per unit length [N/m] +x2_Dh = 0.471 # Hydrodynamic outer diameter [m] +x2_Ca₂ = 1.0 * ρ * π*x2_Dh^2/4 # Transverse added mass coefficients [N/m/(m/s^2)] +x2_Ca₃ = 1.0 * ρ * π*x2_Dh^2/4 +x2_Cq₂ = 1.0 * 0.5 * ρ * x2_Dh # Transverse drag coefficients [N/m/(m/s)^2] +x2_Cq₃ = 1.0 * 0.5 * ρ * x2_Dh +x2_mat = BeamCrossSection(EA=x2_EA, EI₂=x2_EI₂, EI₃=x2_EI₃, GJ=x2_GJ, μ=x2_μ, ι₁=x2_ι₁, w=x2_w, g̃=ramping_gravityField, Ca₂=x2_Ca₂, Cq₂=x2_Cq₂, Ca₃=x2_Ca₃, Cq₃=x2_Cq₃) +```` + +Model SCR, starting from the extremity located on seabed + +````@example DynamicBeamAnalysis +nel = [60, 30, 10] # Number of elements per segment +segLength = [300., 300., 80.] # Segment lengths +xSection = [x1_mat, x2_mat, x1_mat]; # Cross-section type +nothing #hide +```` + +Create Muscade model + +````@example DynamicBeamAnalysis +model = Model(:CatenaryRiser); +nothing #hide +```` + +Create nodes, elements representing the SCR + +````@example DynamicBeamAnalysis +topNode = addnode!(model,[0,0,0]) +nodeList, elementList, nodeCoord = MeshLine!(model, topNode, π, EulerBeam3D, xSection, segLength, nel; orient2=SVector(0.,1,0)); +nothing #hide +```` + +Fix lower extremity + +````@example DynamicBeamAnalysis +[addelement!(model,Hold,[nodeList[1][1]] ;field) for field∈[:t1,:t2,:t3,:r1]]; +nothing #hide +```` + +Loading procedure for the static analysis is as follows +1) start by elongating the line to create geometric stiffness +2) apply the weight (can be done rapidly) +3) move the end of the line to the prescribed displacement +4) contact forces + +Define the prescribed end displacements of the top extremity + +````@example DynamicBeamAnalysis +@functor with(xMotion) horizMove(x,t)= SVector( x[1]-(1.0 - clutch(t,-10.,0,0.,1.,3)*181.0 + xMotion(t)) ) +@functor with(zMotion) vertMove(x,t) = SVector( x[1]-( clutch(t,-10.,0,0.,1.,3)*303.1 + zMotion(t)) ) +addelement!(model,DofConstraint,[nodeList[3][end]]; λclass=:X, xinod=(1,),xfield=(:t1,), λinod=(1,),λfield=(:λt1,), gap=horizMove, mode=Muscade.equal) +addelement!(model,DofConstraint,[nodeList[3][end]]; λclass=:X, xinod=(1,),xfield=(:t3,), λinod=(1,),λfield=(:λt3,), gap=vertMove , mode=Muscade.equal); +nothing #hide +```` + +Contact elements for the bottom segment + +````@example DynamicBeamAnalysis +[addelement!(model,SoilContact,[nodeList[1][idxNod]],z₀=0.,Kh=1.0e3,Kv=1.0e4,Ch=0.,Cv=0.) for idxNod = 1:length(nodeList[1])]; +nothing #hide +```` + +Run the static analysis + +````@example DynamicBeamAnalysis +initialstate = initialize!(model); +staticLoadSteps = (-10:.2:0)*1. +nStaticLoadSteps = length(staticLoadSteps) +staticStates = solve(SweepX{0};initialstate,time=staticLoadSteps,verbose=false,maxΔx=1e-6,maxiter=100); +nothing #hide +```` + +Plot the static analysis sequence + +````@example DynamicBeamAnalysis +fig = Figure(size = (1000,1000)) +ax = Axis3(fig[1,1]) +for stateIdx ∈ 1:nStaticLoadSteps + draw!(ax,staticStates[stateIdx],EulerBeam3D=(;style=:shape)) +end +currentDir = @__DIR__ +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","staticEquilibriumSCR.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"staticEquilibriumSCR.png")),fig) +end +```` + +![Result](assets/staticEquilibriumSCR.png) + +Run the dynamic analysis + +````@example DynamicBeamAnalysis +dynamicLoadSteps = (0.1:0.3:300)*1.0 +nDynamicLoadSteps = length(dynamicLoadSteps) +dynamicStates = solve(SweepX{2}; + initialstate=staticStates[nStaticLoadSteps], + time=dynamicLoadSteps, + verbose=false, + maxΔx=1e-5); +nothing #hide +```` + +Retrieve axial force at top location + +````@example DynamicBeamAnalysis +req = @request gp(resultants(fᵢ)) +out = getresult(dynamicStates,req,[elementList[end]]) +Fgp1_ = [ out[idxEl].gp[1][:resultants][:fᵢ] for idxEl ∈ 1:size(out,2)]; +nothing #hide +```` + +Retrieve bending moments near touch down point + +````@example DynamicBeamAnalysis +req = @request gp(resultants(mᵢ)) +out = getresult(dynamicStates,req,[elementList[64]]) +Mgp1_ = [ out[idxEl].gp[1][:resultants][:mᵢ] for idxEl ∈ 1:size(out,2)]; +nothing #hide +```` + +Retrieve curvature near touch down point + +````@example DynamicBeamAnalysis +req = @request gp(κgp) +out = getresult(dynamicStates,req,[elementList[64]]) +κgp1_ = [ out[idxEl].gp[1].κgp[1] for idxEl ∈ 1:size(out,2)]; +nothing #hide +```` + +Plot comparison between Muscade and RIFLEX results. + +````@example DynamicBeamAnalysis +fig2 = Figure(size = (1000,1000)) +ax1 = Axis(fig2[1, 1],ylabel="Top horiz. disp. [m]") +lines!(ax1,dynamicLoadSteps,xMotion(dynamicLoadSteps)) +ax2 = Axis(fig2[2, 1],ylabel="Top vert. disp. [m]") +lines!(ax2,dynamicLoadSteps,zMotion(dynamicLoadSteps)) +ax3 = Axis(fig2[3, 1],ylabel="Axial force [N]") +lines!(ax3, dynamicLoadSteps, Fgp1_, label="Muscade") +lines!(ax3, df[:,"x"],df[:,"Axial force"], label="RIFLEX") +axislegend() +ax4 = Axis(fig2[4, 1],ylabel="Bending mom. [Nm]") +lines!(ax4, dynamicLoadSteps, getindex.(Mgp1_,3), label="Muscade") +lines!(ax4, df[:,"x"],df[:,"Mom. about local y-axis, end 1"], label="RIFLEX") +ax5 = Axis(fig2[5, 1],ylabel="Curvature [m^{-1}]") +lines!(ax5, dynamicLoadSteps, getindex.(κgp1_,3), label="Muscade") +lines!(ax5, df[:,"x"],df[:,"Curvature about local y-axis, end 1"], label="RIFLEX") +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","dynamicAnalysisSCR.png")),fig2) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"dynamicAnalysisSCR.png")),fig2) +end +```` + +Minor discrepancies might be due to the fact that loads are extracted at Gauss points in Muscade, while they are extracted at nodes in RIFLEX. + +![Result](assets/dynamicAnalysisSCR.png) + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/LICENSE.md b/docs/src/LICENSE.md new file mode 100644 index 0000000..2dc1775 --- /dev/null +++ b/docs/src/LICENSE.md @@ -0,0 +1,21 @@ +# MIT License + +Copyright (c) 2022-2026 SINTEF Ocean + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/docs/src/ModalBeamAnalysis.md b/docs/src/ModalBeamAnalysis.md new file mode 100644 index 0000000..603d46e --- /dev/null +++ b/docs/src/ModalBeamAnalysis.md @@ -0,0 +1,106 @@ +```@meta +EditURL = "../../examples/ModalBeamAnalysis.jl" +``` + +# Modal analysis of a beam + +Beam simply supported at both ends + +````@example ModalBeamAnalysis +using Muscade, StaticArrays, GLMakie +using Muscade.Toolbox + +L = 1.; # Beam length [m] +q = 0.0; # Uniform lateral load [N/m] +EI₂ = 1.; # Bending stiffness [Nm²] +EI₃ = 1.; # Bending stiffness [Nm²] +EA = 1e6; # Axial stiffness [N] +GJ = 1e6; # Torsional stiffness [Nm²] +μ = 1.; +ι₁= 1.; +nothing #hide +```` + +Create model + +````@example ModalBeamAnalysis +nel = 50 +Nnod = nel+1 +nodeCoord = hcat((0:L/nel:L),zeros(Float64,Nnod,2)) +mat = BeamCrossSection(EA=EA,EI₂=EI₂,EI₃=EI₃,GJ=GJ,μ=μ,ι₁=ι₁) +model = Model(:TestModel) +nodid = addnode!(model,nodeCoord) +mesh = hcat(nodid[1:Nnod-1],nodid[2:Nnod]) +eleid = addelement!(model,EulerBeam3D,mesh;mat=mat,orient2=SVector(0.,1.,0.)); +nothing #hide +```` + +Set boundary conditions and constraints + +````@example ModalBeamAnalysis +[addelement!(model,Hold,[nodid[1]] ;field) for field∈[:t1,:t2,:t3,:r1]] # Simply supported end 1 +[addelement!(model,Hold,[nodid[end]];field) for field∈[:t1,:t2,:t3,:r1]] # Simply supported end 2 +[addelement!(model,Hold,[nodid[nodeidx]];field=:t3) for nodeidx∈2:Nnod-1] # Enforce beam motions in one dimension to obtain planar modeshapes +@functor with(q,L,Nnod) val(t) = sin(t)*q*L/Nnod +[addelement!(model,DofLoad,[nodid[nodeidx]];field=:t2,value=val) for nodeidx=1:Nnod]; # Distributed vertical load q +nothing #hide +```` + +Static analysis + +````@example ModalBeamAnalysis +initialstate = initialize!(model); +state = solve(SweepX{0};initialstate,time=[0.]); +nothing #hide +```` + +Solve eigenvalue problem + +````@example ModalBeamAnalysis +nmod = 15 +res = solve(EigX{ℝ};state=state[1],nmod); +nothing #hide +```` + +Analytical solutions for the natural frequency of a simply supported beam +See e.g. https://roymech.org/Useful_Tables/Vibrations/Natural_Vibrations_derivation.html + +````@example ModalBeamAnalysis +fₙ(k) = √(EI₂/μ)*(k^2*π)/(2*L^2) +Φₙ(k,x) = sin.(k*π/L.*x); +nothing #hide +```` + +Display solution and comparison against analytical solution + +````@example ModalBeamAnalysis +fig = Figure(size = (2000,1000)) +axes = [Axis(fig[idxLine,1], yminorgridvisible = false,xminorgridvisible = false ) for idxLine=1:3] +axes[1].title = "Modeshapes of a simply supported beam. Muscade (" *string(nel)*" elements): markers. Analytical solution: lines. " +for idxMod=1:nmod + eigres = increment(state[1],res,[idxMod],[1]); + t2_eig = getdof(eigres;field=:t2,nodID=nodid[1:Nnod]) + δ = sign(Φₙ(idxMod,0:L/nel:L)'*t2_eig) * maximum(t2_eig) + selectAxis = axes[mod(idxMod-1,3)+1] + labelStr= "Mode "*string(idxMod)*", Muscade: "*string(round(res.ω[idxMod]/(2π),digits=3))*" Hz, Analytical: " *string(round(fₙ(idxMod),digits=3))* " Hz" + scatter!(selectAxis,(0:L/nel:L), t2_eig[:]/δ, label=labelStr ); + lines!( selectAxis,(0:L/nel:L), Φₙ(idxMod,0:L/nel:L) ); +end +for ax∈axes; + xlims!(ax,0,1); ylims!(ax, -2,2); axislegend(ax) +end + +currentDir = @__DIR__ +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","beamModes.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"beamModes.png")),fig) +end +```` + +![Result](assets/beamModes.png) + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/Rotations.jl b/docs/src/Rotations.jl new file mode 100644 index 0000000..64c5e45 --- /dev/null +++ b/docs/src/Rotations.jl @@ -0,0 +1,126 @@ +# # 3D rotations +using LinearAlgebra, StaticArrays +using Muscade: sinc1 +""" + Toolbox.scac(x) + +`scac(x) = sinc1(acos(x)),` The function can be differentiated +to the fourth order over ]-1,1] . + +See also [`Toolbox.sinc1`](@ref) +""" +function scac(x) + dx = x-1 + if abs(dx)>1e-3 + sinc1(acos(x)) + else # deliberately a long Taylor series (5th order): this function will be adiffed at least to 2nd order, up to 4th order + evalpoly(dx,(1,1/3,-2/90,0.0052911879917544626,-0.0016229317117234072,0.0005625)) + end +end + +const Mat33{R} = SMatrix{3,3,R,9} +const Vec3{R} = SVector{3,R} + +""" + Toolbox.spin(v::SVector{3}) + +Transform a rotation vector `v` into the cross product matrix `M`, such that +`M ∘₁ a = v × a`. + +See also [`Toolbox.spin⁻¹`](@ref), [`Toolbox.Rodrigues`](@ref), [`Toolbox.Rodrigues⁻¹`](@ref). +""" +spin(v::Vec3) = SMatrix{3,3}(0,v[3],-v[2],-v[3],0,v[1],v[2],-v[1],0) +""" + toolbox.spin⁻¹(M::SMatrix{3,3}) + +Transform a cross product matrix `M` into the rotation vector `v`, such that +`v × a = M ∘₁ a`. + +See also [`Toolbox.spin`](@ref), [`Toolbox.Rodrigues`](@ref), [`Toolbox.Rodrigues⁻¹`](@ref). +""" +spin⁻¹(m::Mat33) = SVector{3}(m[3,2]-m[2,3],m[1,3]-m[3,1],m[2,1]-m[1,2])/2 +""" + trace(v::SMatrix{3,3}) + +Computes the trace of a matrix. +""" +trace(m::Mat33) = m[1,1]+m[2,2]+m[3,3] +""" + Toolbox.Rodrigues⁻¹(v::SVector{3}) + +Transform a rotation matrix `M` into the rotation vector `v`, such that +`|v| < π`. Undefined for rotations of angle `π` + +See also [`Toolbox.spin`](@ref), [`Toolbox.spin⁻¹`](@ref), [`Toolbox.Rodrigues`](@ref), [`Toolbox.adjust`](@ref). +""" +Rodrigues⁻¹(m) = spin⁻¹(m)/scac((trace(m)-1.)/2.) # NB: is necessarily singular for π turn +function LinearAlgebra.norm(v::Vec3) # executes faster, COMPILES MUCH FASTER , and adiffs poorly at origin + n = sqrt(v[1]*v[1]+v[2]*v[2]+v[3]*v[3]) + if n<1e-14 + n = zero(eltype(v)) + end + return n +end + +function spin²(S::Vec3) + ab = S[2,3]*S[3,1] + ca = S[1,2]*S[2,3] + bc = S[3,1]*S[1,2] + ma² = S[3,2]*S[2,3] + mb² = S[3,1]*S[1,3] + mc² = S[2,1]*S[1,2] + return SMatrix{3,3}(mc²+mb²,ab,ca, ab,ma²+mc²,bc, ca,bc,ma²+mb²) +end + +""" + Toolbox.Rodrigues(v::SVector{3}) + +Transform a rotation vector `v` into the rotation matrix `M`. + +See also [`Toolbox.spin`](@ref), [`Toolbox.spin⁻¹`](@ref), [`Toolbox.Rodrigues⁻¹`](@ref), [`Toolbox.adjust`](@ref). +""" +@inline function Rodrigues(v::Vec3{R}) where{R} + # I + spin(v)*sinc1(norm3(v)) + spin²(v)*sinc1(norm3(v)/2)/√2 + θ = norm(v) + s₁ = sinc1(θ ) + s₂ = sinc1(θ/2)*(1/√2) + a,b,c = s₁*v[1],s₁*v[2],s₁*v[3] + vs₂ = v*s₂ + ab ,ac ,bc = vs₂[1]*vs₂[2], vs₂[1]*vs₂[3], vs₂[2]*vs₂[3] + ma²,mb²,mc² = vs₂[1]*vs₂[1], vs₂[2]*vs₂[2], vs₂[3]*vs₂[3] + return Mat33{R}(1. -mc²-mb²,ab+c,ac-b, ab-c,1. -ma²-mc²,bc+a, ac+b,bc-a,1. -ma²-mb²) +end + +""" + Toolbox.adjust(u::SVector{3},v::SVector{3}) + +Compute the matrix of the rotation with smallest angle that transforms `u` into a vector colinear with v. +Fails if |u|=0, |v|=0 or if the angle of the rotation is π. + +See also [`Toolbox.spin`](@ref), [`Toolbox.spin⁻¹`](@ref), [`Toolbox.Rodrigues`](@ref), [`Toolbox.Rodrigues⁻¹`](@ref). +""" +function adjust(u::Vec3{R},v::Vec3{R}) where{R} + u,v = normalize.((u,v)) + c,w = dot(u,v), cross(u,v) + s = norm(w) + θ = atan(s,c) + return w/sinc1(θ) +end +""" + Toolbox.intrinsicrotationrates(rₑ::NTuple{ND,SMatrix{3,3}}) where{ND} + +Transform a `NTuple` containing a rotation matrix and its extrinsic time derivatives, +into a `NTuple` containing a (zero) rotation vector and its intrinsic time derivatives. + +See also [`Toolbox.spin`](@ref), [`Toolbox.spin⁻¹`](@ref), [`Toolbox.Rodrigues`](@ref), [`Toolbox.Rodrigues⁻¹`](@ref). +""" +function intrinsicrotationrates(rₑ::NTuple{ND,SMatrix{3,3}}) where{ND} + vᵢ₀ = (SVector{3,𝕣}(0,0,0), ) + vᵢ₁ = ND<2 ? vᵢ₀ : (vᵢ₀... , spin⁻¹(∂0(rₑ)' ∘₁ ∂1(rₑ)) ) + vᵢ = ND<3 ? vᵢ₁ : (vᵢ₁... , spin⁻¹(∂1(rₑ)' ∘₁ ∂1(rₑ) + ∂0(rₑ)' ∘₁ ∂2(rₑ))) + return vᵢ +end + + + +; \ No newline at end of file diff --git a/docs/src/SCR.csv b/docs/src/SCR.csv new file mode 100644 index 0000000..511c9f1 --- /dev/null +++ b/docs/src/SCR.csv @@ -0,0 +1,3001 @@ +"x" "Curvature about local y-axis, end 1" "Mom. about local y-axis, end 1" "Axial force" "Displacement in x - direction" "Displacement in z - direction" +0.10000000149011612 0.006851674057543278 1103638.0615234375 543760.3759765625 35.35993576049805 3.0999464988708496 +0.20000000298023224 0.006851695012301207 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4.849627494812012 +300.00000447034836 0.006735431496053934 1084914.1845703125 431996.9482421875 31.949846267700195 4.7335686683654785 diff --git a/docs/src/StaticBeamAnalysis.md b/docs/src/StaticBeamAnalysis.md new file mode 100644 index 0000000..381a732 --- /dev/null +++ b/docs/src/StaticBeamAnalysis.md @@ -0,0 +1,218 @@ +```@meta +EditURL = "../../examples/StaticBeamAnalysis.jl" +``` + +# Static analysis of a beam + +Cantilever bend, with out-of-plane load leading to a three-dimensional response mobilizing axial force, bending moment and torque. +Comparison to solutions by Longva (2015) and Crisfield (1990) + +````@example StaticBeamAnalysis +using Muscade, StaticArrays, GLMakie +using Muscade.Toolbox + +R = 100.0; # Radius of the bend [m] +EI₂ = 833.33e3; # Bending stiffness [Nm²] +EI₃ = 833.33e3; # Bending stiffness [Nm²] +EA = 1e9; # Axial stiffness [N] +GJ = 705e3; # Torsional stiffness [Nm²] +nothing #hide +```` + +Create the model + +````@example StaticBeamAnalysis +nel = 8 +nnodes = nel+1 +nodeCoord = hcat( 0 .+ R*cos.(3π/2 .+ ((1:nnodes).-1)/(nnodes-1)*π/4), + 0 .+ zeros(Float64,nnodes,1), + R .+ R*sin.(3π/2 .+ ((1:nnodes).-1)/(nnodes-1)*π/4)) +mat = BeamCrossSection(EA=EA,EI₂=EI₂,EI₃=EI₃,GJ=GJ,μ=1.,ι₁=1.) +model = Model(:TestModel) +nodid = addnode!(model,nodeCoord) +mesh = hcat(nodid[1:nnodes-1],nodid[2:nnodes]) +eleid = addelement!(model,EulerBeam3D,mesh;mat=mat,orient2=SVector(0.,1.,0.)) +[addelement!(model,Hold,[nodid[1]] ;field) for field∈[:t1,:t2,:t3,:r1,:r2,:r3]]; # Clamp at one end +nothing #hide +```` + +Define the loading procedure. First 300 N then 450 and 600 N + +````@example StaticBeamAnalysis +@functor with() function load(t) + t<=1. ? load=t*300. : + t>1. && t<=2. ? load=300. +(t-1)*150. : + load=450. +(t-2)*150. +end +addelement!(model,DofLoad,[nodid[nnodes]];field=:t2,value=load); # Force along axis2 at other +nothing #hide +```` + +Run the static analysis + +````@example StaticBeamAnalysis +initialstate = initialize!(model); +loadSteps = [0.,1.,2.,3.]; +nLoadSteps = length(loadSteps) +state = solve(SweepX{0};initialstate,time=loadSteps,verbose=true,maxΔx=1e-9); +nothing #hide +```` + +Fetch and show beam profile for all load cases + +````@example StaticBeamAnalysis +x_ = [getdof(state[idxLoad];field=:t1,nodID=nodid[1:nnodes]) for idxLoad ∈ 1:nLoadSteps] +y_ = [getdof(state[idxLoad];field=:t2,nodID=nodid[1:nnodes]) for idxLoad ∈ 1:nLoadSteps] +z_ = [getdof(state[idxLoad];field=:t3,nodID=nodid[1:nnodes]) for idxLoad ∈ 1:nLoadSteps] + +fig = Figure(size = (1000,1000)) +ax = Axis3(fig[1,1],xlabel="x [m]", ylabel="y [m]", zlabel="z [m]",aspect=:equal) +clr = [:black,:blue,:green,:red] +for idxLoad ∈ 1:nLoadSteps + draw!(ax,state[idxLoad];EulerBeam3D=(;nseg=10,line_color=clr[idxLoad])) +end +xlims!(ax, 0,70); ylims!(ax, 0,60); zlims!(ax, 0,40); +currentDir = @__DIR__ +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","StaticBeamAnalysis1.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"StaticBeamAnalysis1.png")),fig) +end +```` + +![Result](assets/StaticBeamAnalysis1.png) + +Fetch and display results at the Gauss points for the last load case + +````@example StaticBeamAnalysis +idxLoad = 4; +nothing #hide +```` + +Location of the 4 Gauss points + +````@example StaticBeamAnalysis +xgp1 = (1. /nel)*( (0.5-1/2*sqrt(3/7+2/7*sqrt(6/5))) :1:nel) +xgp2 = (1. /nel)*( (0.5-1/2*sqrt(3/7-2/7*sqrt(6/5))) :1:nel) +xgp3 = (1. /nel)*( (0.5+1/2*sqrt(3/7-2/7*sqrt(6/5))) :1:nel) +xgp4 = (1. /nel)*( (0.5+1/2*sqrt(3/7+2/7*sqrt(6/5))) :1:nel) +xgps = [xgp1;xgp2;xgp3;xgp4]; +nothing #hide +```` + +Axial force + +````@example StaticBeamAnalysis +req = @request gp(resultants(fᵢ)) +out = getresult(state[idxLoad],req,eleid) +Fgp1_ = [ out[idxEl].gp[1][:resultants][:fᵢ] for idxEl ∈ 1:nel] +Fgp2_ = [ out[idxEl].gp[2][:resultants][:fᵢ] for idxEl ∈ 1:nel] +Fgp3_ = [ out[idxEl].gp[3][:resultants][:fᵢ] for idxEl ∈ 1:nel] +Fgp4_ = [ out[idxEl].gp[4][:resultants][:fᵢ] for idxEl ∈ 1:nel]; +nothing #hide +```` + +Bending moments + +````@example StaticBeamAnalysis +req = @request gp(resultants(mᵢ)) +out = getresult(state[idxLoad],req,eleid) +Mgp1_ = [ out[idxEl].gp[1][:resultants][:mᵢ] for idxEl ∈ 1:nel] +Mgp2_ = [ out[idxEl].gp[2][:resultants][:mᵢ] for idxEl ∈ 1:nel] +Mgp3_ = [ out[idxEl].gp[3][:resultants][:mᵢ] for idxEl ∈ 1:nel] +Mgp4_ = [ out[idxEl].gp[4][:resultants][:mᵢ] for idxEl ∈ 1:nel]; +nothing #hide +```` + +Strains + +````@example StaticBeamAnalysis +req = @request ε +out = getresult(state[idxLoad],req,eleid) +εgp1_ = [ out[idxEl].ε[1] for idxEl ∈ 1:nel] +εgp2_ = [ out[idxEl].ε[1] for idxEl ∈ 1:nel]; +εgp3_ = [ out[idxEl].ε[1] for idxEl ∈ 1:nel]; +εgp4_ = [ out[idxEl].ε[1] for idxEl ∈ 1:nel]; +nothing #hide +```` + +Curvatures + +````@example StaticBeamAnalysis +req = @request gp(κgp) +out = getresult(state[idxLoad],req,eleid) +κgp1_ = [ out[idxEl].gp[1].κgp[1] for idxEl ∈ 1:nel] +κgp2_ = [ out[idxEl].gp[2].κgp[1] for idxEl ∈ 1:nel] +κgp3_ = [ out[idxEl].gp[3].κgp[1] for idxEl ∈ 1:nel] +κgp4_ = [ out[idxEl].gp[4].κgp[1] for idxEl ∈ 1:nel]; +nothing #hide +```` + +Plot strains, etc. at Gauss points + +````@example StaticBeamAnalysis +fig = Figure(size = (1000,1000)) +ax=Axis(fig[1,1], ylabel="Forces F [N]", yminorgridvisible = true,xminorgridvisible = true,xticks = (0:1. /nel:1)) +scatter!(ax,xgps, [[Fgp1_[iel][1] for iel=1:nel] ; [Fgp2_[iel][1] for iel=1:nel] ; [Fgp3_[iel][1] for iel=1:nel] ; [Fgp4_[iel][1] for iel=1:nel]], label="F₁"); +ax.title = "Internal loads, elongation and curvature at last load step" +axislegend(); +ax=Axis(fig[2,1], ylabel="Strain [-]", yminorgridvisible = true,xminorgridvisible = true,xticks = (0:1. /nel:1)) +scatter!(ax,xgps, [[εgp1_[iel][1] for iel=1:nel] ; [εgp2_[iel][1] for iel=1:nel] ; [εgp3_[iel][1] for iel=1:nel] ; [εgp4_[iel][1] for iel=1:nel]], label="ε"); +ax=Axis(fig[3,1], ylabel="Moments M [Nm]", yminorgridvisible = true,xminorgridvisible = true,xticks = (0:1. /nel:1)) +scatter!(ax,xgps, [[Mgp1_[iel][1] for iel=1:nel] ; [Mgp2_[iel][1] for iel=1:nel] ; [Mgp3_[iel][1] for iel=1:nel] ; [Mgp4_[iel][1] for iel=1:nel]], label="M₁"); +scatter!(ax,xgps, [[Mgp1_[iel][2] for iel=1:nel] ; [Mgp2_[iel][2] for iel=1:nel] ; [Mgp3_[iel][2] for iel=1:nel] ; [Mgp4_[iel][2] for iel=1:nel]], label="M₂"); +scatter!(ax,xgps, [[Mgp1_[iel][3] for iel=1:nel] ; [Mgp2_[iel][3] for iel=1:nel] ; [Mgp3_[iel][3] for iel=1:nel] ; [Mgp4_[iel][3] for iel=1:nel]], label="M₃"); +axislegend() +ax=Axis(fig[4,1], ylabel="Curvature κ [m⁻¹]", yminorgridvisible = true,xminorgridvisible = true,xticks = (0:1. /nel:1),xlabel="Position along beam / beam length [-]") +scatter!(ax,xgps, [[κgp1_[iel][1] for iel=1:nel] ; [κgp2_[iel][1] for iel=1:nel] ; [κgp3_[iel][1] for iel=1:nel] ; [κgp4_[iel][1] for iel=1:nel]], label="κ₁"); +scatter!(ax,xgps, [[κgp1_[iel][2] for iel=1:nel] ; [κgp2_[iel][2] for iel=1:nel] ; [κgp3_[iel][2] for iel=1:nel] ; [κgp4_[iel][2] for iel=1:nel]], label="κ₂"); +scatter!(ax,xgps, [[κgp1_[iel][3] for iel=1:nel] ; [κgp2_[iel][3] for iel=1:nel] ; [κgp3_[iel][3] for iel=1:nel] ; [κgp4_[iel][3] for iel=1:nel]], label="κ₃"); +axislegend() +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","StaticBeamAnalysis2.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"StaticBeamAnalysis2.png")),fig) +end +```` + +![Result](assets/StaticBeamAnalysis2.png) + +Show comparison with solutions by Longva (2015) and Crisfield (1990) + +````@example StaticBeamAnalysis +# Load 300 N 450 N 600 N +# x,y,z x,y,z x,y,z +# Disp Longva 58.56, 40.47, 22.18 51.99, 48.72, 18.45 46.91, 53.64, 15.65 +# Disp Crisfield 58.53, 40.53, 22.16 51.93, 48.79, 18.43 46.84, 53.71, 15.61 +height = [ nodeCoord[end,1]+x_[2][end], 58.56, 58.53, nodeCoord[end,1]+x_[3][end], 51.99, 51.93, nodeCoord[end,1]+x_[4][end], 46.91, 46.84, + nodeCoord[end,2]+y_[2][end], 40.47, 40.53, nodeCoord[end,2]+y_[3][end], 48.72, 48.79, nodeCoord[end,2]+y_[4][end], 53.64, 53.71, + nodeCoord[end,3]+z_[2][end], 22.18, 22.16, nodeCoord[end,3]+z_[3][end], 18.45, 18.43, nodeCoord[end,3]+z_[4][end], 15.65, 15.61] +colors = [:red, :blue, :green] +tbl = (cat = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9],height, + grp = [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3]) +fig = Figure(size = (2000, 400)) +ax = Axis(fig[1,1], xticks = (1:9, ["x @ 300 N", "x @ 450 N", "x @ 600 N", "y @ 300 N", "y @ 450 N", "y @ 600 N", "z @ 300 N", "z @ 450 N", "z @ 600 N"]), + ylabel = "End node coordinate [m]") +barplot!(ax, tbl.cat, tbl.height, + dodge = tbl.grp, + color = colors[tbl.grp], + bar_labels = :y + ) +ylims!(ax, 0,70); +labels = ["Muscade", "Longva (2015)", "Crisfield (1990)"] +elements = [PolyElement(polycolor = colors[i]) for i in 1:length(labels)] +title = "Method" +Legend(fig[1,2], elements, labels, title) +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","StaticBeamAnalysis3.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"StaticBeamAnalysis3.png")),fig) +end +```` + +![Result](assets/StaticBeamAnalysis3.png) + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/TuningFork.md b/docs/src/TuningFork.md new file mode 100644 index 0000000..9588892 --- /dev/null +++ b/docs/src/TuningFork.md @@ -0,0 +1,256 @@ +```@meta +EditURL = "../../examples/TuningFork.jl" +``` + +# Tuning vibrations of a tuning fork + +Tuning forks should resonate at a desired frequency when subjected to an impulsive load. Achieving the right frequency cab be done by adding material or filing material off the prongs. +We start from a target solution establihed using SweepX. A sprious mass is then introduced, parametrized by an A-dof. We use the SweepXA and DirectXUA solvers to estimate how much mass should be removed. In SweepXA, the excitation is assume to be known, while it is estimated by DirectXUA. + +NB: In several places in this script, `solve` is called with optional `verbose=false`, because this script is part of the generation +of `Muscade`'s online documentation. Setting `verbose=true` would be more relevant in other contexts. + +````@example TuningFork +using Muscade, StaticArrays, GLMakie, Muscade.Toolbox +```` + +Dimensions of the tuning fork and number of elements + +````@example TuningFork +l₀=2.5e-2; n₀=1 # base - length and number of elements +l₁=1e-2; n₁=1 # fork - length and number of elements per side +l₂=9.5e-2; n₂=5 # prong - length and number of elements per side +b = 5e-3; # thickness of the cross-section +E = 210e9; # Young modulus +G = 79.3e9; # shear modulus +μ = 7850. *b^2; # mass per unit length, assumes steel +beamCrossSection = BeamCrossSection(EA=E*b^2, EI₂=1/12*E*b^4, EI₃=1/12*E*b^4, GJ=1/3*G*b^4, μ=μ, ι₁=1/6*μ*b^2,Cl₂=10.,Cl₃=10.); +nothing #hide +```` + +Define an A-parametrized point mass element (inducing an inertia force) + +````@example TuningFork +struct LumpedMass <: AbstractElement; m :: 𝕣; end +LumpedMass(nod::Vector{Node};m::𝕣) = LumpedMass(m) +@espy function Muscade.residual(o::LumpedMass, X,U,A, t,SP,dbg); return (o.m+A[1]).*∂2(X),noFB; end +Muscade.doflist( ::Type{LumpedMass}) = (inod =(1,1,1,2), class=(:X,:X,:X,:A), field=(:t1,:t2,:t3,:mass)); +nothing #hide +```` + +Time step and simulation length settings + +````@example TuningFork +Δt = 3e-4; +nDynamicLoadSteps = 200; +timeVec = Δt:Δt:nDynamicLoadSteps*Δt; +nothing #hide +```` + +Coordinates of the nodes. + +````@example TuningFork +XnodeCoord = vcat( + hcat((0:n₀) * l₀/n₀ , zeros(n₀+1,1) , zeros(n₀+1,1) ), # base nodes + hcat(ones(n₁,1)*l₀ , (1:n₁)*l₁/n₁ , zeros(n₁,1) ), # upper fork + hcat(ones(n₁,1)*l₀ , -(1:n₁)*l₁/n₁ , zeros(n₁,1) ), # lower fork + hcat(l₀ .+ (1:n₂)*l₂/n₂ , ones(n₂,1)*l₁ , zeros(n₂,1) ), # upper prong + hcat(l₀ .+ (1:n₂)*l₂/n₂ , -ones(n₂,1)*l₁, zeros(n₂,1) ) # lower prong +) +```` + +External excitation load at the end of the upper prong + +````@example TuningFork +@functor with() pling(t) = Muscade.FunctionFromVector([-10, 0, 10e-3, 12e-3, 10],[0,0,10,0,0])(t) +plingNode = n₀+2n₁+n₂+1; # where do we hit the tuning fork +nothing #hide +```` + +# Building the model + +````@example TuningFork +model = Model(:TuningFork); +nothing #hide +```` + +Add the nodes describing the beam model, the node that will contain the parameter to optimize + +````@example TuningFork +Xnod = addnode!(model,XnodeCoord) +Anod = addnode!(model,𝕣[]) +```` + +List of nodes that will be used to create the beam elements + +````@example TuningFork +nel = n₀+2n₁+2n₂ +mesh = vcat( + hcat(Xnod[1:n₀],Xnod[2:n₀+1]), # base + hcat(Xnod[n₀+1:n₀+n₁],Xnod[n₀+2:n₀+n₁+1]), # upper fork + [Xnod[n₀+1] Xnod[n₀+n₁+2]], hcat(Xnod[n₀+n₁+2:n₀+2n₁],Xnod[n₀+n₁+3:n₀+2n₁+1]) , # lower fork + [Xnod[n₀+n₁+1] Xnod[n₀+2n₁+2]], hcat(Xnod[n₀+2n₁+2:n₀+2n₁+n₂],Xnod[n₀+2n₁+3:n₀+2n₁+n₂+1]) , # upper prong + [Xnod[n₀+2n₁+1] Xnod[n₀+2n₁+n₂+2]], hcat(Xnod[n₀+2n₁+n₂+2:n₀+2n₁+2n₂],Xnod[n₀+2n₁+n₂+3:n₀+2n₁+2n₂+1]) # lower prong +) +```` + +In the XUA analysis we estimate the distributed load on the element close to the node where the load was applied + +````@example TuningFork +plingElement = n₀+2n₁+n₂; +nothing #hide +```` + +Add beam elements to the model, enabling Udof only for the element that is hit. + +````@example TuningFork +addelement!(model,EulerBeam3D{false}, mesh; mat=beamCrossSection,orient2=SVector(0.,0,1)) +[addelement!(model,Hold,[Xnod[1]] ;field) for field∈[:t1,:t2,:t3,:r1,:r2,:r3]]; +nothing #hide +```` + +# Create variations of the model + +````@example TuningFork +XAmodel = deepcopy(model) +XUAmodel = deepcopy(model); +nothing #hide +```` + +Add parasitic lump mass on both XA and XUA models + +````@example TuningFork +spuriousMass = 4e-3; +spuriousMassLocation = plingNode-1 +addelement!(XAmodel, LumpedMass, [Xnod[spuriousMassLocation] Anod]; m=spuriousMass) +addelement!(XUAmodel, LumpedMass, [Xnod[spuriousMassLocation] Anod]; m=spuriousMass); +nothing #hide +```` + +Add known external force to X and XA models, and an unknown force to XUA model + +````@example TuningFork +addelement!(model, DofLoad,[Xnod[plingNode]];field=:t2,value=pling) +addelement!(XAmodel, DofLoad,[Xnod[plingNode]];field=:t2,value=pling); +@functor with(σᵤₚ=0.1) UcostPling(u,t) = 0.5*(u/σᵤₚ)^2 * clutch(t,0.02,0.04,1.,100.,1) +addelement!(XUAmodel, SingleUdof,[Xnod[plingNode]];Xfield=:t2,Ufield=:t2,cost=UcostPling) +```` + +Establish target response + +````@example TuningFork +initialState = initialize!(model;time=0.); +dynamicStates = solve(SweepX{2};initialstate=initialState, time=timeVec, verbose=false) +vibTarget = getdof(dynamicStates;field=:t2,nodID=[Xnod[plingNode]]) +target = Muscade.FunctionFromVector(timeVec,vibTarget); +nothing #hide +```` + +Add costs on the deviation to target measurements, in the XA and XUA models + +````@example TuningFork +@functor with(σᵥ=1e-6,target) Xcost(x,t)=((x-target(t))/σᵥ)^2 +addelement!(XUAmodel,SingleDofCost,[Xnod[plingNode]]; class=:X, field=:t2, cost=Xcost); +addelement!(XAmodel, SingleDofCost,[Xnod[plingNode]]; class=:X, field=:t2, cost=Xcost) +```` + +Add costs on the correcting mass, in the XA and XUA models + +````@example TuningFork +@functor with(σₘ=5e-3,timeVec) Acost_(a) = 0.5*(a/σₘ)^2/length(timeVec) +addelement!(XUAmodel, SingleAcost,[Anod]; field=:mass, cost=Acost_) +addelement!(XAmodel, SingleAcost,[Anod]; field=:mass, cost=Acost_); +nothing #hide +```` + +# Run analyses + +XA model (before estimating the mass) + +````@example TuningFork +XAinitialState = initialize!(XAmodel;time=0.); +XAdynamicStates = solve(SweepX{2};initialstate=XAinitialState, time=timeVec,verbose=false); +nothing #hide +```` + +XA model (after having estimated the mass) + +````@example TuningFork +optimXAstate = solve(SweepXA{2}; initialstate=XAinitialState, time=timeVec, + maxAiter=20,maxΔa=1e-10,verbose=false); +nothing #hide +```` + +DirectXUA (slack convergence criteria and number of iterations) + +````@example TuningFork +XUAinitialState = initialize!(XUAmodel;time=0.); +optimXUAstate = solve(DirectXUA{2,0,1};initialstate=[XUAinitialState], time=[timeVec], + maxiter=15, maxΔx=5e-2,maxΔλ=Inf,maxΔu=5e-2,maxΔa=1e-4,verbose=false); +nothing #hide +```` + + Gather results for comparison + +````@example TuningFork +analysisResults = (dynamicStates, XAdynamicStates,optimXAstate,optimXUAstate[end]); +t1 = Matrix{Float64}(undef,length(analysisResults),length(timeVec)) +t2 = Matrix{Float64}(undef,length(analysisResults),length(timeVec)) +t3 = Matrix{Float64}(undef,length(analysisResults),length(timeVec)) +transmittedForce = Matrix{Float64}(undef,length(analysisResults),length(timeVec)) +req = @request gp(resultants(fᵢ)) +for idx ∈ 1:length(analysisResults) + t1[idx,:] = getdof(analysisResults[idx];field=:t1,nodID=[Xnod[plingNode]])[:] + t2[idx,:] = getdof(analysisResults[idx];field=:t2,nodID=[Xnod[plingNode]])[:] + t3[idx,:] = getdof(analysisResults[idx];field=:t3,nodID=[Xnod[plingNode]])[:] + end +excEstt2 = getdof(optimXUAstate[end];class=:U,field=:t2,nodID=[Xnod[plingNode]])[:]; +nothing #hide +```` + +Text output + +````@example TuningFork +println("Mass to remove in grams:") +print("- Estimated from DirectXUA analysis (unknown loading) : ") +println(-getdof(optimXUAstate[end];class=:A,field=:mass,nodID=[Anod])[1]*1e3) +print("- Estimated by SweepXA analysis (known loading) : ") +println(-getdof(optimXAstate[end];class=:A,field=:mass,nodID=[Anod])[1]*1e3) +print("- Expected : ") +println(spuriousMass*1e3) +```` + +Plot the axial and lateral displacement of the control node, and the estimated force. + +````@example TuningFork +fig = Figure(size = (1000,1000)) +ax1 = Axis(fig[1,1],ylabel="Axial disp. [mm]") +lines!(ax1,timeVec,t1[1,:]*1e3, label="Target", color=:black, linestyle=:dash) +lines!(ax1,timeVec,t1[2,:]*1e3, label="Detuned config.",color=:black, linestyle=:solid) +scatter!(ax1,timeVec,t1[3,:]*1e3, label="Tuned (XA)", color=:green, markersize = 10) +scatter!(ax1,timeVec,t1[4,:]*1e3, label="Tuned (XUA)", color=:red, markersize = 5) +axislegend(ax1) +ax2 = Axis(fig[2,1],ylabel="Lateral disp. [mm]") +lines!(ax2,timeVec,t2[1,:]*1e3, label="Target", color=:black, linestyle=:dash) +lines!(ax2,timeVec,t2[2,:]*1e3, label="Detuned config.",color=:black, linestyle=:solid) +scatter!(ax2,timeVec,t2[3,:]*1e3, label="Tuned (XA)", color=:green, markersize = 10) +scatter!(ax2,timeVec,t2[4,:]*1e3, label="Tuned (XUA)", color=:red, markersize = 5) +ax3 = Axis(fig[3,1],ylabel="Excitation [N]") +lines!(ax3,timeVec,pling.(timeVec), label="Actual",color=:black, linestyle=:dash) +lines!(ax3,timeVec,excEstt2, label="Estimated (XUA)",color=:red, linestyle=:solid) +axislegend(ax3) + +currentDir = @__DIR__ +if occursin("build", currentDir) + save(normpath(joinpath(currentDir,"..","src","assets","TuningFork.png")),fig) +elseif occursin("examples", currentDir) + save(normpath(joinpath(currentDir,"TuningFork.png")),fig) +end +```` + +![Result](assets/TuningFork.png) + +--- + +*This page was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).* + diff --git a/docs/src/assets/StaticBeamAnalysis1.png b/docs/src/assets/StaticBeamAnalysis1.png index e5f7e24..f564b6d 100644 Binary files a/docs/src/assets/StaticBeamAnalysis1.png and b/docs/src/assets/StaticBeamAnalysis1.png differ diff --git a/docs/src/assets/StaticBeamAnalysis2.png b/docs/src/assets/StaticBeamAnalysis2.png index 4205a35..4940c89 100644 Binary files a/docs/src/assets/StaticBeamAnalysis2.png and b/docs/src/assets/StaticBeamAnalysis2.png differ diff --git a/docs/src/assets/StaticBeamAnalysis3.png b/docs/src/assets/StaticBeamAnalysis3.png index 3e85e58..9d02355 100644 Binary files a/docs/src/assets/StaticBeamAnalysis3.png and b/docs/src/assets/StaticBeamAnalysis3.png differ diff --git a/docs/src/assets/TuningFork.png b/docs/src/assets/TuningFork.png index 977141e..e353f11 100644 Binary files a/docs/src/assets/TuningFork.png and b/docs/src/assets/TuningFork.png differ diff --git a/docs/src/assets/beamModes.png b/docs/src/assets/beamModes.png index 349889e..e0f1afa 100644 Binary files a/docs/src/assets/beamModes.png and b/docs/src/assets/beamModes.png differ diff --git a/docs/src/assets/decay.png b/docs/src/assets/decay.png index 73586f5..ec9d101 100644 Binary files a/docs/src/assets/decay.png and b/docs/src/assets/decay.png differ diff --git a/docs/src/assets/dynamicAnalysisSCR.png b/docs/src/assets/dynamicAnalysisSCR.png index f5bc93c..51f8ced 100644 Binary files a/docs/src/assets/dynamicAnalysisSCR.png and b/docs/src/assets/dynamicAnalysisSCR.png differ diff --git a/docs/src/assets/staticEquilibriumSCR.png b/docs/src/assets/staticEquilibriumSCR.png index 7a7348d..61e502a 100644 Binary files a/docs/src/assets/staticEquilibriumSCR.png and b/docs/src/assets/staticEquilibriumSCR.png differ diff --git a/examples/DecayAnalysis.jl b/examples/DecayAnalysis.jl index 78802b9..d1f59b2 100644 --- a/examples/DecayAnalysis.jl +++ b/examples/DecayAnalysis.jl @@ -11,7 +11,7 @@ using Muscade,StaticArrays,Interpolations,GLMakie fold(x::SVector{6}) = SMatrix{3,3}( x[1],x[2],x[3], x[2],x[4],x[5], x[3],x[5],x[6]) - +solve const floatermotion = (:surge,:sway,:yaw) const idx = (:11,:12,:16,:22,:26,:66) @@ -128,7 +128,7 @@ e7 = addelement!(modelXUA,SingleDofCost,[n1];class=:X,field=:yaw, #Solve inverse problem initialstateXUA = initialize!(modelXUA;time=0.) -stateXUA = solve(DirectXUA{2,0,1};initialstate=[initialstateXUA],time=[T], +stateXUA = solve(DirectXUA{2,0,1};primerstate=[initialstateXUA],time=[T], maxiter=100,saveiter=true,verbose=false, maxΔx=1e-5,maxΔλ=Inf,maxΔu=1e-5,maxΔa=1e-5); diff --git a/examples/StaticBeamAnalysis1.png b/examples/StaticBeamAnalysis1.png new file mode 100644 index 0000000..afc0390 Binary files /dev/null and b/examples/StaticBeamAnalysis1.png differ diff --git a/examples/StaticBeamAnalysis3.png b/examples/StaticBeamAnalysis3.png new file mode 100644 index 0000000..3319259 Binary files /dev/null and b/examples/StaticBeamAnalysis3.png differ diff --git a/examples/TuningFork.jl b/examples/TuningFork.jl index 2147972..d3faad5 100644 --- a/examples/TuningFork.jl +++ b/examples/TuningFork.jl @@ -111,7 +111,7 @@ optimXAstate = solve(SweepXA{2}; initialstate=XAinitialState, time=timeVec, # DirectXUA (slack convergence criteria and number of iterations) XUAinitialState = initialize!(XUAmodel;time=0.); -optimXUAstate = solve(DirectXUA{2,0,1};initialstate=[XUAinitialState], time=[timeVec], +optimXUAstate = solve(DirectXUA{2,0,1};primerstate=[XUAinitialState], time=[timeVec], maxiter=15, maxΔx=5e-2,maxΔλ=Inf,maxΔu=5e-2,maxΔa=1e-4,verbose=false); # Gather results for comparison diff --git a/src/DirectXUA.jl b/src/DirectXUA.jl index 8c6ae1d..0522150 100644 --- a/src/DirectXUA.jl +++ b/src/DirectXUA.jl @@ -428,7 +428,7 @@ A non-linear direct solver for optimisation FEM. An analysis is carried out by a call with the following syntax: ``` -initialstate = initialize!(model) +primerstate = initialize!(model) ``` The solver does not yet support interior point methods. @@ -445,13 +445,15 @@ The solver does not yet support interior point methods. # Named arguments - `dbg=(;)` a named tuple to trace the call tree (for debugging). - `verbose=true` set to false to suppress printed output (for testing). -- `silenterror=false` set to true to suppress print out of error (for testing) . -- `initialstate` an `AbstractVector` of `State`: one initial state for each experiment. - `initialstate` must be with zero time derivatives. It does not - provide initial conditions for the problem, but an initial guess - for the iterative solver. -- `time` an `AbstractVector` (of same length as `initialstate`) of `AbstractRange` - of times at which to compute the steps. Example: 0:0.1:5. +- `silenterror=false` set to true to suppress print out of error (for testing). +- `primerstate` an `AbstractVector` of `State` (1) or an `AbstractVector` of `Vector{State}` (2). `primerstate` must be with zero time derivatives. + It does not provide initial conditions for the problem, but an initial guess for the iterative solver. + As DirectXUA solves multiple experiments at once, the i-th element of `primerstate` is an initial guess for the i-th experiment. + If provided with (1), the i-th initial guess is constructed internally with replicating the i-th input `State` for each timestep of the i-th timerange (see `time`). + If provided with (2), the i-th initial guess is the input `Vector{State}`, if the length corresponds to the i-th timerange (see `time`). + By convention the Adofs for building this primer are taken from the first experiment, first state. +- `time` an `AbstractVector` (of same length as `primerstate`) of `AbstractRange` + of times at which to compute the steps, one for each experiment. Example: 0:0.1:5. - `maxiter=50` maximum number of Newton-Raphson iterations. - `maxΔλ=1e-5` convergence criteria: a norm of the scaled `Λ` increment. - `maxΔx=1e-5` convergence criteria: a norm of the scaled `X` increment. @@ -476,7 +478,7 @@ See also: [`solve`](@ref), [`initialize!`](@ref), [`SweepX`](@ref), [`FreqXU`](@ struct DirectXUA{OX,OU,IA} <: AbstractSolver end function solve(::Type{DirectXUA{OX,OU,IA}},pstate,verbose::𝕓,dbg; time::AbstractVector{AR}, - initialstate::AbstractVector{STATE}, + primerstate::Union{Nothing,AbstractVector}, maxiter::ℤ=50, maxΔλ::ℝ=1e-5,maxΔx::ℝ=1e-5,maxΔu::ℝ=1e-5,maxΔa::ℝ=1e-5, saveiter::𝔹=false, @@ -485,21 +487,35 @@ function solve(::Type{DirectXUA{OX,OU,IA}},pstate,verbose::𝕓,dbg; # Mostly constants local LU nexp,nstep,Δt = length(time),length.(time),step.(time) - length(initialstate)== nexp || muscadeerror("initialstate and time must be of the same length=number of experiments") γ = 0. nder = (1,OX+1,OU+1,IA) - model,dis = initialstate[1].model, initialstate[1].dis if IA==1 Δ², maxΔ² = 𝕣1(undef,4), [maxΔλ^2,maxΔx^2,maxΔu^2,maxΔa^2] else Δ², maxΔ² = 𝕣1(undef,3), [maxΔλ^2,maxΔx^2,maxΔu^2 ] end - + # State storage S = State{1,OX+1,OU+1,@NamedTuple{γ::Float64,iter::Int64}} state = [Vector{S}(undef,nstep[iexp]) for iexp=1:nexp] # state[iexp][istep] - s = initialstate[1] - for (iexp,initialstateᵢ) ∈ enumerate(initialstate) + if eltype(primerstate) <: Muscade.State + model,dis,Ainit = primerstate[1].model, primerstate[1].dis, primerstate[1].A + else + model,dis,Ainit = primerstate[1][1].model, primerstate[1][1].dis, primerstate[1][1].A + end + + # Test input dimension compatibility + length(primerstate)== nexp || muscadeerror("primerstate length must be equal to the number of experiments") + + for (iexp,primerstateᵢ) ∈ enumerate(primerstate) for (istep,timeᵢ) = enumerate(time[iexp]) - state[iexp][istep] = State{1,OX+1,OU+1}(timeᵢ,deepcopy(initialstateᵢ.Λ),deepcopy(initialstateᵢ.X),deepcopy(initialstateᵢ.U),s.A,(γ=0.,iter=1),s.model,s.dis) # all state[iexp][istep].A are === + if eltype(primerstate) <: Muscade.State + # Case where primerstate is an AbstractVector of State. A primer trajectory of state is created from a repetition of this state. + primer = primerstateᵢ + else + # Case where primerstate is already an AbstractVector of Vector{State}. + length(primerstateᵢ)== nstep[iexp] || muscadeerror("trajectories in `primerstate` vector must match their associated time grid in `time` vector") + primer = primerstateᵢ[istep] + end + state[iexp][istep] = State{1,OX+1,OU+1}(timeᵢ,deepcopy(primer.Λ),deepcopy(primer.X),deepcopy(primer.U),Ainit,(γ=0.,iter=1),model,dis) end end @@ -549,6 +565,4 @@ function solve(::Type{DirectXUA{OX,OU,IA}},pstate,verbose::𝕓,dbg; iter