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shapeFunc.jl
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shapeFunc.jl
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#一个单元，前4是u和d，后5只有u。现在只求u的形函数
#Xcoor和Ycoor?
function shapeFuncQ8(node::Array{T2},element::Array{T1}) where {T1<:Int, T2<:Float64}
##1.feisoq
function feisoq(s::T2,t::T2) # s=ξ \xi t=η
#求导derivation,SymEngine
dNds::Array{T2} = [ -0.25*(1 - t)*(1 - s) - 0.25*(-1 - s - t)*(1 - t)
0.25*(1 - t)*(1 + s) + 0.25*(-1 + s - t)*(1 - t)
0.25*(1 + t)*(1 + s) + 0.25*(-1 + s + t)*(1 + t)
-0.25*(1 + t)*(1 - s) - 0.25*(-1 - s + t)*(1 + t)
0.5*(1 - t)*(1 - s) - 0.5*(1 - t)*(1 + s)
0.5*(1 - t)*(1 + t)
0.5*(1 + t)*(1 - s) - 0.5*(1 + t)*(1 + s)
-0.5*(1 - t)*(1 + t)] # ✓
dNdt::Array{T2} = [ -0.25*(1 - t)*(1 - s) - 0.25*(-1 - s - t)*(1 - s)
-0.25*(1 - t)*(1 + s) - 0.25*(-1 + s - t)*(1 + s)
0.25*(1 + t)*(1 + s) + 0.25*(-1 + s + t)*(1 + s)
0.25*(1 + t)*(1 - s) + 0.25*(-1 - s + t)*(1 - s)
-0.5*(1 - s)*(1 + s)
0.5*(1 - t)*(1 + s) - 0.5*(1 + t)*(1 + s)
0.5*(1 - s)*(1 + s)
0.5*(1 - t)*(1 - s) - 0.5*(1 + t)*(1 - s)] # ✓
return dNds, dNdt
end
##2.fejacob
function fejacob(nnel::T1,dNds::Array{T2},dNdt::Array{T2},Xcoor::Array{T2},Ycoor::Array{T2})
jacob=zeros(T2,2,2)
for g=1:nnel #单元内节点数目
jacob[1,1] += dNds[g]*Xcoor[g]
jacob[1,2] += dNds[g]*Ycoor[g]
jacob[2,1] += dNdt[g]*Xcoor[g]
jacob[2,2] += dNdt[g]*Ycoor[g]
end
return jacob
end
##3.federiv
function federiv(nnel::T1,dNds::Array{T2},dNdt::Array{T2},invjacob::Array{T2})
dNdx=zeros(T2,nnel)
dNdy=zeros(T2,nnel)
for s=1:nnel
dNdx[s]=invjacob[1,1]*dNds[s]+invjacob[1,2]*dNdt[s]
dNdy[s]=invjacob[2,1]*dNds[s]+invjacob[2,2]*dNdt[s]
end
return dNdx, dNdy
end
##4.Assemble
nel::T1=size(element,1); nnel::T1 = 8 #单元内节点数目 QX单元
Nu = zeros(T2,2*9,16*nel); Bu = zeros(T2,3*9,16*nel)
detjacob=zeros(T2,9,nel)
for iel=1:nel
nd=zeros(T1,nnel) #节点坐标,8个
Xcoor=zeros(T2,nnel)
Ycoor=zeros(T2,nnel)
for i=1:nnel
nd[i]=element[iel,i]
Xcoor[i]=node[nd[i],1]
Ycoor[i]=node[nd[i],2]
end
gauss = [-0.7746 0.7746;-0.7746 0;-0.7746 -0.7746;
0 0.7746; 0 0; 0 -0.7746;
0.7746 0.7746;0.7746 0;0.7746 -0.7746]
for i=1:9
s::T2=gauss[i,1]; t::T2=gauss[i,2]
N1::T2 = 0.25*(1-s)*(1-t)*(-1-s-t)
N2::T2 = 0.25*(1+s)*(1-t)*(-1+s-t)
N3::T2 = 0.25*(1+s)*(1+t)*(-1+s+t)
N4::T2 = 0.25*(1-s)*(1+t)*(-1-s+t)
N5::T2 = 0.5*(1-s)*(1+s)*(1-t)
N6::T2 = 0.5*(1+s)*(1-t)*(1+t)
N7::T2 = 0.5*(1+s)*(1-s)*(1+t)
N8::T2 = 0.5*(1-s)*(1-t)*(1+t) #✓
dNds,dNdt =feisoq(s,t)
jacob =fejacob(nnel,dNds,dNdt,Xcoor,Ycoor) #雅可比矩阵
detjacob[i,iel]=det(jacob)
invjacob =inv(jacob)
dNdx,dNdy =federiv(nnel,dNds,dNdt,invjacob)
Nu[2*i-1:2*i,16*(iel-1)+1:16*iel] = [[N1 0; 0 N1] [N2 0; 0 N2] [N3 0; 0 N3] [N4 0; 0 N4] [N5 0; 0 N5] [N6 0; 0 N6] [N7 0; 0 N7] [N8 0; 0 N8] ]
Bu[3*i-2:3*i,16*(iel-1)+1:16*iel] = [[dNdx[1] 0 dNdx[2] 0 dNdx[3] 0 dNdx[4] 0 dNdx[5] 0 dNdx[6] 0 dNdx[7] 0 dNdx[8] 0 ]
[0 dNdy[1] 0 dNdy[2] 0 dNdy[3] 0 dNdy[4] 0 dNdy[5] 0 dNdy[6] 0 dNdy[7] 0 dNdy[8] ]
[dNdy[1] dNdx[1] dNdy[2] dNdx[2] dNdy[3] dNdx[3] dNdy[4] dNdx[4] dNdy[5] dNdx[5] dNdy[6] dNdx[6] dNdy[7] dNdx[7] dNdy[8] dNdx[8] ] ]#ReportofLa 2-15-b
end
end
detjacob_u =deepcopy(detjacob)
return Nu,Bu,detjacob_u #d的没确定
end
#--------------------------------------------------------------------------------
function shapeFuncQ4(node::Array{T2},element::Array{T1}) where {T1<:Int, T2<:Float64}
##1.feisoq
function feisoq(s::T2,t::T2)
dNds::Array{T2} = [-(1-t)/4,(1-t)/4,(1+t)/4,-(1+t)/4]
dNdt::Array{T2} = [-(1-s)/4,-(1+s)/4,(1+s)/4,(1-s)/4]
return dNds, dNdt
end
##2.fejacob
function fejacob(nnel::T1,dNds::Array{T2},dNdt::Array{T2},Xcoor::Array{T2},Ycoor::Array{T2})
jacob=zeros(T2,2,2)
for g=1:nnel
jacob[1,1]=jacob[1,1]+dNds[g]*Xcoor[g]
jacob[1,2]=jacob[1,2]+dNds[g]*Ycoor[g]
jacob[2,1]=jacob[2,1]+dNdt[g]*Xcoor[g]
jacob[2,2]=jacob[2,2]+dNdt[g]*Ycoor[g]
end
return jacob
end
##3.federiv
function federiv(nnel::T1,dNds::Array{T2},dNdt::Array{T2},invjacob::Array{T2})
dNdx=zeros(T2,nnel)
dNdy=zeros(T2,nnel)
for s=1:nnel
dNdx[s]=invjacob[1,1]*dNds[s]+invjacob[1,2]*dNdt[s]
dNdy[s]=invjacob[2,1]*dNds[s]+invjacob[2,2]*dNdt[s]
end
return dNdx, dNdy
end
nel::T1=size(element,1); nnel::T1 = 4
Nd = zeros(T2,9,4*nel); Bd = zeros(T2,2*9,4*nel)
Ndp = zeros(T2,16,9*nel); Bdp = zeros(T2,16,9*nel)#d^2
detjacob=zeros(T2,9,nel)
for iel=1:nel
nd=zeros(T1,nnel) #节点坐标
Xcoor=zeros(T2,nnel)
Ycoor=zeros(T2,nnel)
for i=1:nnel
nd[i]=element[iel,i]
Xcoor[i]=node[nd[i],1]
Ycoor[i]=node[nd[i],2]
end
gauss = [-0.7746 0.7746;-0.7746 0;-0.7746 -0.7746;
0 0.7746; 0 0; 0 -0.7746;
0.7746 0.7746;0.7746 0;0.7746 -0.7746]
for i=1:9 #gauss
s::T2=gauss[i,1]; t::T2=gauss[i,2]
N1::T2=0.25*(1-s)*(1-t); N2::T2=0.25*(1+s)*(1-t)
N3::T2=0.25*(1+s)*(1+t); N4::T2=0.25*(1-s)*(1+t)
Nd[i,4*(iel-1)+1:4*iel]=[N1 N2 N3 N4]
dNds,dNdt =feisoq(s,t)
jacob =fejacob(nnel,dNds,dNdt,Xcoor,Ycoor) #雅可比矩阵
detjacob[i,iel]=det(jacob)
invjacob =inv(jacob)
dNdx,dNdy =federiv(nnel,dNds,dNdt,invjacob)
Bd[2*i-1:2*i,4*(iel-1)+1:4*iel] = [[dNdx[1]; dNdy[1]] [dNdx[2]; dNdy[2]] [dNdx[3]; dNdy[3]] [dNdx[4]; dNdy[4]]]
end
Ndp1 = Nd[1,4*(iel-1)+1:4*iel]*Nd[1,4*(iel-1)+1:4*iel]'#4*4
Ndp2 = Nd[2,4*(iel-1)+1:4*iel]*Nd[2,4*(iel-1)+1:4*iel]'
Ndp3 = Nd[3,4*(iel-1)+1:4*iel]*Nd[3,4*(iel-1)+1:4*iel]'
Ndp4 = Nd[4,4*(iel-1)+1:4*iel]*Nd[4,4*(iel-1)+1:4*iel]'
Ndp5 = Nd[5,4*(iel-1)+1:4*iel]*Nd[5,4*(iel-1)+1:4*iel]'
Ndp6 = Nd[6,4*(iel-1)+1:4*iel]*Nd[6,4*(iel-1)+1:4*iel]'
Ndp7 = Nd[7,4*(iel-1)+1:4*iel]*Nd[7,4*(iel-1)+1:4*iel]'
Ndp8 = Nd[8,4*(iel-1)+1:4*iel]*Nd[8,4*(iel-1)+1:4*iel]'
Ndp9 = Nd[9,4*(iel-1)+1:4*iel]*Nd[9,4*(iel-1)+1:4*iel]'
Ndp[:,9*(iel-1)+1:9*iel] = [Ndp1[:] Ndp2[:] Ndp3[:] Ndp4[:] Ndp5[:] Ndp6[:] Ndp7[:] Ndp8[:] Ndp9[:]] #16*9
Bdp1 = Bd[1:2,4*(iel-1)+1:4*iel]'*Bd[1:2,4*(iel-1)+1:4*iel]
Bdp2 = Bd[3:4,4*(iel-1)+1:4*iel]'*Bd[3:4,4*(iel-1)+1:4*iel]
Bdp3 = Bd[5:6,4*(iel-1)+1:4*iel]'*Bd[5:6,4*(iel-1)+1:4*iel]
Bdp4 = Bd[7:8,4*(iel-1)+1:4*iel]'*Bd[7:8,4*(iel-1)+1:4*iel]
Bdp5 = Bd[9:10,4*(iel-1)+1:4*iel]'*Bd[9:10,4*(iel-1)+1:4*iel]
Bdp6 = Bd[11:12,4*(iel-1)+1:4*iel]'*Bd[11:12,4*(iel-1)+1:4*iel]
Bdp7 = Bd[13:14,4*(iel-1)+1:4*iel]'*Bd[13:14,4*(iel-1)+1:4*iel]
Bdp8 = Bd[15:16,4*(iel-1)+1:4*iel]'*Bd[15:16,4*(iel-1)+1:4*iel]
Bdp9 = Bd[17:18,4*(iel-1)+1:4*iel]'*Bd[17:18,4*(iel-1)+1:4*iel]
Bdp[:,9*(iel-1)+1:9*iel] = [Bdp1[:] Bdp2[:] Bdp3[:] Bdp4[:] Bdp5[:] Bdp6[:] Bdp7[:] Bdp8[:] Bdp9[:]]
end
detjacob_d =deepcopy(detjacob)
return Nd,Ndp,Bd,Bdp,detjacob_d
end
#--------------------------------------------------------------------------------
#5.自由度指标
function indices_fields()
nel=size(element,1)
edofMat=zeros(Int,nel,2*8)
for iel=1:nel
edofMat[iel,1]=2*element[iel,1]-1
edofMat[iel,2]=2*element[iel,1]
edofMat[iel,3]=2*element[iel,2]-1
edofMat[iel,4]=2*element[iel,2]
edofMat[iel,5]=2*element[iel,3]-1
edofMat[iel,6]=2*element[iel,3]
edofMat[iel,7]=2*element[iel,4]-1
edofMat[iel,8]=2*element[iel,4]
edofMat[iel,9]=2*element[iel,5]-1
edofMat[iel,10]=2*element[iel,5]
edofMat[iel,11]=2*element[iel,6]-1
edofMat[iel,12]=2*element[iel,6]
edofMat[iel,13]=2*element[iel,7]-1
edofMat[iel,14]=2*element[iel,7]
edofMat[iel,15]=2*element[iel,8]-1
edofMat[iel,16]=2*element[iel,8]
end
iKu::Array{Int} = reshape(kron(edofMat,ones(16,1))',256*nel)
jKu::Array{Int} = reshape(kron(edofMat,ones(1,16))',256*nel)
#存储单元的d自由度指标
dedofMat = element[:,1:4]
iKd::Array{Int} = reshape(kron(dedofMat,ones(4,1))',16*nel)
jKd::Array{Int} = reshape(kron(dedofMat,ones(1,4))',16*nel)
iFd::Array{Int} = reshape(dedofMat',4*nel)
jFd::Array{Int} = ones(4*nel)
return iKu,jKu,iKd,jKd,iFd,jFd,edofMat,dedofMat
end
@info "Formulating the shape functions and indices takes"
@time begin
Nu,Bu,detjacob_u = shapeFuncQ8(node,element)
Nd,Ndp,Bd,Bdp,detjacob_d = shapeFuncQ4(node,element)
iKu,jKu,iKd,jKd,iFd,jFd,edofMat,dedofMat = indices_fields()
end
#------------------------------------------------------------------------------------
#= ##Q9形函数
dNds::Array{T2} = [-0.25*s*t*(1 - t) + 0.25*t*(1 - t)*(1 - s)
-0.25*s*t*(1 - t) - 0.25*t*(1 - t)*(1 + s)
0.25*s*t*(1 + t) + 0.25*t*(1 + t)*(1 + s)
0.25*s*t*(1 + t) - 0.25*t*(1 + t)*(1 - s)
-0.5*t*(1 - t)*(1 - s) + 0.5*t*(1 - t)*(1 + s)
0.5*s*(1 - t)*(1 + t) + 0.5*(1 - t)*(1 + t)*(1 + s)
0.5*t*(1 + t)*(1 - s) - 0.5*t*(1 + t)*(1 + s)
0.5*s*(1 - t)*(1 + t) - 0.5*(1 - t)*(1 + t)*(1 - s)
-2*s*(1 - t^2)] # ✓
dNdt::Array{T2} = [-0.25*s*t*(1 - s) + 0.25*s*(1 - t)*(1 - s)
0.25*s*t*(1 + s) - 0.25*s*(1 - t)*(1 + s)
0.25*s*t*(1 + s) + 0.25*s*(1 + t)*(1 + s)
-0.25*s*t*(1 - s) - 0.25*s*(1 + t)*(1 - s)
0.5*t*(1 - s)*(1 + s) - 0.5*(1 - t)*(1 - s)*(1 + s)
0.5*s*(1 - t)*(1 + s) - 0.5*s*(1 + t)*(1 + s)
0.5*t*(1 - s)*(1 + s) + 0.5*(1 + t)*(1 - s)*(1 + s)
-0.5*s*(1 - t)*(1 - s) + 0.5*s*(1 + t)*(1 - s)
-2*t*(1 - s^2)] # ✓
N1::T2 = 0.25*s*(1-s)*t*(1-t)
N2::T2 = -0.25*s*(1+s)*t*(1-t)
N3::T2 = 0.25*s*(1+s)*t*(1+t)
N4::T2 = -0.25*s*(1-s)*t*(1+t)
N5::T2 = -0.5*(1-s)*(1+s)*t*(1-t)
N6::T2 = 0.5*s*(1+s)*(1-t)*(1+t)
N7::T2 = 0.5*(1-s)*(1+s)*t*(1+t)
N8::T2 = -0.5*s*(1-s)*(1-t)*(1+t)
N9::T2 = (1-s^2)*(1-t^2) #✓
Nu[2*i-1:2*i,18*(iel-1)+1:18*iel] = [[N1 0; 0 N1] [N2 0; 0 N2] [N3 0; 0 N3] [N4 0; 0 N4] [N5 0; 0 N5] [N6 0; 0 N6] [N7 0; 0 N7] [N8 0; 0 N8] [N9 0; 0 N9]]
Bu[3*i-2:3*i,18*(iel-1)+1:18*iel] = [[dNdx[1] 0 dNdx[2] 0 dNdx[3] 0 dNdx[4] 0 dNdx[5] 0 dNdx[6] 0 dNdx[7] 0 dNdx[8] 0 dNdx[9] 0]
[0 dNdy[1] 0 dNdy[2] 0 dNdy[3] 0 dNdy[4] 0 dNdy[5] 0 dNdy[6] 0 dNdy[7] 0 dNdy[8] 0 dNdy[9] ]
[dNdy[1] dNdx[1] dNdy[2] dNdx[2] dNdy[3] dNdx[3] dNdy[4] dNdx[4] dNdy[5] dNdx[5] dNdy[6] dNdx[6] dNdy[7] dNdx[7] dNdy[8] dNdx[8] dNdy[9] dNdx[9] ] ]#ReportofLa 2-15-b
=#
#------------------------------------------------------------------------------------