The given conditions are stiffness matrix,destination function and x from
Ax=b,x is movement ,b is force,A is global stiffness matrix.
J(x)=||x||norm-1
This is a questionable choice because dJ/dx is not defined at x=0.
There is one equation:
dJ(x,θ)/dθ=∂J(x,θ)/∂θ+∂J(x,θ)/∂x
*∂x/∂θ=∂J(x,θ)/∂θ+∂J(x,θ)/∂x*[A^-1(θ))(-∂A(θ)/∂θ)x],because b has no relation
with density.
I don't know how to get [A^-1(θ))(-∂A(θ)/∂θ)x],A is matrix with nxn,if θ is
mx1 vector,∂A(θ)/∂θ is nxnxm matrix.
∂A(θ)/∂θ)x is an nxm matrix. You want to apply A^{-1} to it. The way you do
that is to call the result
Y = A^{-1} [∂A(θ)/∂θ)x]
and then each column of Y, call it y_k (k=1...m) is computed as
y_k = A^{-1} {[∂A(θ)/∂θ)x]_k}
which requires solving a linear system
A y_k = {[∂A(θ)/∂θ)x]_k}
You never need A^{-1} explicitly.
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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