### energy minimization problems -- is there a conjugate gradient method?

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3 months ago by
I am solving a problem in nonlinear elasticity (hyperelasticity).  It can be written as a minimization of the potential energy $\Pi = \int W(\nabla y) - b.y$.

For simpler expressions for the elastic energy density $W$, I am able to use the built-in nonlinear solver in FEniCS and it works fine:
Pi = W(x,y)*dx  - inner(b,y)*dx
dPi = derivative(Pi, y, v)
ddPi = derivative(dPi, y, dy)

solve(dPi == 0, y, bcs=bc, J=ddPi,tol=tol,M=M)​

For more complex expressions for $W$, the Newton method does not converge.  The Newton method is "wasteful" in that it doesn't use the information that this is a minimization problem.  I'm thinking of coding a simple steepest descent (gradient flow) to do this, i.e. I start with an initial guess and go downhill using artificial time-stepping.  This would be robust in finding me a solution but could also be very inefficient if I'm in a narrow valley.  Nonlinear conjugate gradient is likely the best method for this problem, but needs much more effort to code (with line searches and so on) -- at least with my level of knowledge of FEniCS.  Also I would lose the automatic mesh adaptivity.

I wonder if anyone has suggestions on a good way to solve this.  Thanks.

Community: FEniCS Project
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You might want to use the PETScSNESSolver interface instead of the usual solve(). SNES has more options for the solver. Check out the PETSc documentation as well.
written 3 months ago by Miguel
Thanks Miguel (and sorry for the late response).

This is a very useful pointer and I'm looking further into it.
written 3 months ago by K D