Incremental-Iterative Nonlinear FEA with deformation-dependent loads

[Olumide]

Hi Sofa Community,

I am looking to solve a nonlinear problem that requires the incremental-iterative technique as described in the famous video lectures by Klaus-Jurgen Bathe. In addition, the loading is also deformation dependent i.e. the direction of the loading changes with the deformation.

Basically, the incremental-iterative technique involves iteratively (1) solving for the incremental displacement under a given load, (2) based on the incremental displacement, updating the force corresponding to the work done by internal stresses until the (3) “out-of-balance” virtual work becomes zero.

From what I can see the static solver is vaguely similar. Can anyone confirm whether static solver is able to handle incremental-iterative iterative problems, as described above, specially under deformation dependent loads.

Regards,

– Olumide

[jnbrunet]

Hey @olumide,

Yes, the static solver is in fact a Newton-Raphson iterative solver. It can therefore solve any deformation dependent forces, either linear or non-linear with respect to the displacement u(x).

You will have however to implement this load force f(u) manually by creating a new SOFA ForceField“. Moreover, you will have to override three methods of the force field:

addForce(f, x, v): This function fills up the f force “vector”, which is in fact a nx3 matrix containing n nodal force vectors (1×3) for a mesh having n nodes. Here, the parameter x is a nx3 “vector” containing the current positions of the nodes, and “v” contains the nodal velocities.

addKToMatrix(A, k): This function adds the jacobian matrix J of your load force to the matrix A (times a scalar value k, i.e. A += k*J). It therefore computes the derivative of f(u) with respect to the displacement vector field u evaluated at the current displacement u_i: J(u_i) = dF(u_i) / du, where i is the current Newton iteration number.

Finally, addDForce(df, dx) simply computes df = J*dx where againJ is the same jacobian matrix computed in addKToMatrix.

J-N

[Olumide]

Thanks @jnbrunet. I suppose this means that the problem is much too sophisticated to be modeled with an XML scn file.

BTW, the problem that I hope to solve is characterized by multiple non-boundary loads, i.e. acting on nodal points in the interior of the body. I’ve only encountered boundary forces in literature but I assume multiple non-boundary loads will not require special treatment.

[jnbrunet]

Hey @olumide,

Boundary and non-boundary terms are treated the same way in SOFA: it is up to the component to assemble a force vector and its jacobian (in the case the force depends upon the deformation), and feed them to the ODE solver. If your force was constant, then you could have used the ConstantForceField component to give the nodal forces directly in your XML file. However, since yours isn’t constant, you need to provide its jacobian matrix and a way to update the force at each time step (or load increment).

J-N

[Olumide]

Thanks @jnbrunet. Finally, Is Newton-Raphson (NR) the only iterative solver that sofa supports? I can’t see any others e.g. arc length, BFGS, line search etc. Do you know if there any plans to introduce some of these solvers, or is sufficient for soft tissue deformation problems? (Much of the literature on nonlinear incremental-iterative methods that I’ve come across are biased towards structural mechanics and often use the other techniques.)

[jnbrunet]

Yeah, for the moment only a (full) Newton-Raphson is implemented, which means that the system matrix is assembled and inverted at each Newton steps.

There is work being done right now to unify the NR solver for all ODE solvers (static and dynamic). Once this is finished, I’m pretty sure a line search and BFGS could follow closely.

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