How to use arc-length method in MOOSE

[lcontinuum]

Hello,

I'm trying to figure out how to use the arc-length method within MOOSE. But I cannot find any example input file which uses this method.
The method is provided by PETSc as "Newton with Arc Length Continuation", SNESType: "SNESNEWTONAL", Options Name: "newtonal" and was quite recently added to PETSc.

As a toy example I want to solve a snap through problem with one truss element. See the picture below for the setup.

The (quasi-static) force-displacement curve of this problem looks like the blue curve in the graph below. In a force controlled setup ("FunctionNeumannBC") an arc-length method is necessary to follow the blue curve after the first maximum. If the usual Newton solver is used, then one gets the red dashed curve. At the first maximum the Newton solver "jumps" over the unloading path directly to the force-displacement curve after the snap through.

I tried in the input file to activate "Newton with Arc Length Continuation" by

petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -snes_type'
petsc_options_value = ' lu       mumps                         newtonal'

But I get a lengthy error message which ends with No tangent load function or rhs vector has been set.

My input file is

truss_2d_arc_length.i.txt

Any help appreciated.

lcontinuum

Time Step 1, time = 0.1, dt = 0.1
[0]PETSC ERROR: --------------------- Error Message --------------------------------------------------------------
[0]PETSC ERROR: Object is in wrong state
[0]PETSC ERROR: No tangent load function or rhs vector has been set
[0]PETSC ERROR: WARNING! There are unused option(s) set! Could be the program crashed before usage or a spelling mistake, etc!
[0]PETSC ERROR:   Option left: name:-i value: truss_2d_arc_length.i source: command line
[0]PETSC ERROR: See https://petsc.org/release/faq/ for trouble shooting.
[0]PETSC ERROR: Petsc Development GIT revision: unknown  GIT Date: unknown
[0]PETSC ERROR: ../../../pf_frac-opt with 1 MPI process(es) and PETSC_ARCH  on leibniz by user Fri Feb 21 11:03:48 2025
[0]PETSC ERROR: Configure options: --with-64-bit-indices --with-cxx-dialect=C++17 --with-debugging=no --with-fortran-bindings=0 --with-mpi=1 --with-openmp=1 --with-strict-petscerrorcode=1 --with-shared-libraries=1 --with-sowing=0 --download-fblaslapack=1 --download-hpddm=1 --download-hypre=1 --download-metis=1 --download-mumps=1 --download-ptscotch=1 --download-parmetis=1 --download-scalapack=1 --download-slepc=1 --download-strumpack=1 --download-superlu_dist=1 --with-hdf5-dir=/home/user/Local/install/miniforge/envs/moose --with-make-np=16 --COPTFLAGS=-O3 --CXXOPTFLAGS=-O3 --FOPTFLAGS=-O3 --with-x=0 --with-ssl=0 --with-mpi-dir=/home/user/Local/install/miniforge/envs/moose AR=/home/user/Local/install/miniforge/envs/moose/bin/x86_64-conda-linux-gnu-ar RANLIB=/home/user/Local/install/miniforge/envs/moose/bin/x86_64-conda-linux-gnu-ranlib CFLAGS="-march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/user/Local/install/miniforge/envs/moose/include -fdebug-prefix-map=/data/civet2/build/_env/conda-bld/moose-mpi_1735973043613/work=/usr/local/src/conda/moose-mpi-base-2024.12.23 -fdebug-prefix-map=/home/user/Local/install/miniforge/envs/moose=/usr/local/src/conda-prefix " CXXFLAGS="-fvisibility-inlines-hidden -fmessage-length=0 -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/user/Local/install/miniforge/envs/moose/include -fdebug-prefix-map=/data/civet2/build/_env/conda-bld/moose-mpi_1735973043613/work=/usr/local/src/conda/moose-mpi-base-2024.12.23 -fdebug-prefix-map=/home/user/Local/install/miniforge/envs/moose=/usr/local/src/conda-prefix  " CPPFLAGS="-DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/user/Local/install/miniforge/envs/moose/include " FFLAGS="-march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/user/Local/install/miniforge/envs/moose/include -fdebug-prefix-map=/data/civet2/build/_env/conda-bld/moose-mpi_1735973043613/work=/usr/local/src/conda/moose-mpi-base-2024.12.23 -fdebug-prefix-map=/home/user/Local/install/miniforge/envs/moose=/usr/local/src/conda-prefix " FCFLAGS="-march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/user/Local/install/miniforge/envs/moose/include -fdebug-prefix-map=/data/civet2/build/_env/conda-bld/moose-mpi_1735973043613/work=/usr/local/src/conda/moose-mpi-base-2024.12.23 -fdebug-prefix-map=/home/user/Local/install/miniforge/envs/moose=/usr/local/src/conda-prefix " LDFLAGS="-Wl,-O2 -Wl,--sort-common -Wl,--as-needed -Wl,-z,relro -Wl,-z,now -Wl,--disable-new-dtags -Wl,--gc-sections -Wl,--allow-shlib-undefined -Wl,-rpath,/home/user/Local/install/miniforge/envs/moose/lib -Wl,-rpath-link,/home/user/Local/install/miniforge/envs/moose/lib -L/home/user/Local/install/miniforge/envs/moose/lib " --prefix=/home/user/Local/install/miniforge/envs/moose/petsc
[0]PETSC ERROR: #1 SNESNewtonALComputeFunction_NEWTONAL() at /data/civet2/build/_env/conda-bld/moose-petsc_1735974488891/work/src/snes/impls/al/al.c:221
[0]PETSC ERROR: #2 SNESNewtonALComputeFunction() at /data/civet2/build/_env/conda-bld/moose-petsc_1735974488891/work/src/snes/impls/al/al.c:270
[0]PETSC ERROR: #3 SNESSolve_NEWTONAL() at /data/civet2/build/_env/conda-bld/moose-petsc_1735974488891/work/src/snes/impls/al/al.c:331
[0]PETSC ERROR: #4 SNESSolve() at /data/civet2/build/_env/conda-bld/moose-petsc_1735974488891/work/src/snes/interface/snes.c:4837
libMesh terminating:
No tangent load function or rhs vector has been set
Abort(1) on node 0 (rank 0 in comm 0): application called MPI_Abort(MPI_COMM_WORLD, 1) - process 0

PS: A somewhat similar question is raised at [(https://github.com//discussions/24923)] but it end with a reference to a PETSc example src/snes/tutorials/ex16.c

[GiudGiud]

Hello

The merge request @jedbrown mentioned is merged now.
So this capability should be available in moose

however from the error message the solver is expecting additional quantities to be set, so it will require a little bit of development. I expect you will need to tag the contribution of some kernels to the residual and set them as the RHS vector. This has to be done programmatically

@bwspenc @dschwen is there interest in this method at INL?

[lcontinuum]

Thank you for the feedback.
Just for my own understanding:
I think in my example it should be the Neumann boundary condition (force F) which has to be assembled to a separate RHS vector, maybe with vector_tags? And that would be the not set load function or rhs vector from the error message. Because the arc-length method "controls" the load factor lambda by itself to follow the blue snap through curve.
But currently MOOSE has no "interface" to set this function or RHS vector for PETSc's arc-length method. So, one would have to program this "interface". Would that be done as an additional option in [Problem] type = FEProblem. Not that I have the experience to do that, just out of interest.

[GiudGiud]

Hello

I agree the vector tag is the right way to gather that term as a RHS.

Setting it as the load function in petsc would likely be done in the executioner instead of the Problem.
Is this a transient / quasi static problem?

[lcontinuum]

Hello,

Is this a transient / quasi static problem?

Quasi-static problem.
Arc-length methods are used to follow an unstable load path for quasi-static problems. Like snap-through or snap-back behavior of structures. The arc-length method "finds" the necessary load scaling parameter lambda to keep the system in an even unstable equilibrium.

In a transient problem, one would have inertia and thus a mass matrix. Then the above toy problem would start accelerating downwards after the load reaches the first maximum. Reducing the time step should be sufficient to solve this in a dynamic setting. But one would most probably get a different load-displacement curve as one has a real time parameter and inertia forces.

[GiudGiud]

ok then you will want to modify the Transient executioner, while still using the QuasiStatic SolidMechanics physics.

To add the gathering of the RHS, you will likely need to use tag parameters in that Physics or modify the code in it

[lcontinuum]

Thanks you.
I'll give it a try as a side project to learn more about Transient executioner internals.