more about the internal iteration
> details of `SNESSetType(snes, SNESNEWTONAL);` compared to `SNESSetType(snes,
> SNESNEWTONLS);` so that I can more accurately replicate and improve upon the
> arc-length algorithm I have implemented in the attached code using SNES.
I would recommend again referring to the documentation and source for easier
comparison. If you have additional specific questions about the implementation,
please feel free to ask!
Best,
Zach Atkins
From: Jed Brown
Sent: Tuesday, March 18, 2025 11:34 AM
To: David Jiawei LUO LIANG <12431...@mail.sustech.edu.cn>; petsc-dev
Cc: Zach Atkins
Subject: Re: [petsc-dev] Inquiry Regarding Arc-Length Method Implementation in
SNESNEWTONAL Solver
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Hi, David. I'm copying Zach, who developed this implementation and will be able
to more directly answer your questions.
"David Jiawei LUO LIANG"<12431...@mail.sustech.edu.cn> writes:
> Dear PETSc Developers,
>
>
> I am currently using the SNESNEWTONAL nonlinear solver and its arc-length
> method module. I noticed from the tutorial that, besides the standard
> residual and Jacobian definitions:
>
>
> SNESSetFunction(snes, r, FormFunction2nd, &user);
> SNESSetJacobian(snes, J, J, FormJacobian2nd, &user);
>
>
> it also requires:
>
>
> SNESNewtonALSetFunction(snes, ComputeTangentLoad, &user);
>
>
> which is used to define the arc-length constraint function.
>
>
> I have several questions regarding the arc-length implementation, and I would
> greatly appreciate your clarification:
>
>
> 1. According to the tutorial, the arc-length constraint function
> `ComputeTangentLoad` is used to compute ( Q = dF^{ext}/d\lambda ), the
> derivative of the external load with respect to lambda. In the case of a
> linear load, i.e., ( F^{ext} = \lambda F^{ext} ), ( Q = F^{ext} ). In this
> scenario, is it correct to simply return ( F^{ext} ) inside
> `ComputeTangentLoad`?
>
>
> 2. From my experiments, I observed that the `ComputeTangentLoad` function is
> called once before assembling the residual at every iteration. Interestingly,
> it seems that during the first iteration, ( Q ) contributes to the residual
> calculation, but from the second iteration onward, it no longer does. Could
> you help me understand why this happens?
>
>
> 3. I have attached my own buckling example of a Lee Frame structure where I
> implemented the arc-length method as the nonlinear solver. In my
> implementation, I divided the arc-length method into two steps:
> (1) A predictor step, which predicts the intersection of the
> constraint surface and the tangent;
> (2) A corrector step, which uses Newton-Raphson iterations to
> refine the intersection point between the constraint surface and the actual
> equilibrium path.
> This is the standard arc-length algorithm principle. I would
> like to know if `SNESNEWTONAL` follows a similar predictor-corrector
> procedure internally, or if I need to implement an outer load step loop and
> predictor step myself?
>
>
> 4. I remember that the tutorial describes solving two linear systems during
> each corrector step to obtain ( \Delta x^F ) and ( \Delta x^Q ) separately.
> Where in the SNESNEWTONAL workflow are these two solves performed? Do they
> happen right after assembling the residual and Jacobian?
>
>
> 5. Additionally, is the Jacobian matrix used in SNESNEWTONAL the original
> system Jacobian, or is it augmented to include the constraint surface
> function? Similarly, is the input vector expanded to ( n+1 ) dimensions to
> account for ( \lambda ), or does it remain the original size ( n ) as in the
> standard Newton-Raphson implementation?
>
>
> 6. In general, I would like to understand more about the internal iteration
> details of `SNESSetType(snes, SNESNEWTONAL);` compared to `SNESSetType(snes,
> SNESNEWTONLS);` so that I can more accurately replicate and improve upon the
> arc-length algorithm I have implemented in the attached code using SNES.
>
>
> Thank you very much for your time and assistance!
>
>
> Best regards,
>
>
>
> David
>
>
>
>
>
> David Jiawei LUO LIANG
>
>
>
> 南方科技大学/学生/研究生/2024
>
>
>
> 广东省深圳市南山区学苑大道1088号
>
>
>
>
>
