# Error in Newton-Raphson Incremental Iteration

**URL:** <https://community.freefem.org/t/error-in-newton-raphson-incremental-iteration/4137>\
**Category:** General Discussion\
**Created:** [November 18, 2025, 1:57pm UTC](https://community.freefem.org/t/error-in-newton-raphson-incremental-iteration/4137 "2025-11-18T13:57:57Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![AlexanderJFDR](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/alexanderjfdr/32/3092_2.png) [@AlexanderJFDR](https://community.freefem.org/u/AlexanderJFDR)\
**Post date:** [November 18, 2025, 1:57pm UTC](https://community.freefem.org/t/error-in-newton-raphson-incremental-iteration/4137/1 "2025-11-18T13:57:58Z")

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Hi,

I am trying to use FreeFem++ to solve plasticity mechanics problems using an incremental iterative approach, with the specific code documentation being:

[test.edp](https://community.freefem.org/uploads/short-url/97aXbCytZV9kd0XvK2JuPLfka2a.edp) (15.6 KB)

[tension.zip](https://community.freefem.org/uploads/short-url/gNb6C0KGhvTd5fLGHPc9S174paV.zip) (307.4 KB) (it’s the mesh file)

, but I have discovered an issue:

When calculating stress, I use

```auto
Vh [u, v], [u1, v1]；
Eh[int] sigma(3)；
sigma[0] = E/(1-nu^2)*dx(u) + nu*E/(1-nu^2)*dy(v);
sigma[1] = nu*E/(1-nu^2)*dx(u) + E/(1-nu^2)*dy(v);
sigma[2] = 0.5*E/(1+nu)*(dx(v)+dy(u));

```

where u and v represent displacements, and sigma[3] represents stress.

The process of assigning values computed through dx(u) at integration points to the nodes of the sigma variable appears to introduce errors.

To verify this conjecture, I designed an extremely simple numerical example of linear elasticity

[test\_simple.edp](https://community.freefem.org/uploads/short-url/6Id73yNNHRXtJRzwiRPL114QWnf.edp) (3.1 KB)

[tension.zip](https://community.freefem.org/uploads/short-url/gNb6C0KGhvTd5fLGHPc9S174paV.zip) (307.4 KB) (it’s the mesh file)

that should fully converge after the first iteration step. However, when I run the code:

```auto
solve A11([u,v],[u1,v1]) = int2d(Th)(epsilon(u,v)'*Copen(E,nu)*epsilon(u1,v1))
                                -int2d(Th)(epsilon(ubefore,vbefore)'*Copen(E,nu)*epsilon(u1,v1))               
                                -int2d(Th)([sigma[0],sigma[1],sigma[2]]'*epsilon(u1,v1))
                                +on(1,u=0,v=-displacement)+on(2,v=displacement);

```

, it fails to converge. But when I run:

```auto
solve A11([u,v],[u1,v1]) = int2d(Th)(epsilon(u,v)'*Copen(E,nu)*epsilon(u1,v1))
                                -int2d(Th)(epsilon(ubefore,vbefore)'*Copen(E,nu)*epsilon(u1,v1))               
                                -int2d(Th)(epsilon(ubefore,vbefore)'*Copen(E,nu)*epsilon(u1,v1))   
                                +on(1,u=0,v=-displacement)+on(2,v=displacement);

```

it converges completely in the first step, just as expected.

It is strange that these two codes are essentially identical. I am unsure whether this discrepancy is caused by errors introduced when interpolating values from integration points to nodes. I would appreciate any help from friends here to resolve this issue.

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<div class="post-metadata">

**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\
**Post date:** [November 19, 2025, 11:20am UTC](https://community.freefem.org/t/error-in-newton-raphson-incremental-iteration/4137/2 "2025-11-19T11:20:55Z")

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Alexander,  
you are right, the difference in the results obtained by the two codes is due to the interpolation of dx(u) into the space Eh (=P1 with your definitions). If you take Eh=P0 there is no more discrepancy, the code with sigma converges also. Indeed in this case since u is P1,  
dx(u) is P0 thus there is no interpolation error.

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<div class="post-metadata">

**Author:** ![AlexanderJFDR](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/alexanderjfdr/32/3092_2.png) [@AlexanderJFDR](https://community.freefem.org/u/AlexanderJFDR)\
**Post date:** [November 20, 2025, 4:19pm UTC](https://community.freefem.org/t/error-in-newton-raphson-incremental-iteration/4137/3 "2025-11-20T16:19:15Z")

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Thank you for your prompt reply, Professor. It’s really helpful to me.