Hello everyone,

I have a running code that solves two linear equations successively: first I solve for a polarity vector and then I solve for the velocity vector in a force balance equation with some forces depending on the (previously solved) polarity. The second equation has a bi-linear term representing the friction with the form

\int_\Omega \xi \,\vec{v}\cdot\vec{u},

With \xi being the friction coeficcient, \vec{v}, the velocity vector and \vec{u}, the vectorial test function. I want to modify this model so that the fricition depends on the orientation between the polarity and velocity vectors. With that the friction form becomes

\int_\Omega \xi \,\left(1-\frac{\vec{p}\cdot\vec{v}}{||\vec{p}|| ||\vec{v}||}\right)\,\vec{v}\cdot\vec{u},

which adds a non-linear term to the first bi-linear form. At the time that I am solving for the velocity, the polarity \vec{p} is known, so this is just a quadratic form in \vec{v}. I am aware that I should implement a Newton-Raphson to solve this non-linear form, which is not a problem, but I am unable to properly define it in FreeFem.

My code goes as

```
macro p [px,py] //
macro v [vx,vy] //
macro u [ux,uy] //
macro normInner(p,v) ((p'*v)/(sqrt(p'*p)*sqrt(v'*v))) // normalized inner product
```

Then I define the FEM spaces, solve for `p`

and try to define the friction form as

```
// p-v alignment-dependent friction
varf aLinFric(v,u) = int2d(Th)(xi*(v'*u));
varf aNLinFric(v,u) = int2d(Th)(xi*normInner(p,v)*(v'*u));
```

resulting in the error

```
>> error operator * <10LinearCombI7MGauche4C_F0E>, <10LinearCombI7MGauche4C_F0E>
```

at the line where I define `macro normInner(p,v)`

. I’ve also tried with defining the macro more explicitly as

```
macro Inner(px,py,vx,vy) (px*vx + vy*py) //
macro normInner(px,py,vx,vy) (Inner(px,py,vx,vy)/(sqrt(Inner(px,py,px,py))*sqrt(Inner(vx,vy,vx,vy)))) // normalized inner product
```

but I end up obtaining the same error. I understand that I am not properly combining the FEM solutions, but I do not manage to understand how should I define a non-liner term depending on vectors like that.

Thank you so much,

Joan Térmens