Development based on step-25

[Toddy Liu]

Dear Deal.II community,

I'm getting trouble in developing new code based on step-25. In step-25, the equation to be solved is u_tt−∆u=−sin(u) and the tutorial shows me how to solve such a nonlinear problem.

Specifically, the tutorial tells me how to calculate (sin(u),φ_j)_Ω in detail. So I 'm wondering how to solve such a problem which is like u_tt−∆u=∇∙V, where V is a vector and its components(x direction and y direction in 2D) are dependent on u. So could you please give me some advice about doing this or any reference in existing dealii examples.

Thank you very much.

[Wolfgang Bangerth]

The standard point of departure is always the weak formulation of the problem.
In your case, you will have a right hand side of the following form:

(div V(u), phi_i)

for which you can decide whether you want to integrate it by parts to get

-(V(u), grad phi_i) + boundary terms

or whether you want to leave it as is. If the latter, then you have to apply
the chain rule and get something along the lines of

(V'(u) . grad u, phi_i)

which you have to deal with in much the same way as if you just had

(sin(u), phi_i)

except that you also have terms that involve derivatives of u.

Does that make sense?

Best
W.


--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Toddy Liu]

Hello, professor Wolfgang

Thank you very much for your reply and it really helps me. You explained how to deal with the latter case (V'(u) . grad u, phi_i) and I've got the point how to solve this. Addtionally. how about the former case -(V(u), grad phi_i). Specifically, how to construct the vector V(u) and then multiply with grad phi_i or to say fe_values.shape_grad(i,q_point)?

Thank you very much!

Best,

Toddy

[Wolfgang Bangerth]

On 9/15/21 10:47 PM, Toddy Liu wrote:
>
> Thank you very much for your reply and it really helps me. You explained
> how to deal with the latter case (V'(u) . grad u, phi_i) and I've got
> the point how to solve this. Addtionally. how about the former case
> -(V(u), grad phi_i). Specifically, how to construct the vector V(u) and
> then multiply with grad phi_i or to say fe_values.shape_grad(i,q_point)?

You perform the integral via quadrature, and so all you have to know is what
V(u)(x_q) = V(u(x_q))
which you evaluate in the same way as you did
sin(u(x_q))
in step-25 with the only exception that now V(u(x_q)) is a vector
(actually, a Tensor<1,dim>) instead of a scalar quantity.

[Toddy Liu]

Hi, professor Wolfgang

Yes, I can write the components of V(u(x_q)) by known expression, for example V(u(x_q))= (2u(x_q),3u(x_q)) where 2u(x_q) is the x-direction and 3u(x_q) is y-direction. I construct a vector<Tensor> named tmp_vector and assign the values by tmp_vector[0]=2u(x_q), tmp_vector[1]=3u(x_q). Then I do the multiplication tmp_vector*fe_values.shape_grad(i,q_point). But when I ran the program, errors occurred because maybe the operation between these two terms is not permitted. So I define the V(u(x_q)) by Point<dim>(2u(x_q),3u(x_q)) and the program runs without errors.

I want to ask whether the way I define V(u(x_q)) is right and are there any better ways to do this?

Thank you very much!

Best,

Toddy

Wolfgang Bangerth <bang...@colostate.edu> 于2021年9月17日周五 上午8:33写道：

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[Wolfgang Bangerth]

On 9/16/21 7:34 PM, Toddy Liu wrote:
>
> Yes, I can write the components of V(u(x_q)) by known expression, for example
> V(u(x_q))= (2u(x_q),3u(x_q)) where 2u(x_q) is the x-direction and 3u(x_q) is
> y-direction. I construct a vector<Tensor> named tmp_vector and assign the
> values by tmp_vector[0]=2u(x_q), tmp_vector[1]=3u(x_q). Then I do the
> multiplication tmp_vector*fe_values.shape_grad(i,q_point). But when I ran the
> program, errors occurred because maybe the operation between these two terms
> is not permitted. So I define the V(u(x_q)) by Point<dim>(2u(x_q),3u(x_q)) and
> the program runs without errors.

The dot product is defined between two objects of type Tensor<1,dim>. A
Point<dim> is derived from Tensor<1,dim>, so this works. Just
vector<Tensor<1,dim>> doesn't -- it should have been Tensor<1,dim>.

[Toddy Liu]

Professor Wolfgang

Thanks for your professional explanation and patient reply.

Best,

Toddy