I am trying to obtain the weak form of the Navier-Cauchy equation, which is

$$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$

and can be written in the component form

$$-(2 \mu +\lambda) \frac{\partial ^2 U_1}{\partial x_1 ^2} - \mu \frac{\partial ^2 U_1}{\partial x_2 ^2} - (\mu + \lambda) \frac{\partial ^2 U_2}{\partial x_1 \partial x_2} - \rho \omega ^2 U_1 = F_1$$

$$-(2 \mu +\lambda) \frac{\partial ^2 U_2}{\partial x_2 ^2} - \mu \frac{\partial ^2 U_2}{\partial x_1 ^2} - (\mu + \lambda) \frac{\partial ^2 U_1}{\partial x_1 \partial x_2} - \rho \omega ^2 U_2 = F_2$$

The general procedure is to multiply the PDE by a test function $\textbf{v}$ in the space $\textbf{V}$, or $v$ in the space $V$, and integrate it over the domain $\Omega$. I will proceed with the component form, for I believe it is easier for me to understand. Setting $\textbf{F} = 0$ and rearranging the terms

$$-(2 \mu +\lambda) \int_\Omega v \left[ \frac{\partial ^2 U_1}{\partial x_1 ^2} + \frac{\partial ^2 U_2}{\partial x_2 ^2} \right]dxdy - \mu \int_\Omega v \left[ \frac{\partial ^2 U_1}{\partial x_2 ^2} + \frac{\partial ^2 U_2}{\partial x_1 ^2} \right]dxdy -(\mu + \lambda)\int_\Omega v \left[ \frac{\partial ^2 U_2}{\partial x_1 \partial x_2} + \frac{\partial ^2 U_1}{\partial x_1 \partial x_2} \right]dxdy - \rho \omega ^2 \int_\Omega v \left[ U_1+U_2 \right]dxdy = 0$$

From Green's theorem I know that $$ \int_{\Omega} \left(v \frac{\partial ^2 u}{\partial x ^2} \right)dxdy = \int_\Gamma \left(v \frac{\partial u}{\partial x} \hat{n}_x \right)ds - \int_{\Omega} \left( \frac{\partial v}{\partial x} \frac{\partial u}{\partial x} \right)dxdy$$

Which is sufficient to deal with the first and second integrals. However, I do not know how to proceed with the cross derivatives $\partial ^2 / \partial x_1 \partial x_2$ of the third integral. Can someone help me with this?

  • $\begingroup$ I have asked a similar question before, but I realized I have simplified the PDE too much and could not proceed with the given answers. Now I made a more detailed question in the hope this will clarify the problem. $\endgroup$ – Lucas Vieira Jul 3 '20 at 16:16
  • $\begingroup$ There is another way of doing so: fenicsproject.discourse.group/t/… $\endgroup$ – Lucas Vieira Jul 11 '20 at 20:27

The identity you're missing from Gauss' divergence theorem is:

$$ \int_\Omega \nabla \varphi \cdot\mathbf{v} = -\int_\Omega \varphi\nabla\cdot\mathbf{v} +\int_{\partial\Omega}\varphi\mathbf{v\cdot n} $$

where I've written $\varphi$ as an arbitrary scalar field. So, using the divergence of $\mathbf{u}$ as the scalar field you'd get

$$ -\int_\Omega(\lambda+\mu) \nabla(\nabla\cdot\mathbf{u}) \cdot\mathbf{v} = \int_\Omega (\lambda+\mu)(\nabla\cdot\mathbf{u})\nabla\cdot\mathbf{v} -\int_{\partial\Omega}(\lambda+\mu)(\nabla\cdot\mathbf{u})\mathbf{v\cdot n} $$

and you can complete your weak formulation.

Anyway, here's in a few more steps:

Note the divergence of the product (scalar*vector) $$\nabla\cdot(\varphi\mathbf{v})=\nabla\varphi\cdot\mathbf{v}+\varphi\nabla\cdot\mathbf{v}$$ Rearrange to get $$\nabla\varphi\cdot\mathbf{v}=\nabla\cdot(\varphi\mathbf{v})-\varphi\nabla\cdot\mathbf{v}$$ And plug it into that integral $$\int_\Omega\nabla\varphi\cdot\mathbf{v} = \int_\Omega\nabla\cdot(\varphi\mathbf{v})-\int_\Omega\varphi\nabla\cdot\mathbf{v}$$ Apply Gauss' divergence theorem for vector fields in that second integral $$\int_\Omega \nabla\cdot\mathbf{v}=\int_{\partial\Omega}\mathbf{v\cdot n} \quad\Rightarrow\quad\int_\Omega \nabla\cdot\mathbf{\varphi v}=\int_{\partial\Omega}\varphi\mathbf{v\cdot n} \qquad\Rightarrow$$

$$\int_\Omega\nabla\varphi\cdot\mathbf{v} = \int_{\partial\Omega}\varphi\mathbf{v\cdot n}-\int_\Omega\varphi\nabla\cdot\mathbf{v}$$ Remember that $\varphi=\nabla\cdot\mathbf{u}$, enter Lamé's parameters, and voila: $$(\lambda+\mu)\int_\Omega\nabla(\nabla\cdot\mathbf{u})\cdot\mathbf{v} =(\lambda+\mu)\left( \int_{\partial\Omega}(\nabla\cdot\mathbf{u})\mathbf{v\cdot n}-\int_\Omega(\nabla\cdot\mathbf{u})\nabla\cdot\mathbf{v}\right)$$

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    $\begingroup$ Thank you very much! Can I just ask if you have used only Gauss' divergence theorem or something in addition to it? I haven't found this identity the way you wrote it. $\endgroup$ – Lucas Vieira Jul 4 '20 at 23:18
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    $\begingroup$ Oh... I guess I could've broken it down in a few more steps. I'm not sure if the correct term for it is this but, as I understand it, this is a flavor of the product rule for derivatives. One that stems from the divergence of a vector field multiplied by a scalar field: $$\nabla\cdot(\varphi\mathbf{v}) = \varphi\nabla\cdot\mathbf{v}+\mathbf{v}\cdot\nabla\varphi$$ And here I've used it together with the formal presentation of Gauss' divergence theorem. Authors usually skip it, as I did, to save some time... but I agree that it isn't the best way to present it to those who are learning $\endgroup$ – lima Jul 4 '20 at 23:32
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    $\begingroup$ You will find this and more in Gurtin (1981) page 30 equation 2; Kind of old school, but a great book for continuum mechanics, nonetheless $\endgroup$ – lima Jul 4 '20 at 23:35
  • $\begingroup$ Oh, sure, first year of Calculus... Once again, thank you. I appreciate the old books though! $\endgroup$ – Lucas Vieira Jul 5 '20 at 2:06

The general form of the equation is $$ \frac{\partial \sigma_{ij}}{\partial x_j} + F_i = \rho \frac{\partial^2 U_i}{\partial t^2} $$ where the stress is given by $$ \sigma_{ij} = \sigma_{ij}(U) = 2 \mu \varepsilon_{ij} + \lambda \varepsilon_{kk} \delta_{ij}, \qquad \varepsilon_{ij} = \varepsilon_{ij}(U) = \frac{1}{2}\left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i}\right) $$ We are using the Einstein summation convention. It is better to derive the weak form here.

If $V_i$ is the test function $$ \int_\Omega V_i \frac{\partial \sigma_{ij}(U)}{\partial x_j} dx = \int_{\partial\Omega} V_i \sigma_{ij}(U) n_i ds - \int_\Omega \sigma_{ij}(U) \frac{\partial V_i}{\partial x_j} dx $$ In this equation, we have summation over both indices i and j. Since $\sigma$ is symmetric tensor, you can show that $$ \sigma_{ij}(U) \frac{\partial V_i}{\partial x_j} = \sigma_{ij}(U) \varepsilon_{ij}(V) $$ Hence you can use this form $$ \int_\Omega V_i \frac{\partial \sigma_{ij}(U)}{\partial x_j} dx = \int_{\partial\Omega} V_i \sigma_{ij}(U) n_i ds - \int_\Omega \sigma_{ij}(U) \varepsilon_{ij}(V) dx $$ The mathematical analysis of the weak formulation should be done in many books, e.g.,

S. Kesavan, Topics in Functional Analysis and Applications, Section 3.2.4

  • $\begingroup$ NSE is a parabolic system, so there is used only $\rho \frac {\partial U_i}{\partial t}$ and not second derivative. $\endgroup$ – Alex Trounev Jul 4 '20 at 14:49
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    $\begingroup$ @AlexTrounev, the question is about Navier-Cauchy (elasticity) and not about Navier-Stokes (Newtonian fluids) equations. $\endgroup$ – nicoguaro Jul 4 '20 at 16:49
  • $\begingroup$ I was just about to say the same thing @nicoguaro. I understand the Navier-Cauchy equation is the time-harmonic form of the elastic wave equation. $\endgroup$ – Lucas Vieira Jul 4 '20 at 17:56
  • $\begingroup$ Ah, sorry, I didn't pay attention for the topic name. $\endgroup$ – Alex Trounev Jul 4 '20 at 19:18

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