Integrating tensor expressions

Useful theorems

Green-Gauss

The Green-Gauss theorem is often stated as

∫Ωϕ[q‾⋅∇‾]dΩ=∫Γϕq‾⋅n‾dΓ−∫Ω[∇‾ϕ]⋅q‾dΩ\begin{aligned} \int_\Omega \phi \left[\underline{\boldsymbol{ q}}\cdot\underline{\boldsymbol{ \nabla}}\right] \mathrm{d} \Omega &= \int_\Gamma \phi \underline{\boldsymbol{ q}}\cdot \underline{\boldsymbol{ n}} \mathrm{d} \Gamma - \int_\Omega \left[\underline{\boldsymbol{ \nabla}}\phi\right]\cdot \underline{\boldsymbol{ q}} \mathrm{d} \Omega \end{aligned}

where Ω\Omega is a closed region (e.g. a volume) and Γ\Gamma its bounding surface (e.g. an area). n‾\underline{\boldsymbol{ n}} is the outward pointing normal on Γ\Gamma. ϕ\phi is a scalar and q‾\underline{\boldsymbol{ q}} a vector, both functions of the position x‾\underline{\boldsymbol{ x}}. If we write the corresponding expression in index notation, we have

∫Ωϕ∂qi∂xidΩ=∫ΓϕqinidΓ−∫Ω∂ϕ∂xiqidΩ\begin{aligned} \int_\Omega \phi \frac{\partial q_i}{\partial x_i} \mathrm{d} \Omega &= \int_\Gamma \phi q_i n_i \mathrm{d} \Gamma - \int_\Omega \frac{\partial \phi}{\partial x_i} q_i \mathrm{d} \Omega\\ \end{aligned}

Let's set up this equation for a vector instead of the scalar ϕ\phi. By using the index notation, each free index is actually just a scalar so we effectively just write three equations of the same as above, changing ϕ\phi to uku_k with u‾=uke‾k\underline{\boldsymbol{ u}}=u_k\underline{\boldsymbol{ e}}_{ k}.

∫Ωuk∂qi∂xidΩ=∫ΓukqinidΓ−∫Ω∂uk∂xiqidΩ∫Ωu‾⊗[q‾⋅∇‾]dΩ=∫Γu‾⊗q‾⋅n‾dΓ−∫Ω[u‾⊗∇‾]⋅q‾dΩ\begin{aligned} \int_\Omega u_k \frac{\partial q_i}{\partial x_i} \mathrm{d} \Omega &= \int_\Gamma u_k q_i n_i \mathrm{d} \Gamma - \int_\Omega \frac{\partial u_k}{\partial x_i} q_i \mathrm{d} \Omega\\ \int_\Omega \underline{\boldsymbol{ u}} \otimes \left[\underline{\boldsymbol{ q}}\cdot \underline{\boldsymbol{ \nabla}}\right] \mathrm{d} \Omega &= \int_\Gamma \underline{\boldsymbol{ u}} \otimes \underline{\boldsymbol{ q}}\cdot\underline{\boldsymbol{ n}} \mathrm{d} \Gamma - \int_\Omega \left[\underline{\boldsymbol{ u}} \otimes \underline{\boldsymbol{ \nabla}}\right]\cdot \underline{\boldsymbol{ q}} \mathrm{d} \Omega \end{aligned}

Similarly, we can write it exchanging the vector q‾=qie‾i\underline{\boldsymbol{ q}}=q_i\underline{\boldsymbol{ e}}_{ i} with a 2nd order tensor, a=ajie‾j⊗e‾i\boldsymbol{ a}=a_{ji}\underline{\boldsymbol{ e}}_{ j}\otimes\underline{\boldsymbol{ e}}_{ i}

∫Ωuk∂aji∂xidΩ=∫ΓukajinidΓ−∫Ω∂uk∂xiajidΩ∫Ωu‾⊗[a⋅∇‾]dΩ=∫Γu‾⊗a⋅n‾dΓ−∫Ω[u‾⊗∇‾]⋅aTdΩ\begin{aligned} \int_\Omega u_k \frac{\partial a_{ji}}{\partial x_i} \mathrm{d} \Omega &= \int_\Gamma u_k a_{ji} n_i \mathrm{d} \Gamma - \int_\Omega \frac{\partial u_k}{\partial x_i} a_{ji} \mathrm{d} \Omega\\ \int_\Omega \underline{\boldsymbol{ u}} \otimes \left[\boldsymbol{ a} \cdot \underline{\boldsymbol{ \nabla}} \right] \mathrm{d} \Omega &= \int_\Gamma \underline{\boldsymbol{ u}} \otimes \boldsymbol{ a}\cdot\underline{\boldsymbol{ n}} \mathrm{d} \Gamma - \int_\Omega \left[\underline{\boldsymbol{ u}} \otimes \underline{\boldsymbol{ \nabla}}\right]\cdot \boldsymbol{ a}^{\mathrm{T}} \mathrm{d} \Omega \end{aligned}

This expression represents 9 equations (kk and jj). If we sum the equations where k=jk=j, i.e. multiply by δkj\delta_{kj}, we get

∫Ωuj∂aji∂xidΩ=∫ΓujajinidΓ−∫Ω∂uj∂xiajidΩ∫Ωu‾⋅[a⋅∇‾]dΩ=∫Γu‾⋅a⋅n‾dΓ−∫Ω[u‾⊗∇‾]:adΩ\begin{aligned} \int_\Omega u_j \frac{\partial a_{ji}}{\partial x_i} \mathrm{d} \Omega &= \int_\Gamma u_j a_{ji} n_i \mathrm{d} \Gamma - \int_\Omega \frac{\partial u_j}{\partial x_i} a_{ji} \mathrm{d} \Omega\\ \int_\Omega \underline{\boldsymbol{ u}} \cdot \left[\boldsymbol{ a}\cdot \underline{\boldsymbol{ \nabla}} \right] \mathrm{d} \Omega &= \int_\Gamma \underline{\boldsymbol{ u}}\cdot\boldsymbol{ a}\cdot\underline{\boldsymbol{ n}} \mathrm{d} \Gamma - \int_\Omega \left[\underline{\boldsymbol{ u}} \otimes \underline{\boldsymbol{ \nabla}}\right] : \boldsymbol{ a} \mathrm{d} \Omega \end{aligned}
© Knut Andreas Meyer.
Last modified: April 22, 2024.
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