I am developing an Abaqus UEL with 3D 8 nodes brick elements and I need second order derivatives of the shape functions, I have already mapped the first order derivatives from the element coordinates to the physical coordinates through: $\frac{\partial N}{\partial \textbf{X} } = J^{-1} \frac{\partial N}{\partial \textbf{x}} $ where $N$ are the shape functions, $X$ the physical coordinates, $x$ the local coordinates and the Jacobian matrix is $J_{ij}=\frac{\partial N}{\partial x_i} X_j$.

How can I compute and implement the second order derivatives of the shape functions with respect to global coordinates?

  • $\begingroup$ Apply the first order operator twice? $\endgroup$
    – ConvexHull
    Feb 1 at 17:42
  • $\begingroup$ Can you comment a little bit on the problem that you want to solve? $\endgroup$
    – nicoguaro
    Feb 1 at 17:51

You want to compute the following (let me use $\nabla$ instead of your $\frac{\partial}{\partial\textbf{X}}$ for derivatives in real space and $\hat\nabla$ instead of $\frac{\partial}{\partial\textbf{x}}$ for derivatives on the reference cell) $$ (\nabla^2 N)_{ij} = \nabla_i(\nabla_j N) = [J^{-1}]_{ik} \hat\nabla_k ([J^{-1}]_{jl} \hat\nabla_l N). $$ By the product rule, this is $$ (\nabla^2 N)_{ij} = [J^{-1}]_{ik} [J^{-1}]_{jl} (\hat\nabla^2 N)_{kl} + [J^{-1}]_{ik} \hat\nabla_k[J^{-1}]_{jl} \hat\nabla_l \hat N. $$ You know $J^{-1}$ here from computing the first derivatives, and computing the derivatives $\hat\nabla^2 N$ is easy because it happens on the reference cell. The difficulty is the derivative of the inverse of the Jacobian: $$ \hat\nabla [J^{-1}]. $$ This has a simple solution: We know $$ \hat\nabla [JJ^{-1}]=0 $$ and so $$ J \hat\nabla [J^{-1}] + [\hat\nabla J] [J^{-1}] = 0. $$ which you can solve for $$ \hat\nabla [J^{-1}] = -[J^{-1}] [\hat\nabla J] [J^{-1}] $$ where $\hat\nabla J$ is simply the second derivative of the transformation, and is as easily computed as $J$ was in the first place.


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