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Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here.

Problem summary

Given a common¹ set of shape functions defined at master element space $\{\psi_j(\xi,\eta)\}$ and associated to each corner $j$ of that element;

We can compute the affine transformation that maps parametric space coordinates $(\xi,\eta)$ into 'real' space coordinates $(x,y,z)$ as an interpolation of corner coordinates $\mathbf{p_j}$:

$$\bf x(\xi,\eta) = \sum_j\psi_j p_j$$

In similar fashion, we can compute the gradient² of that map as a rectangular matrix and using the partial derivatives of shape functions:

$${\rm grad}_\xi \mathbf{x}= \sum_j \left[ \begin{aligned} \frac{\partial \psi_j}{\partial \xi}p_{jx} && \frac{\partial \psi_j}{\partial \eta}p_{jx} \\ \frac{\partial \psi_j}{\partial \xi}p_{jy} && \frac{\partial \psi_j}{\partial \eta}p_{jy} \\ \frac{\partial \psi_j}{\partial \xi}p_{jz} && \frac{\partial \psi_j}{\partial \eta}p_{jz} \end{aligned}\right]$$

(I define the Jacobian Matrix down below)

Prove the Jacobian matrix of that transformation can be computed through the $QR$ decomposition of ${\rm grad}_\xi\bf x$, where $[R]_{2\times 2}$ is the Jacobian matrix, and $[Q]_{3\times 2}$ is a unitary matrix whose columns are orthonormal vectors that exist in the plane of the mapped element.


Context:

So last semester I took a course in intro to Finite Element approximations. In the course, the professor instructed us to compute the Jacobian Matrix of the geometrical map from the parametric space (of the master element) to 'real' space through the $QR$ decomposition of the 'gradient' of that map, in which

  • $R$ is the Jacobian matrix we seek and
  • $Q$ is a unitary matrix whose columns are orthonormal vectors in the plane of the mapped element.

He justified it by explaining that, in mapping 2D-to-2D, or 3D-to-3D elements, the ${\rm grad}_\xi\bf x$ exactly matches the Jacobian matrix, but that is not the case when mapping 2D master elements to 3D. For those, the linear geometrical map is an affine transformation from their 2-dimensional parametric space (of a triangle or a quadrilateral) into the actual coordinates of the element, which exist in $\mathbb{R}^3$.

Although I trust that he is right and I kinda have an intuition for it, he didn't prove that $R$ is indeed the so-called Jacobian matrix.

When I asked him for a proof, he seemed annoyed and brushed me off, which gave me the impression that this is supposed to be trivial and I was asking a dumb question. Yet, none of my classmates managed to prove it either. Now the course is over I decided to go back to the problem that's been bugging me for a few months.


Traditional Jacobian Matrix definition

(for mapping from 2D isoparametric space to 2D 'real' space as an example)

The Jacobian Matrix $\mathbf{J}$ is the linear transformation that maps line segments $d\xi$ and $d\eta$ in the parametric space $(\hat\Omega)$ to the real space $\mathbb R^2$

$$ \begin{bmatrix} dx \\ dy \end{bmatrix} = \begin{bmatrix} \dfrac{\partial x}{\partial \xi} && \dfrac{\partial x}{\partial \eta} \\ \dfrac{\partial y}{\partial \xi} && \dfrac{\partial y}{\partial \eta} \end{bmatrix} \begin{bmatrix} d\xi \\ d\eta \end{bmatrix} = \mathbf{J} \begin{bmatrix} d\xi \\ d\eta \end{bmatrix} $$ $$ \mathbf{J} = \begin{bmatrix} \dfrac{\partial x}{\partial \xi} && \dfrac{\partial x}{\partial \eta} \\ \dfrac{\partial y}{\partial \xi} && \dfrac{\partial y}{\partial \eta} \end{bmatrix} $$


Notes:

  1. By 'common' I mean the classic FEM linear shape functions. Valued 1 at their corresponding node and 0 on all other nodes.
  2. Pardon me if terms 'gradient' and 'Jacobian Matrix' I used aren't exactly their usual formal definitions, but I'm matching the terms used by the professor. I've put some effort into trying to keep it unambiguous;
  3. I'm a last year engineering student so, if you could keep your proof within the realms of my elementary maths, I'd appreciate it. But I'll take any help I can get;
  4. This is not homework, so feel free to answer a complete proof. I'm just honestly trying to learn.
  5. There's a question in math.stackexchange that seems to hint me in the right direction, but I haven't been able to complete this proof using it either.
  6. There's a closely related question here in which the poster is asking for the solution my professor gave us. They discuss it a bit there (applied to triangles), but it doesn't solve my problem. I'll gladly reply to them once I can actually prove it.
  7. I'll take any help anyone can give me. If you can't prove it but help me prove it, by sharing some knowledge or pointing to a reference, I'll be very much thankful.
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