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I'm programming a beam finite element model by following a book (Nonlinear Finite Element Analysis of Solids and Structures Volume 2, in case you're wondering!).

I've come across the following statement, which I've paraphrased slightly (unless mentioned matrices, denoted by $[A]$, are of dimension $3 \times 3$ and vectors, denoted by $\{a\}$ are of size $3 \times 1$):

$${\{a\}}^T[A]\{\delta\beta\}=\{b\}^T\{c\} \tag{1}\label{1}$$ $${\{d\}}^T[E]\{\delta\beta\}=\{f\}^T\{g\} \tag{2}\label{2}$$

Equations \eqref{1} and \eqref{2} provide two equations in the three components of $\{\delta\beta\}$. If we use the procedure of Rankin and Brogan [R1.16] which was discussed at the beginning of Section 17.1.6, we can then differentiate equation (17.59a) to obtain equation \eqref{3}: $${\{h\}}^T[L]\{\delta\beta\}=\{m\}^T\{n\} \tag{3}\label{3}$$

Using equations \eqref{1}-\eqref{3} one can obtain $[V]^T$ using \eqref{4}, where $[V]^T$ is a $3 \times 12$ matrix and $\{\delta p\}$ is a $12 \times 1$ column vector (both $\{\delta\beta\}$ and $\{\delta p\}$ are known at this stage): $$\{\delta\beta\}=[V]^T\{\delta p\} \tag{4}\label{4}$$

I need to find $[V]^T$. I think I've correctly managed to solve for the components of $\{\delta\beta\}$, but I'm at a loss as to how to get a $3 \times 12$ matrix from a $3 \times 1$ vector and a $12 \times 1$ vector.

In MatLab you can solve for the pseudo inverse using either $\{\delta\beta\}*\text{pinv}(\{\delta p\})$ or $\{\delta\beta\}/\{\delta p\}$ but although these both give $3 \times 12$ matrices they are not correct (based on numerical tests I've done).

Any help would be much appreciated!

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Your system of equations is under-specified (3 equations, 36 unknowns!). You probably will not be able to solve it (i.e. find all components of $[V]$) by purely algebraic means. I suggest instead that you look at the structure of V and more generally of your problem. For instance, if V is a transformation matrix applied to some deformation of a 4-node element, then the deformations at each node are not independent, so the coefficients of V are related in some way. As another example, in continuum mechanics, the spin matrix, although its dimensions are $3\times3$, only has 3 independent components. I suspect the situation here is similar.

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  • $\begingroup$ Thanks @lmsteffan, I'm of the same opinion. The thing I don't understand is how $\{\delta p\}$ can be found, as equations (1), (2) and (3) don't contain all of its components. I've tried using SymPy to solve (1), (2) and (3) symbolically and I get 6 terms in each of the equations for the components of $\{\delta \beta\}$. I think a bit more elucidation from the authors of the paper on what is going on here would have been useful! I cross posted this (naughty, I now know) with more information: math.stackexchange.com/questions/1105357/… $\endgroup$ – Peter Greaves Jan 19 '15 at 7:32

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