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I am used to interpreting each entry of a solid mechanic system's stiffness matrix as a 1D (linear or angular) spring joining one DOF (column index) to another (row index). But this interpretation leaves something to be explained:

  1. What do diagonal, i.e., DOF-to self entries mean? I mean, a spring connectiong a node to itself makes no physical sense right?
  2. What do diagonal SUBMATRICES (relative to a node's DOFs) mean? Those relate each DOF of a node to the other DOFs of the same node, but again, I can't see a physical explanation to that (e.g., why should x-translation of a node influence its z-rotation?)
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  • $\begingroup$ A diagonal stiffness matrix simply means that the operation being discretized is a pointwise operation, such as the operation $u(x)\mapsto \rho(x)u(x)$. As for submatrices, could you elaborate a bit more? If they are principal submatrices then they are basically just a local version of the above case but in general I don't know if they have much meaning $\endgroup$
    – whpowell96
    Commented May 29 at 21:38
  • $\begingroup$ No sorry what I meant is diagonal terms in a matrix that has also non-diagonal terms. With digonal submatrices, I meant the n*n submatrix (n = n. of DOFs per node) which lies on the diagonal of the global matrix and which corresponds to a single node's DOFs, thus correlating those to themselves. I don't really get what you mean with "point-wise operation". What would be the physical meaning of such operation? Can you maybe point out an example? $\endgroup$
    – temporary_pigeon
    Commented May 29 at 21:43
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    $\begingroup$ Many nonlinearities encountered outside of fluid dynamics are pointwise. For example, the function $u(x)\mapsto u^2(x)$ often represents a spatially distributed chemical that self-catalyzes and feeds its own growth. $\endgroup$
    – whpowell96
    Commented May 29 at 22:25
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    $\begingroup$ This question does not make sense unless you define exactly what you mean by a stiffness matrix. The name originates from solid mechanics and analogous spring networks but today this terminology is used to refer to almost any matrix describing the stiffness of any kind of structure and also in other physical contexts these matrices, consisting of inner products of shape function gradients, are referred to as "stiffness matrices". The spring analogy does not mean much in heat transfer, for example. I suggest you edit the question to define what you mean by a stiffness matrix. $\endgroup$
    – knl
    Commented May 30 at 12:54
  • $\begingroup$ Done. I was in fact referring to the case of solid mechanics. $\endgroup$
    – temporary_pigeon
    Commented May 30 at 13:16

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Consider a simple spring with stiffness $k$ lying along the x-axis with the degrees of freedom being the x-displacements at the two ends. We have the matrix equation

\begin{equation} \left\{\begin{array}{c} F_1 \\ F_2 \\ \end{array} \right\} =\left[\begin{array}{cc} k & -k \\ -k & k \\ \end{array} \right] \left\{\begin{array}{c} u_1 \\ u_2 \\ \end{array} \right\} \end{equation}

Suppose we constrain the displacement at either the right or left end to zero. (We must do this to solve the system because the stiffness matrix is singular.) For either constraint set, we have an easily-solvable scalar equation that shows that the diagonal term, $k$, is the force we must apply at the unconstrained node to produce a unit displacement.

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