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Does Finite Element Method use direct stiffness method? Or it is another mathematical method? We definitely use direct stiffness method to solve frames. FEM is based on entirely this or depending on he mesh, it is something else? Direct Stiffness method is for when we have simple lines and nodes, but when we have a complex mesh we use another method in FEM?

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FE solvers are not dependent on the direct stiffness method.

You can introduce conceptually the FE methodology to students using the direct stiffness method.

FEM Solvers use different formulations/schemes. For example, you'd probably use different approaches for a explicit or an implicit solver. Or you might need to use different formulations depending on the type of analysis (eg. static vs eigenmodes).

The most common application nowadays in arguably the linear static analyses. If you have a read at Hughes book on FEM, it starts with the weak formulation. The benefit over the strong formulation (governing differential equations) is that solving the strong formulation

  • is not always efficient
  • there may not be smooth (classical) solutions to a particular problem
  • incorporating boundary conditions in them is a daunting task due to stronger requirement on continuity of the field variables.

UPDATE:

To add to what @alephzero explained, I will try to give an example of a similar relationship (an analogue). If you want to find the speed of a falling object starting from rest you can use either kinematics equation (i.e. ( $u = g\cdot t$) or kinetics (i.e. energy equation $m*g*h = \frac{1}{2} m v^2 \Rightarrow u =\sqrt{2gh}$). Both method yield the right result, with little resemblance between them.

My understanding is that there is a similarity to the relationship of Direct Stiffness method and weak formulation compared to the kinematics and kinetics approach for solving a simple falling object problem. (@alephzero is more experienced in the field and he can validate my analogue)

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  • $\begingroup$ So then what does FEM use? and what it uses has nothing to do with direct stiffness method or is it still ultimately based on it? $\endgroup$ Oct 23 '20 at 9:13
  • $\begingroup$ Please see this link: wiki.csiamerica.com/display/kb/Discretization it talks about a direct stiffness method for node-element model, and then it taalks about a finite element model as if two separate things. I thought FEM used direct stiffness method but then when I saw this I was confused. So the methods you talk about above, do not depend on Direct Stiffness at all? $\endgroup$ Oct 23 '20 at 13:57
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    $\begingroup$ The simplest formulation of FEM uses the direct stiffness method but there are alternatives. For example to model incompressible materials there may be variables representing the hydrostatic pressure in the element, which don't belong to physical "nodes" or "grid points" and don't represent "forces" or "displacements." Also FEM can be used for completely different applications, e.g. electromagnetism, where there are no "displacements" of the structure at all. $\endgroup$
    – alephzero
    Oct 23 '20 at 15:12
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    $\begingroup$ Historically the "weak formulation" of FEM for structural mechanics was derived by minimizing the strain energy for an assumed displacement field within each element. But there are alternative weak formulations which have different physical interpretations. In other applications of FEM the weak formulation may not have any "obvious" physical interpretation at all. $\endgroup$
    – alephzero
    Oct 23 '20 at 15:17
  • $\begingroup$ thanks a lot and the alternatives you mention, also ultimately depend on direct stiffness method? in other words, the direct stiffness method is the main fundamental thing? or, even if there was no direct stiffness method (for the sake of example), the other methods could still exist and work? $\endgroup$ Oct 25 '20 at 14:33

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