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Engineers often use Finite Element Analysis software to solve for stresses inside complicated structures to generate pictures like the one above.

I am familiar with FEA as a method to solve differential equations. So what (differential) equation are we using to describe forces/stresses inside objects?

When it comes to analyzing forces, I know only Newton's laws. But how can we use the laws in here, when the internal stresses are dependent on the geometry of the object?

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  • $\begingroup$ This is a very broad question, but google the Ritz and Galerkin method (only for linear elastic deformations), both are based on density of elastic energy (equations you are searching for), there are plenty of other methods, a basic knowledge of continuum mechanics helps. I don't know if are familiar with Hook's law? $\endgroup$
    – Sam Farjamirad
    Apr 17, 2019 at 17:57
  • $\begingroup$ @Sam Farjamirad Yes, I know Hooke's law. But how is it applied here? It is not a differential equation. I would be very interested in seeing how the weak for is developed for Hookes equation. $\endgroup$
    – S. Rotos
    Apr 17, 2019 at 18:03
  • $\begingroup$ I admit that, the simple linear equations doesn't say much here, but in Ritz method we are searching for a extremums of potential energy, the potential energy is a function of volume forces (gravitation or centrifugal ... ), the contact forces, and the elastic energy density, the latter one is a function of strain which is related to stresses by Hook's law. I would like to write a decent answer, but so many equations ... $\endgroup$
    – Sam Farjamirad
    Apr 17, 2019 at 18:44

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For structural FEA, there are a few approaches. Most commonly differential equations are developed that relate the deformation state and the stress state of each element in the structure in order to satisfy equilibrium.

A good resource for the details of structural FEA is a book by Ed Wilson Called Three Dimensional Static and Dynamic Analysis Of Structures.

Here is a chapter that shows the basis of equilibrium and compatibility.

http://www.edwilson.org/Book/02-equi.pdf

**To be more descriptive: For linear elasticity problems that use FEA to find internal forces and deformations, most FEA solvers do not directly solve the the partial differential equations. Instead FEA is used to approximate the solutions to the PDEs. I would like to note that using this method, solutions can be found to problems which are nearly impossible (if not impossible) to write closed formed elastic solutions.

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  • $\begingroup$ Respectfully, the compatibility conditions together with 15 other equations (at least three differential equations) is too difficult and complex to solve analytically, i guess those have no practical use. $\endgroup$
    – Sam Farjamirad
    Apr 17, 2019 at 18:46
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    $\begingroup$ I'm not sure what you mean? Some basic assumptions about material properties and deformations make most elastic FEA problems solveable... $\endgroup$
    – ShadowMan
    Apr 17, 2019 at 18:48
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    $\begingroup$ But in FEA you set the proper boundary conditions and assumptions on the deformation of each element and you are able to reduce the DE to a system of linear equations. This can be solved by hand using matrix methods. Although it is computationally intense, and provides an approximate solution, this is what most FEA software does. $\endgroup$
    – ShadowMan
    Apr 17, 2019 at 18:59
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    $\begingroup$ @SamFarjamirad I think you are rather missing the point here. Unless you can relate the behaviour of "practical" solution methods back to the behaviour of the underlying differential equations, you are just relying on luck as to whether your numerical solutions converge at all as you refine the mesh, and if they do converge whether they converge to the correct solution. If you use so-called "mixed" energy formulations that don't even have maxima or minima, only stationary values, it's a very good idea to understand what's going on at a deeper level than "the answers look about right!" $\endgroup$
    – alephzero
    Apr 17, 2019 at 19:13
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    $\begingroup$ Furthermore, the idea of FEA is that although approximate, as the mesh is refined the solution will converge with the exact solution for linear elastic solids, so to answer the OP, the solution to the general equilibrium and constitutive differential equations is approximated using the direct method that most FEA solvers employ. $\endgroup$
    – ShadowMan
    Apr 17, 2019 at 21:08
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The laws implied there are complex equations hard to resolve by hand. The computer divides the realm into several little pieces (elements) and applies the equations for those elements, resolving them by iterating.

https://en.wikipedia.org/wiki/Finite_element_method

Here are the equations for the weak formulation. There's also a strong formulation: https://www.researchgate.net/post/What_is_the_difference_between_strong_form_and_weak_form

The weak one is preferred because integrating is generally easier than derivating for computers.

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  • $\begingroup$ You've just described what FEA does in general. What I'm asking is how are forces described in FEA, what kind of equation FEA is solving for me when I do structural analysis? For example, there is the heat equation for heat flow problems. It is a differential equation solved by FEA. What equation am I solving when I'm solving for stresses inside an object? $\endgroup$
    – S. Rotos
    Apr 17, 2019 at 17:40
  • $\begingroup$ This one: simscale.com/blog/2016/10/what-is-finite-element-method $\endgroup$
    – user20096
    Apr 17, 2019 at 17:43

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