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Due to reasons, I can not fully describe the process explicitly so I will be using substitutes to represent the variables.

Nomenclature:

x - an independent variable

y - another independent variable

Fz - Original force value

Fz' - New force value.

*Note: these variables are all vectors. The number of components depend on the number of locations required. For example, a 4 location problem will have x1,x2,x3 and x4.

Goal:

  • Find the value for x that will produce Fz' that's within 100-110% of Fz. Each individual Fz' must be within 10% of its respective Fz such as Fz'1 & Fz1 Often time changing x alone will not produce any results, therefore another variable can be added y to reach convergence of a solution. Such that Fz' = f(x,y}

Constraints

  • The value of for x is explicitly defined. A fixed set. example: {55500 44751 321548 ...} - The value for y is within -100<y<100 with 1 as the min increments.

Difficulty

  1. Given that the relationships come from an FEA tool though the analysis itself is linearly static, the interactions between the locations and their effects on Fz' may not fit cleanly into a linear matrix framework.
  2. The couplings between locations complicates the problem. Such that change x or y or both in location 1 could affect the value for Fz' at other locations.

I was wondering if anyone have an idea on how to tackle this problem. I want to see if there's a mathematical way of solving this. Originally I wanted to solve this through linear algebra, but the relationship between Fz and x is most definitely nonlinear. The brute force iterative methods would be very computationally expensive.

My goal is to find a way to converge the solution and automate the process with code.

Please let me know what you guys have in mind.

Thank you guys for your help.

-Zhihao Liu

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  • $\begingroup$ I’m voting to close this question because it is better posted on Mathematics. $\endgroup$
    – Solar Mike
    Sep 15, 2023 at 4:55
  • $\begingroup$ Sorry about that. The nature of the work is related to structural engineering but I couldn't disclose it. I will post on mathematics instead. $\endgroup$
    – lone_coder
    Sep 15, 2023 at 6:00
  • $\begingroup$ @lone_coder It it is just a particular case that you cannot disclose, maybe you could come up with some similar problem, for which same solution techniques should be applicable. With a concrete example, I think this question might fit here. As it is now, I would agree with Solar Mike, that it is better suited for Mathematics. $\endgroup$ Sep 16, 2023 at 8:15

1 Answer 1

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Have you looked at how nonlinear FEA works? The solution has to be iterative and you just need to solve linear system repeatedly with some corrections (usually of the stiffness matrix) between the iterations. Newton Rhapson method is used most often, but you can also use Picard method which is less complex.

Newton Rhapson method updates stiffness but also starts at displacements from previous iteration. On the other hand, Picard method basically just corrects stiffness matrix, so that in the end, you will get the nonlinear solution (with desired accuracy) in just one solution of the system of linear equations (using coefficients from the modified stiffness matrix). In many cases the correction in Picard method is just application of rule of three (especially for 1D elements).

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