Implemented generalized solution for ILD

[Preypatel2413]

References to other Issues or PRs

Fixes #25714
Fixes #24439

Brief description of what is fixed or changed

• In solve_for_reaction_loads() : Rewriting shear_force and bending_moment in Heaviside function before taking limit.

  • This gives correct reaction forces when there is applied force at a varible distance (Independent from length of beam).

• Changed _solve_for_ild_equations() : this helper function was changed to take care of common code in solve_for_ild_reactions(), solve_for_ild_shear() and solve_for_ild_moment().

  • The current implementation of solve_for_ild_shear() and solve_for_ild_moment() splits ILD curve in 2 segments from the specified section which does not give correct solution for multispan beam.
  • using generalized method to find ILD curves.
    • Applying load at variable distance y (which is removed in the end).
    • Solving reaction forces in terms of y which gives ILD_reactions.
    • Using these reaction forces in shear force and bending moment to find generalized ILD curve. These equations will involve two variables x (self.variable: the variable at which we want to find the value of reaction force, shear force, or bending moment) and y (the distance at which the influence line load (ILD_load) is applied).
    • In the individual functions substituting x = Distance (the input distance of the section at which the ILD curve is to be found) and y = x. This gives ILD equation for shear force or bending moment which represents the shear force or bending moment at chosen distance when ILD load is applied at distance x.

Other comments

• Taking example from 24439 : reaction_loads was {M0 : F1*L , R0 : -F1}

  • After changes : reaction_loads are {M_0: F_1L + F_2dHeaviside(L - d, 1), R_0: -F_1 - F_2Heaviside(L - d, 1)}. Which are correctly generalized.

• Taking example from 25714 :

  • After changes : ILD_moment is
    Piecewise((x*(x**2 - 100)/1000, x <= 10), (3*x**2/100 - 2*x/5 + 1, x <= 15), (3*x**2/100 - 7*x/5 + 16, x <= 20), (0, x > 30), (x/10 - (x - 30)**3/1000 - 3, True))
    and following graph (which is accurate result)

    

First-time Contribution:

• This is my first pull request to SymPy, and I'm excited to contribute to the project! I appreciate any feedback or guidance from the community and look forward to learning and contributing more in the future.

Release Notes

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#### References to other Issues or PRs
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Fixes #25714
Fixes #24439

#### Brief description of what is fixed or changed
* In `solve_for_reaction_loads()` : Rewriting shear_force and bending_moment in Heaviside function before taking limit.
  * This gives correct reaction forces when there is applied force at a varible distance (Independent from length of beam).

* Changed `_solve_for_ild_equations()` : this helper function was changed to take care of common code in `solve_for_ild_reactions()`, `solve_for_ild_shear()` and `solve_for_ild_moment()`.
  * The current implementation of solve_for_ild_shear() and solve_for_ild_moment() splits ILD curve in 2 segments from the specified section which does not give correct solution for multispan beam.
  * using generalized method to find ILD curves.
    * Applying load at variable distance `y` (which is removed in the end).
    * Solving reaction forces in terms of `y` which gives ILD_reactions.
    * Using these reaction forces in shear force and bending moment to find generalized ILD curve. These equations will involve two variables `x` (`self.variable`: the variable at which we want to find the value of reaction force, shear force, or bending moment) and `y` (the distance at which the influence line load (ILD_load) is applied).
    * In the individual functions substituting `x` = `Distance` (the input distance of the section at which the ILD curve is to be found) and `y` = `x`. This gives ILD equation for shear force or bending moment which represents the shear force or bending moment at chosen distance when ILD load is applied at distance `x`.


#### Other comments
* Taking example from [24439](https://github.com/sympy/sympy/issues/24439#issue-1514409011) : reaction_loads was {M0 : F1*L , R0 : -F1}
  * After changes : reaction_loads are {M_0: F_1*L + F_2*d*Heaviside(L - d, 1), R_0: -F_1 - F_2*Heaviside(L - d, 1)}. Which are correctly generalized.

* Taking example from [25714](https://github.com/sympy/sympy/issues/25714#issue-1907206104) :
  * After changes : ILD_moment is
    `Piecewise((x*(x**2 - 100)/1000, x <= 10), (3*x**2/100 - 2*x/5 + 1, x <= 15), (3*x**2/100 - 7*x/5 + 16, x <= 20), (0, x > 30), (x/10 - (x - 30)**3/1000 - 3, True))` 
and following graph (which is accurate result)

    ![ild_moment](https://github.com/Preypatel2413/ML_Assignment1/assets/76490445/01073f96-6050-43f7-b36d-a4cffc72860b)

**First-time Contribution:** 
* This is my first pull request to SymPy, and I'm excited to contribute to the project! I appreciate any feedback or guidance from the community and look forward to learning and contributing more in the future.


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[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

| Change   | Before [a00718ba]    | After [c81c8169]    |   Ratio | Benchmark (Parameter)                                                |
|----------|----------------------|---------------------|---------|----------------------------------------------------------------------|
| -        | 68.6±0.7ms           | 44.0±0.2ms          |    0.64 | integrate.TimeIntegrationRisch02.time_doit(10)                       |
| -        | 67.7±1ms             | 43.7±0.3ms          |    0.65 | integrate.TimeIntegrationRisch02.time_doit_risch(10)                 |
| +        | 18.7±0.2μs           | 30.4±0.07μs         |    1.63 | integrate.TimeIntegrationRisch03.time_doit(1)                        |
| -        | 5.28±0.02ms          | 2.92±0.03ms         |    0.55 | logic.LogicSuite.time_load_file                                      |
| -        | 73.1±1ms             | 29.0±0.2ms          |    0.4  | polys.TimeGCD_GaussInt.time_op(1, 'dense')                           |
| -        | 25.7±0.1ms           | 17.1±0.03ms         |    0.67 | polys.TimeGCD_GaussInt.time_op(1, 'expr')                            |
| -        | 74.5±0.7ms           | 28.9±0.2ms          |    0.39 | polys.TimeGCD_GaussInt.time_op(1, 'sparse')                          |
| -        | 258±1ms              | 125±0.7ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'dense')                           |
| -        | 258±2ms              | 126±0.8ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'sparse')                          |
| -        | 650±4ms              | 373±2ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'dense')                           |
| -        | 659±0.6ms            | 375±1ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'sparse')                          |
| -        | 497±1μs              | 288±1μs             |    0.58 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(1, 'dense')            |
| -        | 1.78±0.01ms          | 1.05±0ms            |    0.59 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(2, 'dense')            |
| -        | 5.87±0.03ms          | 3.11±0.01ms         |    0.53 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(3, 'dense')            |
| -        | 446±1μs              | 229±2μs             |    0.51 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(1, 'dense')               |
| -        | 1.49±0.01ms          | 688±5μs             |    0.46 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(2, 'dense')               |
| -        | 4.88±0.03ms          | 1.68±0.01ms         |    0.34 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(3, 'dense')               |
| -        | 376±2μs              | 206±0.7μs           |    0.55 | polys.TimeGCD_SparseGCDHighDegree.time_op(1, 'dense')                |
| -        | 2.46±0.02ms          | 1.25±0.01ms         |    0.51 | polys.TimeGCD_SparseGCDHighDegree.time_op(3, 'dense')                |
| -        | 10.1±0.06ms          | 4.37±0.04ms         |    0.43 | polys.TimeGCD_SparseGCDHighDegree.time_op(5, 'dense')                |
| -        | 352±2μs              | 168±0.8μs           |    0.48 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(1, 'dense')            |
| -        | 2.48±0.03ms          | 911±3μs             |    0.37 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(3, 'dense')            |
| -        | 9.56±0.03ms          | 2.68±0.01ms         |    0.28 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(5, 'dense')            |
| -        | 1.03±0.01ms          | 426±5μs             |    0.41 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'dense')           |
| -        | 1.73±0.01ms          | 497±3μs             |    0.29 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'sparse')          |
| -        | 5.97±0.05ms          | 1.82±0.02ms         |    0.3  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'dense')           |
| -        | 8.38±0.06ms          | 1.49±0.01ms         |    0.18 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'sparse')          |
| -        | 286±2μs              | 64.0±0.7μs          |    0.22 | polys.TimePREM_QuadraticNonMonicGCD.time_op(1, 'sparse')             |
| -        | 3.42±0.03ms          | 401±3μs             |    0.12 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'dense')              |
| -        | 3.99±0.04ms          | 278±1μs             |    0.07 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'sparse')             |
| -        | 7.14±0.1ms           | 1.27±0.01ms         |    0.18 | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'dense')              |
| -        | 8.73±0.09ms          | 837±2μs             |    0.1  | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'sparse')             |
| -        | 5.04±0.02ms          | 2.96±0.01ms         |    0.59 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(2, 'sparse') |
| -        | 12.2±0.1ms           | 6.64±0.05ms         |    0.54 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'dense')  |
| -        | 22.6±0.09ms          | 9.09±0.04ms         |    0.4  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'sparse') |
| -        | 5.21±0.02ms          | 867±3μs             |    0.17 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(1, 'sparse')    |
| -        | 12.6±0.1ms           | 7.04±0.04ms         |    0.56 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(2, 'sparse')    |
| -        | 104±0.4ms            | 26.2±0.06ms         |    0.25 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'dense')     |
| -        | 169±2ms              | 54.2±0.4ms          |    0.32 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'sparse')    |
| -        | 173±0.6μs            | 112±0.6μs           |    0.65 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'dense')      |
| -        | 357±5μs              | 216±1μs             |    0.61 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'sparse')     |
| -        | 4.30±0.06ms          | 843±4μs             |    0.2  | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'dense')      |
| -        | 5.35±0.03ms          | 384±0.8μs           |    0.07 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'sparse')     |
| -        | 20.4±0.1ms           | 2.81±0.01ms         |    0.14 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'dense')      |
| -        | 22.8±0.2ms           | 626±2μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'sparse')     |
| -        | 476±2μs              | 134±0.7μs           |    0.28 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(1, 'sparse') |
| -        | 4.63±0.06ms          | 634±4μs             |    0.14 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'dense')  |
| -        | 5.39±0.09ms          | 139±2μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'sparse') |
| -        | 13.2±0.06ms          | 1.29±0ms            |    0.1  | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'dense')  |
| -        | 13.8±0.06ms          | 140±0.3μs           |    0.01 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'sparse') |
| -        | 133±0.5μs            | 72.8±0.5μs          |    0.55 | solve.TimeMatrixOperations.time_rref(3, 0)                           |
| -        | 248±0.4μs            | 88.3±0.5μs          |    0.36 | solve.TimeMatrixOperations.time_rref(4, 0)                           |
| -        | 24.0±0.04ms          | 10.2±0.04ms         |    0.42 | solve.TimeSolveLinSys189x49.time_solve_lin_sys                       |

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[Preypatel2413]

@AdvaitPote ,
Dear Sir,
I created this PR a while ago for solving issues related influence line diagram and calculating reaction loads for the Beam object when using undefined variables. I made changes in solve_for_reaction_loads() to obtain correct reaction forces even for variable distances, and generalized algorithm to solve for ILD equations. It would be very helpful if you could review this PR. Your suggestions will be very helpful for me.

[AdvaitPote]

Sure will do. I have my midsems going on right now so I might take some days to get back to you.

[Preypatel2413]

@AdvaitPote a gentle reminder.

[AdvaitPote]

Add something in the docs below for these parameters. Maybe a small definition for each.

[AdvaitPote]

Try not to leave your Release Notes Empty while making a major change like this

[AdvaitPote]

However, since I haven't worked extensively on the beam module, I'd suggest waiting for additional inputs from other members as well