[Backward] Fix trapezoidal rule in EulerImplicitSolver

[hugtalbot]

Fix #5140

Pendulum__r60_0002.mp4

NB: it would be nice to have the image of the issue added (as text) in the doc

[ci-depends-on https://github.com/sofa-framework/Regression/pull/74]

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[alxbilger]

Here is what I have for Backward Euler:

$$ \left( \left( 1 + \alpha \Delta t \right) M - \Delta t B - \Delta t \left( \Delta t + \beta \right) K \right) \Delta v = \Delta t F(x_n, v_n) + \Delta t \left( -\alpha M + \left( \beta + \Delta t \right) K \right) v_n $$

and for the trapezoidal rule:

$$ \left[ \left(1 + \frac{1}{2} \alpha \Delta t \right) M - \frac{1}{2} \Delta t B - \frac{1}{2} \Delta t \left( \frac{1}{2} \Delta t + \beta \right) K \right] \Delta v = \Delta t F(x_n, v_n) + \Delta t \left( -\alpha M + \left( \beta + \frac{1}{2} \Delta t \right) K \right) v_n $$

So, your changes make sense to me

[fredroy]

[ci-build][with-all-tests]

[bakpaul]

Could you make a PR on Regression and add a ci-depends-on ?

[sofabot]

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[hugtalbot]

[ci-build][with-all-tests]

[sofabot]

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