Quarto slide-template

E4H flavour

Alexandre Daby-Seesaram

alexandre.daby-seesaram@polytechnique.edu

LMS, École Polytechnique, Paris, France

Second Author

second.author@polytechnique.edu

LMS, École Polytechnique, Paris, France

Third Author

third.author@polytechnique.edu

LMS, École Polytechnique, Paris, France

September 10, 2025

Slide title

Image caption

• You can also input videos as
  • ![Caption](Figures/Movies/your_video.mp4){ width=80% loop="true" autoplay=true}
  • with the possibility to specify the size and the reading options
    • Loop
    • Autoplay
    • More options here

Default box title

• Use boxes to organise your content
• This is what a default mybox looks like

Second box version

• Here is a second header version

• With different colours

Latex equations

\[ \definecolor{Violet}{RGB}{162, 32, 185} \definecolor{Teal}{RGB}{0, 103, 127} \definecolor{Blue}{RGB}{58, 70, 245} \definecolor{Green}{RGB}{0,103,127} \definecolor{LGreen}{RGB}{62,128,102} \definecolor{red}{RGB}{206,0,55} \]

• You can use latex just in the same way as you would in a beamer

\[ \boldsymbol{u} = \mathop{\mathrm{arg\,min}}_{H^1\left(\Omega \right)} \int_{\Omega} \Psi\left( \boldsymbol{E}\left(\boldsymbol{u}\left(\textcolor{Blue}{\boldsymbol{x}}, \textcolor{LGreen}{\left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}}\right)\right) \right) ~\mathrm{d}\Omega- W_{\text{ext}}\left(\textcolor{Blue}{\boldsymbol{x}}, \textcolor{LGreen}{\left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}}\right) \]

• \(\boldsymbol{F} = \boldsymbol{1} + \boldsymbol{\nabla}\boldsymbol{u}\) - deformation gradient
• \(\boldsymbol{C} = \boldsymbol{F}^T \cdot \boldsymbol{F}\) - right Cauchy-Green tensor
• \(\boldsymbol{E} = \frac{1}{2}\left(\boldsymbol{C} -\boldsymbol{1} \right)\) - Green-Lagrange tensor
  • \(I_1 = \text{tr}\left(\boldsymbol{C} \right)\) - first invariant of \(\boldsymbol{C}\)

Works just as well in boxes

\(\Psi = \frac{\lambda}{2} \text{tr}\left(\boldsymbol{E}\right)^2 + \mu \boldsymbol{E}:\boldsymbol{E}\)

• Where you can add
  • More content
    • Well organised

And you can

• Put equations
  • In lists \(\left(\textcolor{Blue}{\boldsymbol{x}}, \textcolor{LGreen}{\left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}}\right)\)
    • While benefiting from the same macros you use in day-to-day LaTeX use

Part II

You can display your outlines

On the Part slides with a distinct style

To emphasise the transition

Between a part and another

Other styling options

• Still using column to organise content in the slide

\[ \mathcal{U}_h = \left\{\boldsymbol{u}_h \; | \; \boldsymbol{u}_h \in \text{Span}\left( \left\{ N_i^{\Omega}\left(\boldsymbol{x} \right)\right\}_{i \in \mathopen{~[\!\![~}1,N\mathclose{~]\!\!]}} \right)^d \text{, } \boldsymbol{u}_h = \boldsymbol{u}_d \text{ on }\partial \Omega_d \right\} \]

• You can use
  • The .green-bullets environment
  • To highlight positive aspects

• Or
  • The .red-bullets environment
  • For negative aspects

• With different sliding animations

\[ \boldsymbol{u} \left(x_{0,0,0} \right) = \sum\limits_{i = 0}^C \sum\limits_{j = 0}^{N_i} \sigma \left( \sum\limits_{k = 0}^{M_{i,j}} b_{i,j}+\omega_{i,j,k}~ x_{i,j,k} \right) \]

• More animations exist, including
  • Morphing
    • To link a piece of the slide through different slides
  • Fade-in-then-out
    • To replace pieces of slides with new ones

Morphing animation - Slide 1

• You can easily cite items from a .bib file, which appears in the bibliography list.(Chinesta et al., 2011; Ladeveze, 1985)

Some content

• Use .r-stack div to allow selecting several element to be put on top of each other
  • To be used with fragment to incrementally reveal elements
    • see here
• Use data-id="xxx" to name the fragments

\[ \textcolor{VioletLMS_2}{\boldsymbol{u}}\left(\textcolor{Blue}{\boldsymbol{x}}, \textcolor{LGreen}{\left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}}\right) = \sum\limits_{i=1}^m \textcolor{Blue}{\overline{\boldsymbol{u}}_i(\boldsymbol{x})} ~\textcolor{LGreen}{\prod_{j=1}^{\beta}\lambda_i^j(\mu^j)}\]

• Here we show how to morph some content from one slide
• To another

Morphing animation - Slide 2

• Here, the target shape and position of the morphing animation are set
• We also use .fragment fragment and fragment-index to incrementally replace the equations

\[ \boldsymbol{u}\left(\boldsymbol{x}, \left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}\right) = \overline{\boldsymbol{u}}(\boldsymbol{x}) ~\prod_{j=1}^{\beta}\lambda^j(\mu^j) \]

\[ \boldsymbol{u}\left(\boldsymbol{x}, \left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}\right) = \textcolor{Red}{\sum\limits_{i=1}^{2}} \overline{\boldsymbol{u}}_{\textcolor{Red}{i}}(\boldsymbol{x}) ~\prod_{j=1}^{\beta}\lambda_{\textcolor{Red}{i}}^j(\mu^j) \]

\[ \boldsymbol{u}\left(\boldsymbol{x}, \left\{\mu_i\right\}_{i \in \mathopen{~[\!\![~}1, \beta \mathclose{~]\!\!]}}\right) = \sum\limits_{i=1}^{\textcolor{Red}{m}} \overline{\boldsymbol{u}}_i(\boldsymbol{x}) ~\prod_{j=1}^{\beta}\lambda_i^j(\mu^j) \]

Incremental reveal

1. We start with the target of the morphing
2. That can be different from the source

3. We then fade it out
4. And fade-in (then-out) its replacement

5. And again

Python plots

Loss convergence

Loss decay

Using plotly

• Interactive plots
  • Pan, zoom in, explore the plots, click on the legend

• I recommend saving .csv from the results and loading them to create the plots

3D interactive plots

• You can simply use the 3D plotting options of plotly to get interactive 3D plots

• You can also use html iframe to embed 3D html
  • Those can be created from FEM results
  • And stylised in Blender
    • as described in (Mathur et al., 2023)

References

Chinesta, F., Ladeveze, P., & Cueto, E. (2011). A Short Review on Model Order Reduction Based on Proper Generalized Decomposition. Archives of Computational Methods in Engineering, 18(4), 395–404. https://doi.org/10.1007/s11831-011-9064-7

Ladeveze, P. (1985). Sur une famille d’algorithmes en mécanique des structures. Sur Une Famille d’algorithmes En Mécanique Des Structures, 300(2), 41–44.

Mathur, M., Brozovich, J. M., & Rausch, M. K. (2023). A brief note on building augmented reality models for scientific visualization. Finite Elements in Analysis and Design, 213, 103851. https://doi.org/10.1016/j.finel.2022.103851