Numerical solutions
- Lagrangian and Euler description leads to PDE (Partial Differential Equations)
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- In general: no explicit solutions \(\Rightarrow\) approximate numerical solution
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- - Discretize in space on a grid \(\Delta x\) and time \(\Delta t\)
- - Use numerical approximation of derivatives using masks in space and time
- (+) Very general, simple to setup
- (+) Works well with rectangular grid (ex. Euler description)
- (-) Difficult to handle shape boundaries
- (-) Instabilities
- Finite elements method (FEM)
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- - Discretize the shape into simple elements. Build continuous function on each element.
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- In CG: Elements are tetrahedron (in volume). Continuous function are barycentric coordinates (linear interpolation functions).
- - Integrate PDE over each element (weak formulation), leads to a linear system
- (+) Handle boundaries (ex. Deforming solid)
- (+) Guarantee on accuracy
- (-) Complex to set up, and computationally heavy
- (-) Requires good quality meshing
