Dynamic Elastic Simulation with FDM, FVM, and/or FEM

This lecture covers slides 21 .

Reading Material

Here is a list of resources to help you learn more.

  • Chapter 2 and Section 3.4 in Kenny's book notes
  • J. Teran, S. Blemker, V. Ng Thow Hing, and R. Fedkiw. 2003. Finite volume methods for the simulation of skeletal muscle. In Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation (SCA '03). Eurographics Association, Goslar, DEU, 68–74. https://dl.acm.org/doi/10.5555/846276.846285. 
    Eftychios Sifakis and Jernej Barbic. 2012. FEM simulation of 3D deformable solids: a practitioner's guide to theory, discretization and model reduction. In ACM SIGGRAPH 2012 Courses (SIGGRAPH '12). Association for Computing Machinery, New York, NY, USA, Article 20, 1–50. https://doi.org/10.1145/2343483.2343501. 
  • Theodore Kim and David Eberle. 2022. Dynamic deformables: implementation and production practicalities (now with code!). In ACM SIGGRAPH 2022 Courses (SIGGRAPH '22). Association for Computing Machinery, New York, NY, USA, Article 7, 1–259. https://doi.org/10.1145/3532720.3535628.

Idealization of a Dynamic Elastic Beam

  • This video summarizes the physics equations that are used as part of the finite volume method discretization.
  • Covers slides 21 (pages 2-4)

Median Dual Vertex Centred Control Volume

  • This video defines the median dual vertex-centered control volume.
  • The aim is to learn how to create and define this type of volume.
  • Covers slides 21 (pages 5-6)

Control Volume Notation and Terminology

  • This video introduces notation and terminology that is used when working with the median dual vertex-centred control volume.
  • Covers slides 21 (pages 7-8)

Finite Volume Method on The Cauchy Equation

  • This video introduces the steps that need to be done in the spatial discretization process.
  • The aim is to provide an overview of the process such that students can work out the details themselves.
  • Covers slides 21 (pages 9-11)

Computing The Deformation Gradient

  • This video outlines the math behind computing the deformation gradient.
  • The aim is to learn how linear elements and differentials can be exploited to derive an efficient computational formula.
  • Covers slides 21 (pages 12-13)

Picking The Mesh Layout

  • This video outlines a possible mesh layout for how to create a data structure for storing values during simulation.
  • The aim is to learn that vector/control volume quantities are stored at vertices and tensors are stored at triangles and edges only keeping geometry information about themselves.
  • Covers slides 21 (page 14)

The Clever Rewrite

  • This video demonstrates a convenient transformation of the elastic surface (line in 2D) can be conveniently transformed.
  • The aim is to learn the technicalities in the mathematical rewrite and be able to account for how this rewrite is connected to the choice of the control volume.
  • Covers slides 21 (pages 16-18)

Adding Time Discretization

  • This video outlines the math for proper time discretization.
  • The aim is to learn how the semi-implicit first-order time-stepping method is derived.
  • Covers slides 21 (page 19)

The Final Numerical Method

  • This video summarizes all the past derivations into a step-by-step guide on how to implement the computations
  • The aim is to learn how to order the computations and gives an idea for structuring one's own code when implementing a simulator.
  • Covers slides 21 (page 20)

Oh no! The Model Collapse Issue

  • This video analyzes how the Saint Venant Kirchoff model behaves under extreme extension/compression.
  • The aim is to understand when and how material failures may occur with this material model.
  • Covers slides 21 (pages 23-26)