Summary
We have successfully implemented an inversion-free 2D elasticity simulation by discretizing the Neo-Hookean model using linear triangle elements.
By maintaining a linearly varying displacement field within each triangle, we can directly calculate a constant deformation gradient for each triangle using both the material and world space coordinates of the vertices. This foundational setup facilitates the computation of the Neo-Hookean energy, as well as its gradient and Hessian with respect to , by applying the chain rule. These calculations are essential for the optimization-based time integration discussed in previous lectures.
To ensure the simulation remains free of both interpenetration and inversion, we adopt a similar strategy as previously described: the initial step size in the line search is determined by solving a quadratic equation for each triangle. This equation calculates a critical step size that reduces the triangle's volume by 90%. The smallest of these critical step sizes across all triangles is then used to initialize the line search, ensuring robustness against both non-interpenetration and non-inversion.
In the upcoming chapter, we will delve into the derivation of the governing equations for hyperelastic solids, providing a detailed explanation of each step to further solidify understanding.