Elasticity Term
For the elasticity term in the discrete weak form system in Equation (19.1), we can write it as the summation of integrals on each triangle in vector form:
Analogously, this summation also only needs to involve the incident triangles of node .
Recall from Strain Energy, to compute the first Piola-Kirchoff stress , we only need the deformation gradient . From Section Kinematics, we know that . Applying the chain rule with the parameter space variables as intermediates, we have:
which is exactly the same as Equation (15.1.1) from our earlier implementation (Section Inversion-Free Elasticity). Here, we also see that with linear finite elements, the deformation gradient field is piecewise constant in , so is .
Then for , depending on the index of in triangle , we can derive it again using parameter space variables as:
This also allows us to see that is constant within any triangle and it is equivalent to since:
Substituting into Equation (19.3.1) we obtain:
which is exactly how nodal elasticity force is computed in Section Inversion-Free Elasticity. This also indicates that the total elasticity potential can be calculated as , which is before spatial discretization.
Remark 19.3.1. [Linear FEM] Linear FEM refers to being a piecewise linear function of , but the elasticity model can still be nonlinear, i.e. can be a nonlinear function of .