Elasticity Term

For the elasticity term $\int_{Ω^{0}} N_{\overset{a}{^}, j} (X) P_{\hat{i} j} (X, t^{n}) d X$ in the discrete weak form system in Equation (19.1), we can write it as the summation of integrals on each triangle $e$ in vector form:

$\int_{Ω^{0}} N_{\overset{a}{^}, j} (X) P_{\hat{i} j} (X, t^{n}) d X = \int_{Ω^{0}} (P (X, t^{n}) \nabla^{X} N_{\overset{a}{^}} (X))_{\hat{i}} d X = e \in T \sum \int_{Ω_{e}^{0}} (P (X, t^{n}) \nabla^{X} N_{\overset{a}{^}} (X))_{\hat{i}} d X . (19.3.1)$

Analogously, this summation also only needs to involve the incident triangles of node $\overset{a}{^}$ .

Recall from Strain Energy, to compute the first Piola-Kirchoff stress $P (X, t^{n})$ , we only need the deformation gradient $F (X, t^{n})$ . From Section Kinematics, we know that $F = \frac{\partial x}{\partial X}$ . Applying the chain rule with the parameter space variables $(β, γ)$ as intermediates, we have:

$F = \frac{\partial x}{\partial ( β , γ )} (\frac{\partial X}{\partial ( β , γ )})^{- 1} \approx \frac{\partial x ^}{\partial ( β , γ )} (\frac{\partial X}{\partial ( β , γ )})^{- 1} = [x_{2} - x_{1}, x_{3} - x_{1}] [X_{2} - X_{1}, X_{3} - X_{1}]^{- 1}, (19.3.2)$

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 $Ω^{0}$ , so is $P$ .

Then for $\nabla^{X} N_{\overset{a}{^}} (X)$ , depending on the index of $\overset{a}{^}$ in triangle $e$ , we can derive it again using parameter space variables as:

$\nabla^{X} N_{1} (X) \nabla^{X} N_{2} (X) = ([1, 0] [X_{2} - X_{1}, X_{3} - X_{1}]^{- 1})^{T} \nabla^{X} N_{3} (X) = ([0, 1] [X_{2} - X_{1}, X_{3} - X_{1}]^{- 1})^{T} . = \frac{\partial ( 1 - β - γ )}{\partial X} = (\frac{\partial ( 1 - β - γ )}{\partial ( β , γ )} (\frac{\partial X}{\partial ( β , γ )})^{- 1})^{T} = ([- 1, - 1] [X_{2} - X_{1}, X_{3} - X_{1}]^{- 1})^{T} = \frac{\partial β}{\partial X} = (\frac{\partial β}{\partial ( β , γ )} (\frac{\partial X}{\partial ( β , γ )})^{- 1})^{T} = \frac{\partial γ}{\partial X} = (\frac{\partial γ}{\partial ( β , γ )} (\frac{\partial X}{\partial ( β , γ )})^{- 1})^{T}$

This also allows us to see that $P (X, t^{n}) \nabla^{X} N_{\overset{a}{^}} (X)$ is constant within any triangle $e$ and it is equivalent to $\frac{\partial Ψ _{e}}{\partial x _{\overset{a}{^}}}$ since:

$\frac{\partial Ψ _{e}}{\partial x _{\overset{a}{^}}} = \frac{\partial Ψ _{e}}{\partial F} \frac{\partial F}{\partial x ^} \frac{\partial x ^}{\partial x _{\overset{a}{^}}} = P \frac{\partial x ^ / \partial X}{\partial x ^} N_{\overset{a}{^}} = P \nabla^{X} N_{\overset{a}{^}} .$

Substituting $\frac{\partial Ψ _{e}}{\partial x _{\overset{a}{^}}}$ into Equation (19.3.1) we obtain:

$\int_{Ω^{0}} N_{\overset{a}{^}, j} (X) P_{\hat{i} j} (X, t^{n}) d X = e \in T \sum \int_{Ω_{e}^{0}} (P (X, t^{n}) \nabla^{X} N_{\overset{a}{^}} (X))_{\hat{i}} d X = e \in T \sum \int_{Ω_{e}^{0}} (\frac{\partial Ψ _{e}}{\partial x _{\overset{a}{^}}})_{\hat{i}} d X = e \in T \sum A_{e} (\frac{\partial Ψ _{e}}{\partial x _{\overset{a}{^}}})_{\hat{i}},$

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 $\sum_{e \in T} A_{e} Ψ_{e}$ , which is $\int_{Ω^{0}} Ψ (X) d X$ before spatial discretization.

Remark 19.3.1. [Linear FEM] Linear FEM refers to $x$ being a piecewise linear function of $X$ , but the elasticity model can still be nonlinear, i.e. $P$ can be a nonlinear function of $F$ .