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Formulations

The following formulations are available in the PolyFEM list of possible formulations. The constants can be set in params. All formulations support boundary conditions. For the elasticity formulations we output:

Cauchy stress tensor

\[ \sigma = \frac 1 J \frac{\partial \Psi}{\partial F} F^T \]

Frist Piola Kirchhoff stress tensor (Wikipedia)

\[ P = J \sigma F^{-T} \]

Second Piola Kirchhoff stress tensor (Wikipedia)

\[ S = J F^{-1} \sigma F^{-T}, \]

where \(\Psi\) is the energy density, \(F\) the deformation gradient, and \(J=\det(F)\).

Scalar

Laplacian

  • Constants: none
  • Description: solve for \(-\Delta u = f\)

Bilaplacian (mixed)

  • Constants: none
  • Description: solve for \(-\Delta^2 u = f\)

Helmholtz

  • Constants: \(k\)
  • Description: solve for \(-\Delta u - k^2 u = f\)

Tensor

Linear Elasticity

  • Constants: E/nu, lambda/mu
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where
\[ \sigma[u] = 2 \mu \epsilon[u]+ \lambda \text{tr}(\epsilon[u]) I \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T + \nabla u\right) \]

Hooke Linear Elasticity

  • Constants: elasticity_tensor, E/nu, lambda/mu
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where
\[ \sigma[u] = C : \epsilon[u] \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T + \nabla u\right) \]

where \(C\) is the elasticity tensor

Incompressible Linear Elasticity (mixed)

  • Constants: E/nu, lambda/mu
  • Description: solve for
\[\begin{align} -\text{div}(2\mu\epsilon[u] + p I) &= f\\ \text{div}(u) - \lambda^{-1}p &= 0 \end{align}\]

Saint Venant–Kirchoff Elasticity

  • Constants: elasticity_tensor, E/nu, lambda/mu
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where
\[ \sigma[u] = C: \epsilon[u] \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T \nabla u + \nabla u^T + \nabla u\right) \]

where \(C\) is the elasticity tensor

NeoHookean Elasticity

  • Constants: E/nu, lambda/mu
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where
\[ \sigma[u] = \mu (F[u] - F[u]^{-T}) + \lambda \ln(\det F[u]) F[u]^{-T} \qquad F[u] = \nabla u + I \]

Mooney-Rivlin Elasticity

  • Constants: c1/c2/k=1
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where \(\sigma[u]=\nabla_u \Psi[u]\). The energy density \(\Psi\) is
\[ \Psi[u] = c_1 (\widetilde{I_1} - d) + c_2 (\widetilde{I_2} - d) + \frac{k}{2} \ln^2(J) \]

where \(d\) is the dimension (2 or 3),

\[\begin{align} F = \nabla u + I, \quad J = \det(F), \quad \tilde{F} = \frac{1}{\sqrt[d]{J}} F, \quad \widetilde{C} = \widetilde{F} \widetilde{F}^T, \\ \widetilde{I_1} = \text{tr}\left(\widetilde{C}\right), \quad \text{and} \quad \widetilde{I_2} = \frac{1}{2} \left(\left(\trace{\widetilde{C}}\right)^2 - \text{tr}\left(\widetilde{C}^2\right)\right). \end{align}\]

Incompressible Ogden Elasticity

  • Constants: c/m/k
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where \(\sigma[u]=\nabla_u \Psi[u]\). The energy density \(\Psi\) is
\[ \Psi[u] = \sum_{i=1}^N \frac{c_i}{m_i^2} \left( \sum_{j=1}^d \tilde{\lambda}_j^{m_i} - d \right) + \frac{1}{2} K \ln(J)^2 \]

where \(N\), the number of terms, is dictated by the number of coefficients given, \(d\) is the dimension (2 or 3), \(J = \det(F)\) where \(F = \nabla u + I\), and \(\tilde{\lambda}_j = J^{-\frac{1}{d}}\lambda_j\) are the eigenvalues of \(\tilde{F}\) (same as in the Mooney-Rivlin Elasticity).

Unconstrained Ogden Elasticity

  • Constants: mus/alphas/Ds
  • Description: solve for \(-\text{div}(\sigma[u]) = f\) where \(\sigma[u]=\nabla_u \Psi[u]\). The energy density \(\Psi\) is
\[ \Psi[u] = \sum_{i=1}^N \frac{2 \mu_i}{\alpha_i^2} \left( \sum_{j=1}^d \tilde{\lambda}_j^{\alpha_i} - d \right) + \sum_{i=1}^N \frac{\left(J-1\right)^{2 i}}{D_i} \]

where \(N\), the number of terms, is dictated by the number of coefficients given, \(d\) is the dimension (2 or 3), \(J = \det(F)\) where \(F = \nabla u + I\), and \(\tilde{\lambda}_j = J^{-\frac{1}{d}}\lambda_j\) where \(\lambda_j\) are the eigenvalues of \(F.\)

Viscous Damping

  • Constants: phi/psi
  • Description: an extra energy that represents dissipation, adding to the elastic energy in transient problems
\[ R(F,\dot{F})=\psi \|\dot{E}(F,\dot{F})\|^2+\frac{\phi}{2}\text{tr}^2\dot{E}(F,\dot{F}) \]

where \(F[u] = \nabla u + I\) and \(E[u] = \frac{1}{2}(F^TF-I)\).

The above corresponds to the viscous Piola-Kirchhoff stress

\[ P=F(2\psi\dot{E}+\phi\text{tr}(\dot{E})I)=\nabla_2 R(F,\dot{F}). \]

Stokes (mixed)

  • Constants: viscosity (\(\nu\))
  • Description: solve for
\[\begin{align} -\nu\Delta u + \nabla p &= f\\ -\text{div}(u) &= 0 \end{align}\]
  • Constants: viscosity (\(\nu\))
  • Description: solve for
\[\begin{align} u\cdot \nabla u -\nu\Delta u + \nabla p &= f\\ -\text{div}(u) &= 0 \end{align}\]

Implementing New Formulations

Todo

Describe how to implement a new formulation in C++.

Last update: 2023-10-03