Quasistatic Deformable Simulation

Current formulation of simulation

\begin{aligned} \text{min}_{\Omega}\quad & \int_{\Omega} \rho d(x;\Omega_0)^2 dv + \int_{\partial\Omega} 1 ds \\ \text{s.t.} \quad & \psi(x) \geq 0 \quad \forall x\in\Omega\\ & \int_{\Omega} \rho dv = \int_{\Omega_0} dv \end{aligned}

\begin{aligned} d \mathcal{L}(\Omega;V) & = \int_{\partial\Omega} \left( d(x;\omega_0) + H(x) + \mu - \lambda(x)\psi(x) \right) V(0) \cdot n ds \end{aligned}

Formulation

Shape gradients & Optimality conditions

\begin{aligned} \mu \end{aligned}

\begin{aligned} \lambda(x) \quad \forall x \in \Omega \end{aligned}

\begin{aligned} d \mathcal{L}(\Omega;V) & = \int_{\partial\Omega} \left( d(x;\omega_0) + H(x) + \mu - \lambda(x)\psi(x) \right) V(0) \cdot n ds \end{aligned}

\begin{aligned} \lambda(x)\psi(x) = 0 \end{aligned}

\begin{aligned} \lambda(x) \geq 0 \quad \forall x \in\Omega \end{aligned}

\begin{aligned} \psi(x) \geq 0 \quad \forall x \in\Omega \end{aligned}

\begin{aligned} \int_\Omega 1 dv = \int_{\Omega_0} 1 dv \end{aligned}

Duals

Implementation