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

we now have a "ShapeMathematicalProgram" which acts like Drake's MP

Program for mean curvature flow 

2D Results

Initial iteration

Final iteration

2D Results

2D Results

Initial iteration

Final iteration

2D Results

Comparison with PlasticineLab

Real Video

PlasticineLab

Bug / Feature: Slicing 

3D Simulation

Next Steps

Simulation side: 

Planning side:

Hardware & Experiments: 

-  More scenarios to try, figure out limitations of this simulation method 

-  More simulators to set up and compare against - Nvidia PhysX / FleX

-  Can we do better than MCTS....?

-  Can we get a pottery table?

Copy of QuasistaticDeformable

By Terry Suh

Copy of QuasistaticDeformable

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