Level Set#

Level Set Representations#

OpenPisco supports several level set representations adapted to different simulation and optimization workflows.

🟦 Structured Level Sets

Level sets defined on Cartesian grids.

Structured Level Set

🔺 Unstructured Level Sets

Level sets defined on simplicial meshes.

Unstructured Level Set

🧩 Body-Fitted Level Sets

Lagrangian representation relying on mesh conformity to the interface.

Body-fitted Level Set

Level Set Operations#

Regularization#

Purpose#

Extend and smooth a velocity field initially defined on the structural interface so that it can be used reliably during level set transport.

Why is it needed?#

Shape derivatives are naturally computed on the interface only. Regularization extends this information to the surrounding domain while reducing numerical oscillations and improving robustness.

Mathematical formulation#

The regularized velocity field \(V \in H^1(D)\) is obtained by solving

\[\forall w\in H^{1}(D), \:\: \alpha \int_{D}{\nabla V \cdot \nabla w \:dx } + \int_{\partial\Omega}{Vw\:ds} = \int_{\partial\Omega}{v_{\partial\Omega}w\:ds}\]

where \(\alpha > 0\) controls the amount of regularization and \(v_{\partial\Omega}\) denotes the input interface velocity.

Implementation#

Implemented by:

  • OpenPisco.Unstructured.Levelset.Regularize()

The variational problem is solved using the finite element capabilities provided by Muscat 1.

Transport#

Purpose#

Move the level set interface according to a prescribed velocity field.

Why is it needed?#

Transport is the main mechanism used to evolve the geometry during shape and topology optimization iterations.

Mathematical formulation#

The level set function is transported by solving

\[\frac{\partial \phi}{\partial t} + \theta \cdot \nabla \phi = \frac{\partial \phi}{\partial t} - v |\nabla \phi| = 0\]

where \(\theta = -vn\) is the normal velocity field.

Implementation#

Implemented by:

  • OpenPisco.Unstructured.Levelset.Transport()

  • OpenPisco.Unstructured.Levelset.Advect

The advection problem is solved through an interface to the open-source Advection library 2.

Reinitialization#

Purpose#

Restore the signed-distance property of a level set function.

Why is it needed?#

After several transport steps, the level set field progressively deviates from a signed-distance function. Reinitialization improves numerical stability while preserving the interface location.

Mathematical formulation#

The signed-distance function \(d_\Omega\) is defined as

\[\begin{split}d_{\Omega}(x)= \begin{cases} -d(x,\partial\Omega) & x\in\Omega \\ 0 & x\in\partial\Omega \\ d(x,\partial\Omega) & \text{otherwise} \end{cases}\end{split}\]

where

\[d(x,\partial\Omega)= \min_{p\in\partial\Omega}|x-p|.\]