Help with Adaptive Quadrature in Finite Cell Method

[Armored]

Hello everyone,

I am working on a 3D linear elasticity problem and developing a lower order stress simulation framework from scratch. For this, I’m using the Finite Cell Method (a fictitious domain approach) with axis-parallel hexahedral cells. These cells are refined into tetrahedra or smaller hexahedra, categorized as either fully inside or fully outside the domain boundary.

The meshing is working well, but I am stuck on implementing adaptive integration for cells cut by the domain surface. I’ve been using 2x2x2 Gauss quadrature for regular hexahedra, but I am unsure how to properly account for the geometry of intersected cells. My current approach involves weighting the integration based on the volume fraction inside the domain (using a very small weight for outside points), but the results don’t seem correct.

Has anyone worked with similar setups or implemented adaptive integration for cut cells? I’ve attached my routine for stiffness calculation of surface-intersected cells where all elements are inside the domain. Any insights or code review would be greatly appreciated!

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[rb1957]

it does say something of a post that can have 197 views and this is the only reply !? Possibly no one understands the question ?? I suspect this is a thesis project ? and maybe too specialised for work-a-day "oompa loompas" such as us ?? (and yes, I'm braced for the "speak for yourself" replies.)

 

[Coreform_Greg]

You've come upon one of the greatest challenges with FCM (and related methods). I've not read your code, but it's not surprising that you're having issues -- especially if by "the results don't seem correct" you're referring to simulation results rather the correctness of your quadrature routine. In our experience, adaptive quadrature is an extremely inefficient approach to integrating cut-cells. We (Coreform) have some unique things that we're doing, which I can't talk too much about, but in the open literature some of the most promising approaches include:

• Robust numerical integration on curved polyhedra based on folded decompositions
• Efficient cut-cell quadrature based on moment fitting for materially nonlinear analysis

 

[rb1957]

great, now I have some new reading material ...