Lisbon, June 2024

Oriol Colomés* ([email protected]), Jan Modderman ([email protected]), Guglielmo Scovazzi

Abstract

Untted/Immersed or Embedded Finite Element methods have gained signicant attention in the past decade for their eectiveness in simulating problems in complex geometries. However, many existing methods in the literature necessitate ad-hoc modications of standard Finite Element data structures. This often involves tessellation of elements intersected by the embedded boundary, the construction of special quadrature rules, or the denition of geometry-dependent surrogate boundaries. In this work, we propose a novel approach: a generalization of the Weighted Shifted Boundary method [1], which eliminates the need for special data structures or quadrature rules, and enables the solution of untted problems with geometry-independent Finite Element spaces. This newly introduced framework holds particular signicance for problems involving evolving geometries and potential topological changes. Applications extend to scenarios such as topology optimization or fluid-structure interaction problems where the ability to handle dynamic geometries is crucial. In this talk, we will present the formulation of the Generalized Weighted Shifted Boundary method and showcase its application to a variety of problems with evolving domains.

[1] O. Colomés, A. Main, L. Nouveau and G. Scovazzi (2021). A weighted shifted boundary method for free surface flow problems. Journal of Computational Physics, 424, 109837.

Main results

Conclusions:

• Simple, stable, accurate method with embedded moving domains
• Minimizes spurious pressure oscillations in time
• Accuracy in mass/momentum conservation
• Easy to implement in standard FE libraries

What does it enable?

• Parametrized formulation for arbitrary geometries
• Moving domains with topology changes
• Differentiability with respect the time-dependent solution

Drawbacks:

• Integration on all edges
• Solution in the inactive cells (most of them not needed)

Advantages:

• No need to cut cells and use ad-hoc integration rules
• No need to re-define FE spaces (and associated linear algebra structures

Additional references

1. A. Main, G. Scovazzi. The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems.  2018.
2. N.M. Atallah, C. Canuto, and G. Scovazzi. “Analysis of the Shifted Boundary Method for the Stokes problem.” Computer Methods in Applied Mechanics and Engineering 358 (2020): 112609
3. O. Colomés, A. Main, L. Nouveau and G. Scovazzi (2021). A weighted shifted boundary method for free surface flow problems. Journal of Computational Physics, 424, 109837.
4. Xu, OC, A. Main, K. Li, N. Atallah, N. Abboud, G. Scovazzi. A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow. JCP. (2024).
5. Yang, C. H., Saurabh, K., Scovazzi, G., Canuto, C., Krishnamurthy, A., & Ganapathysubramanian, B. Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes. CMAME. (2024)