High Order Unfitted Finite Element Method for Elliptic Interface Problems with Automatic Mesh Generation

핵심 개념

The author presents an adaptive high-order unfitted finite element method for elliptic interface problems, focusing on automatic mesh generation and reliable implementation. The approach involves a posteriori error estimates and competitive performance demonstrations.

초록

The content discusses the development of an advanced unfitted finite element method for solving elliptic interface problems with automatic mesh generation. It introduces key concepts like cell merging, hp a posteriori error estimates, and reliable algorithms for induced mesh generation. The method's performance is illustrated through numerical examples, emphasizing its effectiveness in handling curved geometric singularities.

통계

An hp a posteriori error estimate is derived for a new unfitted finite element method.
The small cut cell problem is addressed by the ghost penalty or merging small cut cells.
The main purpose is to develop a reliable algorithm to automatically generate the induced mesh.
The induced mesh avoids possible small intersections of the interface or boundary with the elements.

인용구

"The widely used newest vertex bisection algorithm can no longer guarantee the shape regularity of the refined meshes."
"The reliability of the algorithm to generate the induced mesh is also established."

핵심 통찰 요약

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

by Zhiming Chen... 게시일 arxiv.org 03-07-2024

https://arxiv.org/pdf/2403.03418.pdf

더 깊은 질문

How does this advanced method compare to traditional finite element approaches

The advanced unfitted finite element method described in the context above offers several advantages over traditional finite element approaches. One key difference is the ability to handle geometrically curved singularities and piecewise smooth interfaces more effectively. By using a fixed background mesh with different finite element functions in cut cells, this method eliminates the need for body-fitted meshes, simplifying the mesh generation process. Additionally, the adaptive high-order capabilities of this method allow for efficient refinement strategies based on a posteriori error estimates, leading to improved accuracy and convergence rates. The use of hp-quasi-interpolation operators further enhances the approximation quality of the solution.

What are potential limitations or challenges when applying this method in practical engineering simulations

While the advanced unfitted finite element method presents many benefits, there are also potential limitations and challenges when applying it in practical engineering simulations. One challenge lies in ensuring proper implementation and validation of complex algorithms involved in automatic mesh generation for piecewise smooth interfaces. The reliability and efficiency of merging small cut cells with surrounding elements can impact computational performance significantly. Moreover, controlling stability issues related to interface penalty terms and boundary conditions requires careful consideration to avoid numerical instabilities or inaccuracies that may arise during simulations.

How might advancements in automatic mesh generation impact other fields beyond mathematics

Advancements in automatic mesh generation driven by methods like unfitted finite element techniques have broader implications beyond mathematics into various fields such as engineering, physics, biology, and computer science. In engineering disciplines like structural analysis or fluid dynamics simulations, automated meshing tools can streamline model preparation processes while improving simulation accuracy and efficiency. In physics research areas involving complex geometries or material interfaces, automatic mesh generation can facilitate detailed modeling without manual intervention. Furthermore, advancements in this technology could enhance interdisciplinary collaborations by enabling researchers from different domains to leverage sophisticated numerical methods for their specific applications efficiently.