🔢 mathematics

Hybrid CutFEM-IFEM model for embedded and immersed interfaces with triply-cut cells for the 2D Poisson problem

This paper presents a hybrid CutFEM-IFEM method for 2D Poisson problems that combines unfitted finite elements with immersed interface techniques to efficiently handle complex fluid-fluid and fluid-solid interfaces on a single mesh, utilizing composite polynomial representations for triply-cut cells and enriched basis functions to achieve optimal convergence and high accuracy comparable to fitted meshes.

Original authors: Darsh Nathawani, Christopher Kees

Published 2026-10-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Darsh Nathawani, Christopher Kees

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to simulate how water flows around a complex rock formation while simultaneously tracking the boundary between two different types of water, perhaps fresh and salt, or oil and water. In the world of engineering and physics, these interactions happen constantly, from the waves crashing against a seawall to the movement of air and water around a ship. To understand these phenomena, scientists use mathematical models to predict how fluids behave. However, a major hurdle has always been the mesh, a grid of tiny shapes used to divide the space for calculation. Traditionally, this grid had to be perfectly shaped to fit every curve of the rock and every line where the fluids met. If the rock moved or the water line shifted, the entire grid had to be rebuilt, a process that is slow, fragile, and computationally expensive.

To solve this, researchers developed "unfitted" methods, where the grid remains fixed and the objects simply cut through it. Two powerful techniques emerged to handle these cuts: one method, known as the immersed interface method, is excellent at capturing sharp jumps in properties between two fluids, while another, called the cut finite element method, is ideal for handling solid boundaries where fluid stops. For a long time, these two approaches worked in separate lanes. The challenge was to combine them into a single, robust system that could handle a solid object sitting inside a domain where two different fluids are also interacting, all without rebuilding the grid.

In this work, researchers at Louisiana State University have successfully merged these two techniques into a single hybrid model. They created a new way to simulate a three-phase problem: two different fluids and one solid object, all interacting on a single, unchanging background grid. The key innovation lies in how they handle the tiny, messy pieces of the grid that get sliced by both the fluid boundary and the solid object simultaneously. When a single grid cell is cut by two different lines, it creates a complex, three-part shape. The team developed a mathematical trick using "equivalent polynomials" to calculate the physics inside these tricky, triply-cut cells without needing to break them down into smaller, simpler shapes. This allows the computer to integrate the equations accurately, just as it would on a standard grid, even when the geometry is highly irregular.

The researchers tested their method on a scenario involving a circular solid object and a straight line separating two fluids. They ran simulations using both linear and quadratic elements, which are different levels of mathematical complexity used to approximate the solution. The results showed that their hybrid method achieved the highest possible accuracy for the type of math they used, matching the precision of a traditional method that requires a perfectly fitted mesh. Crucially, this high accuracy was maintained even when the difference in properties between the two fluids was extreme, with the ratio of their material constants reaching up to one thousand. This suggests the method is robust enough to handle real-world scenarios where materials differ vastly, such as air and water or oil and water.

The study also explored how the method handles different types of boundaries. When the solid object had a standard "no-slip" condition, the method performed perfectly. However, when they introduced a more complex boundary condition that models heat transfer, the accuracy for the most complex mathematical elements was slightly limited. This limitation was not due to a flaw in the new method, but rather because the solid object's curved edge was represented by straight lines in the grid. The researchers noted that to achieve perfect accuracy in such cases, a more advanced representation of the curve would be needed in future work.

Ultimately, this research provides a powerful tool for engineers and scientists who need to model complex, moving interfaces without the burden of constantly remeshing. By combining the strengths of two existing methods and solving the specific problem of triply-cut cells, the team has demonstrated that it is possible to achieve optimal accuracy and efficiency in simulations involving multiple phases. The method handles strong discontinuities and complex geometries with a stability that rivals traditional approaches, offering a reliable path forward for simulating the dynamic interactions of fluids and solids in coastal and environmental engineering.

Technical Summary: Hybrid CutFEM-IFEM Model for Embedded and Immersed Interfaces

Problem Statement
The paper addresses the numerical solution of the 2D Poisson problem involving three distinct phases: two fluid phases ( $\Omega_a, \Omega_b$ ) and one solid phase ( $\Omega_s$ ). The domain contains two types of interfaces:

Immersed Interface ( $\Gamma_f$ ): Separates the two fluid phases, characterized by discontinuities in material properties ( $\mu$ ), solution ( $u$ ), and flux ( $\mu \partial u / \partial n$ ).
Embedded Interface ( $\Gamma_s$ ): Represents the boundary of the solid region, which is removed from the computational domain, requiring the imposition of boundary conditions (Dirichlet or Robin).

The challenge lies in simulating these interactions on a single, fixed background mesh where both interfaces may cut through the same elements, creating "triply-cut cells." Traditional mesh-fitting approaches are computationally expensive and fragile when interfaces move or deform. Unfitted methods are preferred, but existing techniques often struggle to simultaneously handle strong discontinuities (via Immersed Finite Element Method, IFEM) and embedded boundaries (via CutFEM) within the same cut cell, particularly regarding integration and consistency.

Methodology
The authors propose a hybrid CutFEM–IFEM formulation that leverages the strengths of both methods on a single unfitted mesh:

Basis Functions and Enrichment:
- For cells cut by the fluid-fluid interface ( $\Gamma_f$ ), the method employs IFEM basis functions. These are constructed to be continuous across the interface but discontinuous across element edges, exactly satisfying the jump conditions for the solution and flux.
- To handle non-homogeneous jumps (prescribed jumps $J_s$ and $J_f$ ), the trial function is augmented with specific "lift" functions ( $j_h$ for solution jumps and $q_h$ for flux jumps). These enrichment functions reproduce the exact jumps, ensuring the error does not depend on the magnitude of the discontinuity.
- For the embedded solid boundary ( $\Gamma_s$ ), Nitsche's method is used to weakly enforce boundary conditions (Dirichlet or Robin) without adding degrees of freedom.
Weak Formulation and Stability:
- The weak form includes standard Nitsche terms for the embedded boundary and ghost penalty terms to stabilize the system against small cut fractions.
- A critical modification is introduced for triply-cut cells. Because the solid boundary removes a portion of the edge shared by two fluid-cut elements, standard IFEM edge integrals become inconsistent. The authors derive specific consistency correction terms ( $c(u_h, v_h)$ ) that account for the flux on the remaining fluid portion of the cut edges, restoring optimal convergence.
Integration on Triply-Cut Cells:
- The paper extends the equivalent polynomial technique to handle integration over cells cut by multiple interfaces.
- Instead of subdividing cells into sub-triangles, the method replaces Heaviside and Dirac delta functions with equivalent polynomials.
- For triply-cut cells, the authors categorize the eight possible topological configurations of the cuts (based on which nodes are isolated and whether a triple point exists). They derive analytical formulas to compute the right-hand side vectors for the moment-fitting process for each phase (solid, fluid $a$ , fluid $b$ ) within these complex geometries.

Key Contributions

Hybrid Formulation: A unified framework combining IFEM for fluid-fluid interfaces and Nitsche-based CutFEM for fluid-solid boundaries on a single mesh, applicable to both linear ( $p=1$ ) and quadratic ( $p=2$ ) elements.
Triply-Cut Integration: An extension of the equivalent polynomial integration method to cells intersected by two distinct interfaces, handling all topological configurations including triple points.
Consistency Correction: A novel correction term for triply-cut cells that accounts for the truncated edges caused by the embedded boundary, ensuring the method remains consistent and achieves optimal convergence.
Exact Jump Representation: The use of enrichment functions to handle non-homogeneous solution and flux jumps, decoupling the error magnitude from the size of the jumps.

Numerical Results
The method was validated against manufactured solutions with varying coefficient contrasts (up to $\mu_b/\mu_a = 1000$ ) and jump conditions (homogeneous and non-homogeneous).

Convergence: The method achieves optimal convergence rates of order $p+1$ in the $L^2$ norm for both linear and quadratic elements.
Accuracy: The accuracy is comparable to that of a mesh explicitly fitted to both interfaces, despite using a single background mesh.
Robustness: The method remains stable and accurate for high coefficient contrasts.
Limitations: When a Robin condition is applied to a curved embedded boundary, the quadratic convergence rate ( $p=2$ ) degrades to second order. The authors attribute this to the linear representation of the curved boundary in the level set function, noting that higher-order boundary representations are required to restore optimal rates for $p=2$ .

Significance and Claims
The paper claims to provide a robust and efficient numerical tool for complex multi-phase problems where interfaces move and interact. By combining IFEM and CutFEM, the method avoids the need for mesh regeneration while maintaining high accuracy and handling strong discontinuities. The authors emphasize that this is the first work to couple an IFEM treatment of an immersed interface with a Nitsche treatment of an embedded boundary within the same cut cell, specifically addressing the integration and consistency challenges of triply-cut cells. The results demonstrate that the hybrid approach offers the accuracy of fitted meshes with the flexibility of unfitted methods, making it suitable for applications involving moving fluid-fluid interfaces and complex solid boundaries.

Technical Summary: Hybrid CutFEM-IFEM Model for Embedded and Immersed Interfaces

More like this