How do you compare and contrast different FEM methods for coupled problems, such as FSI, TSI, and MFSI?

1 FSI methods

FSI problems occur when the deformation of a solid structure affects the flow of a fluid, or vice versa. For example, the vibration of a bridge under wind load, or the blood flow in an artery. FSI methods can be classified into two categories: monolithic and partitioned. Monolithic methods solve the fluid and solid equations as a single system, while partitioned methods solve them separately and exchange information at the interface. Monolithic methods are more robust and accurate, but also more complex and computationally expensive. Partitioned methods are more flexible and efficient, but also more prone to instability and convergence issues.

2 TSI methods

TSI problems occur when the temperature of a solid structure affects its mechanical behavior, or vice versa. For example, the thermal expansion of a metal rod, or the heat generation by friction. TSI methods can be similar to FSI methods, either monolithic or partitioned, depending on how the thermal and mechanical equations are coupled. However, TSI problems are usually less nonlinear and less dynamic than FSI problems, which makes them easier to solve. TSI methods can also benefit from some simplifications, such as assuming linear thermal expansion, or neglecting the thermal inertia.

3 MFSI methods

MFSI problems occur when the interaction of fluid, solid, and thermal domains is significant and cannot be ignored. For example, the cooling of a turbine blade by air flow, or the melting of ice by water flow. MFSI problems are the most challenging and realistic among the coupled problems, as they involve multiple physics and multiple scales. MFSI methods can be either monolithic or partitioned, but they also need to account for the coupling of all three domains, not just two. MFSI methods can also use different FEM formulations for different domains, such as Lagrangian for solid, Eulerian for fluid, and ALE for interface.

4 Comparison criteria

To compare and contrast different FEM methods for coupled problems, you need to consider several criteria, such as accuracy, stability, efficiency, flexibility, and complexity. Accuracy refers to how well the method approximates the true solution of the problem. Stability refers to how well the method avoids numerical errors and oscillations. Efficiency refers to how fast the method converges and how much computational resources it requires. Flexibility refers to how easily the method can adapt to different problem settings and boundary conditions. Complexity refers to how difficult the method is to implement and understand.

5 Advantages and disadvantages

Each FEM method for coupled problems has its own advantages and disadvantages, depending on the criteria and the problem characteristics. For example, monolithic methods are more accurate and stable than partitioned methods, but also more complex and inefficient. Partitioned methods are more flexible and efficient than monolithic methods, but also less accurate and stable. Similarly, FSI methods are more nonlinear and dynamic than TSI methods, but also more realistic and relevant. TSI methods are more linear and static than FSI methods, but also more simplified and idealized. MFSI methods are the most comprehensive and realistic among the coupled problems, but also the most challenging and demanding.

6 Tips and tricks

To choose the best FEM method for your coupled problem, you need to balance the trade-offs between the criteria and the problem characteristics. Here are some tips and tricks to help you make a good decision. First, identify the dominant physics and scale of your problem, and select the appropriate FEM formulation for each domain. Second, evaluate the strength and frequency of the coupling effects, and decide whether to use a monolithic or a partitioned method. Third, test the accuracy and stability of your method with some benchmark problems and validation experiments. Fourth, optimize the efficiency and flexibility of your method with some advanced techniques, such as adaptive mesh refinement, parallel computing, or substructuring.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?