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A Moving Mesh Finite Element Method for Solving Bernoulli and Other Free Boundary Problems


Core Concepts
A moving mesh finite element method based on pseudo-transient continuation is presented for efficiently solving Bernoulli free boundary problems and other types of free boundary problems.
Abstract
The content describes a moving mesh finite element method for solving Bernoulli free boundary problems and other types of free boundary problems. The key highlights are: The method is based on the pseudo-transient continuation approach, where a time-dependent moving boundary problem is constructed and solved until steady state is reached. The steady-state solution is then taken as the solution of the underlying free boundary problem. The moving boundary problem is solved in a split manner at each time step: the moving boundary is updated using the Euler scheme, the interior mesh points are moved using a moving mesh method, and the corresponding initial-boundary value problem is solved using the linear finite element method. The moving mesh method, specifically the MMPDE (moving mesh PDE) method, is used to generate a new mesh for the updated domain at each time step. This method can handle domains of various geometries, including convex and concave shapes, and guarantees a non-singular mesh. Two approaches, area-averaging and quadratic least squares fitting, are discussed for reconstructing the gradient of the finite element solution and the normal to the boundary at the boundary vertices, which are needed for the Bernoulli condition but not directly available in the standard finite element approximation. Numerical examples are presented for Bernoulli free boundary problems with constant and non-constant Bernoulli conditions, as well as for nonlinear free boundary problems involving the p-Laplacian and an obstacle problem. The results demonstrate the accuracy, robustness, and flexibility of the proposed moving mesh finite element method.
Stats
The Bernoulli condition is given by: -∂u/∂n = λ, on Γ2. The p-Laplace equation is given by: ∇·(|∇u|^(p-2)∇u) = 0, in Ω.
Quotes
"The method can take full advantages of both the pseudo-transient continuation and the moving mesh method. Particularly, it is able to move the mesh, free of tangling, to fit the varying domain for a variety of geometries no matter if they are convex or concave." "The results obtained here are comparable with those in literature and particularly those obtained in [15] using a shape optimization method."

Deeper Inquiries

How can the proposed moving mesh finite element method be extended to solve free boundary problems in three dimensions

The proposed moving mesh finite element method can be extended to solve free boundary problems in three dimensions by adapting the methodology to account for the additional spatial dimension. In three dimensions, the mesh movement and adaptation algorithms would need to consider the complexities of volume meshes instead of just surface meshes. This would involve updating the mesh vertices in three dimensions based on the movement of the boundary and the solution field. The moving mesh PDEs would need to be formulated in three dimensions to ensure the mesh remains free of tangling and adapts effectively to the changing domain geometry. Additionally, the finite element discretization would need to be extended to three-dimensional elements, such as tetrahedra or hexahedra, to accurately represent the domain and solution field in three dimensions.

What are the potential challenges and limitations of using the pseudo-transient continuation approach for free boundary problems with more complex geometries or topological changes

The pseudo-transient continuation approach, while effective for many free boundary problems, may face challenges and limitations when dealing with more complex geometries or topological changes. Some potential challenges include: Mesh Tangling: In cases where the domain undergoes significant deformations or topological changes, ensuring that the mesh remains free of tangling can be challenging. The mesh movement algorithms may need to be carefully designed to handle such scenarios. Convergence Issues: Complex geometries or topological changes can lead to convergence issues in the pseudo-transient continuation method. The iterative nature of the approach may struggle to converge for highly nonlinear or rapidly changing problems. Computational Cost: More complex geometries often require finer meshes for accurate representation, leading to increased computational cost. The pseudo-transient continuation method may require more computational resources to handle such cases efficiently. Boundary Reconstruction: Topological changes may require the reconstruction of the boundary or the mesh, which can introduce additional complexities in the solution process.

What other types of free boundary problems, beyond the Bernoulli and nonlinear examples shown, could benefit from the moving mesh finite element framework presented in this work

The moving mesh finite element framework presented in this work can benefit various types of free boundary problems beyond the Bernoulli and nonlinear examples shown. Some other types of free boundary problems that could benefit from this framework include: Stefan Problems: Problems involving phase change, such as solidification or melting, where the location of the phase boundary is a free boundary that evolves over time. Shape Optimization: Problems where the shape of a domain needs to be optimized to minimize a certain objective function, leading to free boundary constraints. Fluid-Structure Interaction: Problems involving the interaction between a fluid and a deformable structure, where the fluid-structure interface acts as a free boundary that needs to be accurately captured. Optimal Control: Problems where the control inputs affect the evolution of the system and the optimal control strategy leads to free boundary conditions. The flexibility and adaptability of the moving mesh finite element method make it suitable for a wide range of free boundary problems with varying complexities and physical phenomena.
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