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A Shape-Newton Method for Free-boundary Problems Subject to The Bernoulli Boundary Condition

Yiyun Fan, John Billingham, Kristoffer van der Zee

TL;DR

This paper develops a shape-Newton method for solving free-boundary problems where the free boundary is governed by a Bernoulli-type condition. By formulating two weak statements and applying Hadamard shape derivatives, the authors derive a Newton-like iteration that updates the free surface and the boundary velocity through a linearised boundary-value problem at each step. A key theoretical contribution is the shape derivative of the Bernoulli condition, expressed via the normal derivative of velocity magnitude in a form that can be computed using tangential quantities and curvature, enabling a robust linearisation. The approach is validated on a flow problem over a submerged triangular obstacle, where the scheme demonstrates superlinear convergence and consistency with known results, highlighting its potential for general free-boundary problems with nonlinear boundary conditions.

Abstract

We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the Bernoulli equation. The Newton-like scheme is developed by employing shape derivatives in the weak forms, which allows us to update the position of the free surface and the potential on the free boundary by solving a boundary-value problem at each iteration. To validate the effectiveness of the approach, we apply the scheme to solve a problem involving the flow over a submerged triangular obstacle.

A Shape-Newton Method for Free-boundary Problems Subject to The Bernoulli Boundary Condition

TL;DR

This paper develops a shape-Newton method for solving free-boundary problems where the free boundary is governed by a Bernoulli-type condition. By formulating two weak statements and applying Hadamard shape derivatives, the authors derive a Newton-like iteration that updates the free surface and the boundary velocity through a linearised boundary-value problem at each step. A key theoretical contribution is the shape derivative of the Bernoulli condition, expressed via the normal derivative of velocity magnitude in a form that can be computed using tangential quantities and curvature, enabling a robust linearisation. The approach is validated on a flow problem over a submerged triangular obstacle, where the scheme demonstrates superlinear convergence and consistency with known results, highlighting its potential for general free-boundary problems with nonlinear boundary conditions.

Abstract

We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the Bernoulli equation. The Newton-like scheme is developed by employing shape derivatives in the weak forms, which allows us to update the position of the free surface and the potential on the free boundary by solving a boundary-value problem at each iteration. To validate the effectiveness of the approach, we apply the scheme to solve a problem involving the flow over a submerged triangular obstacle.
Paper Structure (28 sections, 2 theorems, 92 equations, 8 figures, 2 tables)

This paper contains 28 sections, 2 theorems, 92 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Free-boundary Problem with Bernoulli or Dirichlet free-boundary condition
  3. Free-boundary Problem With Bernoulli Condition
  4. Free-boundary Problem with Dirichlet Boundary Condition
  5. The Weak Form
  6. Shape Derivatives
  7. Linearisation
  8. Linearisation of R1
  9. Linearisation of R2 with Dirichlet condition
  10. Linearisation of R2 with Bernoulli condition
  11. N dimensional case
  12. Three dimensional case
  13. Two dimensional case
  14. Newton-Like Schemes
  15. Weak form of the problem with Dirichlet Boundary condition
  16. ...and 13 more sections

Key Result

Theorem 4.1

Suppose $\phi\in W^{1,1}\left(\mathbb{R}^N\right)$, where and $\Omega$ is an open and bounded domain with boundary $\Gamma = \partial \Omega$ of class $C^{0,1}$. Consider the domain integral Then its shape derivative with respect to the perturbation $\delta\boldsymbol \theta\in C^{0,1}\left(\mathbb{R}^N;\mathbb{R}^N\right)$ is given by where $\boldsymbol n$ denotes the outward normal derivative

Figures (8)

  • Figure 1: The sketch of the parametrization of the free boundary $\Gamma_F$ by the displacement $\boldsymbol\theta\left(\boldsymbol x_0\right)$ with respect to the reference boundary $\Gamma_0$.
  • Figure 1: The initial domain and the change of the domain in three following Newton-like iterations. The free surface is updated vertically.
  • Figure 2: The Dirichlet error $||\phi - h||_{L_2}$ and surface error $|| \eta-\hat{ \eta}||_{L_2}$ on $\Gamma_F$ measured in $L_\infty$-form against the number of iterations. The upper plot shows the Dirichlet error, and the lower shows the surface error. The values of $N+1$ are the number of the nodes along the $x$-axis.
  • Figure 3: The sketch of the domain we used for the second test case. $\alpha$ is denoted as the angle and $w_0$ as the half width of the triangle.
  • Figure 4: An example of the domain and the triangulation with $\alpha = \frac{\pi}{4}$, $F=2$ and the half width of the triangle $w_0 = 0.5$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Remark 2.1: Fixed free boundary at inflow
  • Remark 2.2: Data compatibility I
  • Remark 2.3: Data compatibility II
  • Theorem 4.1: Shape derivative of domain integral
  • Theorem 4.2: Shape derivative of boundary integral
  • Remark 4.3
  • Remark 4.4: Piecewise-smooth free boundary
  • Remark 6.1: Solving directly for $\phi^{k+1}$
  • Remark 6.2: Solvability of the shape-linearized systems
  • Remark 7.1: Mesh deformation
  • ...and 2 more