# First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Navigation Functions

**Definition**

**1.**

- 1.
- Analytic on the interior of $\mathcal{F}$,
- 2.
- Polar on $\mathcal{F}$ with a minimum at an interior point ${q}_{d}\in \circ \mathcal{F}$,
- 3.
- Admissible on $\mathcal{F}$,
- 4.
- Morse on Ω.

#### 2.2. Semi-Analytical Approximation of the Green’s Function for the Laplacian Operator

#### 2.2.1. Green’s Function

#### 2.2.2. Green’s Function from Conformal Mappings

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

#### 2.2.3. Harmonic Function Exterior to Non-Overlapping Disks

#### 2.3. First Order Variation of the Harmonic Solution

## 3. Results and Discussion

#### 3.1. Numerical Validation of the Green’s Function Approximation

#### 3.1.1. Convergence of the Method

#### 3.1.1.1. Reconstruction of the Harmonic Solution for Constant Boundary Conditions

#### 3.1.1.2. Reconstruction for Non-Constant Boundary Conditions

#### 3.1.2. Time Complexity

#### 3.1.3. Possible Applications of the Solution

#### 3.1.3.1. Green’s Function of Sphere Worlds

#### 3.2. Numerical Validation of the Hadamard Variation of the Green’s Function

- by varying the center of the obstacle disc, which is representative of the effect of a locally constant wind on an aircraft.
- and by varying the radius of the obstacle disc.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A sphere world bounded by an outer circle of center ${c}_{1}$ and three non overlapping inner circles of centers ${c}_{2}$, ${c}_{3}$ and ${c}_{4}$, respectively.

**Figure 2.**Mapping of the sphere world $\mathsf{\Omega}$ by ${T}_{{z}_{0}}$. ${\mathsf{\Omega}}^{T}$ is the space exterior to three circles.

**Figure 3.**$log|1-{\int}_{\partial \mathsf{\Omega}}\frac{\partial g}{\partial n}(x,y)\phantom{\rule{4pt}{0ex}}\mathrm{d}y|$ with $x=-2$ and a fixed N for values of ${n}_{c}$ ranging from 2 to 64 in the configuration described in Section 3.1.1.1. The x-axis is the number of collocation points ${n}_{c}$.

**Figure 4.**Domain on which an exact solution is computed. The outer boundary is the unit circle. The inner boundary is the circle of center ${c}_{2}=0.4$ and of radius ${r}_{2}=0.4$. The Dirichlet boundary conditions used are such that the value is ${c}_{1}=0$ on the inner boundary and ${r}_{1}=1$ on the outer boundary.

**Figure 5.**$log{\u03f5}_{1}$ with our method based on the Green’s function compared to that obtained with the finite elements method. (

**a**) $log{\u03f5}_{1}$ as a function of ${n}_{c}$ for a fixed number of terms in the Laurent expansion $N=2$, $N=5$ or $N=10$; (

**b**) $log{\u03f5}_{1}$ as a function of the number of mesh vertices for the finite element method with Lagrangian elements of order 2.

**Figure 6.**Normalized execution as a function of ${n}_{c}$ for a fixed number of terms in the Laurent expansion $N=2$, $N=5$, or $N=10$.

**Figure 7.**Approximation g of the Green’s function on unit disc with obstacle of radius $0.1$ centered at $-0.2-0.5i$ and destination disc of radius $0.2$ centered at $0.5i$.

**Figure 8.**Navigation functions with trajectories from starting points $40-40i$, $35-55i$, $50-25i$, $-80-40i$ and $-80+50i$ represented.

**Figure 9.**Relative error of the variation of the Green’s function deduced from the Hadamard variation for a perturbation of the radius of the inner disc (displacement component $\overrightarrow{\delta c}$ is null). The x-axis is the logarithm of the inverse of the step by which the inner disc is modified.

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**MDPI and ACS Style**

Santos, I.; Puechmorel, S.; Dufour, G. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. *Math. Comput. Appl.* **2018**, *23*, 48.
https://doi.org/10.3390/mca23030048

**AMA Style**

Santos I, Puechmorel S, Dufour G. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. *Mathematical and Computational Applications*. 2018; 23(3):48.
https://doi.org/10.3390/mca23030048

**Chicago/Turabian Style**

Santos, Isabelle, Stéphane Puechmorel, and Guillaume Dufour. 2018. "First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World" *Mathematical and Computational Applications* 23, no. 3: 48.
https://doi.org/10.3390/mca23030048