物理代写|电磁学代写electromagnetism代考|ELEC3104

物理代写|电磁学代写electromagnetism代考|Truncated Exterior Problem

Let us consider the case of an exterior problem, such as a diffraction problem around a perfectly conducting object. In this case, to perform computations, one adjusts the domain (Sect. 1.6.1): this results in a truncated exterior problem, set in a computational domain $\Omega$ that has a boundary $\Gamma$ equal to $\overline{\Gamma_P} \cup \overline{\Gamma_A}$, with $\partial \Gamma_P \cap$ $\partial \Gamma_A=\emptyset$. Here, $\Gamma_P$ is the “physical” part on which the perfect conductor boundary condition is imposed, and $\Gamma_A$ is purely “artificial”. For instance, let us choose $\Gamma_A$ to be a sphere, on which an absorbing boundary condition (referred to as an $\mathrm{ABC}$ from now on) is imposed, such as the Silver-Müller $\mathrm{ABC}$ (1.137) or (1.138). One usually assumes that the medium is homogeneous ${ }^2$ in a neighborhood of $\Gamma_A$, so it writes:
$$\boldsymbol{E}(t) \times \boldsymbol{n}+\sqrt{\frac{\mu}{\varepsilon}} \boldsymbol{H}_{\top}(t)=\boldsymbol{g}^{\star}(t) \text { on } \Gamma_A,$$

where we recall that $\boldsymbol{H}{\top}(t)$ denote the tangential components of $\boldsymbol{H}(t)$ on the boundary and $g^{\star}$ is the data on $\Gamma_A$. On the other hand, for the truncated exterior problem, one finds the relation below, using the integration-by-parts formula (3.5): $$\frac{d W}{d t}(t)-\gamma_A\left\langle\boldsymbol{E}(t) \times \boldsymbol{n}, \boldsymbol{H}{\top}(t)\right\rangle_{\pi_A}=-(\boldsymbol{J}(t) \mid \boldsymbol{E}(t)), \quad t>0 .$$
Above, the duality bracket reduces to $\Gamma_A$, because $\boldsymbol{E} \times \boldsymbol{n}=0$ on $\Gamma_P$. Hence, the index $_A$. Note that there is no need to use the theory summarized in Theorem 3.1.29, because in the present case, $\partial \Gamma_P \cap \partial \Gamma_A=\emptyset$.
It is possible to address uniqueness as before. Indeed, one now obtains that
$$\delta \boldsymbol{E}(t) \times \boldsymbol{n}+\sqrt{\frac{\mu}{\varepsilon}} \delta \boldsymbol{H}{\top}(t)=0 \text { on } \Gamma_A,$$ together with the relation \begin{aligned} \frac{d}{d t}\left[\frac{1}{2}{(\mathbb{\propto} \delta \boldsymbol{E}(t) \mid \delta \boldsymbol{E}(t))+(\mu \delta \boldsymbol{H}(t) \mid \delta \boldsymbol{H}(t))}\right] \ -\gamma_A\left\langle\delta \boldsymbol{E}(t) \times \boldsymbol{n}, \delta \boldsymbol{H}{\top}(t)\right\rangle_{\pi_A}=0, \quad t>0 . \end{aligned}

物理代写|电磁学代写electromagnetism代考|Truncated Interior Problem

At first glance, it appears that one can tackle the case of a truncated interior problem similarly. The first difference with the previous study is that it can happen that $\Gamma=$ $\overline{\Gamma_P} \cup \overline{\Gamma_A}, \Gamma_P \cap \Gamma_A=\emptyset, \partial \Gamma_P \cap \partial \Gamma_A \neq \emptyset .{ }^3$ In this situation, one needs to use the integration-by-parts formula of Theorem 3.1.29, to find
$$\frac{d W}{d t}(t)-\gamma_A^0\langle\boldsymbol{E}(t) \times \boldsymbol{n}, \boldsymbol{H} \top(t)\rangle_{\pi_A}=-(\boldsymbol{J}(t) \mid \boldsymbol{E}(t)), \quad t>0 .$$
In other words, the duality product has been modified, to take into account the fact that $\partial \Gamma_A \neq \emptyset$. We consider from now on that $\partial \Gamma_A$ is piecewise curvilinear. Let $v$ be the unit outward normal vector to $\partial \Gamma_A$, and $\tau$ the unit tangent vector to $\partial \Gamma_A$ so that $(\tau, v)$ is direct. As before, to prove uniqueness, we build a relation like (5.22). The obvious difficulty in the present situation is to obtain some decompositions of the traces, with boundary conditions on $\partial \Gamma_A$. We propose below a constructive proof (for the magnetic field), thus complementing the process we described in Sect. $3.1$.
First, thanks to Proposition 3.1.27, we can write on $\Gamma_A$ (for a given $t>0$ )
$$\delta \boldsymbol{E}(t) \times \boldsymbol{n}=\operatorname{curl}{\Gamma} \phi^{-}+\operatorname{grad}{\Gamma} \psi^{+}, \phi^{-} \in \widetilde{H}^{1 / 2}\left(\Gamma_A\right), \psi^{+} \in \mathcal{H}\nu\left(\Gamma_A\right) .$$ Note that we have, in a weak sense, $t_p\left(\delta \boldsymbol{E}(t) \times \boldsymbol{n}{\mid \Gamma_A}\right)=0$ on $\partial \Gamma_A$, where we recall that $t_v(\boldsymbol{f}):=\boldsymbol{f} \cdot \boldsymbol{v}{\mid \partial \Gamma_A}$, and similarly for $\operatorname{grad}{\Gamma} \psi^{+}$(see the definition of $\mathcal{H}_v\left(\Gamma_A\right)$ ). Hence, we have at hand some boundary conditions for the trace of the electric field.

物理代写|电磁学代写electromagnetism代考|Truncated Exterior Problem

$$\boldsymbol{E}(t) \times \boldsymbol{n}+\sqrt{\frac{\mu}{\varepsilon}} \boldsymbol{H}{\top}(t)=\boldsymbol{g}^{\star}(t) \text { on } \Gamma_A,$$ 我们记得 $\boldsymbol{H} \top(t)$ 表示切向分量 $\boldsymbol{H}(t)$ 在边界和 $g^{\star}$ 是关于 $\Gamma_A$. 另一方面，对于截断外部问题，可以使用分部 积分公式 (3.5) 找到以下关系: $$\frac{d W}{d t}(t)-\gamma_A\langle\boldsymbol{E}(t) \times \boldsymbol{n}, \boldsymbol{H} \top(t)\rangle{\pi_A}=-(\boldsymbol{J}(t) \mid \boldsymbol{E}(t)), \quad t>0 .$$

物理代写|电磁学代写electromagnetism代考|Truncated Interior Problem

$\overline{\Gamma_P} \cup \overline{\Gamma_A}, \Gamma_P \cap \Gamma_A=\emptyset, \partial \Gamma_P \cap \partial \Gamma_A \neq \emptyset .{ }^3$ 在这种情况下，需要使用定理 $3.1 .29$ 的分部积分公式，求
$$\frac{d W}{d t}(t)-\gamma_A^0\langle\boldsymbol{E}(t) \times \boldsymbol{n}, \boldsymbol{H} \top(t)\rangle_{\pi_A}=-(\boldsymbol{J}(t) \mid \boldsymbol{E}(t)), \quad t>0 .$$

$t_p\left(\delta \boldsymbol{E}(t) \times \boldsymbol{n} \mid \Gamma_A\right)=0$ 上 $\partial \Gamma_A$ ， 我们记得 $t_v(\boldsymbol{f}):=\boldsymbol{f} \cdot \boldsymbol{v} \mid \partial \Gamma_A$ ，同样对于 $\operatorname{grad} \Gamma \psi^{+}$(见定义 $\mathcal{H}_v\left(\Gamma_A\right.$ )
）。因此，我们手头有一些电场轨迹的边界条件。

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: