case_fdtd_3d_cartesian_wave

测试目标

case_fdtd_3d_cartesian_wave 是 3D Cartesian Yee-FDTD 的可视化案例。它用中心 z 向电场源激发波，并输出三个正交中心平面裁切后的 Ez 幅值 cutaway 图，用于直观检查三维传播形态。 run.sh 运行 fdtd_3d_cartesian_wave.py 生成图片，不编译、也不调用 E_Maxwell 目录中的 Fortran 子程序。目录中还保留了一张 z=0 单平面参考图，用于和三维切片图互相对照。

对应算法

该脚本的内部六分量 curl 更新与 E03 3D Cartesian 中 sub_E03_fdtd_3d_cartesian_H 和 sub_E03_fdtd_3d_cartesian_E 的差分形式一致。中心 Ez 软源、Gaussian 空间包络、RMS 累积图和 sponge layer 边界都是 Python 脚本里的展示设置，不属于 E03 Fortran 子程序本身。

文件说明

文件	作用
`fdtd_3d_cartesian_wave.py`	Python 3D FDTD 演示脚本。
`fdtd_3d_cartesian_ez_slices.png`	当前 `run.sh` 生成的主图：三个正交中心平面 `x=0,y=0,z=0` 裁切后的 `Ez` RMS cutaway 图。
`fdtd_3d_cartesian_wave_slice.png`	目录中保留的单平面参考图：`z=0` 截面上的场幅值快照，便于初学者先从二维切片理解波前。
`run.sh`	运行 Python 脚本并重新生成图片。

运行方式

source ~/.venv/bin/activate
cd tests/005_maxwell/case_fdtd_3d_cartesian_wave
bash run.sh

主流程

建立 3D Cartesian Yee 网格。
交替更新 Hx/Hy/Hz 和 Ex/Ey/Ez。
在电场更新后加入中心 z 向电场源，并施加 sponge layer。
取 x=0、y=0、z=0 三个中心平面。
绘制 Ez RMS 幅值 cutaway 图并保存 PNG；目录中另有 z=0 单平面参考图可对照阅读。

典型例子：3D 中心源和三正交切片

fdtd_3d_cartesian_wave.py 用 100 x 100 x 100 网格演示三维 Yee 更新。网格间距取为波长的十分之一，时间步长按三维 CFL 公式计算。源项不是单点脉冲，而是一个很窄的 Gaussian 空间包络，加到 Ez 上，这样可以减少单格点源带来的尖峰。

nx = ny = nz = 100
dx = dy = dz = lam0 / 10.0
dt = s / (c0 * np.sqrt((1.0/dx**2) + (1.0/dy**2) + (1.0/dz**2)))

ic, jc, kc = nx // 2, ny // 2, nz // 2
src_profile = np.exp(-(((ii-ic)**2 + (jj-jc)**2 + (kk-kc)**2) /
                       (2.0 * src_sigma * src_sigma)))
src_profile /= np.max(src_profile)

主循环中六个分量都按 curl 交替更新，源项在电场更新后加入。边界不是 CPML，而是轻量的 sponge layer；脚本还在最后 80 步累积 Ez 的 RMS 幅值，避免单步快照刚好落在相位零点。

ez[1:, 1:, :] += ce * (
    (hy[1:, 1:, :] - hy[:-1, 1:, :]) / dx
    - (hx[1:, 1:, :] - hx[1:, :-1, :]) / dy
)

src = 0.25 * ramp * np.sin(w0 * t)
ez += src * src_profile

ex *= damp; ey *= damp; ez *= damp
hx *= damp; hy *= damp; hz *= damp

重点调用方式

update Hx, Hy, Hz from Ex, Ey, Ez
update Ex, Ey, Ez from Hx, Hy, Hz
add centered Ez source and sponge damping
plot half-width vertical center planes and one horizontal quadrant

结果和图像

这页有两张图，建议先看下面的二维 z=0 单平面图理解中心源的环状波前，再看三维三切片图判断 x=0、y=0、z=0 三个中心平面上的整体传播形态。两张图的色标和统计量不同：主图是最后 80 步的 Ez RMS，单平面图是保留的参考快照，因此只能比较形态，不能逐点比较数值。

../../_images/fdtd_3d_cartesian_ez_slices.png — `fdtd_3d_cartesian_ez_slices.png`：当前脚本输出的主图。它把 `x=0`、`y=0`、`z=0` 三个中心平面裁成一个开角 cutaway 视图：两个竖直中心面保留完整 `z` 范围，水平 `z=0` 面只显示一个象限，避免完整平面互相遮挡。颜色表示最后 `80` 步累积得到的 `Ez` RMS 幅值；这张图主要帮助观察中心源附近三维剖面的整体形态。

../../_images/fdtd_3d_cartesian_wave_slice.png — `fdtd_3d_cartesian_wave_slice.png`：单平面参考图，显示 `z=0` 截面上的场幅值快照。它更接近普通二维热图，适合先看中心源附近的环状波前、径向对称性和边界吸收后的剩余场分布。注意这张图的色标和量纲来自该参考快照，不能直接与上方 RMS 三切片图逐数值比较。

常见误读

Ez 切片并不代表完整电磁能量；它只是一个便于观察的分量。这个 Python 图适合检查传播形态，数值通过与否仍应看 single-step、MMS、稳定性和 CPML 测试。

Test Goal

case_fdtd_3d_cartesian_wave is a 3D Cartesian Yee-FDTD visualization case. It excites a centered z-directed electric source and outputs Ez magnitude cutaway slices from three orthogonal center planes for qualitative inspection. run.sh runs fdtd_3d_cartesian_wave.py to generate the figures; it does not compile or call any Fortran routine under E_Maxwell. The directory also keeps a single z=0 plane reference image, which is useful for reading the 3D result from a simpler 2D view first.

Related Algorithm

The script’s interior six-component curl update matches the finite-difference form used by sub_E03_fdtd_3d_cartesian_H and sub_E03_fdtd_3d_cartesian_E in E03 3D Cartesian. The centered Ez soft source, Gaussian spatial profile, RMS accumulation figure, and sponge boundary are Python-side display choices, not parts of the E03 Fortran routines.

Files

File	Role
`fdtd_3d_cartesian_wave.py`	Python 3D FDTD demo.
`fdtd_3d_cartesian_ez_slices.png`	Main figure regenerated by `run.sh`: a cutaway `Ez` RMS view from the three center planes `x=0,y=0,z=0`.
`fdtd_3d_cartesian_wave_slice.png`	Retained single-plane reference: a field-amplitude snapshot on the `z=0` section.
`run.sh`	Run the Python script and regenerate the figures.

Run Command

source ~/.venv/bin/activate
cd tests/005_maxwell/case_fdtd_3d_cartesian_wave
bash run.sh

Main Flow

Build a 3D Cartesian Yee grid.
Alternate Hx/Hy/Hz and Ex/Ey/Ez updates.
Add the centered z-directed electric source after the electric update and apply the sponge layer.
Extract center planes at x=0, y=0, and z=0.
Plot and save the Ez RMS cutaway image; compare it with the retained z=0 reference slice when useful.

Example: 3D Center Source and Orthogonal Slices

fdtd_3d_cartesian_wave.py demonstrates a 3D Yee update on a 100 x 100 x 100 grid. The spacing is one tenth of the wavelength and the time step is computed from the 3D CFL condition. The source is a narrow Gaussian spatial profile added to Ez rather than a single-cell spike, which keeps the visualization cleaner.

nx = ny = nz = 100
dx = dy = dz = lam0 / 10.0
dt = s / (c0 * np.sqrt((1.0/dx**2) + (1.0/dy**2) + (1.0/dz**2)))

ic, jc, kc = nx // 2, ny // 2, nz // 2
src_profile = np.exp(-(((ii-ic)**2 + (jj-jc)**2 + (kk-kc)**2) /
                       (2.0 * src_sigma * src_sigma)))
src_profile /= np.max(src_profile)

The main loop alternates the six curl updates and adds the source after the electric update. The edges use a lightweight sponge layer rather than CPML. The plotted amplitude is accumulated as an Ez RMS value over the last 80 steps, avoiding a misleading single-phase snapshot.

ez[1:, 1:, :] += ce * (
    (hy[1:, 1:, :] - hy[:-1, 1:, :]) / dx
    - (hx[1:, 1:, :] - hx[1:, :-1, :]) / dy
)

src = 0.25 * ramp * np.sin(w0 * t)
ez += src * src_profile

ex *= damp; ey *= damp; ez *= damp
hx *= damp; hy *= damp; hz *= damp

Core Pattern

update Hx, Hy, Hz from Ex, Ey, Ez
update Ex, Ey, Ez from Hx, Hy, Hz
add centered Ez source and sponge damping
plot half-width vertical center planes and one horizontal quadrant

Results and Figures

This page has two figures. It is often easiest to read the retained 2D z=0 slice first to understand the ring-like wavefront from the centered source, then read the 3D triple-slice figure to inspect the overall shape on the x=0, y=0, and z=0 center planes. The color scales and plotted quantities differ: the main figure is an Ez RMS over the final 80 steps, while the single-plane image is a retained reference snapshot.

Common Pitfall

Ez slices are not total electromagnetic energy; they are one component chosen for visualization. This Python figure is useful for checking the propagation shape; use single-step, MMS, stability, and CPML tests for numerical validation.