added potential derivation for homogeneous piezo

4dcc5c91 · Sam Katiraee-Far · 24a90100 · 4dcc5c91 · 4dcc5c91 · 4dcc5c91
Commit 4dcc5c91 authored 3 years ago by Sam Katiraee-Far
--- a/.DS_Store
+++ b/.DS_Store
--- a/Outline of results.ipynb
+++ b/Outline of results.ipynb
@@ -123,18 +123,29 @@
    "    * Show the corresponding potential \n",
    "\n",
    "        * Just like in the homogeneous piezo, we get a oscillating potential in the region with finite piezo coefficient. When the piezo coefficient is zero, there is no electro mechanical coupling and we get the expected solution to the homogeneous poisson equation: linearly decreasing line\n",
+    "        \n",
+    "        * Explain why it seems like the oscillations linearly change in the same direction as the linear line. \n",
    "\n",
    "    * Take a few different piezo coefficient\n",
    "\n",
    "        * Show that if we increase the piezo-electric coefficient, the relative amplitudes of the potential and the mechanical modes will increase, because the coupling is stronger.\n",
    "\n",
-    "2. Varying piezo + varying density \n",
-    "\n",
-    "    * Show the combined effect of varying density + varying piezo\n",
-    "\n",
-    "3. Varying piezo + varying stiffness\n",
+    "2. Varying permitivitty\n",
+    "    \n",
+    "    * Show the dispersion \n",
+    "       \n",
+    "       * Same curinving and kink, same story, dont have to repeat it, just refer back to previous findings.\n",
+    "   \n",
+    "   * Show a specific mechanical mode\n",
+    "       \n",
+    "       * Discuss how it changes (didn't do rigorous testing yet, but seems like the wavelength differs again)\n",
+    "   \n",
+    "   * Show corresponding potential \n",
+    "   \n",
+    "       * Discuss how it changes\n",
+    "       \n",
+    "   \n",
    "\n",
-    "    * Show the cobined effect of varying stiffness + varying piezo\n",
    "\n",
    "4. AlN - Sapphire (50/50 or 1-500?)\n",
    "\n",

 %% Cell type:markdown id: tags:


 # Results


 ### Homogeneous Acoustic

 1. Show the dispersion relation

    * Compare to the analytic dispersion relation

        * We see a curving. We are getting numerical errors because the frequency is to rapid for the grid spacing.

        * Can show an example of (a few) modes where we show that it is going wrong.

        * If we want to have a good physical interpretation, we should limit ourselves to modes below 100 or so.

    * Then show a few numerical modes along with analytical solution

        * Say that the spatial frequency matches the expectation.



 ### Inhomogeneous Acoustic

 1. Comparison to the homogeneous numerical method (no plots in report) <br/>

 2. Inhomogeneous density

    * First show the dispersion relation

        * Say that we discussed the curving before. We are getting numerical errors because the frequency is to rapid for the grid spacing.

        * Can show an example of (a few) modes where we show that it is going wrong.

        * Now this happens earlier in the dispersion, because we have two different wave-vectors, of which one is higher.

        * If we want to have a good physical interpretation, we should limit ourselves to modes below 70 or so. Here we can get the wavelength represented in both sides.

    * Then show mode 15

        * We have two regions with same frequency but different wavelengths due to different densities. The combination of the boundary conditions at the outside and that the solution must be continuous and have a continuous derivative, gives the different amplitudes

    * Then show index number vs amplitude

        * This should oscillate

        * We could compare it to the analytical solution

            * Gary said this doesn't matter that much



 3. Inhomogeneous stifness

    * Show mode 15 again

        * We see a kink

        * Compute the derivative numerically again. We indeed see that there is a discontinuity

            * Lead discussion back to the relevant formula

        * However, if we compute c * the derivative, it becomes continuous.

    * Same analysis of the peak to peak amplitude





 ### Homogeneous Piezo

 1. Show the dispersion

    * Compare to the analytic solution

        * We get the expected result as they overlap

        * Explain that we get the same curving since we used exactly the same method, just with a different constant.

    * Compare it to the homogeneous acoustic

        * show the oscillation frequency has increased, this is due to the electromechanical coupling

 2. Show a few modes

    * Since we used the same method, we also get the same modes, with the same spatial frequencies.

 3. Show the potential

    * Compare to the analytic results

    * Mention that indeed since we have mechanical oscillations, they also induce oscillations of the potential.





 ### Inhomogeneous Piezo

 1. Varying piezo coefficient

    * Show the dispersion

        * Once again it we get the same curving.

        * Show a few modes to show where it goes wrong.

        * It fails more rapidly as we have different spatial frequencies in the two regions.

        * We should limit ourselves (this is pretty much identical to inhomogeneous acoustic)

    * Show a specific mechanical mode

        * The wavelength is different in the two sides of the piezo, due to the renormalized sound velocity, but equal oscillation frequency.

    * Show the corresponding potential

        * Just like in the homogeneous piezo, we get a oscillating potential in the region with finite piezo coefficient. When the piezo coefficient is zero, there is no electro mechanical coupling and we get the expected solution to the homogeneous poisson equation: linearly decreasing line

+        * Explain why it seems like the oscillations linearly change in the same direction as the linear line.
+
    * Take a few different piezo coefficient

        * Show that if we increase the piezo-electric coefficient, the relative amplitudes of the potential and the mechanical modes will increase, because the coupling is stronger.

-2. Varying piezo + varying density
+2. Varying permitivitty
+
+    * Show the dispersion
+
+       * Same curinving and kink, same story, dont have to repeat it, just refer back to previous findings.
+
+   * Show a specific mechanical mode
+
+       * Discuss how it changes (didn't do rigorous testing yet, but seems like the wavelength differs again)
+
+   * Show corresponding potential
+
+       * Discuss how it changes

-    * Show the combined effect of varying density + varying piezo

-3. Varying piezo + varying stiffness

-    * Show the cobined effect of varying stiffness + varying piezo

 4. AlN - Sapphire (50/50 or 1-500?)

    * Say that using realisic variables gives very big issues at first, due to very huge difference in magnitude of the elements in the different matrices

        * Hence, we should do a change of variables by scaling with appropriate constants. We use trial and error to get it to work.

    * Show the modes

        * Explain all of the combined effects above



--- a/Simulations/.DS_Store
+++ b/Simulations/.DS_Store
--- a/Simulations/Archive/Inhomogeneous_piezo_testing_Gary.ipynb
+++ b/Simulations/Archive/Inhomogeneous_piezo_testing_Gary.ipynb
--- a/Simulations/Final/Acoustic_homogeneous.ipynb
+++ b/Simulations/Final/Acoustic_homogeneous.ipynb
--- a/Simulations/Final/Acoustic_inhomogeneous.ipynb
+++ b/Simulations/Final/Acoustic_inhomogeneous.ipynb
--- a/Simulations/Final/Inhomogeneous_piezo.ipynb
+++ b/Simulations/Final/Inhomogeneous_piezo.ipynb
--- a/Simulations/Final/acoustic_hom_dispersion1.pdf
+++ b/Simulations/Final/acoustic_hom_dispersion1.pdf
--- a/Simulations/Final/acoustic_hom_increase_N.pdf
+++ b/Simulations/Final/acoustic_hom_increase_N.pdf
--- a/Simulations/Final/acoustic_hom_increase_N.png
+++ b/Simulations/Final/acoustic_hom_increase_N.png
--- a/Simulations/Final/inhom_density_ampl_oscillations.pdf
+++ b/Simulations/Final/inhom_density_ampl_oscillations.pdf
--- a/Simulations/Final/inhom_density_ampl_oscillations.png
+++ b/Simulations/Final/inhom_density_ampl_oscillations.png
--- a/Simulations/Final/inhom_density_dispersion_1.pdf
+++ b/Simulations/Final/inhom_density_dispersion_1.pdf
--- a/Simulations/Final/inhom_density_dispersion_1.png
+++ b/Simulations/Final/inhom_density_dispersion_1.png
--- a/Simulations/Final/inhom_density_ex_problems.pdf
+++ b/Simulations/Final/inhom_density_ex_problems.pdf
--- a/Simulations/Final/inhom_density_ex_problems.png
+++ b/Simulations/Final/inhom_density_ex_problems.png
--- a/Simulations/Final/inhom_density_specific_mode.pdf
+++ b/Simulations/Final/inhom_density_specific_mode.pdf
--- a/Simulations/Final/inhom_density_specific_mode.png
+++ b/Simulations/Final/inhom_density_specific_mode.png
--- a/Simulations/Final/inhom_density_vary_amplitude.pdf
+++ b/Simulations/Final/inhom_density_vary_amplitude.pdf
--- a/Simulations/Final/inhom_density_vary_amplitude.png
+++ b/Simulations/Final/inhom_density_vary_amplitude.png