Topic 7f Time Domain FDM

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Bruno Wilson
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Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.

1 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) E Mail: rcrumpf@utep.edu Topic 7f Time Domain FDM EE 4386/5301 Computational Methods in EE Topic 7f Time Domain FDM 1 Outline Introduction Time Domain Solution of the Heat Equation Time Domain Solution of Maxwell s Equations Topic 7f Time Domain FDM 1

2 Introduction Topic 7f Time Domain FDM 3 Basic Algorithm Formulation 1. Identify governing equation. Approximate equation using finite differences 3. Solve finite difference equation for the function at the future time value 4. Collect constants into update coefficients. 5. Write final update equation. Implementation 1. Dashboard. Calculate grid 3. Build device on grid 4. Calculate update coefficients 5. Initial unknown function(s) 6. Main loop iterate over time 7. Post process data 8. Visualize data Topic 7f Time Domain FDM 4

3 Notes Time domain FDM does not require linear algebra. Scales nearly linearly Excellent technique for very large scale simulations Excellent for simulating transient response Excellent for incorporating nonlinear devices and materials Excellent technique for visualizing motion and waves Excellent technique for learning about devices Topic 7f Time Domain FDM 5 Time Domain Solution of the Heat Equation Topic 7f Time Domain FDM 6 3

Define Problem A long bar of length L is a temperature 0 C. At time t = 0, a 350 C heat source is applied to the far end of the bar while the other end is held at 0 C.

4 Define Problem A long bar of length L is a temperature 0 C. At time t = 0, a 350 C heat source is applied to the far end of the bar while the other end is held at 0 C. Calculate and show how the temperature evolves with time over the length of the bar if the thermal diffusivity (x) is x 0 x L x x5 50sin 0 x L L Topic 7f Time Domain FDM 7 Governing Equation The governing equation is the time domain form of the heat equation. T t T 0 x T x temperature thermal diffusivity t time del operator Material Metal 10 4 Water 10 7 Air 10 5 Nylon 10 7 Wood 10 7 Topic 7f Time Domain FDM 8 4

5 Reduce to 1D T x t t T t x T 0 T x, t T x, t T x, t T x, t x 0 t x y z,, T x t x 0 Topic 7f Time Domain FDM 9 Approximate with Finite Differences Over governing equation is T x t t,, x T x t x 0 Let s make a guess at the finite difference approximation x T x, tt T x, t T xx, t T x, t T xx, t x t Exists at (x, t + t/) Exists at (x, t) 0 Exists at (x,t) Topic 7f Time Domain FDM 10 5

6 Fixing the Finite Difference Equation (1 of ) Solution 1 Interpret the time finite difference as a backward finite difference. T x, tt T x, t T xx, t T x, t T xx, t x t x Exists at xt, 0 Solution Interpret the time finte difference as a forward finitedifference. T xt, T xt, t T xxt, T xt, T xxt, x 0 t x Exists at xt, Topic 7f Time Domain FDM 11 Fixing the Finite Difference Equation ( of ) Solution 3 We interpolate the second term in the equation to exist at t + t/. T x xt, t T xt, t T x xt, t T x xt, T xt, T x xt, x x,, T x t t T x t x x t 0 Exists at (x, t + t/) Exists at (x, t + t/) This is called the Crank Nicholson scheme. Very often, we can get away using Solution 1 or, but the Crank Nicholson scheme is unconditionally stable. Topic 7f Time Domain FDM 1 6

7 Solve for Future Value We will proceed with Solution 1 because it is the simplest, but not necessarily the most accurate or stable. T x, tt T x, t T xx, t T x, t T xx, t x t x Exists at xt, We now write this equation in terms of array indices. T T T T T t x k1 k k k k i i i1 i i1 i 0 Solve for future value of T. t T T T T T k1 k k k k i i i i i i 1 1 x 0 Topic 7f Time Domain FDM 13 Write Update Equation The update equation is T T c T T T k1 k k k k i i i i1 i i1 When performing the update, calculate the new values based only on old values. Do not mix the two! We will have to store the old and the new values in two different arrays to do this. The update coefficient is c i it x The update coefficient does not change with time so it should be calculated before the main time loop. Topic 7f Time Domain FDM 14 7

8 Stability Condition Our finite difference equations couples only adjacent points on the grid. Thus, it is numerically impossible for a number to travel farther than one grid cell in one time step. We need to make sure that the speed of the heat front is slower than what is numerically possible. The stability condition that ensures this is t i 1 x t x i Typically we calculate the time step based on this stability condition. Topic 7f Time Domain FDM 15 Revised Algorithm 1. Initialize MATLAB. Dashboard define all simulation parameters 3. Calculate grid: Nx and dx. 4. Build device on grid: alpha 5. Calculate a stable time step (and number of iterations): dt and NT 6. Calculate update coefficients: [c 1 c c 3 c N ] 7. Initialize arrays: T, T, etc. 8. Main loop iterate over time t a. Update T i all the way across grid (loop over x) b. Enforce boundary conditions c. Record intermediate results (if needed) d. Visualize intermediate results (if needed) 9. Post process results 10. Visualize results 11. Finished! Topic 7f Time Domain FDM 16 8

9 Time Domain Solution of Maxwell s Equations Topic 7f Time Domain FDM 17 Transient Vs. Steady State Topic 7f Time Domain FDM Slide 18 9

Time Domain Solution of Maxwell s Equations H t E t t A circulating E field induces a change in the H field at the center of circulation in proportion to the permeability.

10 Time Domain Solution of Maxwell s Equations H t E t t A circulating E field induces a change in the H field at the center of circulation in proportion to the permeability. E t H t t A circulating H field induces a change in the E field at the center of circulation in proportion to the permittivity. Topic 7f Time Domain FDM 19 Fields are Staggered in Both Space and Time H t E t t E t H t t E H tt tt Topic 7f Time Domain FDM 0 t H tt E H t E tt t t 10

11 Update Equations E t H H tt tt t t H H E tt tt t H tt E E tt t t t E E H tt t tt Topic 7f Time Domain FDM 1 Sequence of Code Development Step 1 Basic FDTD Algorithm Basic update equations Topic 7f Time Domain FDM Slide 11

12 Sequence of Code Development Step Add Simple Soft Source Basic update equations Add a soft source Topic 7f Time Domain FDM Slide 3 Sequence of Code Development Step 3 Add Absorbing Boundary Basic update equations Add a soft source Add perfect boundary condition Topic 7f Time Domain FDM Slide 4 1

13 Sequence of Code Development Step 4 Add TF/SF Source Basic update equations Add a soft source Add perfect boundary condition Incorporate TF/SF oneway source Topic 7f Time Domain FDM Slide 5 Sequence of Code Development Step 5 Move Source and Add T & R Basic update equations Add a soft source Add perfect boundary condition Incorporate TF/SF oneway source Move position of source Calculate transmittance and reflectance Topic 7f Time Domain FDM Slide 6 13

Sequence of Code Development Step 6 Add Device (Algorithm Done) Basic update equations Add a soft source Add perfect boundary condition Incorporate TF/SF oneway source Move position of source

14 Sequence of Code Development Step 6 Add Device (Algorithm Done) Basic update equations Add a soft source Add perfect boundary condition Incorporate TF/SF oneway source Move position of source Calculate transmittance and reflectance Add a real device Topic 7f Time Domain FDM Slide 7 Summary of Code Development Sequence Step 1 Implement basic FDTD algorithm Step Add the source Step 3 Add absorbing boundary Step 4 Add one way source Step 5 Calculate transmittance and reflectance Step 6 Add a device Topic 7f Time Domain FDM Slide 8 14

15 Movie of Simple Hard Source Topic 7f Time Domain FDM 9 Movie of Simple Soft Source Topic 7f Time Domain FDM 30 15

16 Movie of TF/SF Soft Source Topic 7f Time Domain FDM 31 Calculating Transmission & Reflection FDTD Simulation Transmission Response time Reflection Response time Topic 7f Time Domain FDM Slide 3 16

17 Animation of Numerical Dispersion Simulation with near zero numerical dispersion... Simulation with strong numerical dispersion... Topic 7f Time Domain FDM Slide 33 D Code Development Sequence Step 1 Basic Update Equations Dirichlet Boundary Condition Dirichlet Boundary Condition Dirichlet Boundary Condition Dirichlet Boundary Condition Topic 7f Time Domain FDM Slide 34 17

D Code Development Sequence Step Periodic Boundary Conditions Periodic Boundary Condition Periodic Boundary Condition Periodic Boundary Condition Periodic Boundary

18 D Code Development Sequence Step Periodic Boundary Conditions Periodic Boundary Condition Periodic Boundary Condition Periodic Boundary Condition Periodic Boundary Condition Topic 7f Time Domain FDM Slide 35 D Code Development Sequence Step Perfectly Matched Layer y lo PML x lo PML x hi PML y hi PML Topic 7f Time Domain FDM Slide 36 18

19 D Code Development Sequence Step 4 TF/SF Source Periodic Boundary Condition Periodic Boundary Condition Topic 7f Time Domain FDM Slide 37 D Code Development Sequence Step 5 Transmission & Reflection Topic 7f Time Domain FDM Slide 38 19

20 D Code Development Sequence Step 6 Simulate Device Topic 7f Time Domain FDM Slide 39 Summary of D Code Development Sequence Step 1 Basic Update + Dirichlet Step Basic Update + Periodic BC Step 3 Add PML Step 4 TF/SF Step 5 Calculate Response Step 6 Add a Device and Benchmark Topic 7f Time Domain FDM Slide 40 0

21 Real FDTD Simulation Topic 7f Time Domain FDM 41 1

Nonlinear dynamics for signal identification T. L. Carroll Naval Research Lab

Nonlinear dynamics for signal identification T. L. Carroll Naval Research Lab Multiple radars: how many transmitters are there? Specific Emitter Identification Older transmitters Modern transmitters Transients