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**ECE6340 Lecture 17-4: Mur Boundary Condition for 1D FDTD**

Solve a Dirichlet Problem for the Laplace Equation. Solve a Dirichlet Problem for the Helmholtz Equation. Solve a Basic Sturm — Liouville Problem.

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Construct a Complex Analytic Function. All rights reserved. Enable JavaScript to interact with content and submit forms on Wolfram websites.Updated 12 Mar Authors :- Sathya Swaroop Ganta, B. Electrical Engg. Kayatri, M. Engineering Design Pankaj, M.

Sumantra Chaudhuri, M. Projesh Basu, M. Nikhil Kumar CS, M. Instructor :- Dr. A 2D TM wave containing the xy-plane polarized magnetic field having components Hy and Hx and z-polarized electric field Ez. The fields are updated at every time step, in a space, where all physical parameters of free space are not normalized to 1 but are given real and known values.

The field points are defined in a grid described by Yee's algorithm. The H fields are defined at every half coordinate of spacesteps. More precisely, the Hx part is defined at every half y-coordinate and full x-coordinate and the Hy part is defined at every half x-coordinate and full y-coordinate and E fields i.

Also here, the space-step length is taken as 1 micron instead of 1 unit in unitless domain assumed in previous programs.

Also, the time update is done using Leapfrog time-stepping. Here, H-fields i. This is shown by two alternating vector updates spanning only a part of spatial grid where the wave, starting from source, has reached at that particular time instant avoiding field updates at all points in the grid which is unnecessary at that time instant.

These spatial updates are inside the main for-loop for time update, spanning the entire time grid. Also, here, the matrices used as multiplication factors for update equations are initialized before the loop starts to avoid repeated calculation of the same in every loop iteration, a minor attempt at optimization. The boundary condition here is Mur's Absorbing Boundary Condition ABC where the fields at the grid points have electric field values formulated using Engquist Majda one way wave equations where the boundaries give a sense of absorbing the total field incident on them and reflecting none back to the domain.

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Follow Download. Overview Functions. Comments and Ratings 0. Tags Add Tags 2d fdtd computational ele Discover Live Editor Create scripts with code, output, and formatted text in a single executable document.Updated 09 Feb Authors :- Sathya Swaroop Ganta, B. Electrical Engg. Kayatri, M. Engineering Design Pankaj, M. Sumantra Chaudhuri, M. Projesh Basu, M. Nikhil Kumar CS, M. Instructor :- Dr. The time update in Yee Algorithm is done using Leapfrog time-stepping. Here, the H fields are updated every half time-step and E fileds are updated every full time-step.

This is shown by two alternating vector updates spanning entire spatial grid inside a main for-loop for time update spanning the entire time-grid.

Yes, there are No For Loops except for time steps. The vector updates span only a part of spatial grid where the wave, starting from source, has reached at that particular time instant exploiting sparse vectors avoiding field updates at all points in the grid which is unnecessary at that time instant. The spatial and temporal parameters are not unit less and are given real values. The boundary condition here is Mur's Absorbing Boundary Condition ABC where the fields at the grid points have electric field values formulated using Engquist Majda one way wave equations where the boundaries give a sense of absorbing the total field incident on them and reflecting none back to the domain.

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Comments and Ratings 1.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems.

It only takes a minute to sign up. I am trying to solve a 2D wave equation implicitly using FD with central approximations with the following boundary conditions. The general equation is given as. The Mur boundary condition can be expressed mathematically as. The boundary conditions were evaluated analytically and these are the 7 equations following the entire domain:. Problem - I have managed to write some code, including setting up the [A] matrix the LHS of the equationsbut I have problems with formulating the [b] matrix RHS and solving it through the time-steps.

I would appreciate if someone show me how to write it for the first equation - then I will be able to do it dor the remaining 6. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Asked 4 years, 11 months ago. Active 4 years, 11 months ago. Viewed times. The general equation is given as The Mur boundary condition can be expressed mathematically as The boundary conditions were evaluated analytically and these are the 7 equations following the entire domain: Problem - I have managed to write some code, including setting up the [A] matrix the LHS of the equationsbut I have problems with formulating the [b] matrix RHS and solving it through the time-steps.

Can you provide the non discrete formulation of the Mur BCs? Also, can you use MathJax for the equations? I already formatted the first set of them. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog. Q2 Community Roadmap. The Overflow How many jobs can be done at home? Featured on Meta. Community and Moderator guidelines for escalating issues via new response….

Feedback on Q2 Community Roadmap. Related 5.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only takes a minute to sign up. Note: I'm trying to implement a Discontinuous Galerkin method, as kind of a way to learn about these things.

My question is: What are some good absorbing boundary conditions differential equations I can use for the acoustic wave equation in a 2-dimensional or 3-dimensional rectangular domain? The problem with ABC in the form of derivative operators at the boundary is that the exact ABCs are non-local in space and often in time, which makes them hard to use in numerical simulations.

You can derive approximate ABCS based on series expansions, but this can be tedious and it may be difficult to get stable schemes. This approach was made famous by a very widely cited paper by Enquist and Majda.

An easier and widely used approach are perfectly matched layers. This modifies the dispersion relation of your system such that traveling wave modes in the exterior layer are damped. Note that PML provides an approximate ABC, unless the layer would be infinitely long, which is of course impossible in a numerical simulation.

However, the amplitude of the waves usually decays exponentially, so that small layers can already damp your reflections to below the discretization error of your DG method in the interior. Originally, PML was introduced by Berenger although this paper might not be the best way to start. There is also a well written Wikipedia entry and searching for perfectly matched layer certainly will provide you with good material - as said, the method is very widely used.

There exist Absorbing Boundary Conditions for the wave equation that are stable and that go up to any order of accuracy limited only by the accuracy of discretization of your modelso that they are good competition for PMLs. Derivatives are implied. If you need higher accuracy I suggest the Hagstrom-Warburton boundary condition: this condition is based on the definition of auxiliary variables on the boundary, which also satisfy the wave equation there.

Corners require special treatment. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Absorbing boundary conditions for acoustics in Discontinuous Galerkin Ask Question. Asked 4 years, 8 months ago. Active 3 years, 9 months ago.Documentation Help Center. Before you create boundary conditions, you need to create a PDEModel container. Suppose that you have a container named modeland that the geometry is stored in model.

Examine the geometry to see the label of each edge or face. Now you can specify the boundary conditions for each edge or face. If you have a system of PDEs, you can set a different boundary condition for each component on each boundary edge or face. If the boundary condition is a function of position, time, or the solution uset boundary conditions by using the syntax in Nonconstant Boundary Conditions.

If you do not specify a boundary condition for an edge or face, the default is the Neumann boundary condition with the zero values for 'g' and 'q'. The Dirichlet boundary condition implies that the solution u on a particular edge or face satisfies the equation. You can specify Dirichlet boundary conditions as the value of the solution u on the boundary or as a pair of the parameters h and r.

Suppose that you have a PDE model named modeland edges or faces [e1,e2,e3]where the solution u must equal 2. Specify this boundary condition as follows. If you do not specify 'r'applyBoundaryCondition sets its value to 0. If you do not specify 'h'applyBoundaryCondition sets its value to 1.

Suppose that you have a PDE model named modeland edge or face labels [e1,e2,e3] where the first component of the solution u must equal 1while the second and third components must equal 2. The 'u' and 'EquationIndex' arguments must have the same length. If you exclude the 'EquationIndex' argument, the 'u' argument must have length N.

If you exclude the 'u' argument, applyBoundaryCondition sets the components in 'EquationIndex' to 0. The 'r' parameter must be a numeric vector of length N. If you do not specify 'r'applyBoundaryCondition sets the values to 0.

The 'h' parameter can be an N -by- N numeric matrix or a vector of length N 2 corresponding to the linear indexing form of the N -by- N matrix. If you do not specify 'h'applyBoundaryCondition sets the value to the identity matrix. Generalized Neumann boundary conditions imply that the solution u on the edge or face satisfies the equation.

The coefficient c is the same as the coefficient of the second-order differential operator in the PDE equation. If you do not specify 'g'applyBoundaryCondition sets its value to 0.

If you do not specify 'q'applyBoundaryCondition sets its value to 0. For each edge or face segment, there are a total of N boundary conditions. The 'g' parameter must be a numeric vector of length N.

If you do not specify 'g'applyBoundaryCondition sets the values to 0. The 'q' parameter can be an N -by- N numeric matrix or a vector of length N 2 corresponding to the linear indexing form of the N -by- N matrix. If you do not specify 'q'applyBoundaryCondition sets the values to 0.Sign in to comment. Sign in to answer this question.

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## PDE with non-linear boundary conditions

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