ZR-03-38
Contents
1. e FEM Type string finite element type FEM Type is linear for linear finite elements and quadratic for quadratic finite elements e TBC Method string specifies the method of the transparent boundary con dition The only method presently available in HelmPole is PML the perfectly matched layer e TBC PML structure parameters for the PML TBC PML k local wavenumber in each segment described by rays TBC PML xi vector discretization along the rays TBC PML xi has to be sorted in ascending order starting from 0 The last value gives the thickness of the PML TBC PML xi corresponds to the ray like variable in 6 5 As the solution inside the PML decreases exponentially one does not gain much accuracy of the solution by increasing the PML largely The distribution of discretization points inside the PML is arbitrary The distance of two points in the PML close to the boundary should be chosen at the same order maybe a factor of 2 less as the distance of two points in the interior domain A linear distribution will work well in most cases It was observed that a cubic distribution often gives better results That means there are less discretization points required at the end of the PML In most cases it is sufficient to have around twenty points along one ray in your PML However if you observe reflections at the outer boundary you should use more points TBC PML DampFun m file string The damping funct2. 4 3 Acoustic scattering ioa ee ek Nee E Ed Gee dp 4 4 Gau beam zou Poste cy Sou ere tet ee ee a scie a Schk 5 Function References 5 1 The main program MAIN sus es eo cee a bo te A ec SE 5 2 User interface PROBLEMNAME 0 0 0000000088 5 9 Geometry description COARSEGRID 5 4 Material description MATERIALFILE 5 5 Incident wave file INCWAVE 0 0 0 0 00008 eG 5 6 Discretization of the exterior MAKERAYS bar dormehlet daba e hee te 2 ay take Lb eR 58 Exact Solution mi euo x ean ho Box ee ee a Be oa ae te 6 Theory 6 1 Helmholtz equation llle 6 2 The PML method in ID eee 6 3 Diserebization of the exterior uu us o9 RR XR RES 7 Grid Generation 7 1 Conversion interface TRIANGLE2MAT 8 License and Copyright NNO Ol ji 13 15 19 20 23 23 25 29 3l 32 33 36 37 38 38 39 40 42 42 44 Chapter 1 Introduction HelmPole is a Matlab program for the solution of scattering problems on un bounded domains in two dimensions written by L Zschiedrich 11 The al gorithm is based on the finite element method and the perfectly matched layer method PML method originally introduced by Ber nger cf l In contrast to the interpretation of the PML as an absorbing material given by Ber nger the interpretation here is a complex continuation of the solution in the exterior domain This ansatz can
3. IX X X X X XX LX X XX XX VAX XAXA VAAKXX XAAT Uf X X X X X XX VN Konrad Zuse Zentrum E D 14195 Bor Danem f r Informationstechnik Berlin Germany SVEN BURGER ROLAND KLOSE ACHIM SCHADLE AND LIN ZSCHIEDRICH HelmPole A finite element solver for scattering problems on unbounded domains Implementation based on PML ZIB Report 03 38 November 2003 Abstract The solution of scattering problems described by the Helmholtz equation on un bounded domains is of importance for a wide variety of applications for example in electromagnetics and acoustics An implementation of a solver for scattering problems based on the programming language Matlab is introduced The solver relies on the finite element method and on the perfectly matched layer method which allows for the simulation of scattering problems on complex geometries surrounded by inhomogeneous exterior domains This report gives a number of detailed examples and can be understood as a user manual to the freely accessible code of the solver HelmPole Contents 1 Introduction 2 Installation 3 Quick Start od Getting Starved tee eri trm to ee BGDEUEM X XS S 3 2 The problem description file Waveguide m 3 3 Running the prOSFADieieatbtsy demeure tede EC ae ER 4 Examples 4 1 Waveguide revisited sus dus Se tat as eS e e eut eed Ss See oes 4 2 Resonant coupling of optical waveguides
4. 6 8 is exponentially de creasing and obeys the radiation condition To obtain the PML replace 6 7 by Os upuL k 1 io upm 0 UpML p 0 The error introduced by replacing pj with up r decreases exponentially with the thickness p of the PML Next we will discuss the discretization of the exterior using rays This will lead to a system of ODEs The derivation of the PML for the one dimensional case can then be extended to this discretization Details on the PML in the two dimensional case can be found in 4 For details on the implementation of the PML the reader is referred to 9 6 3 Discretization of the exterior Turning back to the two dimensional Helmholtz equation Suppose the computa tional domain is a convex polygon the exterior is then decomposed into a finite number of segments QUE j 1 N as shown in Figure 6 2 The vertices of the polygonal boundary OQ are connected with infinity by non intersecting rays There are different possibilities for constructing these rays cf 8 HelmPole provides several routines for that which are explained in Section 5 6 Each segment QU is the image of a semi infinite rectangle Qu mapping or Qu QUE The construction is described in 8 The local mappings are combined to a global mapping under the Q QEM oe 6 11 ext ext 40 x Figure 6 2 Mapping from the semi infinite rectangular domain to the exterior do
5. For the specified rays shown in the left of 34 A V Figure 5 3 Exterior rays for a problem with waveguides entering the computa tional domain Figure 5 3 the whole set of exterior rays is shown in the right part of the same Figure Please see also the example described in Section 4 1 Caution RAY is not the variable that is passed internally to INCWAVE 35 5 7 Dirichlet data Setting Dirichlet data at interior boundaries Syntax U DIRICHLETFILE P PARA Arguments e P matrix x y coordinates of points on the interior boundary e PARA structure parameters passed to DIRICHLETFILE Description U DIRICHLETFILE P PARA returns the Dirichlet data U on points P of the interior boundary Please note It is not sufficient for quadratic finite elements to return the Dirichlet values only on the grid points of the interior boundary Rather DIRICHLETFILE has to return the Dirichlet data in all degrees of freedom on the interior boundary 36 5 8 Exact Solution Function to calculate the exact solution pointwise on a grid Syntax U EXAKTFILE P PARA Arguments e P 2 x np matrix x y coordinates of points degrees of freedom in the interior of the com putational domain e PARA structure parameters passed to EXAKTFILE Description U EXAKTFILE P PARA calculates the exact solution U in points P Here P are not only gridpoints but degrees of freedom 3T Chapter 6 Theo
6. at these points The exact solution is calculated with a slightly modified formula taken from 5 cf p 29 31 Example 2 4 A uniform refinement of the grid decreases the error by a factor of 1 2 linear finite elements or 1 4 quadratic finite elements respectively In Example Cylinder2 the incoming wave is specified by prescribing Dirichlet and generalized Neumann data at the boundary of the computational domain Pert At the interior boundary homogenous Dirichlet data is set In this case for quadratic finite elements a uniform refinement of the grid decreases the error only in the first refinement The reason for this is that the circular geometry is not resolved by successive uniform refinements of the coarse mesh To decrease the error one has to remesh the geometry 4 4 Gauf beam This examples calculates the field of a Gauf beam in a homogenous material To run it type you will need about 1 5 GB memory to run this example Para k 4 Para matb 3 5 Para matg 3 5 main gauss Para 20 According to the book by Unger 10 p 304 the field of a Gauf beam is given by E r z Ure exp ei is i un arctan 4 5 where A 27 k is the wavelength 44 2 E wok R z 22 4 E 4 6 and 4z w z 41 w2 m 4 7 j C Figure 4 4 Gauf beam The focus is at the point r 0 z 0 In the problem description file the parameters for the Gauf beam are IncWave Para k Material Para ma
7. atan2 2 1 The incident wave is described using the function Waveguide_income2 in sub directory fun This function can be used to describe the incident field for many different geometries The parameters passed to this function are the wavenum ber Para k the parameters Para a Para b and Para c describing the axis of the waveguide ax by c 0 The thickness of the waveguide is set by Para d see Fig 4 1 Para WaveGuideIndexin and Para WaveGuideIndexout are the relative refractive indices of the waveguide material and of the substrate material respectively Waveguide_income2 calculates the incident wave at the left boundary of the computational domain which is the lowest mode of this waveguide As we are calculating a guided wave we can associate a direction with this wave This wave is said to travel from left to right If we take the normal to the waveguide through the middle of the waveguide this normal cuts the boundary of the computational domain in two pieces The one that lies left to the normal is the left boundary of the computational domain Prescribing the incident field only on the left boundary of the computational domain and setting is to zero for the right part of the boundary an error is introduced by the jump in the boundary condition This error becomes visible if you chance the Int PpW in Waveguide2a to 10 and it is prominent if you choose Int PpW equal to 20 Choosing 20 instead of 10 points per w
8. important substructure is PROBLEM GEO with the fields P the point data of the triangulation T the triangle data of the triangulation and E the edge data All three are matrices as described in Section 5 3 SOLUTION a structure with the fields SOLUTION U SOLUTION UL and SOLUTION UG representing the solution U is the solution vector in the interior domain and UL is the solution restricted to points PROBLEM GEO P If you use linear finite elements UL and U will be the same UG is the solution vector of the complete system including the 23 Example PML layer If nn is the number of degrees of freedom in the interior do main UG 1 nn equals to U Setting the number of degrees of freedom in the PML layer equal to npm1 UG nn 1 nn npm1 is the solution in the PML layer With the number of degrees of freedom on the outer boundary nbn the generalized Neumann data of the scattered field on the boundary is UG nn npm1 1 nn npml nbn ERROR If EXACT FILE in the problem description file is empty ERROR is empty too Else ERROR has four fields a ERROR FIELD a vector of the same size as SOLUTION U giving the pointwise error in the interior b ERROR L2 the relative l error c ERROR LINF the relative error in the maximum norm and d ERROR REF the exact solution in points PROBLEM GEO P BOUNDARY a structure describing the boundary We will not describe it in detail here BOUNDARY P are xy coordinates of the boundayr points of the tr
9. opening windows show properties of the solution of the waveguide problem Figure 3 2 shows the distribution of the incoming light field along the left boundary Figure 3 3 shows the real part and the absolute value of the 11 Real part Absolute value 6 0 15 6 f 4 0 1 4 2 0 05 2 0 0 0 OOO eu i 2 0 05 2 TT 4 0 1 4 6 0 15 6 6 4 2 0 2 4 6 Figure 3 5 Real part and absolute value of the error of the solution Fig 3 3 of the problem Waveguide m discrete solution Figure 3 4 shows the exact solution of the problem obtained semi analytically and Figure 3 5 shows the real part and the absolute value of the error of the discrete solution Additionally Figure 3 5 shows the real part and the absolute value of the error of a more accurate solution obtained from a run of Waveguide m using Int UniRefine 1 and Int PpW 10 Real part Absolute value 0 02 0 02 6 i 0 015 0 018 4 0 01 0 016 0 014 0 005 2 0 012 0 0 BE 0 01 0 005 2 0 098 0 006 0 01 4 7 0 004 0 015 0 002 6 3 6 4 2 0 2 4 6 Figure 3 6 Real part and absolute value of the error of the problem Waveguide m obtained with Int UniRefine 1 and Int PpW 10 On the command line some information is given on the progress of the main driver If the exact solution is provided relative errors are printed 12 Chapter 4 Examples In this chapter we discuss some more examples Section 4 1 shows how to de scribe waveguides i
10. pml jump 1 1 0 1 0 Real 1 Imag 0 Abs 1 Log 0 Contour 0 not edit beyond this line First geometry material sources and incoming waves are set In the first lines the wavenumber KO is set and the searched light field is chosen to be TM polarized Then the string w the name of the binary mat file w mat is spec ified as the name of the file with the geometry description of the computational domain In this case our computational domain is a square with a horizontal waveguide For details of the geometry description see Section 5 3 The file w mat is located in the folder geo and has been created previously by a triangulation tool see Chapter 7 Next different refractive indices are attributed to different regions in the geometry of the interior domain by setting Material File Waveguide material and the above defined wavenumber is attributed to the interior domain For the syntax of the Material File see Section 5 4 Material Para is a structure that is passed to the Material File The file Waveguide material m is located in the directory fun 3x4 x x X 0 8 4 0 6F A 0 4 4 0 2 4 x x Ox sR te hoe d 9X4 44 OO xk 6 4 2 0 2 4 6 Figure 3 2 Amplitude distribution of the incoming wave along the left part of the outer boundary for the problem Waveguide m With IncWave File Waveguide_income the incident wave is specified In the following li
11. user is adviced to use convex computational domains e intBound vector of edge markers indicating that the edge is an interior boundary Caution There are two restrictions regarding the inner bound aries The inner boundaries have to be closed curves as well At the inner boundaries only Dirirchlet boundary conditons can be set Example By typing load rect21 pdemesh p e t at the matlab prompt the most simple example of a triangulation of a rectangle with four points and two triangles is loaded to the workspace and plotted Other example files are located in geo 2We plan to overcome these restrictions in future releases 30 5 4 Material description MATERIALFILE The local properties of the material are specified in MATERIALFILE Syntax K MATERIALFILE MATERIALINDEX PARA Arguments e MATERIALINDEX vector MATERIALINDEX is the material index specified in t 4 and e 4 e PARA structure parameters passed to MATERIALFILE Description K MATERIALFILE MATERIALINDEX PARA returns the local wavenumber K for a triangle or edge corresponding to the material with index MATERIALINDEX PARA is a structure of auxiliary parameters The same syntax is used for the interior material description Material File for the material description of the boundary IncWave Material File and for the material description of the exterior Ext Ma terial File Example Waveguides can be described with the
12. 0 4 and w 0 073 The waveguide material and the cavity have a relative refractive index of 3 4 and the substrate has relative refractive index of 1 45 The problem description file is mcr2 m located in examples Let s have a look at it first Using the argument Para we can pass different wavenumbers to the problem description file Additionally the first eigenmode of the waveguide and the relative refractive indices of the waveguide material and the substrate are passed to the problem description file The function mcr2 income calculates the incident wave at port A It is important that no further outgoing fields are given anywhere Using the function makerays waveguide rays for the discretiza tion of the exterior are calculated See Section 4 1 for a detailed description of that function The script mcr2analyser m in examples is essentially a loop over different wavelengths In the first part the first eigenmode of the waveguide is calculated As the two waveguides are of the same material and width w this is done once With a call to main the problem is solved The rest of the script analyses the 16 scattered field and calculates the power flux along the waveguides at each port The result is shown in Figure 4 3 Let s have a closer look at the physics of this problem We start with some elementary considerations about guided waves for the Helmholtz equation Au x y K y u x y 0 in a two dimensional space Here the loc
13. 1 The file consists of lists of points edges holes and regional attributes We also include an interface to triangle which converts the triangle output files to a single file as input for HelmPole geometry file see chapter 5 3 This inter face consists of the function triangle2mat m located in the trianglefiles directory 7 1 Conversion interface TRIANGLE2MAT Syntax P E T TRIANGLE2MAT NAME Arguments e NAME Name of the output files of triangle located in the direc tory HelmPole trianglefiles without the file extensions 1 node l ele 1 poly Description This function reads the files HelmPole trianglefiles name 1 node name 1 poly name 1 ele and writes a geometry file HelmPole geo name mat 42 Please note the following convention for edge identifiers for boundaries to the exterior id lt 100 for interior boundaries 100 lt id lt 200 for inner edges not on a boundary id gt 200 43 Chapter 8 License and Copyright HelmPole is copyright 2003 Commic Group ZIB Zuse Institute Berlin Takustr 7 D 14195 Berlin Germany commic zib de HelmPole may be freely redistributed under the condition that the copyright notices are not removed and no compensation is received Private research and institutional use is free You may distribute modified versions of HelmPole under the condition that the code and any modifications made to it in the same file remain under copyright of the or
14. Approach to Coupled Interior Exterior Helmholtz Type Problems Theory and Algorithms Habilitationsschrift Konrad Zuse Zentrum Berlin 2002 F Schmidt L Zschiedrich R Klose and A Schadle A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal shaped domains in 2d to be submitted 2003 H G Unger Obptische Nachrichtentechnik Teil 1 Optische Wellenleiter Studientexte Elektrotechnik H thig 2 edition 1990 46 11 L Zschiedrich Transparent boundary conditions for time harmonic scat tering problems and time dependent Schroedinger equations PhD thesis Fachbereich Mathematik und Informatik FU Berlin in preparation 2003 AT
15. Computational domain 2 within the physical domain Q The bound aries of Q are Pert Pine and L ij 45 Boundary values are set on Tinti Thus restricted to Equation 6 2 reads Au k z u z 20 in Q u x u x usx on OQ 6 3 Onul L Onui O u m on OQ Here denotes the outward normal derivative with the normal n I In the subsequent sections only Equation 6 2 is considered 6 2 The PML method in 1D Consider the one dimensional Helmholtz equation on the semi infinite interval Q 1 00 Orzu k u 0 u 1 21 6 4 O u iku for x oo We want to restrict this equation to the interval 1 0 Thus the exterior is 0 oo The idea of the PML method is to transform the coordinate T y x 6 5 where for example y x 1 z 6 6 As 7 0 0 and 9 0 1 ei the solution of the transformed equation pyr x u y x coincides with u at 0 Assuming that k is constant for x gt 0 39 the solution of Equation 6 4 can be extended analytically to the complex plane hence the normal derivative at 0 is O u 0 O pyr 0 1 4 ic and the solution py is given by Os pyr t k 14 ic pyr 0 py 0 u 0 6 7 O t p x ik pyr 1l 4 ic The two fundamental solutions of the PDE in Equation 6 7 are xr exp ik 1 io x exp ikx kox 6 8 and z c exp ik 1 io x exp ikz koz 6 9 Solution 6 9 is exponentially increasing and Solution
16. al wavenum ber k does not depend on x Recall that the electrical fields z component of a TM polarized electromagnetic wave satisfies a scalar Helmholtz equation A guided wave is a field distribution which varies harmonically in x direction ug z y v y e Since ug satisfies the Helmholtz equation we conclude that Oyyth y k y v y Kv y Hence w is an eigenfunction to the operator O k y Using results from spectral theory we decompose a function w y as X c y where the y are eigenfunctions of the operator Oy k y The coefficients cj are given by the integral c f vr y w y dy Here our notation is formal since for the continuous part of the spectrum the sum must be replaced by an integral The eigenfunctions v are orthogonal to each other vivi dy dw 4 1 For any splitting c c cj the function y 5 E e e cL y ettet l l is a solution to the Helmholtz equation The left sum corresponds to left traveling waves and the right sum to right traveling waves To compute the splitting assume that also v y O u x y is given for x 0 Using the orthogonality relation we get JZ p v y dy ik ci c Hence given any solution u to the Helmholtz equation Au x y k y u z y 0 we are able to compute the above splitting by the two formulas n CL a ypu 0 y dy iks Ci c65 L Vi Oru 0 y We now consider the case of a TM polarized time harmo
17. at fit waveguides in the exterior The parameters are listed below vertices nv x2 matrix containing the x y coordinates of the vertices of the polygon in clockwise order Here nv is the number of vertices vray nv x2 matrix containing the x y coordinates of the rays starting a the vertices of the polygon They have to be chosen in accordance with the rule given above In case vray is empty normal rays at vertices are chosen i e the rays are the mean value of the normals of the two adjacent segments vray are the arrows in the corners of the rectangle shown in the left of Figure 5 3 Waveguide nwr x 3 matrix of the rays describing the waveguide Each row in Waveguide specifies the x y coordinates of the starting points of a ray and the angle of this ray with the normal to the face of the polygon on which this starting point lies Each of the nwr 2 waveg uides is specified by its two enclosing rays In the example shown in Figure 5 3 four parallel vectors are given indicating the positions and the directions of the waveguides entering the computational domain on the left and on the right In this case the waveguides are continued horizontally through the computational domain The function makerays_waveguide calculates all other rays for the dis cretization of the exterior from the rays specified in vertices vray and Waveguide It uses weighted averages of the given rays for the determi nation of the remaining rays
18. ave length the maximum error decreases only negligible Note that we discretize the PML according to the interior to see this behavior The rays for the discretization of the exterior are calculated using the function makerays waveguide The rays have to be cho sen such that the refractive index is constant in each segment of the exterior i e between each pair of rays Therefore the rays have to be adjusted to the waveguide in the exterior In addition the rays have to form an admissible dis cretization of the exterior see Section 5 6 and Chapter 6 The parameters passed to makerays_waveguide are Para vertices Para vray and Para Waveguide Para vertices are the x y coordinates of the polygonal computational domain in clockwise order Para vray are z y coordinates of the rays v prescribed at the 14 lt l V Ww az by c 0 a V y d T y V a V a V V Figure 4 1 Geometry of waveguide in example Waveguide2a edges which have to be chosen to be admissible Para vray may be empty in which case the rays are chosen to be the mean of the normal vectors of the two adjacent faces at each vertex If we would not specify Para vray in this example the rays at the vertices would intersect with the waveguide and we could not obtain a valid discretization of the exterior Para Waveguide gives the position and direction of the waveguide Each row of Para Waveguide specifies one ray by giving the z and y coordinate of the root
19. be found for example in the papers by Lassas and Som ersalo 7 6 and Collino and Monk 2 for homogenous exterior Based on the work of F Schmidt 8 HelmPole however allows for certain types of inhomoge neous exterior domains including waveguide structures A detailed description of the implementation can be found in 9 Convergence issues of the PML are discussed in 4 This manual gives a detailed description of HelmPole s user interface which allows to adapt geometries and material properties as well as the characteristics of the incoming wave The two main applications of HelmPole are electromagnetic and acoustic scattering Typical electromagnetic scattering problems are modeled by Maxwell s equa tions V B 0 V D p 1 1 OB OD D RE E VOIE CP pm 1 2 with permeability u 1 vanishing charge density p vanishing electric current density j and a spatial dependent relative permittivity e x Assuming a har monic time dependency equations 1 2 take the form V x E x poiwH x V x A x eoe z iu E a 1 3 3 Decoupling these equations yields the photonic wave equations Vx xsv x Hs pom 1 4 V x V x E a e 5 E a 1 5 HelmPole assumes problems homogeneous in z direction For light propagating only in xy direction it is possible to separate modes in two different polarizations In the case of TE polarization the H field takes the form 0 0 H and the photonic wave equation 1 4 simplif
20. dgements We thank P Deuflhard and F Schmidt for support and discussions We acknowledge support by the initiative DFG Research Center Mathematics for key technologies of the Deutsche Forschungsgemeinschaft German Research Foundation and by the German Federal Ministry of Education and Research BMBF under contract no 13N8252 HiPhoCs 45 Bibliography 1 2 3 10 J P B renger A perfectly matched layer for the absorption of electromag netic waves J Comput Phys 114 2 185 200 1994 F Collino and P Monk The perfectly matched layer in curvilinear coordi nates SIAM J Sci Comput 1998 M Hammer Resonant coupling of dielectric optical waveguides via rectan gular microcavities the coupled guided mode perspective Optics Commu nications 214 1 6 155 170 2002 T Hohage F Schmidt and L Zschiedrich Solving time harmonic scattering problems based on the pole condition Convergence of the PML method Preprint 01 23 Konrad Zuse Zentrum ZIB 2001 F Ihlenburg Finite Element Analysis of Acoustic Scattering volume 132 of Applied Mathematical Sciences Springer Verlag New York 1998 M Lassas J Liukkonen and E Somersalo Complex Riemannian metric and absorbing boundary condition J Math Pures Appl 80 7 739 768 2001 M Lassas and E Somersalo On the existence and convergence of the solu tion of PML equations Computing 60 3 229 241 1998 F Schmidt A New
21. ect the two rays From these intersections one draws again parallels to the adjacent faces and so on The rays are admissible if in the end the last two parallels and the last ray intersect in one point Formally this means that the mapping Q given in Equation 6 11 is continuous There are three general purpose methods supplied in fun e makerays_radial calculates radial rays from an origin given by the vec tor PARA LineCenter This gives very good discretizations for circles and squares if LineCenter is chosen to be the center of the circle or square It also works for rectangles where the ratio of the two different sides is not too large See left of Fig 5 2 in this examples the center of the rays is the center of the rectangle e makerays_normal calculates normal rays to the boundary This works well as long as the boundary does not have too sharp corners In this case PARA is empty See right of Fig 5 2 At corners the normal direction is the average of the normals of the two adjacent faces makerays_normal can lead to very 33 Figure 5 2 Radial left and normal right rays for the exterior surrounding a rectangular domain bad discretizations of the exterior as a refinement of the interior will not improve the approximation of the exterior at the corners e makerays waveguide is an elaborate tool to calculate rays for polygonal domains It allows to calculate rays th
22. ectory examples This file is the interface in which the user can choose from the different features of the program and set variables The file looks as follows function problem AlgPara waveguide Para template for problem description file for data structure and syntax read README AA geometry material sources and incoming waves kO 1 TM 1 CoarseGrid w Material File Waveguide material Material Para k k0 Material Para ni 1 45 relativ refraction index of substrate material Material Para n2 6 6 relativ refraction index of waveguide material IncWave File Waveguide_income IncWave Para Material Para IncWave Para TM TM IncWave Para WaveGuidePos 6 6 0 2 0 2 Exterior Material File Waveguide material 7 Exterior Material Para Exterior Ray File Exterior Ray Para Exact File Exact Para Exact Para lambda Int Material Para makeRays normal Waveguide_exact IncWave Para 5 04973 UniRefine tol o o PreRefine PpW 5 Int Int FEM quadrat Type TBC TBC TBC TBC TBC Method PML PML k k0 PML xi PML DampFun PML d Triangulation Material 1 Solution Real Solution Imag solution Abs solution Log ExactSolution Do Solution Contour ExactSolution ExactSolution ExactSolution ExactSolution ic 0 2 279 9 2793
23. erial File may be empty Exterior Ray File m file string a function describing the rays of the exterior discretization see Section 5 6 There are three files already provided in fun namely makeRays normal m makeRays radial m and makeRays_waveguide m Exterior Ray Para structure parameters passed to Exterior Ray File The structure Exterior Ray Para may be empty e Dirichlet structure Dirichlet boundary data This structure may be omitted if there are no interior edges cf Section 5 3 Dirichlet file m file string Dirichlet data on interior bound aries Dirichlet file is the name of the file that calculates the Dirichlet boundary values on the interior boundary Dirichlet Para structure parameters passed to Dirichlet file may be empty e Exact File m file string name of the file to calculate the exact solution in the interior The parameters passed to Exact File are Exact Para e Int UniRefine integer number of uniform refinement steps of the coarse grid useful for convergence studies e Int PreRefine integer number of local refinement steps to obtain a bal anced grid i e the grid is refined where the local wavenumber is high obsolete use Int PpW instead The algorithm picks the triangles with the top 20 of the largest triangles weighted by the local wavenumber 26 e Int PpW real local refinement to obtain at least Int PpW discretization points per wavelength locally
24. following material file function k waveguide_material index Para kO Para k matb 1 45 matg 6 6 k ones 1 length index matg iti find index 1 index 3 k iti matb k k k0 With this a relative refractive index of 1 45 is attributed to triangles and edges with material indices 1 or 3 The rel refractive index of 6 6 is attributed to all others 3l 5 5 Incident wave file INCWAVE Calculates the Dirichlet data and generalized Neumann data on the outer bound ary generated by an incoming wave Syntax U DU INCWAVEFILE P RAY PARA Arguments e P 2 x nedof matrix boundary points x y coordinates of points on the boundary where nedof is the number of degrees of freedom which in case of quadratic finite elements is 2 ne e RAY 2 x nedof matrix rays discretizing the exterior see Section 5 6 Each column represents one ray The rays are not normalized e PARA structure parameters that are passed to INCWAVEFILE Description U DU INCWAVEFILE P RAY PARA calculates the Dirichlet values U and the directional derivative of the incoming wave in direction of the rays DU on the degrees of freedom on the outer boundary DU VU RAY U and DU are column vectors Example Consider a plane wave exp ikr given boundary points p rays ray and the parameters of the structure IncWave Para with fields k and angle Then EW income calculates the Dirichlet values u of the i
25. iangulation BOUNDARY RAY are the rays used to discretize the exterior The transpose of BOUNDARY PMP is the projection for function values given on points of the triangulation POEBLEM GEO P onto boundary points BOUNDARY P p s e b main Waveguide calls the example Waveguide described in Section 3 There quadratic finite elements are used With quiver b p 1 b p 2 b ray 1 b ray 2 you can plot the rays plot3 b p 1 b p 2 abs b PMP s ul x hold on plots the absolute value of the linear part of solution on the boundary 24 5 2 User interface PROBLEMNAME The problem description file is located in the directory HelmPole Examples of problem description files can be found in examples It is the the main user interface where the parameters of the program run are set Syntax PROBLEM ALGPARA PROBLEMNAME PARA Argument e PARA structure this is the structure of parameters passed to MAIN Description PROBLEMNAME returns two structures PROBLEM describes the problem parameters and ALGPARA describes the algorithmic parameters The options for the different parameter choices are listed below e kO real wavenumber The wavenumber k0 specifies the wavenumber in equations 1 7 and 1 6 e TM integer equation type The TM switch specifies which equation is to be solved TM 1 corresponds to equation 1 7 TM polarization TM 0 corresponds to equation 1 6 TE p
26. ies to 1 w V xz vitse a Hy U 1 6 In the case of TM polarization the E field takes the form 0 0 E and the photonic wave equation 1 5 simplifies to AB 0 y Sela y E ns 0 1 7 For an introduction to acoustic scattering we refer to 5 where a derivation of the Helmholtz equation 1 8 from linearized equations for compressible fluids may be found The stationary pressure p in a compressible fluid is given by Ap k p 0 1 8 with the wavenumber T s 1 9 C where w is the circular frequency of the wave and c is the speed of sound The plan of this manual is as follows Chapter 2 describes the installation of the Matlab package HelmPole tar Chapter 3 guides the user through a test run of HelmPole with the parameters of the wave guide example In Chapter 4 several further examples give the user a feeling for the different problems which can be approached by HelmPole Chapter 5 provides a detailed explanation of all variables which can be modified by the user Chapter 6 gives a short introduction to the theory of the PML method and describes the coordinate system used to discretize the exterior Chapter 7 shortly presents a conversion interface to the triangular grid generation tool triangle by J Shewchuk Chapter 2 Installation To install HelmPole extract the tarred file HelmPole tar gz by typing tar xzvf HelmPole tar gz This creates a directory HelmPole containing the subdirectories exam
27. iginal authors the code is made freely available without charge and clear notice is given of the modifications Distribution of HelmPole as part of a commercial system is permissible only by direct arrangement with the Commic group at ZIB If you are not directly supplying this code to a customer and you are instead telling them how they can obtain it for free then you are not required to make any arrangement with us Referencing We kindly ask you to reference the program and its authors in any publi cation for which you used HelmPole The preferred citation is this manual S Burger R Klose A Sch dle and L Zschiedrich HelmPole A finite element solver for scattering problems on unbounded domains Implementation based on PML ZIB Report 03 38 Zuse Institute Berlin 2003 Or in BibTeX format techreport HelmPole2003 author S Burger and R Klose and A Sch adle and L Zschiedrich title HelmPole A finite element solver for scattering problems on unbounded domains 44 Implementation based on PML institution Zuse Institute Berlin number 03 38 year 2003 Updates Please note that the program HelmPole is intended to be constantly up dated Updated versions of the code and of this manual will be available from the homepage of the Computational Microwave Technology group at the Konrad Zuse Center for Information Technology Berlin http www zib de Commic Acknowle
28. ion is the function y introduced in 6 6 TBC PML d real damping factor for the PML It is save to set this factor to 1 TBC PML d equals o in 6 6 e Plots structure specifies which figure to open and what to plot Plots Triangulation boolean plot the triangulation Plots Material boolean plot the local wavenumber Plots Solution structure specifies which features of the solution to plot Plots Solution Real boolean plot the real part of the solution 27 Plots Solution Imag boolean plot the imaginary part of the solution Plots Solution Abs boolean plot the absolute value of the so lution Plots Solution Log boolean plot the logarithm of the absolute value of the solution Plots Solution Contour boolean plot contour lines Contour lines are only plotted if the Matlab PDE toolbox is avail able e Plots ExactSolution structure specifies what features of the exact so lution to plot if it is available It also plots the pointwise error The usage is the same as for Plots Solution Example See the example problem description files in examples 28 5 9 Geometry description COARSEGRID The geometry description file stores the triangulation of the computational do main and specifies which of the boundaries are outer transparent boundaries and which are interior boundaries where Dirichlet data is given Syntax CoarseGrid Description CoarseGrid mat file contains the ar
29. main Its Jacobian is denoted by J The exterior problem can be formulated in the n coordinate system The transformed Helmholtz equation is Ven JII TJ Venus J k u 0 6 12 The details on the discretization of Equation 6 12 can be found in 9 We will only give a sketch here The ansatz m gt tis Owl 6 13 with w V where V is a discrete space periodic on Nmin max yields a system of second order ODEs cf 9 Av Aide Abd E 0 6 14 It is shown in 4 that provided k is constant in each segment Equation 6 14 can be extended analytically to the complex plane k being constant in each segment allows to model infinite waveguide structures entering the computational domain To obtain the PML the transformed equation is replaced by an equation on a finite domain with e g Dirchlet boundary condition A Chapter 7 Grid Generation We recommend the use of the triangulation tool triangle Triangle needs a description of the geometry as input file poly file and creates files containing the nodes elements and boundary of a triangulation respectively The han dling of triangle and the description of the input file is found on the web site http www 2 cs cmu edu quake triangle html We include several examples of input files for triangle in the folder trianglefiles e g w2 poly which is used for the grid generation for the waveguide problem described in section 4
30. n different geometries and how to obtain rays that give a better discretization of the exterior Section 4 2 deals with the resonant coupling of optical waveguides via rectangular microcavities with an example originally discussed in 3 In Section 4 3 an example from acoustic scattering is discussed We present two ways how to calculate the field scattered by a soft object Section 4 4 discusses a Gauf light beam incident on the computational domain 4 1 Waveguide revisited The example Waveguide2a is run by typing the command main Waveguide2a at the Matlab prompt In this case the computational domain is a rectangle given by 0 1 x 2 2 with an infinite waveguide directed along an axis given by y 2r 1 cf Fig ure 4 1 Besides the different geometry of the problem the main differences to the example Waveguide are the way how to describe the incident wave and the way how to define the rays for the discretization of the exterior This is done by setting IncWave File Waveguide_income2 IncWave Para Material Para IncWave Para a 2 IncWave Para b 1 IncWave Para c 1 IncWave Para d 0 4 sqrt 1 5 IncWave Para Plot 1 and 13 Exterior Ray File makeRays_waveguide Exterior Ray Para vertices 0 2 1 2 1 2 0 2 Exterior Ray Para vray 1 2 1 2 1 2 1 2 Exterior Ray Para Waveguide 0 1 5 atan2 2 1 0 0 5 atan2 2 1 1 1 5 atan2 2 1 1 0 5
31. ncoming wave and the gener alised Neumann data us us Vu ray on the boundary function u us EW income p ray Para k Para k alpha Para angle u exp i k cos alpha p 1 sin alpha p 2 dxu i k cos alpha exp i k cos alpha p 1 sin alpha p 2 dyu i k sin alpha exp i k cos alpha p 1 sin alpha p 2 us sum dxu dyu ray 32 5 6 Discretization of the exterior MAKERAYS MAKERAYS is used to calculate the rays for the discretization of the exterior Syntax RAY MAKERAYS P S PARA Arguments e P 2xnbp matrix of x y coordinates of boundary points nbp is the number of boundary points e S structure describing the segments S p_i is a 2 x nbp matrix of point indices of vertices on the boundary e PARA structure any additional parameters for MAKERAYS Description RAY MAKERAYS P S PARA calculates a 2 x nbp array of rays Each column represents one ray The rays are normalized to euclidian length 1 The rays have to form an admissible discretization of the exterior To put it simple The discretization of the computational domain transforms the boundary of the computational domain to a polygon faces The exterior is modelled by quadrilateral elements each defined by a face of the computational domain and by the two rays starting in the adjacent vertices of the face and going outwards If you start drawing a parallel to one face this has to inters
32. nes parameters for the incoming field are set These are the material parameters plus some additional parameters The parameters are used by the file Waveguide income m in fun to calculate the Dirichlet data and the Neumann data at the left boundary of the computational domain See Section 5 5 for a detailed description of the syntax of IncWave File Then the exterior is specified The exterior is divided into segments be tween rays starting at the computational domain and going outwards to infinity Exterior Material File describes the refractive indices in the exterior It has the same syntax as Material File and in this case it is the same file Thus we have to pass the parameters Material Para to Exterior Material Para The Exterior Ray File specifies the discretization of the exterior By spec ifying makeRays normal normal rays to the boundary are chosen At corners the normal is defined to be the mean of the two normal vectors to the adjacent Real part Absolute value 2 4 6 4 2 0 2 4 6 Figure 3 3 Real part and absolute value of the solution of the problem Wave guide m edges How to define rays in general is described in Section 5 6 In this case the structure Exterior Ray Para is empty however it cannot be omitted The use of makeRays normal should be limited to smooth computational domains In the present example it gives a bad discretization of the exterior because the outgoing segments at corners of
33. ng or absorbing boundary conditions at Fest This can formally be written using the linear operator DtN that given Dirichlet data on the boundary I for the Helmholtz equation in the exterior returns the Neumann data of the solution O p DtN p on Pest One way to evaluate the operator DtN is the PML method Additionally we have a Dirichlet boundary condition at Ding 19 The total field p can be split into the known incident field p and the scattered field pse All three are solutions of 4 3 At Ding we have that p 0 pin psc and hence pin Psc Now there are two ways to formulate the problem e The first one is to derive an equation for the scattered field Pse Pse has to be a solution of the Helmholtz equation Apsc k ps 0 4 4 It has to fulfil the Dirichlet condition soft scattering Pse g on Tint where 9 pis r and it has to fulfil the Robin condition Qp DtN psc at T aes e The other way is to derive an equation for the total field p p has to satisfy Equation 4 3 At Ting we have p 0 and at the boundary Pert the scattered field p has to satisfy Psc p Pin O p Pin DtN p pa Two examples for the formulations are provided Both model the scattering of a plane sound wave by a soft circle Iry the examples Cylinder m and Cylinder2 m In Cylinder the incoming wave is set to zero the Dirichlet values on the interior boundary are set to the values of an incoming wave
34. nic electromagnetic wave in a waveguide The electrical field is given as 0 E x y 0 E x y The time average power flux along the waveguide direction is given by the z component of the Poynting vector 1 oS z E x H Making use of H iy curl E with y J ue k we get Ce IA iE 0 Et To compute the total power flux P in x direction we integrate this expression over y Splitting E as above E S a ih y e m So ui y ete and using the orthogonality relation 4 1 we end up with P 1 ket t gt kzl C l i a po P Hence for each l in the sum above P kz cj is that part of the left going power flux P 4 which is transported by the lth guided mode In the example above we want to compute the coupling of the scattered field with the fundamental modes of the waveguides At each port the scattered field excites the fundamental mode of the waveguides Let us restrict the analysis to Port C Assume that the solution field E x y is given at every point x y R Since there is no incoming wave from the right co 0 the overlap integral Po Eye 4 2 is equal to P o The simulation only yields the total field E within the com putational domain But since the fundamental mode Y is evanescent outside the waveguide we replace the infinite integral in 4 2 by an integration over the right boundary of the computational domain Figure 4 3 shows the power at the four port
35. olarization e CoarseGrid mat coarse grid file string geometry description The variable CoarseGrid is a string specifying the name of the mat file with the description of the geometry of the problem The file is located in the directory geo The contents of this file is described in Section 5 3 e Material This specifies the nature of the materials in the interior compu tational domain Material File m file string local wavenumber Material File specifies the name of the corresponding file located in the directory fun The contents of this file is described in section 5 4 Material Para structure parameters for Material File Mate rial Para may be empty e IncWave This specifies the incident wave 25 IncWave File m file string incident wave on the outer boundary IncWave File specifies the name of the file with the description of the incident field The file is located in the directory fun IncWave Para structure parameters for IncWave File IncWave Para may be empty e Exterior structure to describe the exterior Exterior Material File m file string local wavenumber for the exterior Exterior Material File specifies the name of the file with the de scription of the material distribution in the exterior The wavenumber is constant in each segment described by the rays This allows to simulate waveguides Exterior Material Para structure parameters for Exterior Ma t
36. ples fun geo trianglefiles the function startup m and the file main d11 The subdirectory examples contains examples of problem description files see chapter 5 2 User defined functions for the description of materials bound aries incoming fields etc are located in the subdirectory fun The subdi rectory geo contains geometry description files see Section 5 3 which have been created using the files located in the subdirectory trianglefiles sce chapter 7 To run HelmPole you need the programming language Matlab http www mathworks com For applications with complex user defined geometries we recommend the triangulation tool triangle by J Shewchuk freely accessible under http www 2 cs cmu edu quake triangle html see chapter 7 To check if your version is working go through the example described in Chap ter 3 Chapter 3 Quick Start In this chapter we go through a sample session of HelmPole step by step This should give a basic idea of the program and its possibilities A more complete description of the supported features is given in Chapter 5 In Chapter 4 a detailed description of various further examples is given In our example we simulate the propagation of light in an infinite straight waveguide in space R To obtain a bounded computational domain we introduce an artificial so called outer boundary The computational domain we get this way in the example is a square region with a s
37. point of the ray and the angle between the ray vector w and the normal vector v of the face of the polygon The geometry in example Waveguide2a seems to be very disadvantageous The reason for that is not clear Maybe the acute angle in the finite elements in the PML causes this problem Other examples with waveguides are Waveguide2 and Waveguide2b 4 2 Resonant coupling of optical waveguides This example deals with the resonant coupling of optical waveguides via a rect angular microcavity see Ref 3 To run the simulation invoke the script mcr2analyser m which is located in the folder fun In the problem description file mcr2 the number of points per wavelength is set to 15 to obtain accurate results However this means that there are about 100 000 unknows in the linear 15 system The coputational domain of the problem is shown in Figure 4 2 Two hori zontal infinite straight waveguides are coupled by a rectangular microcavity The outer boundary is again a rectangle At port P4 we prescribe the incident wave traveling in the waveguide At ports P4 Pg Po and Pp the response of the microcavity is measured We want to plot the power flux at the ports depending on the wavenumber respectively wavelength of the incoming wave Fnac ifs Po E 4 y W Pa gt Pp a 3Z a Figure 4 2 Resonance coupler In this example we choose the parameters W L 1 451 g
38. rays p e t outBound and intBound e p Points 2 x np matrix containing the Euclidian coordinates of the points where np is the number of points e e Edges 4 x ne matrix containing the boundary edges of the coarse grid where ne is the number of edges Each column of e corresponds to one edge e 1 index of start point e 2 index of end point e 3 edge marker to determine whether the edge is belonging to the inner or to the outer boundary e 4 material index used by the file Ext Material File The computational domain lies on the left of an edge i e all edges are orientated in counter clockwise order e t Triangles 4 x nt matrix containing the triangles where nt is the number of triangles t 1 index of first vertex t 3 index of third vertex x t 2 index of second vertex FAS od material marker to be used by the file Material File For the orientation see Figure 5 3 e outBound vector of edge markers indicating that the edge is belonging to the outer boundary Caution Please note that the outer boundary has to be a closed curve The computational domain bounded by the outer lIn future releases this restriction will be overcome 29 Figure 5 1 Orientation of the degrees of freedom on a triangle boundary has to be starshaped However for starshaped non convex do mains great care has to be taken to chose a physically meaningfull problem Hence the
39. ry This chapter provides a short introduction to the theory It serves as a reference for the parameters that can be set in HelmPole The PML method is derived for the simple case of the one dimensional Helmholtz equation The interpretation of the PML used here is a complex continuation of the solution in the exterior domain This ansatz can be found for example in the papers by Lassas and Somersalo 7 6 and Collino and Monk 2 Next the discretization of the exterior of a bounded convex polygonal two dimensional domain in a ray like manner is explained A much more detailed description of the theory can be found in Ref 9 6 1 Helmholtz equation The basic equations we consider are of Helmholtz type To be more precise two types of equations are considered V Vu z keu a 0 6 1 and Au z kee z u a 0 6 2 These equations hold for Q where Q is in general infinite In addition these equations have to be completed by boundary conditions on the interior boundary Ilint of Q and radiation boundary conditions at infinity The region of interest is denoted by 2 see Figure 6 1 We define the outer boundary Text to be the part of 0 that is not part of the interior boundary Pint On Pert U is the sum of the known incoming field u and the scattered field us The aim is to obtain a relation between us and yus on Tex The wavenumber k is defined by k z Kee at 38 L5 Figure 6 1
40. s vs the wavelength of the incoming eigenmode For the parameters see the problem description file mcr2 m in examples 18 x lower left port e lower right port upper right port upper left port power at port eo Cc T 1 56 08 es A e amp rt 1 54 1 542 1544 1 546 1 548 1 55 1 552 1 554 1 556 1 558 wavelength Figure 4 3 Normalized power at the four ports A marker is plotted at every 10 point 4 3 Acoustic scattering This example treats acoustic scattering by soft infinite cylindrical obstacles in R Problems of this type can be reduced to two dimensional scattering from closed curves Our computational domain is a rectangular box with a hole In order to have the analytic solution at hand we take a circular hole The boundary of Q naturally splits in two parts One is the interior boundary Iin the boundary of the hole The other is the exterior boundary Tl 444 where non reflecting boundary conditions have to be imposed For a derivation of the Helmholtz equation 4 3 as a mean to model acoustic scattering see the introduction in the book by F Ihlenburg 5 Apt k p 0 4 3 Here p is the stationary wave and k is the wavenumber The Helmholtz equation has to be supplemented by boundary conditions at infinity i e radiation condi tions for the scattered field If we want to bound the computational domain we have to impose non reflecti
41. tb kO IncWave Para a 1 IncWave Para b 0 5 IncWave Para c 0 IncWave Para wO 1 This means in x y coordinates the main axis is given by ax by c 0 and the focus is the cut of the main axis with the orthogonal axis br ay c 0 The exact solution is the Gauf beam in the computational domain which is only an approximate solution to the Helmholtz equation Thus the error is mainly due to modeling a wave by a Gauf beam Typing 21 Para k 4 Para matb 3 5 Para matg 1 main gauss Para calculates a Gauf beam as before incident on a photonic crystal with 3x7 holes on a hexagonal grid The error calculated for this setting of the material parameters is meaningless 22 Chapter 5 Function References 5 1 The main program MAIN Syntax PROBLEM SOLUTION ERROR BOUNDARY PROBLEM SOLUTION ERROR BOUNDARY MAIN PROBLEMNAME MAIN PROBLEMNAME PARA Argument e PROBLEMNAME is a string specifying the name of a MATLAB function that describes the equation to be solved the geometry and the trans parent boundary condition cf 5 2 Some example files with problem de scriptions are included in examples An example template is given in example template e PARA is structure it allows to pass any parameters to the main program Description e PROBLEM SOLUTION ERROR BOUNDARY MAIN PROBLEMNAME returns four structures PROBLEM describing the problem The most
42. the computational domain cannot be improved by choosing a finer computational grid in the interior In the example described in Section 4 1 we will explain how to obtain good rays Exact File is the name of the m file located in fun that computes the exact solution for a given set of points This function is supplied to the program in order to facilitate the study of the convergence of the discrete solution towards the exact solution Parameters are passed to Exact File with the structure Exact Para Except for the parameter Para la these are the same as for the incoming wave In our example Para 1a is the first eigenmode of the waveguide This eigenmode is calculated in IncWave File Then algorithmic parameters follow With Int UniRefine 0 we choose the number of uniform refinement steps of the coarse grid Also a pre refinement that locally equalizes the number of discretization points per wavelength is omit ted Int PreRefine 0 Instead by setting Int PpW 5 we refine the coarse grid locally such that there are at least 5 points per wavelength By FEM Type quadratic quadratic finite elements are chosen With TBC Method PML the type of transparent boundary conditions is chosen to be the method of perfectly matched layers TBC PML k kO sets the wavenumber in the exterior domain The discretization of each segment in the exterior domain is characterized by TBC PML xi This is a vector giving the dis cretization of
43. the variable in 6 5 TBC PML DampFun pml jump defines the damping function for the PML pml_jump y cf 6 6 and TBC PML d 10 1 sets the damping factor 1 in 6 6 More details on how to set the algorithmic parameters are provided in Section 5 2 For a brief introduction to PML see Chapter 6 In the following lines different features are chosen to be plotted during the program execution Contour lines can only be plotted if the Matlab PDE toolbox is available Some more lines follow in the problem description file which should not edited by the user and which are not described here Real part Absolute value eo 2 4 6 Figure 3 4 Real part and absolute value of the exact solution to problem Wave guide m 3 3 Running the program Now we are ready to start the program by typing main Waveguide at the matlab prompt During the run a number of windows opens to show some information to the user The first opening window is shown in Fig 3 1 The left part of this figure shows the geometrical description of the problem The interior domain is triangulated and the PML layer is divided into stripes bordered by outgoing arrows Obviously the exterior domain is not well resolved at the corners by the normal rays A refinement of the interior domain would not improve that The right part of the picture shows the distribution of the material refractive index in the interior domain The other
44. traight waveguide of high refractive index material and a low index background see Figure 3 1 The light field reaching the computational domain from the exterior from left is the lowest mode of the waveguide Milli 6 S S 7 2 Q K K SI K CNS A N PSS Fa al SS CNN RRR ORS KOO ZD ZNN Z pra IA VINNNN Z 2 BONG M MAH CASS N NDARI p M fi O e A XA NIS Re SX A D 0 gum ERE Se SS ee SSeS EUN VERS VN SSS NERS SEATS 2 WAV ZZ SAMT DE s5 Xe AA Sess 4 AZ M DANI HEREIN 0 6 4 2 2 4 6 28 4 2 0 2 4 6 vid SZ D S ex 2 A AO ZA R Ni ya e IL ISTIS PSS By DO CX 4 NAAN KNSS KS NA So TS KISS eS SI SL v Co K p o Figure 3 1 Geometry and material distribution of the problem Waveguide m 3 1 Getting started Start Matlab and change to your HelmPole directory Within this directory you locate the folders examples fun geo trianglefiles the function startup m and the file main dll Set the necessary environment variables by typing Startup Else you can start Matlab from the directory HelmPole Matlab will then automatically set the necessary environment variables by examining the file startup m 3 2 The problem description file Waveguide m Start an editor and load the file Waveguide m which is located in the dir
Download Pdf Manuals
Related Search
ZR 03 38 zr038-l