Digests from his work is presented in this chapter partly rewritten/restructured

Transcription

1 FEMLAB - for løsning av partielle differensial ligninger This chapter is based on the diploma thesis written by siv. ing. Helge Hansen in He evaluated the MATLAB-FEMLAB software and concluded that such software is well suited for university studies in which both mathematical modelling and advanced numerical solution of PDE problems are needed. 3.1 Acknowledgement Digests from his work is presented in this chapter partly rewritten/restructured by Robert Nilssen Problems involving partial differential equations (PDE s) appear in several areas of science and engineering. In general only a few of those equations can be solved by traditional approaches. These traditional approaches include analytical methods applied to the differential equations. 3.2 Summary 45

2 FEMLAB - for løsning av partielle differensial ligninger The PDE problems are for instance field and potential problems, including temperature gradient fields, electric potentials and fluid flow. In the recent decades, there has been a tremendous development of constantly more and more powerful computers. These incidents have opened for new strategies to approach a PDE problem. These new strategies are the applications of numerical methods. On the other hand, the demands to the software are numerous. Both the input of the problem data and the output of the solution have to be userfriendly and understandable. This chapter encounters MATLAB and the MATLAB-toolbox called FEMLAB. MATLAB is a product of the MathWorks Inc., and is an advanced interactive software package designed for scientific and engineering numerical computations. This program package is available on personal computers (PC s) as well as on mainframe computers. FEMLAB is one of several toolboxes made by MATLAB s rather intuitive programming language. The Toolbox is specially designed to solve two-dimensional PDE problems with a spesial version for electromagnetic problems. The basic use of the FEMLAB is presented to a certain extent, including some solved sample problems. In these sample problems, some of the possibilities and limits of the FEMLAB have been illuminated. An important subject discussed in this work, is FEMLABs potential of contributing to a students physical understanding of field related problems. To gain such understanding, it is important, that the PDE problem can be loaded into the computer in a fast and easy way, and that the presentation of the solution is of high quality. Furthermore, MATLAB is not the FEMLAB only. Some of the possibilities of using MATLAB to create own routines, which can solve PDE problems or parts of PDE problems, has also swiftly been looked upon. This mainly with the following question in mind: Is there any benefit to a student, trying to understand the numerical methods of PDE solving by the use of MATLAB. The basic material from electromagnetic theory is also provided, in Appendix A, as well as some theory of thermal fields. 46

3 Seksjon 3.3 MATLAB FEMLAB applies the finite elements method (FEM) to solve the PDE problems. This requires a treatment of the FEM as well. The two approaches presented are the Galerkin method and the variational formulation, which are presented in Appendix C. An overview of the symbols and the notation used in this work is given in Appendices J and K respectively Introduction MATLAB is a software package designed specially to solve scientific and engineering computational problems [10], [11], [12] and [23]. The software package is highly evolved and interactive. MATLAB is available on a wide spectrum of machines, ranging all the way from PC s to supercomputers. Basically MATLAB is written in FORTRAN, but subsequent versions are written in optimized C. 3.3 MATLAB The main features of MATLAB are: advanced algorithms for high-performance numerical computations, particularly in the field of matrix algebra a large assortment of predefined mathematical functions, and the ability to construct user-defined functions toolboxes available for advanced problem solving in various application areas two- and three-dimensional graphics for plotting and displaying data, and for pedagogical and scientific illustrations and visualizations puissant matrix/vector-orientated high-level programming language for individual applications capability to cooperate with programs written in other languages and for importing and exporting data Graphics MATLAB is one of the few program packages which can provide the computational capabilities of both generating data and displaying it in a variety of graphical representations. MATLAB is not just a computa- GRUNNLAG FOR ELKRAFT TEKNIKK 47

4 FEMLAB - for løsning av partielle differensial ligninger tion and plotting package. It is a flexible and versatile tool which allow users with rather elementary programming capabilities to produce sophisticated graphics and graphical user interfaces (GUI s). The process of learning how to create different plots and graphical objects utilizing MATLAB is rather fast. This is partly because most of the MATLAB graphics commands are straight-forward and intuitive. Plotting of data takes little time due to MATLAB s high-level language with its natural notation. MATLAB handles two- and three-dimensional plotting in an acceptable way. Several types of plots can be made, like for instance 3Dmesh, shaded surface plots, contour plots and quiver plots to mention some of the three-dimensional plot types. Even 4D-data can be plotted relatively easily as three-dimensional plots. MATLAB graphics also include commands that offers a possibility to make animated graphics. This feature could be of importance when time dependent problems are encountered Numerical Mathematics MATLAB is a powerful tool when working with numerical computations. Numerical computations often involves relatively large matrices. One of the advantages of MATLAB, is that the matrix is the basic element. The matrix does not even require predimensioning, when utilizing MATLAB. The way that MATLAB handles matrices and vectors, makes awkward indexed loops almost dispensable. MATLAB also includes a huge library of advanced routines for solving numerical problems. As an example the MATLAB ODE Suite can be mentioned, which is a feature used by FEMLAB. The m-files, which are discussed later, make it possible to produce numerical routines and save them as different m-files. The m-files are easy to call when one of the specific routines are wanted Communication With Other Programs MATLAB can be used in cooperation with other programs. A FOR- TRAN- or a C program can be called by MATLAB, or MATLAB can be called by a FORTRAN- or a C program. The first option can be valu- 48

5 Seksjon 3.3 MATLAB able if a MATLAB program is slow. Since MATLAB is an interpreting language, the commands are interpreted as they are executed, which sometimes may result in a rather slow execution. It should be noted, that it is not recommended to write FORTRAN or C routines unless it is indisputably necessary. The libraries included within MATLAB are written in FORTRAN 77 and C, but it is possible to link them with FORTRAN 90 and C++ as well. A compiled routine which can be called from MATLAB, is termed a mex-file. MATLAB keeps track of how they are called, and the mexfiles can, when compiled be utilized like an m-file M-files An m-file is a file which contains a number of consecutive MATLAB commands. It may refer to another m-file, and even be recursive. There is a number of predefined m-files supported by the MATLAB program package. The m-files can be divided into two types, which are: Function m-files Command m-files A function m-file is a user-defined function, which is a special kind of m-file. A function can have one or several arguments or parameters. The functions in MATLAB have a relatively strong resemblance to functions in FORTRAN or C. The function files must have certain properties: The first line in a function file must embrace the word function The first line must specify the function name, the input arguments and the output arguments A function may have zero, one or several input parameters and return values A function m-file is called by writing filename(input arguments) on the MATLAB command line. It is recommended that the name of the function is the same as filename. GRUNNLAG FOR ELKRAFT TEKNIKK 49

6 FEMLAB - for løsning av partielle differensial ligninger A command file does not have to contain the word function in its first line. Function files and command files are executed just like ordinary MAT- LAB commands. The statements in the file are executed when the name of the file is written on the command line, together with the arguments, if there are any. All m-files are common ASCII-files and can be created in a text editor as for instance Notepad. 3.4 FEMLAB The FEMLAB Toolbox (presented in more detaile later and in separate tutorilas) is built up of several files called m-files. These files are function- and command-files. The reason that they are called m-files, is that their names have the extension.m. The FEMLAB uses the finite elements method to solve PDE problems of various types. Several types of PDE problems can be solved by using FEMLAB, including physical problems as problems involving electromagnetic fields or thermal fields. A typical session of solving a PDE problem is made by a process, which includes three stages. These stages are: Preprocessing Calculation Postprocessing In the following, an orientation on how these three stages are gone through by utilizing FEMLAB will be presented Preprocessing The term preprocessing could be substituted by the term modeling the problem. Before a PDE problem can be solved, a certain amount of information must be loaded into the computer. The part of the software which handles this task is called the preprocessor. 50

7 Seksjon 3.4 FEMLAB In FEMLAB, there is no specific software module taking care of the preprocessing only. A requirement to the preprocessor is that it must be a very flexible tool, which in an efficient way, can enable the user to specify a PDE problem. The preprocessing includes the following steps: Specify the geometry of the PDE problem Assign boundary conditions to the problem Define the material properties Generate the mesh How FEMLAB attends these tasks will be presented in the succeeding. The Geometry. A geometrical model can be built and modified by applying the geometry commands. These commands consist of m-files, which execute the actions specified. There are two ways to construct a geometry: Specify a geometry by use of the GUI Specify a geometry from the MATLAB Command Line The GUI, Graphical User Interface, is entered by the MATLAB command pdetool. It is a command m-file making the necessary calls to other m-files in order to take care of various tasks. Inside the FEMLAB GUI (which is to be used mostly in this course), preprocessing, calculation and postprocessing are taken care of by making the right calls to the different m-files required. The GUI enables the user to work with the mouse, in a graphical sense, combined by the keyboard for entering such as coordinates, names etc. when necessary. To draw a geometry in the GUI, the draw mode has to be selected. The MATLAB Command Line, from now on referred to as the command line, is the main workspace of MATLAB. Entering the names of GRUNNLAG FOR ELKRAFT TEKNIKK 51

8 FEMLAB - for løsning av partielle differensial ligninger command- and function- m-files at the prompt on the command line, will commence the desired action. A solid rectangle can for instance be drawn by the command pderect. If this command, included its required parameters, is typed on the command line, the GUI will automatically start, such that the solid rectangle drawn can be viewed. The geometry of the PDE problem is built up by solid objects. There are four types of solid objects: Circle solid object Rectangle solid object Ellipse solid object Polygon solid object These solid objects are allowed to overlap. The geometry made by a combination of such solid objects is called a Constructive Solid Geometry (CSG). The model is termed a CSG-model ( see Appendix D.3.4.4???). A simple CSG-model can be seen in Figure 1. A cons. This geometry could be the geometry of a three-phase conductor just above the surface of earth. It is emphasized that this example is by no means anything else than a stylistic illustration SQ1 F i g u r e 1. A C1 C2 C3 c o n s

9 Seksjon 3.4 FEMLAB Each of the solid objects are given a unique name by the program. It is also possible for the user to specify a user-defined name as well. Figur 3 1: onstrua constructive solid geometry The CSG-model is internally represented by three data structures, which are: The Geometry Description Matrix The Set Formula The Name Space Matrix All of these three data structures, which are matrices, can be exported from the GUI to the command line. The geometry description matrix describes the CSG-model drawn. Each column in the geometry description matrix corresponds to a solid object in the CSG-model. The geometry description matrix consists of numerical entries. The set formula is a variable which describes how the solid objects are combined to comprise a main solid object. This main solid object could constitute the domain of definition for the PDE, for instance. In the set formula, three set operators are optionally included: Union, intersection and difference. The name space matrix is a text matrix which relates the columns in the geometry description matrix to the variable names of the set formula. The constructive solid geometry of Figure 1. A cons, consists of a square solid, SQ1, and three circle solids, C1, C2 and C3. To exclude the circle solids, that is making holes in the square, the following set formula is applied: SQ1-(C1+C2+C3). To figure out a set formula that picks out the valid domain, could be a bit cumbersome, but when some experience is gained, this would be a practicable operation. It could be advisable to draw the geometry on paper before drawing it on the screen, planning the composition of the solid objects, to make the desired geometry. It is always important to make a good sketch of the problem by hand first. To make the CSG understandable to other FEMLAB functions, it has to be decomposed. This means that the CSG is transformed into a set of GRUNNLAG FOR ELKRAFT TEKNIKK 53

10 FEMLAB - for løsning av partielle differensial ligninger disjoint minimal regions bounded by boundary segments and border segments. Optionally the set formula can be evaluated. The GUI uses the decsg procedure to execute this transformation. The decsg function may be entered at the command line, including the required parameters, as well as it may be called from the GUI. The decomposed geometry is represented by the decomposed geometry matrix. Decsg returns minimal regions which evaluates to true by the set formula. The decomposed geometry matrix contains a representation of the decomposed geometry in terms of disjoint minimal regions. Each edge segment of the minimal regions correspond to a column in the decomposed geometry matrix. It should be noted that the edge segments between minimal regions are referred to as border segments, and the outer boundaries are referred to as boundary segments. The decomposed geometry description matrix of the geometry in Figure 1. A cons, can be inspected in An example of a geometry description matrix. Further the decomposed geometry matrix of the same geometry can be viewed in A decomposed geometry matrix. For more information about representation of the geometry, see Appendices D.3.4.4, D and D Inside the GUI in FEMLAB there are ten??? application modes. That is eight application modes in addition to the generic scalar mode (the default mode) and the generic system mode. The generic scalar mode handles generic PDE s, and the generic system mode handles a system of to coupled PDE s. It should be noted that changing application mode causes all PDE-coefficients and boundary conditions to be reset to their default value of the specific application mode. gd =

11 Seksjon 3.4 FEMLAB f a geometry description matrix gd 0 = A n e x a m p l e o dl = Columns 1 through Columns 8 through Columns 15 through 16 GRUNNLAG FOR ELKRAFT TEKNIKK 55

12 FEMLAB - for løsning av partielle differensial ligninger Figur 3 2: A decomposed geometry matrix When employing an application mode, the generic PDE-coefficients are replaced by application-specific parameters such as the magnetic permeability µ in magnetostatics application mode. For more about the application modes, see Appendix D.3.6 and [10]. The Boundary Conditions. The boundary conditions taken care of by FEMLAB is the following three types (see Appendix D.3Diric.3): Dirichlet boundary conditions Neumann boundary conditions Mixed boundary conditions The boundary conditions can be specified either as a boundary condition matrix, or as a boundary m-file. A boundary m-file equivalent to a boundary condition matrix can be created by the wbound function. To specify boundary conditions inside the GUI, the boundary mode has to be selected. A boundary condition matrix is created internally in pdetool (the GUI). To each column in the decomposed geometry matrix there must be a corresponding column in the boundary condition matrix. The entries of the boundary condition matrix are real numbers. A boundary condition matrix could also be exported to the command line. In the GUI, the edges, which are assigned boundary conditions are colored according to the following scheme: Red color: Dirichlet boundary condition Blue color: Neumann boundary condition Green color: Mixed boundary conditions 56

13 Seksjon 3.4 FEMLAB These edges are oriented as well; An arrowhead displays in which direction they are oriented. As an example the following boundary conditions are assigned to the geometry in Figure 1. A cons: On the three boundary edges comprising the upper part of the square: Neumann boundary conditions: u/ n=0 On the lower boundary edge of the square: Dirichlet boundary condition: u=0 On the four boundary edges comprising the left circle: Dirichlet boundary conditions: u= 50 On the four boundary edges comprising the central circle: Dirichlet boundary conditions: u= 50 On the four boundary segments comprising the right circle: Dirichlet boundary conditions: u=100 Now the PDE problem could represent a three phase conductor just above the surface of earth in sinusoidal steady state, analyzed when the voltage is maximum in the right-most phase. The resulting picture on the GUI display, is reviewed in Figure 4. Boundary c. It should be noted that a whole circle consists of minimum four edges F i g u r e 4. B o u n d a r y c 3 3:Figur GRUNNLAG FOR ELKRAFT TEKNIKK 57

14 FEMLAB - for løsning av partielle differensial ligninger Boundary condions of the PDE problem It is often wise to include solid objects in the CSG as helping objects. For instance when rounding corners, small solid circles can be placed in the corners where the rounding is going to take place. To fulfill the rounding of the corner, some of the circle segments must be removed in boundary mode, which is the mode where the boundary conditions are assigned. This removal can be done by utilizing the mouse. The Material Properties. The material properties are entered as the coefficients of the PDE (see Appendix D.3.5.4). These coefficients are entered in the PDE-mode inside the GUI, and can be exported to the command line as well. The data structures taking care of the PDE-coefficients are either the coefficient matrix or the coefficient m-file. A coefficient m-file is a user-defined m-file. When using a coefficient m-file, the PDE- coefficients are not entered through the GUI. To enter the PDE-coefficients when working inside the GUI, the PDE mode has to be selected. In the GUI, the PDE-coefficients can be entered by simply doubleclicking on the desired subdomain. At the same time as the PDE-coefficients are entered, the type of PDE is chosen out of four possible choices (see Appendix D.3.2 and Appendix B): Elliptic PDE Parabolic PDE Hyperbolic PDE Eigenvalue problem These PDE s are written in standard FEMLAB form as: 58

15 Seksjon 3.4 FEMLAB ( c u) + au = f (3-1) (3-2) u d ( c u) au f t + = 2 u d ( c u) au f 2 t + = ( c u) + au = λdu (3-3) (3-4) As a continuation on the example, some PDE coefficients will be assigned. The coefficients assigned are: c=1.0, a=0.0 and f=0.0. This makes the PDE become an inhomogeneous elliptic PDE (see Appendix B). A view of the subdomains, actually only one domain, since the circle solids were excluded, can be seen in Figur 3 4:. The option of showing the subdomain label is chosen, which enforces the display of the 1. The choice of PDE-coefficients makes the following PDE: ( u) = 0 (3-5) The coefficients can be scalar complex valued functions of u. They may also be time dependent. There are some exceptions, which can be reviewed in Appendix D.3. GRUNNLAG FOR ELKRAFT TEKNIKK 59

16 FEMLAB - for løsning av partielle differensial ligninger Figur 3 4: Figure 5. Domain of the PDE problem The Mesh Generation Mesh generation. The only type of mesh that can be generated by FEMLAB, consists of linear triangular elements. The mesh data is described by the following three matrices (see Appendix D or [9]): The Point Matrix The Edge Matrix The Triangle Matrix The point matrix contains one column for each point in the mesh. Each column has two rows containing the x- and y-coordinate of the point or node. The edge matrix contains one column for each edge. Each column has seven rows containing information, as numbers, of an edge. The triangle matrix contains one column for each triangle of the mesh. Each column consists of four rows, of which the three first contains the node (point) numbers in a counterclockwise order, and the last row contains the subdomain number. When generating a mesh, the minimal regions are triangulated into subdomains, and the border segments and the boundary segments are broken down into edges. It should be noted, that the word edge means a part of either a boundary segment or a border segment. The mesh data is created from the decomposed geometry by the function initmesh. The mesh can be altered by the functions refinemesh and jigglemesh, which refines and jiggles the mesh respectively. 60

17 Seksjon 3.4 FEMLAB There also exists a function, adaptmesh, which creates a mesh as a part of the solution process. The initial mesh generation is done automatically by FEMLAB. The user has several options interfering this process, both before initializing and before an eventual refining or jiggling. The mesh data can also be exported from the GUI to the command line. In, an initialized mesh is shown. The mesh consists of 231 nodes, 408 triangles and 58 edges. When this mesh was generated neither jiggling nor refinement was done. Enabling the jiggle option generally increases the triangle quality. For the definition of the triangle quality, see equation (E 5-9) in Appendix E.5. The maximum edge size for this mesh was chosen to 0.6. An example of a point matrix, an edge matrix and a triangle matrix for a simple mesh, viewed in Figur 3 5:, is given in, Point matrix of a simple mesh, Edge matrix of a simple mesh and Triangle matrix of a simple mesh respectively. The options to show triangle number and node number was enabled here. The region which is meshed is the ellipse with center (0,0), semi major axis 2.4 and semi minor axis 1.4. Figur 3 5: Initialized mesh GRUNNLAG FOR ELKRAFT TEKNIKK 61

18 FEMLAB - for løsning av partielle differensial ligninger Figur 3 6: Very simple mesh of an elliptic domain To generate such a simple mesh, the maximum edge size was set to nf, which mean infinite p = Columns 1 through Column Figur 3 7: Point matrix of a simple mesh 62

19 Seksjon 3.4 FEMLAB e = Columns 1 through Column Figur 3 8: Edge matrix of a simple mesh t = Figur 3 9: Triangle matrix of a simple mesh GRUNNLAG FOR ELKRAFT TEKNIKK 63

20 FEMLAB - for løsning av partielle differensial ligninger Calculation The calculation or analysis, is done by applying the finite element method to the, now defined, PDE problem (see Appendix D.3.7). The calculation consists of assembling the matrices which build up the huge sparse matrix equation. This assembling, involves solution of several integrals. Solving this huge sparse matrix equation is also a part of the calculation. To solve huge sparse matrix equations is one of MAT- LAB s specialties. Before the calculation actually starts, there must be made a choice, whether the adaptive solver or not is to be applied. Furthermore, if the PDE problem is nonlinear, the nonlinear solver has to be invoked. It should be noted, that when adaptive solving is enabled, the PDE-coefficients can not be time dependent. When adaptive solving is applied, several parameters concerning the mesh have to be set. If these parameters are omitted, the problem will be solved with default values set by the program [9]. The nonlinear solver in FEMLAB is called by pdenonlin. A damped Newton iteration process is applied (see Appendix D.3.7.7). Several options can be enabled before the calculation is started, included specification of tolerance, and how the Jacobian is to be computed. The pdenonlin also calls required functions from the MATLAB environment outside FEMLAB. A scalar PDE can be solved, or a system consisting of two PDE s can be solved from inside the GUI. When the system contains more than two PDE s, the command line has to be used. The basic function of FEMLAB, is the function assempde (see Appendix D.4.1). This function assembles the stiffness matrix and the right hand side of an elliptic PDE. The command can optionally produce a solution to the PDE. assempde calls among other functions, the two functions assema and assemb. The function assema, assembles the area integral contributions to the matrix equation. The integrals involved are calculated analytically, due to the finite elements simplicity. 64

21 Seksjon 3.4 FEMLAB The function assemb, assembles the contributions from the boundary conditions to the matrix equation. When considering parabolic PDE problems, the function parabolic is invoked. This is a function which makes calls to advanced MATLAB functions outside of FEMLAB in addition to calling functions inside FEMLAB environment. Parabolic PDE problems will generate a time derivative term in the matrix equation. The matrix equation can be considered as a matrix ODE (Ordinary Differential Equation). To solve this ODE, FEMLAB uses the MATLAB ODE solver outside FEM- LAB. The functions assema, assemb and assempde are all called from parabolic (see Appendix D.3.7.5), to assist assembling the matrix equation. To hyperbolic problems, the function hyperbolic (see Appendix D.3.7.6) applies much in the same way as parabolic does to parabolic PDE problems. Now the matrix equation will be an ODE of second order. The solution, u, of a PDE problem can be exported from the GUI to the command line, where further operations can be performed on it. The solution is a vector. Below in Solution vector of the simplified PDE problem, the solution to a PDE problem defined on the domain of which the mesh is shown in Figur 3 6:, is revealed. The boundary conditions are: Dirichlet condition u=1 to the edges in the first, second and fourth quadrant, and Neumann condition u/ n=1 on the part of the boundary in the third quadrant. The PDE-coefficients are chosen to be c=a=f=2. The mesh is not further refined due to that the solution vector is to be listed as an illustration. u = Figur 3 10: Solution vector of the simplified PDE problem GRUNNLAG FOR ELKRAFT TEKNIKK 65

22 FEMLAB - for løsning av partielle differensial ligninger Postprocessing The term postprocessing could as a matter of fact be replaced with the term presentation of results The postprocessing part of the solving strategy can be divided into two parts as follows: Graphical presentation of the results Analysis of the results The graphical presentation of the results should be able to provide a view of the geometry and the results, in an informative way to the user. This is a very important property due to the understanding of the achieved results. The result analysis should provide the user with quantities which are derived from the solution. Such quantities could be capacitances or energy losses in an electrical problem for instance. The graphical presentation options of FEMLAB are numerous. Plotting can be done from inside the GUI, as well as from the command line. There are several functions that produce various plot types from the command line. The main solution presentation options are the following: Color plots Contour plots (isolines) Arrow plots (quiver) Height plots (3D-plot) Animations The animation option is only possible to choose when time dependent PDE problems are considered. There is possible to specify how many times the movie should run, and how many times in an interval a picture should be taken. This interval is specified by the user. The time steps within the interval is also user-specified, and may be chosen in a logarithmic manner such that the movie focuses on, for instance, the first moments of the time interval [9]. Several of these plot options can be chosen in the same presentation plot. In addition, the mesh can be plotted into these plots. 66

23 Seksjon 3.4 FEMLAB In the generic scalar application mode the following quantities can be plotted in the following various plot types: Plot type Color / contour Arrows Deformed mesh Height (3D-plot) Property u, u, c u, user entry u, c u, user entry u, c u, user entry u, u, c u, user entry Tabell 3 1: Plot options in generic scalar mode When a potential (solution) is plotted as a color plot, there are two options to choose from to display the colors: Flat shading Interpolated shading Flat shading means that the color is constant over the finite element (triangle), and interpolated shading means that the color varies with the solution inside the triangle. The colors can be changed by choosing from several optional colormaps. When contour plots are made, the contour of the geometry is shown as well. Contours can be plotted alone, combined with the arrow plot, or combined with the color plot. If the contour plot contains only the contour, or is combined with the arrow plot option, the color of the contours follows the colormap chosen. If the plot is a combination of a color- and a contour plot, the contours become black. The number of contours shown, can be specified by the user. Arrows in arrow plots are always visualized in red. There are two options attached to the arrow plot: Normalized GRUNNLAG FOR ELKRAFT TEKNIKK 67

24 FEMLAB - for løsning av partielle differensial ligninger Proportional The term normalized means that the arrows only show the direction of the vector field plotted, and the term proportional suggests that the length of the arrow shown are reflecting the strength of the vector field as well. Choosing the height plot option gives the ability to plot the same quantities as with the color plots. The height plot is a three-dimensional plot in a separate window. This plot can be either continuous or discontinuous. The height plot can include a plot of the mesh and the arrows as well, and it can be viewed from different angles rotating the plot using the mouse. The Figures below shows a few plot options connected to the PDE problem introduced in section. The solution presented is based on a mesh generation which includes jiggling and one regular refinement. To be able to plot the absolute value of the negative gradient of the solution, the user entry sqrt(ux.^2+uy.^2) was entered as plot option, see Plot options in generic scalar mode.the example has been made, using the electrostatic application mode (see Appendix D.3). 100 Color: V Vector field: E Figur 3 11: Solution in colors, and the negative gradient of the solution as arrows 68

25 Seksjon 3.4 FEMLAB Contour: sqrt(ux. 2 +uy. 2 ) Vector field: E Figur 3 12: The negative gradient of the solution and its absolut value Contour: sqrt(ux. 2 +uy. 2 ) Vector field: E Figur 3 13: A zoom up of the details around the central and right-most circles from GRUNNLAG FOR ELKRAFT TEKNIKK 69

26 FEMLAB - for løsning av partielle differensial ligninger Color: sqrt(ux. 2 +uy. 2 ) Vector field: E Figur 3 14: The absolute value of u as colors, and - u as arrows Figur 3 15: Heigth plot of u with the absolute value of u as colors. 70

27 Seksjon 3.4 FEMLAB Figur 3 16: Same as fig.3 15 from a different space angle with other colors The quality of the wholly colored images is better on a computer screen than it is imported into a text editor. GRUNNLAG FOR ELKRAFT TEKNIKK 71

28 FEMLAB - for løsning av partielle differensial ligninger 72