Finite Element Method Introduction, 1D heat …

1 Finite Element MethodIntroduction, 1d heat conduction1 Siemens Wind Power A/SExternallecturer, Jesper Winther St rdahlFinite Element MethodIntroduction, 1d heat conduction2 Post doc. Johan ClausenFinite Element MethodIntroduction, 1d heat conduction3 Lectureplan FiniteElement Element MethodIntroduction, 1d heat conduction4 Form and expectations To give the participants an understanding of the basic elements of the Finite Element Method as a tool for finding approximate solutions of linear boundary value problems. These will be exemplified with examples within stationary heat conduction .

2 After the course the participants should be able to set up their own Finite Element solution of linear boundary value problems. We expect that you participate in the given lectures and exercises and ask questions when something is unclear. Many of the exercises are programming problems, which should be solved on a laptop. Preferably you should work in groups of two. Each group should bring a laptop with MatLabinstalled. We expect minor prior knowledge of scientific programming. Basic MatLabprogramming will be repeated during the lectures You should bring pen and paper, and a copy of the lecture Element MethodIntroduction, 1d heat conduction5 Choices we have made The lecture material is mainly slides.

3 The referenced book is additional information. The book is not a beginners book, but can be used for additional information and also for advanced Finite Element problems. We believe the basics can be understood from the slides and attending the lectures. You will be programming into a rather extensive matlabtoolbox developed by the lecturers. the overview might be lost in the beginning but you will have a working program, which also will be used in later courses. You will program a simple Finite Element program from nothing during the final lectures.

4 We do everything in matlab. You will have your own program with the source code and all the problems this gives. We believe that in order to understand the Finite Element Method you need to do some programming instead of just using commercial solvers 4 full lectures including exercise time and 1 self study, instead of 5 short lectures without exercisesFinite Element MethodIntroduction, 1d heat conduction6 Literature Lectures will be given from the slides. The theory and derivations are not directly based on the chapters in the book The book is chosen as a good overall Finite Element book which can be used also for more advanced problems During the lectures references will be given to relevant chapters and pages where the present subject is presented with Cook in parenthesis.

5 (Cook: chapter 4). These pages are summarized in the lecture plan. Relatively cheap! Finite Element MethodIntroduction, 1d heat conduction7 Beginners literature The formulations are based on this book, however, it is expensive and does not cover more advanced issues. What is covered in this book is essential what we cover during the lectures and on the slides. References to this book will be referenced in parenthesis with OP. (OP: chapter 4) Finite Element MethodIntroduction, 1d heat conduction8 Whatis the FiniteElement Method (FEM)?

6 A nummericalapproach for solvingpartialdifferentialequation, boundaryvalueproblems Finiteelement Method Finitedifference Method Finitevolumemethod Boundaryelement Method An approxmativesolution. Simplificationof geometry Meshdependent Convergenceissues Widelyusedwithinstructuralanalysis deformation heat conductionFinite Element MethodIntroduction, 1d heat conduction9 WhyuseFEM? General approach for solvingcomplexstructures. Staticundeterminedsystems causesnoproblems unsteadyproblems canbesolved(not includedin thiscourse) Fewbasicbuildingblocks( elements ) cansimulatecomplexproblems Verysuiteablefor computer solution.

7 Fast solutions Manydifferentload cases Finite Element MethodIntroduction, 1d heat conduction10 Basic steps of the Finite - Element Method (FEM) strong formulation Partial differential weak formulation Multiply with arbitrary field and integrate over space Mesh shape and weight functions Element stiffness matrix Local and global global system stiffness nodal boundary conditions temperature/flux/forces/forced global system of equations Solve for nodal values of the primary variables (displacements/temperature) temperature/stresses/strains etc.

8 Within the Element Using nodal values and shape functionsFinite Element MethodIntroduction, 1d heat conduction11 MatLabFE-program (main program runs until "return" plot functions located at the bottom) (defines node coordinates, done by user) (defines the Element topology, done by user) (defines the BC, done by user) (determines the global dofnumbering, done by program) (assemble the global stiffness matrix, done by program) (solve the system of equations for primary dofand forces, done by program) Plot functions (visualize the 1d heat problem with a temperature curve) Visualize2D (visualize all 2D problems including 2D heat conduction and 2D structural elements ) (visualize the 3D beam problem with geometry, deformation, beam normals, node and dofnumbering) (visualize the 3D beam problem with geometry, deformation, beam normals, node numbering and section forces) Finite Element MethodIntroduction, 1d heat conduction12 Exercise: Test thatthe program canrun.

9 Start Run the plotting of 1d heat conduction Plotting functionsareat the bottomof the program Output: u = [0 2]T K = [1 0 ; 0 1] Hints: Openthe m-filein the editor F5 runs the program untilit reachesa "return"-commandora red circlenextto the line numbers(debugmode) To debuguseF11 to advanceoneline (willentersub-functions). UseF10 to advanceoneline withoutenteringsub-functions F9 runs the marked section Enablecell-mode. Thenctrl-enterwillrun the present cell. Cellsaredividedby %% Finite Element MethodIntroduction, 1d heat conduction13 Advancedplotting in MatLabusinghandles Whena plot is generatedin matlabcorrespondinghandles arecreated.

10 A handle for the figure a handle for the axis a handle for eachplot onthe figure In thishandle everyinformation aboutthe plot is defined Directhandles arecreatedby puttinga variable beforethe plot The information in the handle canbeaccessedby The information canbechangedby>> hp = plot([0 1],[0 1])>> get(hp)information is put between ' '>> set(hp,'linewidth',3,'marker','o')new value, is put between ' ' if it is none numericalFinite Element MethodIntroduction, 1d heat conduction14 Handle for figuresand axescanbeaccessedeitherby puttinga handle whengeneratingthemorby gcf(get-current-figure) orgca(get-current-axis) as handles Changingvaluesworksas for the plot Position information, remembercorrectunits position = [lower_left_corner_xlower_left_corner_yw idthheight] Activatinga figurefor addingaxesoraxesfor plotting >> get(gca)>> get(gcf)>> hf= figure>> get(hf)>> ha = axes>> get(ha)