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Outline :

OUTLINES. Outline :. 1) PDE toolbox DEMOS. 2) Reference Triangle. 3) Finite Element triple. 4) Quadratic on Triangle. 5) Bilinear on Rectangle. 6) Linear on Tetrahedron. 7) some common finite element. 8) delaunay3 Matlab command. 9) Line Command (PDETOOL).

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Outline :

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  1. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  2. PDE DEMOS >> pdedemos C:\Program Files\MATLAB\R2008b\toolbox\pde

  3. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  4. Reference Triangle Local basis functions on reference triangle Maps nodes and midpoints and centroid Change of variables will change any integral over an element K into an integral over the refrence triangle.

  5. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  6. Finite Element Finite Element Example is a triple Linear polynomial on triangle is a geometric object, for example a triangle is triangle Finite dimensional linear space of functions defined on K Linear function on K Set of degrees of freedon Is the values at the vertices of K

  7. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  8. Quadratic on Triangle Example Quadratic on triangle is triangle Local basis functions on reference triangle quadraticon K Is the values at the vertices and midpoints on edges of K REMARK: local stiffness matrix is of size 6X6 matrix.

  9. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  10. bilinear on rectangle Example bilinear on rectangle Is a rectangle Local basis functions on reference rectangle Bilinear on K Is the values at the vertices of the rectangle K REMARK: local stiffness matrix is of size 4X4 matrix.

  11. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  12. Linear on Tetrahedron Example Quadratic on triangle is tetrahedral Linear on K Local basis functions on reference triangle Is the values at the vertices of K REMARK: local stiffness matrix is of size 4X4 matrix.

  13. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  14. some common finite element

  15. some common finite element Common and Unusual Finite Elements By Robert C. Kirby, Anders Logg, and Andy R. Terrel

  16. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  17. delaunay3 How many elements? x = [-1 -1 -1 -1 1 1 1 1 0]; y = [-1 -1 1 1 -1 -1 1 1 0]; z = [-1 1 -1 1 -1 1 -1 1 0]; T = delaunay3( x, y, z, {'Qt', 'Qbb', 'Qc', 'Qz'} ); XX = [x(:),y(:),z(:)]; tetramesh(T,XX); d = [-1 1]; [x,y,z] = meshgrid(d,d,d); % A cube x = [x(:);0]; y = [y(:);0]; z = [z(:);0]; % [x,y,z] are corners of a cube plus the center. Tes = delaunay3(x,y,z); XX = [x(:),y(:),z(:)]; tetramesh(Tes,XX); 3d.zip

  18. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  19. Line Command (PDETOOL) g=[ 2 2 2 2; -1 1 1 -1; 1 -1 1 -1; % g='squareg'; -1 1 -1 1; % The [-1,1]x[-1,1] square -1 1 1 -1; 1 1 1 1; 0 0 0 0 ]; % pdecoeff c=1; a=0; f = '-2*((x.^2 - 1) + (y.^2 - 1))'; % right hand side % Mesh [p,e,t]=initmesh(g,'Hmax',0.1); pdemesh(p,e,t) % Solve [A,rhs,B,ud] = assempde(b, p, e, t, c, a, f); [u1] = pcg(A,rhs,1e-5,100); %u1=A\rhs; uh = B*u1+ud; %uh = assempde(b, p, e, t, c, a, f); % visualize x=linspace(-1,1,31);y=x; [uu]=tri2grid(p,t,uh,x,y); Figure; surf(x,y,uu); b=[ 1 1 1 1 ; 1 1 1 1 ; 1 1 1 1 ; 1 1 1 1 ; % b='squareb1'; 1 1 1 1 ; % Dirichlet data 1 1 1 1 ; '0' '0' '0' '0'; '0' '0' '0' '0'; '1' '1' '1' '1'; ‘0' ‘0' '0' '0‘ ]; u=0 u=0 Sample.zip

  20. Line Command (PDETOOL) For each column in the Decomposed Geometry matrix there must be a corresponding column in the Boundary Condition matrix. The format of each column is according to the following list: b=[ 1 1 1 1 ; 1 1 1 1 ; 1 1 1 1 ; 1 1 1 1 ; % b='squareb1'; 1 1 1 1 ; % Dirichlet data 1 1 1 1 ; '0' '0' '0' '0'; '0' '0' '0' '0'; '1' '1' '1' '1'; ‘0' ‘0' '0' '0‘ ]; u=0 u=0

  21. OUTLINES Outline: 1) PDE toolbox DEMOS 2) Reference Triangle 3) Finite Element triple 4) Quadratic on Triangle 5) Bilinear on Rectangle 6) Linear on Tetrahedron 7) some common finite element 8) delaunay3 Matlab command 9) Line Command (PDETOOL) 10) Resources: (videos,Books,Workshop,Conf)

  22. Resources http://www.mathworks.com/access/helpdesk/help/pdf_doc/pde/pde.pdf

  23. Resources http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/video-lectures/lecture-18-finite-elements-in-1d-part-2/ Gilbert Strang Videos (MIT OpenCourseWare)

  24. Resources

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  27. Resources • Adina R&D, Inc. • Algor (Professional) • Alstom (SLIM) • Altair • AMPS • Amtec (Mesh Generator) • Anker-Zemer (LINFLOW) • Ansoft Corporation (HFSS) • ANSYS • ATIR (STRAP) • Argus Interware, Inc. (Argus ONE) • AutoFEA Software Co. (JL Analyzer) • Automated Analysis Corporation (COMET) • CAE PowerTools • CADFEM • CAST3M • CEETRON (GLview) • CIMNE (GiD) • CivilFEM • Cobus (FesaWin) • Computational Mechanics Co. (ProPHLEX) • Computational Mechanics Australia (Meshing) • Computers and Structures Inc. • COMSOL (FEMLAB) • COMSOL (PDE Toolbox) • Computing Objects (CM2) • Concurrent Analysis Corporation (CAEFEM) • CrossX • Cyberlogic (FEMdesigner) • DIANA • D.I.E Software • Dr. Software • EDS (FEMAP) • EDS PLM Solutions (I-DEAS Simulation) • Electric Power Research Institute (Merlin, Spider) • Engineering Mechanics Research Corporation • ESI Group (PAM-SYSTEM) • Fastflo • ESRD (StressCheck) • FEA Ltd. (LUSAS) • FEMCOS (COSAR) • Femsys (FEMGV) • Field Precision (TriComp/AMaze/GamBet) • FinitES-Plast • Flexcom-3D (MCS) • FLUENT (FLUENT/FIDAP) • ForDyn animation • FrameMaster • GEO-SLOPE • Hibbitt, Karlsson and Sorensen (ABAQUS) • HOMARD • ICEM CFD Engineering • Integrated Design and Analysis Consultants (IDAC) • Inter-CAD Ltd (AxisVM) • Intuition Software (VisualFEA) • INFOLYTICA • Intelligent Light (FieldView) • Intes (PERMAS) • Laboratoire Central des Ponts et Chaussees (CESAR) • Lawrence Livermore National Laboratory (DYNA3D/NIKE3D) • LEAP (ANSYS) • LIRA Finite Element Analysis Software • LISA • Livermore Software Technology Corp. (LS-DYNA3D) • MacNeal-Schwendler Corporation (MSC/NASTRAN, MSC/PATRAN) • MACRO Industries (MI/NASTRAN) • Macsyma (PDEase) • Magsoft (ATILA, Flux2D) • Makros-A • Maple • MARC • Mathematica • Mathworks (MATLAB) • Mathworks (Simulink) • MIDAS • MSTR Technology (FEWaves) • NM SESES (Centre for Computational Physics ZHW) • Noran Engineering, Inc. (NEiNastran) • Northwest Numerics and Modelling (Zebulon) • Numerical Objects (Diffpack) • Open Engineering (Oofelie) • PDE2D • PDE Solutions Inc (FlexPDE) • Parametric Technology Corporation (Pro/MECHANICA) • PLAXIS • Pointwise, Inc (Gridgen/FIELDVIEW) • PVM Corp (FEVA) • PVM Corp (SAGE) • Radioss Consulting Corp. • Research Engineers (STAAD) • Robobat (Robot 97) • Rockfield (ELFEN) • Roensch & Associates • SAGE CRISP • SAMTECH (samcef) • SDTools (Structural Dynamics Toolbox) • SEPRA (SEPRAN) • SIMULOG (TetMesh-GHS3D) • Softing (Nolian) • Softek Services Ltd. (P-FRAME, S-FRAME) • Soilvision (SVFlux) • SOLVIA • SRT (FEA-Crack) • Steer Engineering (AFEMS, FemForm, FemTool) • Strand7 • Structural Analysis Inc (mTAB*STRESS) • Structural Research & Analysis Corp (COSMOS/M) • GT STRUDL • Synaps (Epogy) • T-Systems (MEDINA) • Tera Analysis (QuickField) • ThermoAnalytics (RadTherm) • Thin-Walled Structures Software • Third Wave Systems (AdvantEdge) • TrueGrid • Utah State University (VABS) • Vector Fields (TOSCA, OPERA) • VectorSpace Programming (vs.lib) • Visual Kinematics Inc. (DevTools) • Visual Numerics (PV-WAVE) • VMA Engineering (GENESIS) • Weidlinger Associates Inc. • ZACE Services Ltd. • Zentech International (Zencrack)

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