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Steady-state heat conduction on triangulated planar domain May, 2002

Steady-state heat conduction on triangulated planar domain May, 2002. Bálint Miklós (miklosb@student.ethz.ch) Vilmos Zsombori (v.zsombori@gold.ac.uk). Overview. about physical simulations 2D NURB curves finite element method for the steady-state heat conduction

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Steady-state heat conduction on triangulated planar domain May, 2002

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  1. Steady-state heat conduction on triangulated planar domainMay, 2002 BálintMiklós (miklosb@student.ethz.ch) Vilmos Zsombori (v.zsombori@gold.ac.uk)

  2. Overview • about physical simulations • 2D NURB curves • finite element method for the steady-state heat conduction • mesh generation (Delaunay triangulation) • conclusions, further development

  3. CAD system Mesh generation FEM BEM FDM Visualisation Results Physical simulations • Object • Shape • Material and other properties • Phenomenon • Transient • Balance • Modell • Results • Analytical • Numerical Definition of material , data, loads …

  4. FEM - overview • equation: • method: finite element method (FEM) • transform into an integral equation • Greens’theorem - >reduce the order of derivatives • introduce the finite element approximation for the temperature field with nodal parameters and element basis functions • integrate over the elements to calculate the element stiffness matrices and RHS vectors • assemble the global equations • apply the boundary conditions

  5. FEM – equation, domain • the integral equation: • after Greens’theorem: • the triangulation of the domain:

  6. FEM – element (triangle) • triangle – coordinate system, basis functions: • integrate, element stiffness matrix

  7. FEM – assembly • assembly - >sparse matrix • boundary conditions - >the order of the system will be reduced • the solution of the system: • direct - „accurate”, „slow” • iteratív – „approximate”, „faster”

  8. FEM - … the goal • and finally the results: Kx=10E-10; Ky=10E+10 Kx=10E+10; Ky=10E-10

  9. NURBs – about curves • planar domains - >bounded by curves • curves - >functions: • explicit • implicit • parametrical • goal: a curve which • can represent virtually any desired shape, • can give you a great control over the shape, • has the ability of controlling the smoothness, • is resolution independent and unaffected by changes in position, scale or orientation, • fast evaluation.

  10. NURBs - properties • NURB curves: (non uniform rational B-splines) • defines: • its shape – a set of control points (bi ) • its smoothness – a set of knots (xi ) • its curvature – a positive integer - >the order(k) • properties: • polynomial– we can gain any point of the curve by evaluatingknumber ofk-1 degree polynomial • rational – every control point has a weight, which gives its contributions to the curve • locality - > control points • non uniform – refers to the knot vector - > possibility to control the exact placement of the endpoints and to create kinks on the curve

  11. NURBs – basis, evaluation, locality • basis functions: • evaluation: ;equation: • locality of control points:

  12. NURBs – uniform vs. non-uniform basis • uniform quadric basis functions: • non-uniform quadric basis functions:

  13. Mesh – the problem • Triangulation • Desired properties of triangles • Shape – minimum angle: convergence • Size: error • Number: speed of the solving method • Goal • Quality shape triangles • Bound on the number of triangles • Control over the density of triangles in certain areas.

  14. Mesh – Delaunay triangulation • Delaunay triangulation • input: set of vertices • The circumcircle of every triangle is “empty” • Maximize the minimum angle • Algorithm • Basic operation: flip • incremental

  15. Mesh – constrained Delaunay triangulation • constrained Delaunay triangulation • Input: planar straight line graph • Modified empty circle • Input edges belong to triangulation • Algorithm • Divide-et-impera • For every edge there is one Delaunay vertex • Only the interior of the domain is triangulated

  16. Mesh – Delaunay refinement • General Delaunay refinement • Steiner points • Encroached input edge - > edge splitting • Small angle triangle - >triangle splitting • Guaranteed minimum angle (user defined) • Custom mesh • Certain areas: smaller triangles • Boundary: obtuse angle -> input edge encroached - > splitting • Interior: near vertices -> small local feature - > splitting

  17. Conclusions • Approximation errors • spatial discretization: mesh • nodal interpolation • Further development • Improve accuracy vs. speed by quadric/cubic element basis • Transient equation • Same mesh generator, introduce time discretization • Other equation • Same mesh generator, improve solver • 3-Dimmension • New mesh generator, minimal changes on the solver • Running time • Parallelization using multigrid mesh

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