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Steady-state heat conduction on triangulated planar domain May, 2002 Bálint Miklós (miklosb@student.ethz.ch) Vilmos Zsombori (v.zsombori@gold.ac.uk)

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

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Physical simulations Object Shape Material and other properties Phenomenon Transient Balance Modell Results Analytical Numerical CAD system Mesh generation Definition of material, data, loads … FEMBEMFDM Visualisation Results

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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

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FEM – equation, domain °the integral equation: °after Greens theorem: °the triangulation of the domain:

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FEM – element (triangle) °triangle – coordinate system, basis functions: °integrate, element stiffness matrix

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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

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FEM - … the goal °and finally the results: K x =10E-10; K y =10E+10 K x =10E+10; K y =10E-10

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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.

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NURBs - properties °NURB curves: (non uniform rational B-splines) °defines: its shape – a set of control points (b i ) its smoothness – a set of knots (x i ) its curvature – a positive integer - > the order (k) °properties: polynomial – we can gain any point of the curve by evaluating k number of k-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

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NURBs – basis, evaluation, locality °basis functions: °evaluation: ; equation: °locality of control points:

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NURBs – uniform vs. non-uniform basis °uniform quadric basis functions: °non-uniform quadric basis functions:

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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.

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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

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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

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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

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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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