Presentation on theme: "Steady-state heat conduction on triangulated planar domain May, 2002"— Presentation transcript:
1Steady-state heat conduction on triangulated planar domain May, 2002 Bálint MiklósVilmos Zsombori
2Overviewabout physical simulations2D NURB curvesfinite element method for the steady-state heat conductionmesh generation (Delaunay triangulation)conclusions, further development
3Definition of material , data, loads … Physical simulationsObjectShapeMaterial and other propertiesPhenomenonTransientBalanceModellResultsAnalyticalNumericalCAD systemMesh generationDefinition of material , data, loads …FEMBEMFDMVisualisationResults
4method: finite element method (FEM) FEM - overviewequation:method: finite element method (FEM)transform into an integral equationGreens’ theorem - > reduce the order of derivativesintroduce the finite element approximation for the temperature field with nodal parameters and element basis functionsintegrate over the elements to calculate the element stiffness matrices and RHS vectorsassemble the global equationsapply the boundary conditions
5FEM – equation, domainthe integral equation:after Greens’ theorem:the triangulation of the domain:
6FEM – element (triangle) triangle – coordinate system, basis functions:integrate, element stiffness matrix
7FEM – assembly assembly - > sparse matrix boundary conditions - > the order of the system will be reducedthe solution of the system:direct - „accurate”, „slow”iteratív – „approximate”, „faster”
8and finally the results: FEM - … the goaland finally the results:Kx=10E-10; Ky=10E+10Kx=10E+10; Ky=10E-10
9planar domains - > bounded by curves curves - > functions: NURBs – about curvesplanar domains - > bounded by curvescurves - > functions:explicitimplicitparametricalgoal: a curve whichcan 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.
10NURB curves: (non uniform rational B-splines) defines: NURBs - propertiesNURB 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 evaluating k number of k-1 degree polynomialrational – every control point has a weight, which gives its contributions to the curvelocality - > control pointsnon uniform – refers to the knot vector - > possibility to control the exact placement of the endpoints and to create kinks on the curve
11NURBs – basis, evaluation, locality basis functions:evaluation: ; equation:locality of control points:
13Desired properties of triangles Mesh – the problemTriangulationDesired properties of trianglesShape – minimum angle: convergenceSize: errorNumber: speed of the solving methodGoalQuality shape trianglesBound on the number of trianglesControl over the density of triangles in certain areas.
14Mesh – Delaunay triangulation input: set of verticesThe circumcircle of every triangle is “empty”Maximize the minimum angleAlgorithmBasic operation: flipincremental
15Mesh – constrained Delaunay triangulation Input: planar straight line graphModified empty circleInput edges belong to triangulationAlgorithmDivide-et-imperaFor every edge there is one Delaunay vertexOnly the interior of the domain is triangulated
17Conclusions Approximation errors Further development spatial discretization: meshnodal interpolationFurther developmentImprove accuracy vs. speed by quadric/cubic element basisTransient equationSame mesh generator, introduce time discretizationOther equationSame mesh generator, improve solver3-DimmensionNew mesh generator, minimal changes on the solverRunning timeParallelization using multigrid mesh