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Steady-state heat conduction on triangulated planar domain May, 2002
Bálint Miklós Vilmos Zsombori

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

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

method: finite element method (FEM)
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

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

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

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”

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

planar domains - > bounded by curves curves - > functions:
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.

NURB curves: (non uniform rational B-splines) defines:
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 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

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

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

Desired properties of triangles
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.

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

Mesh – 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

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

Conclusions Approximation errors Further development
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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