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

• mesh generation (Delaunay triangulation)

• conclusions, further development

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

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

),( ),,(),(

)( ,0 2

yxyxfyxu

RSuy

uk

yx

uk

x yx

FEM – equation, domain

° the integral equation:

° after Greens’ theorem:

° the triangulation of the domain:

0)( wduk

wdn

ukwduk

FEM – element (triangle)

° triangle – coordinate system, basis functions:

° integrate, element stiffness matrix

))(())((

)()(1

11

)()(1

)()(1

12131312

1221122121213

31133113222

23322332111

yyxxyyxx

yxyxyxxxyy

yxyxyxxxyy

yxyxyxxxyy

),(),(),(),( 321 yxuyxuyxuyxu qpk

dn

uk

yyxxku m

i

mnmnn

oldal Jobb

3/23/13/1

3/13/23/1

3/13/13/2

3

2

1

u

u

u

k

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”

FEM - … the goal

° and finally the results:

Kx=10E-10; Ky=10E+10

Kx=10E+10; Ky=10E-10

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.

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:

t)} {X(t), Y(Q(t)

1

1,1

1

1,,

11,

)()()()()(

otherwise ,0

if ,1)(

iki

kiki

iki

kiiki

iii

xx

tNtx

xx

tNxttN

xtxtN

1

0,

1

0,

)(

)()(

n

ikii

n

ikiii

tNw

tNwBtQ

NURBs – uniform vs. non-uniform basis

° uniform quadric basis functions:

° non-uniform quadric basis functions:

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

° 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

° 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• 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