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

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. 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

4. 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

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

9. 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.

10. 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

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. 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.

14. Mesh – 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
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 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