Transcript Steady-state Heat Conduction on triangulated planar domain

Steady-state heat conduction on
triangulated planar domain
May, 2002
Bálint Miklós ([email protected])
Vilmos Zsombori ([email protected])
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
•
•
CAD system
Shape
Material and other properties
Mesh generation
• Phenomenon
•
•
Definition of
material , data,
loads …
Transient
Balance
FEM
• Modell
BEM
Visualisation
• Results
•
•
Analytical
Numerical
Results
FDM
FEM - overview
• equation:
  
u   
u 
2



k


k


0
,
u

S
(
R
 )



x
y


x  y 
y 
 x 
u ( x, y )  f ( x, y ), ( x, y )  

• 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:
  (k  u)wd  0

u
° after Greens’ theorem:  ku  wd   k n wd


° the triangulation of the domain:
FEM – element (triangle)
° triangle – coordinate system, basis functions:
1

( y 2  y3 ) x  ( x3  x2 ) y  x2 y3  x3 y 2 






1
1
1



      1 ( y  y ) x  ( x  x ) y  x y  x y 
2
2
2
3
1
1
3
3 1
1 3



1
 3  1  1   2  1   1   2  ( y1  y 2 ) x  ( x 2  x1 ) y  x1 y 2  x 2 y1 


  ( x 2  x1 )( y 3  y1 )  ( x3  x1 )( y 2  y1 )
u( x, y)  uk 1 ( x, y)  u p 2 ( x, y)  uq3 ( x, y)
° integrate, element stiffness matrix
  n  m  n  m 
u
u
k
i n   x x  y y    k n  m d
 
 
 2 / 3  1 / 3  1 / 3  u1 
k  1 / 3 2 / 3  1 / 3 u2   Jobb oldal
 1 / 3  1 / 3 2 / 3  u3 
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

1, if xi  t  xi 1
N
(
t
)


 i ,1

0, otherwise

 N (t )  (t  xi ) N i ,k 1 (t )  ( xi  k  t ) N i 1,k 1 (t )
 i ,k
xi  k 1  xi
xi  k  xi 1

° basis functions:
n 1
° evaluation:
Q(t ) 
B w N
i 0
n 1
i
i
w  N
i 0
i
° locality of control points:
i ,k
i ,k
(t )
;
(t )
equation:
Q(t)  {X(t), Y( t)}
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