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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 - PowerPoint PPT Presentation
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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• 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