introduction to element based computing --- finite volume and … · 2020. 3. 9. · smolarkiewicz,...
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![Page 1: Introduction to element based computing --- finite volume and … · 2020. 3. 9. · Smolarkiewicz, PK and Szmelter, J (2009) Iterated upwind schemes for gas dynamics, Journal of](https://reader036.vdocuments.mx/reader036/viewer/2022071517/613a385a0051793c8c00eb21/html5/thumbnails/1.jpg)
Introduction to element based computing ---
finite volume and finite element methods.
Mesh generation
Joanna Szmelter
Loughborough University, UK
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Traditional Discretisation Methods
•Finite Difference•Finite Element •Finite Volume
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Finite Difference Method
i-1,j
i,j+1i-1,j+1
i,j
i+1,j+1
i+1,j
x
uu
x
u jiji
ji
−
−+
2
))(,)(()( ,11
,
x
02
))(,)((
2
))(,)((
0)()(
1,1,11
, =
−+
−+
=
+
+
−+−+
y
vv
x
uu
t
y
v
x
u
t
jijijiji
ji
y)( 2xO
central
x
uu
x
u jiji
ji
−
+ ))(,)(()( ,1
,
forward
)( xO
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Finite Volume Method
0)()(
0)()(
=
+
+
=
+
+
d
y
vd
x
ud
t
y
v
x
u
t
0)()(
0)()(
=++
=++
y
j
jx
j
jii
yx
SvSuVt
dnvdnudt
From Gauss Divergence Theorem:
xn
yn
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Finite Element Method
15 165
45
Element Nodes
68 15 165 4 5
0)()(
0)()(
=
+
+
=
+
+
d
y
vd
x
ud
t
y
v
x
u
t
Weighted residual analysis:
68
i
0))(
())(
()(
0)()(
=
+
+
=
+
+
iji
iji
iji
jii
jii
jii
vdWy
NudW
x
NdWN
t
dWy
NvdW
x
NudWN
t
15151651654455 NNNN
Nii
+++
=
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Finite Element Method Weighted residual analysis:
If W is chosen to be the same as N the method is called Galerkin method.
0))(
())(
()( =
+
+
iji
iji
iji vdWy
NudW
x
NdWN
t
0)()())(
())(
()( =++
−
−
iyjiixjii
j
ii
j
iiji vdnNNudnNNvdy
NNud
x
NNdNN
t
For easy implementation of boundary conditions this is integrated by parts.
0))(
())(
()( =
+
+
iji
iji
iji vdNy
NudN
x
NdNN
t
0=++
iYelemiXelemielem vu
t BBM
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Finite Element Method
1) Divides computational space into elements
For every element constructs
2) Reconnects elements at nodes
Agglomerates for the whole domain
3) As a result a set of algebraic equations is formed and its solution follows
0=++
iYelemiXelemielem vu
t BBM
elemelemelemYelemXelemelem vρuρρBBM ,,;,,
elem
e
elem
e
elem
e
Yelem
e
YXelem
e
Xelem
e
vρvρuρuρρρ
BBBBMM
===
===
,,
,,
0vρBuρBρM =++
YX
t
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Matrix Agglomeration
a
5
42
1
3
Element a
a22 a23 a24 ρ2
a32 a33 a34 ρ3
a42 a43 a44 ρ4
Element b
b11 b12 b13 ρ1
b21 b22 b23 ρ2
b31 b32 b33 ρ3
Element c
c11 c12 c15 ρ1
c21 c22 c25 ρ2
c51 c52 c55 ρ5
1 b
c
Global matrix is a sum of all element matrixes
b11+c11 b12+c12 b13 0 c15 ρ1
b21+c21 a22+b22+c22 a23+b23 a24 c25 ρ2
b31 a32+b32 a33+b33 a34 0 ρ3
0 a42 a43 a44 0 ρ4
c51 c52 0 0 c55 ρ5
0vρBuρBρM =++
YX
t
e.g. For the linear triangular element the consistent mass matrix
a22 a23 a24 2 1 1
a32 a33 a34 = 1 2 1
a42 a43 a44 1 1 2
Area
Element”a”/12.
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The tetrahedron family of elements
i
i
iN
And when derivatives are of interest:
i
i
i
dx
dN
dx
d
We know functions N,
they are frequently polynomials
-obtaining their derivatives is easy.
SHAPE FUNCTIONS
After Zienkiewicz et al FEM 2000
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Example: a linear interpolation of a scalar T in a triangle. The value of T in an arbitrary point alpha is approximated by:
332211 NNN ++
elementAAN /11 =
where A is an area
SHAPE FUNCTIONS
1
2
3
3A2A
1A
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Curvilinear elements can be formed using transformations
Isoparametric elements
SHAPE FUNCTIONS
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Typical Examples of Data Structures
Structured
Point based --- I,J,K indexing
Set of coordinates and connectivities
Naturally map into the elements of a matrix
i,ji,j
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Typical Examples of Data Structures
Structured
Binary trees
Source: Ullrich et al GMD 2017
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Typical Examples of Data Structures
Unstructured
Element based connectivity
13
2
13
41021
20 100
11
Element 1 10 100 20 21
Element 2 21 11 13 10
Element 3 4 100 10
+ information related to shape functions
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Typical Examples of Data Structures
Unstructured
Edge 8 10 100
10 1008
+ geometrical information
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Edge based data
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Selected Mesh Generation
Techniques
Unstructured Meshes
Direct triangulation
Advancing Front Technique
Delaunay Triangulation
Others
Structured Meshes
Cartesian grids with mapping and/or
immersed boundaries
Variants of icosahedral meshes
Others
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Primary mesh
Dual mesh
5 days
14 days
Rossby-Haurwitz Wave
An example of the direct triangulation
Reduced Gaussian Grid
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Oct.N80 and reduced meshes
Oct.N80 fine and ‘odd’ reduced meshes
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Reduced Mesh – Baroclinic Instability
Day 8, horizontal velocity and potential temperature
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Advancing Front Technique
Simultaneous mesh point
generation and connectivity
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Delaunay Triangulation
How to connect a given set of points?
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Delaunay Triangulation
Create Voronoi polygons, i.e.
The construct that assigns to each point the area
of the plane closer to that point than to any other point
in the set. A side of a Voronoi polygon must be midway between the two points
which it separates
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Delaunay Triangulation
If all point pairs of which have some boundary in common are joint by straight
lines, the result is a triangulation of the convex hull of the points.
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Delaunay Triangulation mesh constructed from the reduced Gaussian grid points
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solid
Fluxes can be
constructed
for the surfaces
which cut cells
Geometry conforming meshes
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Meshing techniques for mesh adaptivity
-for lower order elements are:
point enrichment (h-refinement),
mesh movement
and automatic regeneration
Dynamic regenerations for Meshes utilising a mappingare particularly efficient
-for higher order elements (p-refinement) increases order of shape function that is the order of polynomial used for elemental interpolation
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The Edge Based Finite Volume Discretisation
Edges Median dual computational mesh
Finite volumes
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Geospherical framework
(Szmelter & Smolarkiewicz, J. Comput. Phys. 2010)
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A stratified 3D mesoscale flow past an isolated hill
Reduced planets (Wedi & Smolarkiewicz, QJR 2009)
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Fr=2
Fr=1
Fr=0.5
Stratified (mesoscale) flow past an isolated hill
on a reduced planet
4 hours
Hunt & Snyder J. Fluid Mech. 1980; Smolar. & Rotunno, J. Atmos. Sci. 1989 ; Wedi & Smolar., QJR 2009
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Smith, Advances in Geophys 1979; Hunt, Olafsson & Bougeault, QJR 2001
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V-Cycle
Smoother / Solver, varying number of
Multigrid techniques
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Multigrid meshes using Atlas
Octahedral 16 mesh:
computational domain
Remove odd latitudes
Single level of mesh coarsening
Multigrid using Atlas
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Nonhydrostatic Boussinesq mountain waveSzmelter & Smolarkiewicz , Comp. Fluids, 2011
Comparison with the EULAG’s (structured mesh) results --- very closewith the linear theories (Smith 1979, Durran 2003):
over 7 wavelenghts : 3% in wavelength; 8% in propagation angle; wave amplitude loss 7%
NL/Uo = 2.4
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Gravity wave breaking
in an isothermal stratosphere
Lipps & Hemler
Durran
(Prusa et al JAS 1996,
Smolarkiewicz & Margolin, Atmos. Ocean
1997
Klein, Ann. Rev. Fluid Dyn., 2010,
Smolarkiewicz et al Acta Geoph 2011)
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EULAG CV/GRID
Lipps & Hemler
Durran
Gravity wave breaking
in an isothermal stratosphere
Nonhydristatic Edge-Based NFT
Smolarkiewicz et al
Acta Geoph 2011
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Coarse initial mesh 80x45 =3600 points and solution
Adapted mesh 8662 points and solution
Static mesh adaptivity with MPDATA based error indicator
(Recommended mesh ca10000 points)
Schär Mon. Wea. Rev. 2002
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1121192 Cartesian dx=100
692533 Distorted prisms dx=100-400
441645 tetra dx=50 -450
Stratified flow past a steep isolated
hill
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Stratified flow past a steep isolated hill
Hunt & Snyder JFM 1980
Smolarkiewicz & Rotuno
JAS 1989
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Triangular surface mesh
Prismatic mesh
Szmelter et. al. JCP 2015
The effect of critical levels on stratified flows past an axisymmetric mountain
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Grubisic and Smolarkiewicz J. Atmos Sci 1997
The effect of critical levels on stratified flows past an axisymmetric mountain
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The effect of critical levels on stratified flows past an axisymmetric mountain
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Low Froude Number Fr=0.3 Flow Past a Sphere
Hanazaki , JFM 1989
Hybrid meshSzmelter et. al. JCP 2019
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Hanazaki , JFM 1989
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Convective Planetary Boundary Layer Schmidt & Schumann JFM 1989
Edge-based T
Edge-based C
EULAG
SS LES
Observation
64x64x51
Smolarkiewicz et al JCP 2013
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Selected references for further details
Szmelter J, Smolarkiewicz PK, Zhao Z, Cao Z (2019) Non-oscillatory forward-in-time integrators for viscous
incompressible flows past a sphere, Journal of Computational Physics, 386,pp.365-383.
Smolarkiewicz, PK, Szmelter, J, Xiao, F (2016) Simulation of all-scale atmospheric dynamics on unstructured
meshes, Journal of Computational Physics, 322,pp.267-287
Szmelter, J, Zhang, Z, Smolarkiewicz, PK, (2015) An unstructured-mesh atmospheric model for nonhydrostatic
dynamics: Towards optimal mesh resolution, Journal of Computational Physics, 249,pp.363-381
Smolarkiewicz, PK, Szmelter, J, Wyszogrodzki, AA (2013) An unstructured-mesh atmospheric model for
nonhydrostatic dynamics, Journal of Computational Physics, 254,pp.184-199
Smolarkiewicz, PK and Szmelter, J (2011) A Nonhydrostatic Unstructured-Mesh Soundproof Model for
Simulation of Internal Gravity Waves, Acta Geophysica, 59(6),pp.1109-1134
Szmelter, J and Smolarkiewicz, PK (2011) An edge-based unstructured mesh framework for atmospheric flows,
Computers and Fluids, 46(1), pp.455-460.
Szmelter, J and Smolarkiewicz, PK (2010) An edge-based unstructured mesh discretisation in geospherical
framework, Journal of Computational Physics, 229, pp.4980-4995.
Smolarkiewicz, PK and Szmelter, J (2009) Iterated upwind schemes for gas dynamics, Journal of Computational
Physics, 228(1), pp.33-54
Smolarkiewicz, PK and Szmelter, J (2008) An MPDATA-based solver for compressible flows, International
Journal for Numerical Methods in Fluids, 56(8), pp.1529-1534
Szmelter, J and Smolarkiewicz, PK (2006) MPDATA error estimator for mesh adaptivity, International Journal for
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