nm ln5 6 pde compatibility mode
TRANSCRIPT
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Accuracy/consistency
The discretised equations are notthereal ones
x
u
real
e sc eme oes notso ve t e reaequations !
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approximatex
u
Important Properties of Numerical Schemes
Convergence
numerical scheme solution isconvergent if it comes closer and
closer to the analytical solution of the real ODE/PDE when the
ConsistencyConvergence
Stability
me s ep ecreases;
LaxTheorem: 2 conditions needed for convergence
Consistency
A scheme isconsistent if it gives a correct approximationof the ODE/PDE as the time/space step is decreased
verified using Taylor Series expansion
Stability
A scheme is stable if any initially finite perturbation
remains bounded as time grows
Verification: Matrix method, Fourier method, Domain of
dependence
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Accuracy/consistencyTo reduce the truncation error :
Decrease both t and x
ConsistencyConvergence
Stability
. .reasonable range)
If the truncation error is small:
the discretised equation is consistent
x
taCr
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with the real one.
Accuracy and suitability
Consistency of schemes for PDEs
nnnUCrCrUU
110
Ua
U
ConsistencyConvergence
Stability
xt
uInitial profile of u
Exact solution, (Cr=1)
Cr=0.5, dx=100m
Cr=0.5, dx=50m
umerical diffusion
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x
Numerical diffusion
causes amplitude error
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What you want to solve :
uu
Accuracy/consistency ConsistencyConvergence
Stability
What the scheme sees :
xt
3
2
2
uuuu
Numerical
diffusion
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...32 xxxt
Truncation errorNumerical
dispersion
Accuracy and suitability
Stability of schemes for PDEs
nnnUCrCrUU
110
Ua
U
ConsistencyConvergence
Stability
xt
uInitial profile of u
Exact solution, (Cr=1)
Cr=0.5, dx=100m
Cr=0.5, dx=50m
umerical diffusion
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x
Numerical diffusion
causes amplitude error
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Accuracy and suitability Stability of schemes for PDEs
Because of numerical diffusion we try to use a better scheme,
like for instance Preismann scheme with psi=0.5;
a dis ersion e uation
ConsistencyConvergence
Stability
0
x
Ua
t
U3
3
x
Uk
x
Ua
t
U
Dispersion equationu
Initial
profile
Analytical profile,
Advected
downstream
Numerical dis ersion
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x
Computational
result
Due to derivatives estimation
Sharper profiles and oscilations
Create phase errors umerical dispersion
Numerical diffusion : profile smearing
1.2 Initial
Analytical
Accuracy/consistency ConsistencyConvergenceStability
0.4
0.6
0.8
1
u
Numerical
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0
0.2
0 2 4 6 8 10 12 14 16 18 20
x
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Numerical dispersion : oscillations
1.2 Initial
Accuracy/consistency ConsistencyConvergence Stability
0.2
0.4
0.6
0.8
1
u
Numerical
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-0.4
-0.2
0
0 2 4 6 8 10 12 14 16 18 20
x
Accuracy and suitability
Stability of explicit schemes Von Neumann stability method (Using Fourier
analysis)
ConsistencyConvergence
Stability
Same principle: amplitude factor is less than 1
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Stability Stability of explicit schemes
Von Neumann stability method (Using Fourier analysis)
Same principle: amplitude factor is less than 1
xijtnnj eUU
00 xjixjtnitntnUU iir
n
j sincossincosexp0
0
n
j
n
j
n
j
n
j
NU
UCrCr
U
UA
1
1
1
i
1-1
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xixijtnU
xjitnU
U
Un
j
n
j
exp
exp
1exp0
0
0
01
CrxixCrAN sincos1
Cr
Unit
circle
-i
-
Amplitude and phase portraits
Wave amplitude
Amplification factor =1
Any difference between the numerical phase speed
and true phase speed is the phase error
The graph that shows how the Fouriercomponents are amplified is called anamplitude portrait.
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The graph that shows at what speed theFourier components travel is called aphaseportrait.
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Amplitude and phase portraits
x
2
M- The wave number - represents the
number of grid intervals needed to cover
0.75
0.8
0.85
0.9
0.95
1
1.05
1 10 100
A
1.2
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2
MAArg
Cr
aAArgc N
NN
CrM
iM
CrAN
2
sin2
cos1
M
0
0.2
0.4
0.6
0.8
1 10 100
M
c/u
Phase and amplitude errors
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(B)Parabolic PDEs
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Parabolic Equations Initial Value Problems
A 1D time-dependent parabolic eqn (b=c=0)
2u
au
Lx0
t
Com utational
T
xt t
)()0,( xfxu
)(),()(),0(
thtLutgtu
x
domain
0 L
With I.C
With B.C
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x
u
0 L
t=0
f(x)
t
u
0 T
x=0
g(t)
t
u
0 T
x=L
h(t)
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2
2
x
ua
t
u
nnnnn 1
t
n
1nn
iu n
iu 1n
iu 1
1ni
u
Parabolic PDE:Solution Methods Explicit Methods2
iiiii uuuruu 11
x
1niu
i 1i1i
1n
I.C.: )(0
xifui B.C.:
tnun
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)( tnhunN We can calculate the unknown values of from the known
values of starting from the initial condition
1niun
iu
0
iu
Parabolic PDE: Stability of the Explicit Method
The explicit method is unstable if the timestep is too large.
conditions is
Stability condition for derivative boundaryconditions is
210 r
a
xt
2
2
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for )(fuu
n
uk
x
rk
2
10
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Parabolic Equation An example calculation
Solve
for2
2u
au
x 10
(Where u represents temperature)
with initial condition
t
0.15.0for)1(2
5.00for2)0,(
xx
xxxu
u
1
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and boundary conditions
0),1(
0),0(
tu
tu
x10.5
Parabolic Equation An example calculation2
Solve
for
2u
au
x 10
(Where u represents temperature)
with initial condition
xt t
0.15.0for)1(2
5.00for2)0,(
xx
xxxu
u
1
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and boundary conditions
0),1(
0),0(
tu
tu
1 21
222
2sinsinx
exp8),(
n xnn
tnntxu
Analytical (exact) solut ion
x1.
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Parabolic Example : Case 1 (r
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1Evolution of temperature distribution
Parabolic Example : Case 1 (r
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Evolution of temperature distribution
Parabolic Example : Case 3 (r>0.5)
Plot of the temperature distribution at 0 and 18time steps
0.5
0.6
0.7
0.8
0.9
1
temperature
t=0
t=0.1
0055.0
1.0
t
x
55.02
x
tar
analytic
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
x
UNSTABLE
solution is
meaningless
Parabolic Example :Evolution of maximum temperature (r=0.1)
0.9
1Evolution of maximum temperature
exact solution
r=0.1 solution
0.5
0.6
0.7
0.8
maximum
temperature
STABLE
solution
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3
0.4
time
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0.9
1Evolution of m aximum temperature
exact solution
r=0.5 solution
Parabolic Example :
Evolution of maximum temperature (r=0.5)
0.5
0.6
0.7
0.8
maximumt
emperature
STABLE
solution but
not v.
accurate
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3
0.4
time
Parabolic Example :Evolution of maximum temperature (r=0.55)
0.9
1Evolution of maximum temperature
exact solution
r=0.55 solution
0.5
0.6
0.7
0.8
maximumt
emperature UNSTABLE
solution is
meaningless
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3
0.4
time
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Parabolic Example : Explicit Method Error
r M u % error
Comparison of temperature values at x=0.5 and t=0.1 (for N=11)
analytic - 0.3021 -
0.001 10000 0.3071 1.65
0.01 1000 0.3070 1.60
0.1 100 0.3056 1.16
0.5 20 0.3071 1.64
need to
increase N
to reduce
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0.55 18 0.6336 109
0.6 16 7.2340 2294
error
furtherunstable
(C)Eliptic PDEs
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Elliptic Equations No time variable
),,(),(2
2
2
2
yxufyxux
u
y
u
y
x
Ly
Lx
0
0
- Poisson equation
- Laplace equation
),,(02
2
2
2
yxufx
u
y
u
0002
2
2
2
x
u
y
ufand
With different typs of B.C.:
Dirichlet : u is specified at the boundary
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t
Neumann : erivative o u is speci ie at t e oun ary
Mixed(robin): both u and its derivative is specified atthe boundary
Example : Laplace equation
Equation is:
Lx0
2
2
2
2
x
u
y
u
y
1, jiu1j
j
1jjiu , jiu ,1jiu ,1
1, jiu
NLyy
M
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xi 1i1iApproximate solution determined at all grid pointssimultaneously by solving single system of algebraicequations
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Elliptic PDE:Solution Methods Explicit Methods
y
j
1jjiu , jiu ,1jiu ,1
1, jiu
02
2
2
2
x
u
y
u
2
,1,,1
2
2 2
x
uuu
x
u jijiji
x
1, jiu
i 1i1i
1j
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04 ,1,1,,1,1 jijijijiji uuuuu2
1,,1,
2
2 2
y
uuu
y
u jijiji
02
1,,1,
2
,1,,1
yx
jijijijijiji
Holds for all interior points of the domain
Finite difference methods
What you should remember
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What you should remember Numerical solutions to PDEs can be
obtained by discretising both space andtime.
Explicit numerical schemes for PDEs aresubject to stability constraints.
Implicit numerical schemes for PDEs arealways stable.
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45
solution of non-linear PDEs.
The notions of consistency, stability andconvergence also hold for numericalschemes for PDEs.
What you should remember
First-order accurate schemes producenumerical diffusion; numerical profilesobtained are smoothed and may lead topeak underestimation. Numericaldiffusion leads to amplitude error.
Second-order accurate schemes producenumerical dispersion; numerical profilesexhibit artificial oscillations. Undesirable
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behaviours (such as negativeconcentrations) may appear. Numericaldispersion causes phase error.
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What you should remember Decreasing t or x alone is not sufficient to
reduce the truncation error : tand xshouldbe reduced together.
T e MOC is a speci ic in o numerica met oused for advection modelling. Its maindrawback is that it is generally not conservative(some water, pollutant, or energy, may be lostartificially).
At least three points in space are needed to
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solve a diffusion equation.
The design of 2-D and 3-D computational gridshould be carried out with care, long andnarrow grids should be avoided.
6 Finite volume methods
(FVM)
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FVM principle (1)FVMs are applicable to conservative equations, i.e. equations of the
form
problems)(1D0
x
F
t
U
problems)(3D0
problems)(2D0
z
H
y
G
x
F
t
U
y
G
x
F
t
U
F, G, H : Fluxes in x, y and z
U : Conserved variable
nnn
(5.1)
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n
n
n
n
n
n
n
n
n
n
n
n
nnn
z
U
y
U
x
U
z
U
y
U
x
UUHH
z
U
y
U
x
U
z
U
y
U
x
UUGG
zyxzyxUFF
,,,,,,
,,,,,,
,,,,,,
Eqs. (5.1) express the conservation of U over any bounded volume of
space
(5.2) 0
zHyGxF
t
U
FVM principle (2)
: Divergence operator
By definition of the divergence, Eq. (5.2) can be rewritten as
0d.ddd nzHyGxFzyxU
t
n
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F
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Application:
1) Discretise space into volumes
2) Compute the fluxes at the edges between the volumes
FVM principle (3)
e erm ne e c anges n v a a a ance equa on
(5.3) Vt
UU nn
outin1 FluxFlux
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V
The conservation law together with piecewise constant data
having a single discontinuity is known asthe Riemann problem.
FVM application (1)
0
if
l
x
r
q xq
q x
husQus
hds
husQus
hds
h-discontinuity
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Qds Qds
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husQus
hds
husQus
hds
h-discontinuity
FVM application (2)
ds ds
Three possible patterns:
zones of constant state (depth and velocity are
homogeneous over such zones),
a shock wave (information coming from upstream catches
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and information downstream),
a rarefaction wave (information downstream travels faster
than the information upstream).
Transport equations
2/12/11
jjj
nj
nj FF
x
tCC0
x
CDuC
xt
C
FVM applications (3)
C
xj
Fj-1/2 Fj+1/2
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0if2
0if2
1
1
1
1
1
2/1
uxx
CCDuC
uxx
CCDuC
F
jj
n
j
n
jn
j
jj
n
j
n
jn
j
j
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Applications in 1DDiscontinuous flows
PDE)(scalar0
x
U
t
U
FVM applications (4)
When F [F] is a non-linear function of U [U], the solution
may become discontinuous
Ex. Burgers equation
PDEs)of(system0
xt
FU
form)ion(conservat0
2uu
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invariant)Riemannais(d
dalong0
D
D
form)istic(character0
uut
x
t
u
x
uu
t
u
Applications in 1DBurgers Eq. (continued): formation of shocks from initially
smooth profiles
FVM applications (5)
t
U
x
A
B
A
B
Shock (u travels faster
behind than ahead)
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x
t0
t1
A B
dx/dt = u
u = Cst
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Applications in 1D (4)In finite volume methods: the flux is calculated from the
solution of a Riemann problem
FVM applications(6)
U
Initial
Final
U = Cst here
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t
x
The characteristics are straight
lines in the phase space
Applications in 1DThe solution of the Riemann problem exists even though the
initial profile is discontinuous
FVM applications (7)
Algorithm
1) At each interfacej-1/2, define the Riemann problem
2) Solve it => solutionUj-1/2
3) Compute the flux
njnj UU ,1
2/12/1 jj UFF
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4) For each cell, carry out balance forU
2/12/11
jjj
nj
nj FF
x
tUU
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Multidimensional problemsWave splitting (scalar equations)
0
Uv
x
Uu
t
U
FVM applications(8)
M (xi1/2,yM)
ut
M(xi1/2 ut,yM)
Cell (i,j)Cell (i 1,j)
Interface (i 1/2,j)
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vt
x
M(xi1/2ut,yMvt)
Multidimensional problems (2)
Wave splitting (systems of equations)
FVM applications (9)
Decomposed into
0
yxt
byfollowed
00
xtxtUAUFU
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=> Decompose the Riemann problem into 2 R.Ps: 1 along x
and 1 along y
00
ytyt
B
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Multidimensional problems (3)
Wave splitting (2)
FVM applications(10)
tpx )(
tp
y )1(
P(p, 1)
N(p) M
B
U(p)
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tp
y )2,(
P(p, 2)A
What you should remember
Finite Volume Methods (FVMs) are well-suited for the solution of conservativePDEs. The weak solution of the PDE issought.
FVMs ensure mass conservation
automaticly and can handle shocks anddiscontinuities.
The solution of the advection PDE b the
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Godunov-type FVMs involves thedefinition and the solution of a Riemannproblem.
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7. Finite Element Method
(FEM) - useful for problems withcomplicated geometries and
discontinuities, where analytical
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solutions can not be obtained
What is it FEM?
The finite element method is a numerical methodfor solving problems of engineering andmathematical physics, useful for problems withcomp icate geometries an iscontinuities,where analytical solutions can not be obtained.
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Principle of the method -(3)Discretization
Model the domain by dividing it into an equivalent system ofsmaller domains or units (finite elements) interconnected atpoints common to two or more elements (nodes or nodal points)and/or boundary lines and/or surfaces.
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Principle of the method -(4)
Discretization
eN
jjj
1
,
N1 and N2 are called Shape Functions or Interpolation Functions.
They express the shape of the assumed U.
For a linear representation of 1D elements:
N =1 N =0 at node 1
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N1 =0 N2 =1 at node 2
N1 + N2 =1
1 2
N1
L1 2
N2
L
1 2
2
L
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MethodologyObtain a set of algebraic equations to solve for unknownnodal quantity (displacement).
Principle of the method -(5)
Secondary quantities (stresses and strains) are expressedin terms of nodal values of primary quantity
History
Hrennikoff [1941] - Lattice of 1D bars
McHenry [1943] - Model 3D solids
Courant [1943] - Variational form
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Levy [1947, 1953] - Flexibility & Stiffness
Argryis and Kelsey [1954] - Energy Prin. for MatrixMethods
Turner, Clough, Martin and Topp [1956] - 2D elements
Clough [1960] - Term Finite Elements
Applications
Fluid Flow
Heat Transfer
Structural/Stress Analysis
Electro-Magnetic Fields
Soil MechanicsAcoustics
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Application -1D advection equation
0
x
Ua
t
U),(0.
0
yxwdxwx
yc
t
yL
0.0
*11
L
ji
in
ii
n
i
n
i dxNx
NUc
t
UU
0..0 0
*1*1
L L
ji
in
ijii
n
i
n
i dxNx
NUcdxNN
t
UU
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11110
1
2
1
n
j
n
j
L
ij
in
i UUdxNx
NU
Application - 1D advection equation
0
x
Ua
t
U
111 1 nnL in N
110 2
jji ji x
j
n
jj
n
jj
n
jj dycybya
1
1
11
1
12/1
xa
j xx jj 2/12/12 12/1
xc
j
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3 tc tcj 3 3 tc
nj
jn
j
jjn
j
j
j Ux
Uxx
Ux
tcd 1
2/12/12/1
1
2/1
636
2
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Modeling Considerations
Solve the tridiagonal system [A]Y=B Sy m m et r y -means correspondence in size, shape and position of U and
boundary conditions that are on opposite sides of a dividing line orplane;
Use of symmetry allows us to consider a reduced problem instead of the actual
problem.
The order of the total (global) stiffness matrix and the total number of equations can be
reduced.
Solution time is reduced!
Ba n d w i d t h- An envelope that begins with the firstnonzero component in each column of the [A] matrix
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741
2
2PL
8
6
5
3
2
1
4
5
L
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L L
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41
2
LP
5
3
2
1
4
L
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L L
00000X0XX0XX
000000XX0XX0
0000000XXXXX
00000000X0XX
00X0XX000000
XX0XX0X000000XXXXXXX0000
00X0XX0XX000
000XX0XX0X00
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X00X00000000
0X0XX0000000
X is a nonzero 2 x 2 block nb
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Bandwidth
nb = ndof( m + 1 )
ere:
nb is the semibandwidth
ndof is the number of degrees of freedom per node.
m is the maximum difference in node
numbers for any element.
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Poor Shapes
b
h
b >> h
>>
very small cornerangles
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Poor Shapes
uadrilateral de eneratin
h1h2
h1 >> h2
into triangular shape
Quadrilateral approachingtriangular shape
Finite Element Program.
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START
Input Data
Zero [K] and {F}
Do JE=1,NELE
Compute element stiffness [k]
Assemble Global stiffness [K] and forces {F}
Apply B.C.s
Solve [K]{d}={f}
Compute element quantities
Output Results
END
INPUT
Control parameters
Number of Elements
Number of B.C.s
Geometry
x,y,z location of each node Element connectivity (which nodes are
associated with which elements)
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INPUT
Element Properties
Area
Moment of Inertia
Thickness
Location of Neutral Axis
Physical parameters Information
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Programs
ALGOR
COSMOS/M
STARDYNE
IMAGES-3DMSC/NASTRAN
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ADINA
NISA
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What you should remember Finite Element Methods (FEMs) seek a
weaksolution to the PDEs.
set of basis or shape functions. Thosecan be piecewise linear, or parabolic, etc.
The Galerkin technique uses a weightingof the solution by functions that are the
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