O metodě konečných prvkůLect_9.ppt
M. Okrouhlík
Ústav termomechaniky, AV ČR, PrahaPlzeň, 2010
Disperzní vlastnosti
Dispersion
due to• space discretization
• time discretization• geometry
• material
It is known thatthe spatial and temporal discretizations always
accompany the finite element modelling of transient problems,
the topic is of academic interest but has important practical consequences as well.
In the contributionthe spatial dispersive properties of typical
Lagrangian and Hermitian elements will be presented,the dispersion due to time discretization will be
briefly mentioned andexamples of spatial and temporal dispersions
will be shown.
If )(k ... a so called dispersion relation ... is a linear function of k ... the wavenumber, then the system is called NONDISPERSIVE, otherwise it is DISPERSIVE. For non-dispersive systems the velocity of propagation is constant – does not depend on frequency.
L1 … constant strain element
consistent
diagonal
Ec
l
c
kl
kl
kl
00
00
0
*
0*
0
0*
withand where
matrix mass diagonalfor ))cos(1(2
matrix mass consistentfor )cos(2
))cos(1(6
Dispersion for 1D constant strain elements with consistent and diagonal mass matrix formulations
Podrobné odvozenípro L1C a L1D
See: c/telemachos/vyukove texty/disperse_c3.doc
% disp_L1
% dispersive properties of L1 element
clear
i = 0;
gamma_range = 0.01:0.01:pi;
length_gr = length(gamma_range);
for gamma = gamma_range
i = i + 1;
n = (1 - cos(gamma));
n_c = 6*n;
n_d = 2*n;
den = 2 + cos(gamma);
om_c_star(i) = sqrt(n_c/den);
w_c_star(i) = om_c_star(i)/gamma;
om_d_star(i) = sqrt(n_d);
w_d_star(i) = om_d_star(i)/gamma;
lambda(i) = 2*pi/gamma;
end
for i = 1:length_gr
lambda_rev(i) = lambda(length_gr - i + 1);
w_c_star_rev(i) = w_c_star(length_gr - i + 1);
w_d_star_rev(i) = w_d_star(length_gr - i + 1);
om_c_star_rev(i) = om_c_star(length_gr - i + 1);
om_d_star_rev(i) = om_d_star(length_gr - i + 1);
end
% points
figure(1)
subplot(3,1,1); plot(gamma_range,om_c_star, gamma_range,om_d_star)
title('frequency vs. gamma'); axis([0 3.14 0 4])
subplot(3,1,2); plot(gamma_range,lambda)
title('lambda vs. gamma')
v = [0 20 0 4];
subplot(3,1,3); plot(lambda_rev,om_c_star_rev, lambda_rev,om_d_star_rev); axis(v)
title('frequency vs. lambda')
xx = [0 100]; yy1 = [1.01 1.01]; yy2 = [0.99 0.99];
ww = [1 1];
figure(2)
subplot(2,1,1); plot(gamma_range,w_c_star, gamma_range,w_d_star, xx,ww)
title('phase velocity vs. gamma'); axis([0 3.14 0 1.3])
legend('consistent', 'diagonal', 'continuum', 3)
% subplot(3,1,2); plot(gamma_range,lambda)
% title('lambda vs. gamma')
v = [2 15 0.6 1.3];
subplot(2,1,2); plot(lambda_rev,w_c_star_rev, lambda_rev,w_d_star_rev, xx,yy1, xx,yy2); axis(v)
title('phase velocity vs. lambda')
xlabel('lambda is non-dimensional wave length - how many element lengths into the wave length of a given harmonics')
legend('consistent', 'diagonal', '+1%', '-1%', 4)
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
phase velocity vs. gamma
consistentdiagonalcontinuum
2 4 6 8 10 12 140.6
0.7
0.8
0.9
1
1.1
1.2
1.3phase velocity vs. lambda
lambda is non-dimensional wave length - how many element lengths into the wave length of a given harmonics
consistentdiagonal+1%-1%
0 0.5 1 1.5 2 2.5 3 3.50.6
0.7
0.8
0.9
1
1.1
1.2
phase velocity vs. frequency
consistentdiagonalcontinuum
1D continuum is a non-dispersivemedium, it has infinite number of frequencies, phase velocity is constantregardless of frequency.
Discretized model is dispersive,it has finite number of frequencies,velocity depends on frequency, spectrum is bounded,there are cut-offs.
overestimated consistentFrequency (velocity) is with mass matrix.
underestimated diagonal
L1 element
Eigenvectors, unconstrained rod, 24 L1C elements
Notice the rigid body mode
Higher modes havewrong shapes
Not in accordance with assumed harmonic solution
Nodes in opposition
The system cannot be forced to vibrate more violently
1D – velocity vs. frequency for higher order elements: L2, L3, H3
Band-pass filter
A mixed blessing of a longer spectrum
1D Hermitian cubic element, H3
Eigenvectors of a free-free rod modeled by H3C elements
transversal,shear,S-wave
primary,longitudinal,P-wave
Von Schmidt wavefront
2D wavefronts, Huygen’s principleMaterial points having been hitby primary wave becomesources of both types of wavesi.e. P and S
dp007
2D isotropic medium, plane stress, FE simulation
Dispersive properties of a uniform mesh(plane stress) assembled of equilateral elements, full integrationconsistent and diagonal mass matrices
wavenumberHow many elements to a wavelenght
Direction of wave propagation
Dispersion effects depend alsoon the direction of wave propagation
Artificial (false) anisotropy
Dispersive properties of a uniform mesh (plane stress) assembled of square isoparametric elements, full integration, consistent mass matrix
Hodograph of velocities
meshsize
wavelength
Hodograph of velocities
Hermitian elements are disqualified due to their ‘long’ spectrum
Eigenmodes and eigenvalues of a four-node bilinear element
Eigenmodes and eigenvalues of a four-node bilinear element
Time and space discretization errors vs. frequency
Mother nature is kind to us
No algorithmic damping
Small algorithmic damping
The effect of space and time discretization cancels out
0 0.2 0.4 0.6 0.8 1-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1dis
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
vel
0 0.2 0.4 0.6 0.8 1-50
0
50
100
150
200acc
L1C, 100 elements, h = 0.01, gamma = 0.6, beta = 1/6
‘Dispersionless’ Newmark
E:\edu_mkp_liberec_2\mtl_prog\mtlstep\linbad_c1_dispersionless_newmark.m
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
dis
0 0.2 0.4 0.6 0.8 1
0
0.5
1
vel
0 0.2 0.4 0.6 0.8 1
-50
0
50
acc
L1C, 100 elements, h = 0.01, gamma = 0.5, beta = 0.25*(0.5+gamma)^2
Newmark, no algorithmic damping
0 0.2 0.4 0.6 0.8 1-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
dis
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
vel
0 0.2 0.4 0.6 0.8 1-30
-20
-10
0
10
20acc
Newmark, with small algorithmic damping
L1C, 100 elements, h = 0.01, gamma = 0.6, beta = 0.25*(0.5+gamma)^2
0 0.2 0.4 0.6 0.8 1-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
dis
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
vel
0 0.2 0.4 0.6 0.8 1
-20
-10
0
10
acc
Newmark, with medium algorithmic damping
L1C, 100 elements, h = 0.01, gamma = 0.7, beta = 0.25*(0.5+gamma)^2
0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5dis
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1vel
0 0.2 0.4 0.6 0.8 1-8
-6
-4
-2
0
2
4acc
Newmark, with too big algorithmic damping
L1C, 100 elements, h = 0.01, gamma = 2, beta = 0.25*(0.5+gamma)^2
0 0.2 0.4 0.6 0.8 1-0.8
-0.6
-0.4
-0.2
0eps t = 0.7
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
dis
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8vel
0 0.2 0.4 0.6 0.8 1-3
-2
-1
0
1
2acc
Newmark – s vaničkou jsme vylili i dítě
L1C, 100 elements, h = 0.01, gamma = 10, beta = 0.25*(0.5+gamma)^2