O metodě konečných prvků Lect_9.ppt M. Okrouhlík Ústav termomechaniky, AV ČR, Praha Plzeň, 2010 Disperzní vlastnosti

Dispersion due to space discretization time discretization geometry material

It is known that the 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 contribution the spatial dispersive properties of typical Lagrangian and Hermitian elements will be presented, the dispersion due to time discretization will be briefly mentioned and examples of spatial and temporal dispersions will be shown.

L1 … constant strain element

consistent diagonal

Dispersion for 1D constant strain elements with consistent and diagonal mass matrix formulations

Podrobné odvození pro 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([ ]) subplot(3,1,2); plot(gamma_range,lambda) title('lambda vs. gamma') v = [ ]; 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 = [ ]; yy2 = [ ]; 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([ ]) legend('consistent', 'diagonal', 'continuum', 3) % subplot(3,1,2); plot(gamma_range,lambda) % title('lambda vs. gamma') v = [ ]; 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)

1D continuum is a non-dispersive medium, it has infinite number of frequencies, phase velocity is constant regardless of frequency. Discretized model is dispersive, it has finite number of frequencies, velocity depends on frequency, spectrum is bounded, there are cut-offs. overestimatedconsistent Frequency (velocity) is withmass matrix. underestimateddiagonal L1 element

Eigenvectors, unconstrained rod, 24 L1C elements Notice the rigid body mode Higher modes have wrong 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 principle Material points having been hit by primary wave become sources of both types of waves i.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 integration consistent and diagonal mass matrices wavenumber How many elements to a wavelenght Direction of wave propagation Dispersion effects depend also on 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

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

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

L1C, 100 elements, h = 0.01, gamma = 0.5, beta = 0.25*(0.5+gamma)^2 Newmark, no algorithmic damping

Newmark, with small algorithmic damping L1C, 100 elements, h = 0.01, gamma = 0.6, beta = 0.25*(0.5+gamma)^2

Newmark, with medium algorithmic damping L1C, 100 elements, h = 0.01, gamma = 0.7, beta = 0.25*(0.5+gamma)^2

Newmark, with too big algorithmic damping L1C, 100 elements, h = 0.01, gamma = 2, beta = 0.25*(0.5+gamma)^2

Newmark – s vaničkou jsme vylili i dítě L1C, 100 elements, h = 0.01, gamma = 10, beta = 0.25*(0.5+gamma)^2