Non-Intrusive von Karman Shell

Shallow-curved shell structure with geometric nonlinearities
generate model
Dynamical system setup
Linear Modal analysis and SSM setup
Forced response curves using SSMs

Shallow-curved shell structure with geometric nonlinearities

Finite element model used in the following reference:

Jain, S., & Tiso, P. (2018). Simulation-free hyper-reduction for geometrically nonlinear structural dynamics: a quadratic manifold lifting approach. Journal of Computational and Nonlinear Dynamics, 13(7), 071003. <https://doi.org/10.1115/1.4040021>

Finite element code taken from the following package:

Jain, S., Marconi, J., Tiso P. (2020). YetAnotherFEcode (Version v1.1). Zenodo. <http://doi.org/10.5281/zenodo.4011282>

clear all; close all; clc
run ../../install.m
% change to example directory
exampleDir = fileparts(mfilename('fullpath'));
cd(exampleDir)

system parameters

nDiscretization = 10; % Discretization parameter (#DOFs is proportional to the square of this number)
epsilon = 0.1;

generate model

[M,C,K,~,f_0,outdof] = build_model_nonIntrusive(nDiscretization);

Building FE model
Assembling M,C,K matrices
Applying boundary conditions
Solving undamped eigenvalue problem
Using Rayleigh damping
Assembling external force vector
Getting nonlinearity coefficients

Construction of M,C,K leads to excessive data stored in Assembly class which in turn increases memory requirements during computation when fint is called -> construct fint independently

[fint,dfint] = getfint(nDiscretization,K);


n = length(M); % number of degrees of freedom
disp(['Number of degrees of freedom = ' num2str(n)])
disp(['Phase space dimensionality = ' num2str(2*n)])

Creating Nonlinear Function Handles
Number of degrees of freedom = 1320
Phase space dimensionality = 2640

Dynamical system setup

We consider the forced system

$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}+\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})=\epsilon\mathbf{f}^{ext}(\mathbf{\Omega}t),$

which can be written in the first-order form as

$\mathbf{B}\dot{\mathbf{z}}=\mathbf{Az}+\mathbf{F}(\mathbf{z})+\epsilon\mathbf{F}^{ext}(\mathbf{\phi}),\\\dot{\mathbf{\phi}} =\mathbf{\Omega}$

where

$\mathbf{z}=\left[\begin{array}{c}\mathbf{x}\\\dot{\mathbf{x}}\end{array}\right],\quad\mathbf{A}=\left[\begin{array}{cc}-\mathbf{K} & \mathbf{0}\\\mathbf{0} & \mathbf{M}\end{array}\right],\mathbf{B}=\left[\begin{array}{cc}\mathbf{C} & \mathbf{M}\\\mathbf{M} & \mathbf{0}\end{array}\right],\quad\quad\mathbf{F}(\mathbf{z})=\left[\begin{array}{c}\mathbf{-\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})}\\\mathbf{0}\end{array}\right],\quad\mathbf{F}^{ext}(\mathbf{z},\mathbf{\phi})=\left[\begin{array}{c}\mathbf{f}^{ext}(\mathbf{\phi})\\\mathbf{0}\end{array}\right]$ .

DSorder = 2;
DS = DynamicalSystem(DSorder);
set(DS,'M',M,'C',C,'K',K,'fnl_non',fint,'dfnl_non',dfint );

set(DS.Options,'Emax',5,'Nmax',10,'notation','multiindex')
set(DS.Options,'Intrusion','none')
% set(DS.Options,'Emax',5,'Nmax',10,'notation','tensor')

We assume periodic forcing of the form

$\mathbf{f}^{ext}(\phi) = \mathbf{f}_0\cos(\phi)=\frac{\mathbf{f}_0}{2}e^{i\phi} + \frac{\mathbf{f}_0}{2}e^{-i\phi}$

Fourier coefficients of Forcing

kappas = [-1; 1];
coeffs = [f_0 f_0]/2;
DS.add_forcing(coeffs, kappas,epsilon);

Linear Modal analysis and SSM setup

[V,D,W] = DS.linear_spectral_analysis();

Due to high-dimensionality, we compute only the first 5 eigenvalues with the smallest magnitude. These would also be used to compute the spectral quotients
Assuming a proportional damping hypthesis with symmetric matrices
modal damping ratio for 1 mode is 2.000000e-03
modal damping ratio for 2 mode is 2.000000e-03
modal damping ratio for 3 mode is 2.269789e-03
modal damping ratio for 4 mode is 2.721500e-03
modal damping ratio for 5 mode is 2.909530e-03
the left eigenvectors may be incorrect in case of asymmetry of matrices

 The first 10 nonzero eigenvalues are given as 
   1.0e+02 *

  -0.0029 + 1.4745i
  -0.0029 - 1.4745i
  -0.0063 + 3.1598i
  -0.0063 - 3.1598i
  -0.0094 + 4.1319i
  -0.0094 - 4.1319i
  -0.0148 + 5.4515i
  -0.0148 - 5.4515i
  -0.0173 + 5.9601i
  -0.0173 - 5.9601i

Choose Master subspace (perform resonance analysis)

S = SSM(DS);
set(S.Options, 'reltol', 0.1,'notation','multiindex')
% set(S.Options, 'reltol', 0.1,'notation','tensor')
masterModes = [1,2];
S.choose_E(masterModes);

(near) outer resonance detected for the following combination of master eigenvalues
     4     0
     0     4

These are in resonance with the follwing eigenvalues of the slave subspace
   1.0e+02 *

  -0.0173 + 5.9601i
  -0.0173 - 5.9601i

sigma_out = 5
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     1     2
     2     3

These are in resonance with the follwing eigenvalues of the master subspace
   1.0e+02 *

  -0.0029 + 1.4745i
  -0.0029 + 1.4745i
  -0.0029 - 1.4745i
  -0.0029 - 1.4745i

sigma_in = 5

Forced response curves using SSMs

Obtaining forced response curve in reduced-polar coordinate

order = 3; % Approximation order

setup options

set(S.Options, 'reltol', 1,'IRtol',0.02,'notation', 'multiindex','contribNonAuto',false,'COMPtype','first')
set(S.FRCOptions, 'nt', 2^7, 'nRho', 200, 'nPar', 200, 'nPsi', 100, 'rhoScale', 2 )
set(S.FRCOptions, 'outdof',outdof)

set(S.FRCOptions, 'method','level set')
%set(S.FRCOptions, 'method','continuation ep', 'z0', 1e-4*[1; 1]) % 'level set'

choose frequency range around the first natural frequency

omega0 = imag(S.E.spectrum(1));
omegaRange = omega0*[0.9 1.1];

extract forced response curve

FRC = S.extract_FRC('freq',omegaRange,order);
figFRC = gcf;

*****************************************
Calculating FRC using SSM with master subspace: [1  2]
(near) outer resonance detected for the following combination of master eigenvalues
     2     0
     3     0
     3     1
     4     1
     0     2
     0     3
     1     3
     1     4
     2     0
     3     0
     3     1
     4     1
     0     2
     0     3
     1     3
     1     4
     3     0
     4     0
     4     1
     0     3
     0     4
     1     4
     4     0
     5     0
     0     4
     0     5

These are in resonance with the follwing eigenvalues of the slave subspace
   1.0e+02 *

  -0.0063 + 3.1598i
  -0.0063 + 3.1598i
  -0.0063 + 3.1598i
  -0.0063 + 3.1598i
  -0.0063 - 3.1598i
  -0.0063 - 3.1598i
  -0.0063 - 3.1598i
  -0.0063 - 3.1598i
  -0.0094 + 4.1319i
  -0.0094 + 4.1319i
  -0.0094 + 4.1319i
  -0.0094 + 4.1319i
  -0.0094 - 4.1319i
  -0.0094 - 4.1319i
  -0.0094 - 4.1319i
  -0.0094 - 4.1319i
  -0.0148 + 5.4515i
  -0.0148 + 5.4515i
  -0.0148 + 5.4515i
  -0.0148 - 5.4515i
  -0.0148 - 5.4515i
  -0.0148 - 5.4515i
  -0.0173 + 5.9601i
  -0.0173 + 5.9601i
  -0.0173 - 5.9601i
  -0.0173 - 5.9601i

sigma_out = 5
(near) inner resonance detected for the following combination of master eigenvalues
     2     1
     3     2
     1     2
     2     3

These are in resonance with the follwing eigenvalues of the master subspace
   1.0e+02 *

  -0.0029 + 1.4745i
  -0.0029 + 1.4745i
  -0.0029 - 1.4745i
  -0.0029 - 1.4745i

sigma_in = 5
Due to (near) outer resonance, the exisitence of the manifold is questionable and the underlying computation may suffer.
Attempting manifold computation
Manifold computation time at order 2 = 00:00:00
Estimated memory usage at order  2 = 2.23E+00 MB
Manifold computation time at order 3 = 00:00:00
Estimated memory usage at order  3 = 2.39E+00 MB
gamma = 
  -7.2120e+00 - 2.9595e+02i

Total time spent on FRC computation upto O(3) = 00:00:03