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fe_time

• Solvers and options
• TimeOpt
• Input and output options
• Cleanup
• newmark
• dg
• NLNewmark
• staticNewton
• numerical damping for Newmark, HHT-alpha schemes
• Theta
• Euler

Purpose

Computation of time and non linear responses.

Syntax

  def=fe_time(model)
  def=fe_time(TimeOpt,model)
  [def,model,opt]=fe_time(TimeOpt,model)
  model=fe_time('TimeOpt...',model)
  TimeOpt=fe_time('TimeOpt...')

Description

fe_time groups static non-linear and transient solvers to compute the response of a FE model given initial conditions, boundary conditions, load case and time parameters. Note that you may access to the fe_time commands graphically with the simulate tab of the feplot GUI. See tutorial (section 4.8) on how to compute a response.

Solvers and options

Three types of time integration algorithm are possible: the Newmark schemes, the Theta-method, and the time Discontinuous Galerkin method. Implicit and explicit methods are implemented for the Newmark scheme, depending on the Newmark coefficients β and γ, and non linear problems are supported.

The parameters of a simulation are stored in a time option data structure TimeOpt given as input argument or in a model.Stack entry info,TimeOpt. Initial conditions are stored as a curve,q0 entry.

The solvers selected by the string TimeOpt.Method are

• newmark linear Newmark
• NLNewmark non linear Newmark (with Newton iterations)
• staticNewton static Newton
• Theta Theta-Method (linear)
• Euler method for first order time integration.
• dg Discontinuous Galerkin
• back perform assembly and return model,Case,opt.

Here is a simple example to illustrate the common use of this function.

 model=fe_time('demo bar'); % build the model

 % Define time options and use structure directly
 opt=fe_time('TimeOpt Newmark .25 .5 0 1e-4 100');
 def=fe_time(opt,model); % compute the response

 % Store as model.Stack entry {'info','TimeOpt',opt}
 model=stack_set(model,'info','TimeOpt',opt);
 def=fe_time(model); % compute the response

TimeOpt

The TimeOpt data structure has fields to control the solver

• Method selection of the solver
• Opt numeric parameters of solver if any. For example for Newmark [beta gamma t0 deltaT Nstep]
  
• MaxIter maximum number of iterations.
• nf optional value of the first residual norm. The default value is norm(fc) where f_c=[b]{u(t)} the instant load at first time step. This is used to control convergence on load.
• IterInit,IterEnd callbacks executed in non linear solver iterations. This is evaluated when entering and exiting the Newton solver. Can be used to save specific data, implement modified solvers, ...
• Jacobian string to be evaluated to generate a factored jacobian matrix in matrix or ofact object ki. Defaults are detailed for each solver, see also NLJacobianUpdate if you have the non-linear vibration tools.
• JacobianUpdate controls the update of Jacobian in Newton and quasi-Newton loops. Use 1 for updates and 0 for a fixed Jacobian (default).
• Residual Callback evaluated for residual. The default residual is method dependent.
• InitAcceleration optional field to be evaluated to initialize the acceleration field.
• IterFcn string or function handle iteration (inner loop) function. When performing the time simulation initialization, the string will be replaced by the function handle (e.g. @iterNewton). Iteration algorithms available in fe_time are iterNewton (default for basic Newton and Newmark) and iterNewton_Sec which implements the Newton increment control algorithm.
  
• RelTol threshold for convergence tests. The default is the OpenFEM preference

  getpref('OpenFEM','THRESHOLD',1e-6);

• TimeVector optional value of computed time steps, if exists TimeVector is used instead of deltaT,Nstep.
  
• AssembleCall optional callback for assembly, see nl_spring('AssembleCall'). When model and Case are provided as fully assembled, one can define the AssembleCall field as empty to tell fe_timenot to perform any assembly. Description of assemble calls can be found in section 4.8.8.

to control the output

• OutInd DOF output indices (see 2D example). This selection is based on the state DOFs which can be found using fe_case(model,'GettDof').
• OutputFcn string to be evaluated for post-processing or time vector containing the output time steps. Examples are given below.
• FinalCleanupFcn string to be evaluated for final post-processing of the simulation
  
• c_u, c_v, c_a optional observation matrices for displacement, velocity and acceleration outputs. See section 4.6.4 for more details on observation matrix generation.
• lab_u, lab_v, lab_a optional cell array containing labels describing each output (lines of observation matrices)
• NeedUVA [NeedU NeedV NeedA], if NeedU is equal to 1, output displacement, etc. The default is [1 0 0] corresponding to displacement output only.
• OutputInit optional string to be evaluated to initialize the output (before the time loop). The objective of this call is to preallocate matrices in the out structure so that data can be saved efficiently during the time integration. In particular for many time steps out.def may be very large and you want the integration to fail allocating memory before actually starting.
• SaveTimes optional time vector, saves time steps on disk
• Follow implements a timer allowing during simulation display of results. A basic follow mechanism is implemented (opt.Follow=1; to activate, see NLNewmark example below)). One can also define a simple waitbar with remaining time estimation, with: opt.Follow='cingui(''TimerStartWaitBar-title"Progress bar example..."'')'; More elaborate monitoring are available within the SDT optional function nl_spring (see nl_spring Follow).

Input and output options

This section details the applicable input and the output options.

Initial conditions may be provided in a model.Stack entry of type info named q0 or in an input argument q0. q0 is a data structure containing def and DOF fields as in a FEM result data structure (section 4.8). If any, the second column gives the initial velocity. If q0 is empty, zero initial conditions are taken. In this example, a first simulation is used to determine the initial conditions of the final simulation.

 model=fe_time('demo bar');
 TimeOpt=fe_time('TimeOpt Newmark .25 .5 0 1e-4 100');
 TimeOpt.NeedUVA=[1 1 0];
 % first computation to determine initital conditions
 def=fe_time(TimeOpt,model); 

 % no input force
 model=fe_case(model,'remove','Point load 1');

 % Setting initial conditions
 q0=struct('def',[def.def(:,end) def.v(:,end)],'DOF',def.DOF);
 model=stack_set(model,'curve','q0',q0);

 def=fe_time(TimeOpt,model); 

An alternative call is possible using input arguments

 def=fe_time(TimeOpt,model,Case,q0) 

In this case, it is the input argument q0 which is used instead of an eventual stack entry.

You may define the time dependence of a load using curves as illustrated in section 7.9.

You may specify the time steps by giving the 'TimeVector'

 TimeOpt=struct('Method','Newmark','Opt',[.25 .5 ],...
                'TimeVector',linspace(0,100e-4,101));

This is useful if you want to use non constant time steps. There is no current implementation for self adaptive time steps.

To illustrate the output options, we use the example of a 2D propagation. Note that this example also features a time dependent DofLoad excitation (see fe_case) defined by a curve, (see fe_curve), here named Point load 1.

 model=fe_time('demo 2d'); 
 TimeOpt=fe_time('TimeOpt Newmark .25 .5 0 1e-4 50');

You may specify specific output by selecting DOF indices as below

 i1=fe_case(model,'GettDof'); i2=feutil('findnode y==0',model)+.02;
 TimeOpt.OutInd=fe_c(i1,i2,'ind');
 model=stack_set(model,'info','TimeOpt',TimeOpt);
 def=fe_time(model); % Don't animate this (only bottom line)

You may select specific output time step using TimeOpt.OutputFcn as a vector

 TimeOpt.OutputFcn=[11e-4 12e-4]; 
 TimeOpt=feutil('rmfield',TimeOpt','OutInd');
 model=stack_set(model,'info','TimeOpt',TimeOpt);
 def=fe_time(model); % only two time steps saved

or as a string to evaluate. In this case it is useful to know the names of a few local variables in the fe_time function.

• out the structure preallocated for output.
• j1 index of the current step with initial conditions stored in the first column of out.def so store the current time step in out.def(:,j1+1).
• u displacement field, v velocity field, a acceleration field.

In this example the default output function (for TimeOpt.NeedUVA=[1 1 1]) is used but specified for illustration

 TimeOpt.OutputFcn=['out.def(:,j1+1)=u;' ...
                    'out.v(:,j1+1)=v;out.a(:,j1+1)=a;']; 
 model=stack_set(model,'info','TimeOpt',TimeOpt);
 def=fe_time(model); % full deformation saved

This example illustrates how to display the result (see feplot) and make a movie

  cf=feplot(model,def);
  fecom('ColorDataEvalA'); 
  fecom(cf,'SetProp sel(1).fsProp','FaceAlpha',1,'EdgeAlpha',0.1);
  cf.ua.clim=[0 2e-6];fecom(';view2;AnimTime;ch20;scd1e-2;');
  st=fullfile(getpref('SDT','tempdir'),'test.gif'); 
  fecom(['animMovie ' st]);fprintf('\nGenerated movie %s\n',st);

Note that you must choose the Anim:Time option in the feplot GUI.

You may want to select outputs using observations matrix

 model=fe_time('demo bar'); Case=fe_case('gett',model);
 i1=feutil('findnode x>30',model);
 TimeOpt=fe_time('TimeOpt Newmark .25 .5 0 1e-4 100');
 TimeOpt.c_u=fe_c(Case.DOF,i1+.01);          % observation matrix
 TimeOpt.lab_u=fe_c(Case.DOF,i1+.01,'dofs'); % labels
 
 def=fe_time(TimeOpt,model);

If you want to specialize the output time and function you can specify the SaveTimes as a time vector indicating at which time the SaveFcn string will be evaluated. A typical TimeOpt would contain

 TimeOpt.SaveTimes=[0:Ts:TotalTime];
 TimeOpt.SaveFcn='My_function(''Output'',u,v,a,opt,out,j1,t);';

Cleanup

The field FinalCleanupFcn of the TimeOpt can be used to specify what is done just after the time integration.
fe_simul provides a generic clean up function which can be called using
opt.FinalCleanupFcn='fe_simul(''fe_timeCleanup'',model)';
If the output has been directly saved or from iiplot  it is possible to load the results with the same display options than for the fe_timeCleanup using fe_simul('fe_timeLoad',fname)';
Some command options can be used:

• -cf i stores the result of time integration in the stack of iiplot or feplot figure number i. i=-1 can be specified to use current iiplot figure and i=-2 for current feplot figure. Displacements are stored in curve,disp entry of the stack. Velocities and accelerations (if any) are respectively stored in the curve,vel and curve,acc stack entries. If command option -reset is present, existent stack entries (disp, vel, acc, etc. ...) are lost whereas if not stack entries name are incremented (disp(1), disp(2), etc. ...).
  
• '-ExitFcn"AnotherCleanUpFcn"' can be used to call an other clean up function just after fe_simul('fe_timeCleanUp') is performed.
• -fullDOF performs a restitution of the output on the unconstrained DOF of the model used by fe_time.

  -restitFeplot adds a .TR field to the output to allow deformation on the fly restitution in feplot. These two options cannot be specified simultaneously.

• Command option -rethrow allows outputting the cross reference output data from iiplotor feplotif the option -cf-1 or -cf-2 is used.

newmark

For the Newmark scheme, TimeOpt has the form

 TimeOpt=struct('Method','Newmark','Opt',Opt)

where TimeOpt.Opt is defined by

   [beta gamma t0 deltaT Nstep]

beta and gamma are the standard Newmark parameters [37] ([0 0.5] for explicit and default at [.25 .5] for implicit), t0 the initial time, deltaT the fixed time step, Nstep the number of steps.

The default residual is r = (ft(j1,:)*fc'-v'*c-u'*k)'; (notice the sign change when compared to NLNewmark).
This is a simple 1D example plotting the propagation of the velocity field using a Newmark implicit algorithm. Rayleigh damping is declared using the info,Rayleigh case entry.

 model=fe_time('demo bar'); 
 data=struct('DOF',2.01,'def',1e6,...
             'curve',fe_curve('test ricker dt=1e-3 A=1'));
 model = fe_case(model,'DOFLoad','Point load 1',data);
 TimeOpt=struct('Method','Newmark','Opt',[.25 .5 3e-4 1e-4 100],...
                'NeedUVA',[1 1 0]);
 def=fe_time(TimeOpt,model);

 % plotting velocity (propagation of the signal)
 def_v=def;def_v.def=def_v.v; def_v.DOF=def.DOF+.01;
 feplot(model,def_v);
 if sp_util('issdt'); fecom(';view2;animtime;ch30;scd3');
 else; fecom(';view2;scaledef3'); end

dg

The time discontinuous Galerkin is a very accurate time solver [51] [52] but it is much more time consuming than the Newmark schemes. No damping and no non linearities are supported for Discontinuous Galerkin method.

The options are [unused unused t0 deltaT Nstep Nf], deltaT is the fixed time step, Nstep the number of steps and Nf the optional number of time step of the input force.

This is the same 1D example but using the Discontinuous Galerkin method:

 model=fe_time('demo bar');
 TimeOpt=fe_time('TimeOpt DG Inf Inf 0. 1e-4 100');
 TimeOpt.NeedUVA=[1 1 0];
 def=fe_time(TimeOpt,model);

 def_v=def;def_v.def=def_v.v; def_v.DOF=def.DOF+.01;
 feplot(model,def_v);
 if sp_util('issdt'); fecom(';view2;animtime;ch30;scd3'); ...
 else; fecom(';view2;scaledef3'); end

NLNewmark

For the non linear Newmark scheme, TimeOpt has the same form as for the linear scheme (method Newmark). Additional fields can be specified in the TimeOpt data structure

Following example is a simple beam, clamped at one end, connected by a linear spring at other end and also by a non linear cubic spring. The NL cubic spring is modeled by a load added in the residual expression.

 % Get simple test case for NL simulation in sdtweb demosdt('BeamEndSpring')
 model=demosdt('BeamEndSpring'); % simple example building
 opt=stack_get(model,'info','TimeOpt','GetData');
 disp(opt.Residual)
 opt.Follow=1; % activate simple monitoring of the 
 %               number of Newton iterations at each time step
 def=fe_time(opt,model);

staticNewton

For non linear static problems, the Newton solver iterNewton is used. TimeOpt has a similar form as with the NLNewmark method but no parameter Opt is used.

An increment control algorithm iterNewton_Sec can be used when convergence is difficult or slow (as it happens for systems showing high stiffness variations). The Newton increment Δ q is then the first step of a line search algorithm to optimize the corrective displacement increment ρ Δ q, ρ∈ R in the iteration. This optimum is found using the secant iteration method. Only a few optimization iterations are needed since this does not control the mechanical equilibrium but only the relevance of the Newton increment. Each secant iteration requires two residual computations, which can be costly, but more efficient when a large number of standard iterations (matrix inversion) is required to obtain convergence.

Fields can be specified in the TimeOpt data structure

Below is a demonstration non-linear large transform statics.

%  Sample mesh, see script with sdtweb demosdt('LargeTransform')
model=demosdt('largeTransform'); %

% Now perform the Newton loop 
model=stack_set(model,'info','TimeOpt', ...
   struct('Opt',[],'Method','StaticNewton',...
   'Jacobian','ki=basic_jacobian(model,ki,0.,0.,opt.Opt);',...
   'NoT',1, ... % Don't eliminate constraints in model.K
   'AssembleCall','assemble -fetimeNoT -cfield1', ...
   'IterInit','opt=fe_simul(''IterInitNLStatic'',model,Case,opt);'));
model=fe_case(model,'setcurve','PointLoad', ...
    fe_curve('testramp NStep=20 Yf=1e-6')); % 20 steps gradual load
def=fe_time(model);
cf=feplot(model,def); fecom(';ch20;scc1;colordataEvalZ'); % View shape
ci=iiplot(def);iicom('ch',{'DOF',288.03}) % View response

numerical damping for Newmark, HHT-alpha schemes

You may want to use numerical damping in a time integration scheme, the first possibility is to tune the Newmark parameters using a coefficient α such that β = (1+α)²/4 and γ=1/2+α. This is known to implement too much damping at low frequencies and is very depending on the time step [37].

A better way to implement numerical damping is to use the HHT-α method which applies the Newmark time integration scheme to a modified residual balancing the forces with the previous time step.

For the HHT-α scheme, TimeOpt has the form

 TimeOpt=struct('Method','nlnewmark','Opt',Opt,...
                'HHTalpha',alpha)

where TimeOpt.Opt is defined by

   [beta gamma t0 deltaT Nstep]

beta and gamma are the standard Newmark parameters  [37] with numerical damping, t0 the initial time, deltaT the fixed time step, Nstep the number of steps.

The automatic TimeOpt generation call takes the form [alpha unused t0 deltaT Nstep] and will compute the corresponding β, γ parameters.

This is a simple 1D example plotting the propagation of the velocity field using the HHT-α implicit algorithm:

 model=fe_time('demo bar'); 
 TimeOpt=fe_time('TimeOpt hht .05 Inf 3e-4 1e-4 100');
 TimeOpt.NeedUVA=[1 1 0];
 def=fe_time(TimeOpt,model);

The call

 TimeOpt=fe_time('TimeOpt hht .05 Inf 3e-4 1e-4 100');

is strictly equivalent to

TimeOpt=struct('Method','nlnewmark',...
               'Opt',[.275625 .55 3e-4 1e-4 100],...
               'HHTalpha',.05);

Theta

The θ-method is a velocity based solver, whose formulation is given for example in [53, 54]. It considers the acceleration as a distribution, thus relaxing discontinuity problems in non-smooth dynamics. Only a linear implementation is provided in fe_time. The user is nevertheless free to implement a non-linear iteration, through his own IterFcn.

This method takes only one integration parameter for its scheme, θ set by default at 0.5. Any values between 0.5 and 1 can be used, but numerical damping occurs for θ>0.5.

The TimeOpt.Opt takes the form [theta unused t0 deltaT Nstep].

This is a simple 1D example plotting the propagation of the velocity field using the θ-Method:

model=fe_time('demo bar');
TimeOpt=fe_time('TimeOpt theta .5 0 3e-4 100');
def=fe_time(TimeOpt,model);

Euler

This method can be used to integrate first order problem of the form . One can use it to solve transient heat diffusion equation (see p_heat).

Integration scheme is of the form
θ can be define in opt.Opt(1). Explicit Euler (θ=0) is not implemented at this time. Best accuracy is obtained with θ=1/2 (Crank-Nicolson).

See also

fe_mk, fe_load, fe_case