Hi Michael A general response.
The projected internal variables will, generally, result in a projected solution that is not in equilibrium. You would then need to perform an equilibrating step on the new mesh prior to imposing the actual load increment (this is the approach used here). I know of one approach for damage proposed by Ron Peerlings wherein the projection scheme ensures that the Kuhn-Tucker constraints are satisfied after projection (the K-T constraints govern the evolution damage). This is inline with your idea of doing a clever projection. I quote from an abstract of theirs presented at ECF18 "A second computational difficulty is the transfer of state variables which is necessary after each remeshing step. Data which is available in the nodes and Gauss points of the old mesh must be mapped to those of the new mesh before the simulation can be continued. If this transfer operation is not done carefully, numerical instability may result due to inconsistencies in the governing equations. For this purpose, the strategy proposed by Mediavilla et al. [2] has been adapted to the new finite element formulation. This strategy consists in transferring a minimum set of variables and reconstructing the others via the governing constitutive and balance equations. [2]J. Mediavilla, R.H.J. Peerlings, M.G.D. Geers, A robust and consistent remeshing-transfer operator for ductile fracture simulations, Computers & Structures, 2006, 84, 604-623." If the internal variables are constant on the element then projection becomes simple as there is no interpolation (see eg the approach by Carstensen). if the internal variables are elevated to the status of dof by interpolating them element wise, then the projection becomes standard, but presumably you might still need to perform an equilibrating step. So, yes there can be hassles extrapolating to a new mesh. My experience so far is based on problems in viscoelasticity which is a lot easier than plasticity! You need to be careful, as you say. I know other deal.II users have used similar projection schemes for inelastic problems and it would be interesting to hear what they have to say. Andrew On 07.02.2011, at 11:42, Michael Rapson wrote: > Hey there Andrew, > > Thanks for replying. So in your case there is no problem with > extrapolating the points? > > The situation that makes me uneasy with it as a general process is the > example of nonlinear cochlear models. The fluid itself is linear and > easy, but at the boundary quadrature points the relationship between > pressure and acceleration (for example) is based on a nonlinear ODE in > time. There is a well defined space that can be used to project > pressures and therefore accelerations, but it is not safe to project > the ODE states in the same manner. > > A simple case: assume a single cell with a constant pressure gradient > across it such that the pressures at the first and second quadrature > point and the mid point are about 0.2, 0.8 and 0.5. Assume some state > at each quadrature point evolves as: x'(t) = p(t)^3. If the values > start at zero, 100s later we have 0.8, 51.2 and 12.5 as the values. > Presuming we did not know the value at the center of the cell, 12.5, > projecting it from the values at the quadrature points, 0.8 and 51.2 > using our linear shape functions gives 26 != 12.5. > > So even for this highly contrived example, knowing the value at the > quadrature points does not imply knowledge of the value elsewhere. > Arguably if the error introduced by making the assumption that we can > linearly project these values are significant then there are larger > problems because we implicitly assume this at times with FEM. An > example is if we assume x(t) represents the normal gradient on the > boundary and try to form Neumann boundary conditions with it. > > I have been in two minds about posting this to the list or replying to > Andrew privately, especially because Wolfgang had summarized > everything so nicely, but since it might clarify my previous comments > with an example of my concern and because I would be interested in any > comments on the paragraph above I am sending it to the list as well. > > Thanks! > > Michael > > On Mon, Feb 7, 2011 at 11:42 AM, Andrew McBride > <[email protected]> wrote: >> >>> I have one comment about Wolfgang's summary: >>> >>> Before expanding to a FE field and then interpolating back to the new >>> quadrature point locations, you first need to make sure that a >>> mathematically sound FE field can be found. It is one of those cases >>> where it is easy to do but not necessarily correct to. For example, in >>> the worst case of an unknown nonlinear function, the value at a >>> quadrature point might have no formal relationship to the value at >>> another point. The actual situation would probably not be this bad, >>> because if the original FEM problem was properly posed then presumably >>> there must be some assumptions already made that can be applied. >> >> The reason that you can construct the evolution equations for internal >> variables locally at the quadrature points is that they impose no continuity >> requirements. One could evaluate them weakly as well. You could imagine a >> finite element space where you have a kronecker relationship at the >> quadrature points as opposed to the nodes, or you could simply use DG >> elements at the element level. One does run however run into problems when >> you have inequalities governing the evolution equation (e.g. plastic strain) >> which have sign restrictions. The easy way to handle all of these problem is >> to have the internal variable field constant at the element level. >> >> Andrew >> >> >>> >>> Cheers, >>> Michael >>> >>> On Sat, Feb 5, 2011 at 8:35 AM, Wolfgang Bangerth >>> <[email protected]> wrote: >>>> >>>>> There are various other options to transfer quadrature point data in >>>>> problems where one classically has an evolution equation for an internal >>>>> variable that is satisfied at the level of the quadrature point (e.g. >>>>> return mapping algorithms in plasticity). I would be keen to help with >>>>> these once you need them. >>>> >>>> And Andrew is the person to go to with these sort of questions :-) >>>> >>>> Just for completeness (and in order to leave this in the mailing list >>>> archives): When you have internal variables and want to get them from one >>>> mesh >>>> to the next, the natural way to do this would be >>>> - to expand these variables that are only defined at quadrature points to >>>> a FE field that is defined everywhere >>>> - move this field by interpolation to the next mesh >>>> - restrict the field again to quadrature points on individual cells >>>> >>>> There are functions and classes for each of these steps: >>>> - FETools::compute_projection_from_quadrature_points_matrix >>>> - SolutionTransfer >>>> - FETools::compute_interpolation_to_quadrature_points_matrix >>>> >>>> Hope this helps! >>>> W. >>>> >>>> ------------------------------------------------------------------------- >>>> Wolfgang Bangerth email: [email protected] >>>> www: http://www.math.tamu.edu/~bangerth/ >>>> _______________________________________________ >>>> dealii mailing list http://poisson.dealii.org/mailman/listinfo/dealii >>>> >>> _______________________________________________ >>> dealii mailing list http://poisson.dealii.org/mailman/listinfo/dealii >> >>
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