Finite element exterior calculus: a new approach to the stability of
finite elements
The finite element method is a vastly developed technology which is
surely one of the most important tools of computational mechanics.
Nonetheless fundamental challenges remain in the design and understanding
of finite element methods for certain important classes of problems,
including in key areas like electromagnetism and elasticity. A powerful
new approach--known as the finite element exterior calculus--has recently
enabled substantial advances to long standing open problems such
as the development of stable mixed finite elements for elasticity in two
and three dimensions. The key to the new development is the achievement
of stability by developing discretizations which are compatible with
the geometrical and topological structures which mathematicians have
developed to explore the well-posedness of the PDE problem being solved.
Yannis Kevrekidis
Equation-free modeling and computation for complex/multiscale systems
In current modeling practice for complex reacting systems, the best available descriptions often come at a fine level
(atomistic, stochastic, microscopic, individual-based) while the questions asked and the tasks required by the modeler
(prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level. Traditional modeling approaches start by first deriving macroscopic evolution equations from the microscopic
models, and then bringing our arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions.
Over the last few years, and with several collaborators, we have
developed and validated a mathematically inspired, computational enabling technology that allows the modeler to
perform macroscopic tasks acting on the microscopic models directly.
We call this the ``equation-free'' approach, since it circumvents the
step of obtaining accurate macroscopic descriptions. We will argue that the backbone of this approach is the design of
(computational) experiments. Traditional continuum numerical algorithms can thus viewed as
protocols for experimental design (where experiment means a computational
experiment set up and performed with a model at a different level of description).
Ultimately, what makes it all possible is the ability to initialize computational experiments at will.
Short bursts of appropriately initialized computational experimentation
through matrix-free
numerical analysis and systems theory tools like variance reduction and estimation- bridges
microscopic simulation with macroscopic modeling.
I will also discuss some recent developments in data mining algorithms, exploring large complex data sets to find good "reduction coordinates".
Pierre Ladeveze
Model verification through strict upper error bounds
Today, more than ever, modeling and simulation are central to any mechanical
engineering activity. A constant concern both in industry and in research has
been the verification of models, which can reach very high levels of complexity
today. There are numerous sources of error: modeling, space and time
discretization, iteration stopping. The novelty of today's situation
is that over the last twenty-five years truly quantitative tools for assessing
the quality of a FE model have appeared. Here, we will consider that in
model verification, as the subject is now called, the original continuum
mechanics model remains the reference. One of the key topics is the quality
assessment of calculated outputs of interest obtained, for example, by finite
element analysis. The objective goes beyond that of earlier error estimators,
which provided only global information. This was totally insufficient for
dimensioning purposes in mechanical design, where the dimensioning criteria
involve local values of the stresses, displacements, stress intensity
factors, etc. Since most of the available error estimators are nonconservative,
the derivation of efficient and guaranteed upper error bounds for calculated
outputs of interest is currently a challenge. The central questions discussed
here are how to get efficient and guaranteed error bounds and how to
calculate them. This presentation describes the current
state-of-the-art, then introduces a general and recent answer both for linear
problems and for time-dependent nonlinear problems, such as (visco)plasticity
problems under quasi-static or dynamic conditions. Usual convexity properties
are assumed through the standard thermodynamic framework with internal
variables. This involves nonclassical concepts such as the
``dissipation error'' or the "mirror problem", which take the place
of the adjoint problem. Nonintrusive error calculation methods are also
introduced thanks to partition-of-unity techniques and other methods
which have already been used for years.
Mary Wheeler
Institute for Computational Engineering and Sciences,
The University of Texas at Austin
Multiscale Discretizations for Flow, Transport and Mechanics in Porous Media
A fundamental difficulty in understanding and predicting large-scale
fluid movements in porous media is that these movements depend upon
phenomena occuring on small scales in space and/or time. The
differences in scale can be staggering. Aquifers and reservoirs
extend for thousands of meters, while their transport properties can
vary across centimeters, reflecting the depositional and diagenetic
processes that formed the rocks. In turn, transport properties depend
on the distribution, correlation and connectivity of micron sized
geometric features such as pore throats, and on molecular chemical
reactions. Seepage and even pumped velocities can be extremely small
compared to the rates of phase changes and chemical
reactions. The coupling of flow simulation with mechanical
deformations is also important in addressing the response of
reservoirs located in structurally weak geologic formations.
We will focus on the mortar mixed finite element method (MMFE) which
was first introduced by Arbogast, Cowsar, Wheeler, and Yotov for
single phase flow and later extended to multiphase flow by Lu,
Pesyznska, Wheeler, and Yotov for multiphase flow. The MMFE method is
quite general in that it allows for non-matching interfaces and the
coupling of different physical processes in a single simulation. This
is achieved by decomposing the physical domain into a series of
subdomains (blocks) axnd using independently constructed numerical
grids and possibly different discretization techniques in each
block. Physically meaningful matching conditions are imposed on block
interfaces in a numerically stable and accurate way using mortar
finite element spaces. The mortar approach can be viewed as a subgrid
or two scale approach. Moreover, the use of mortars allows one to
couple MFE and discontinuous Galerkin approximations in adjacent
subdomains. In this presentation we will discuss theoretical a priori
and a posteriori results and computational results will be presented.
Kaspar Willam, Ben Blackard, Carlo Citto and Keun Lee
University of Colorado at Boulder
The presentation highlights fundamental mechanics issues of
continuum and discrete interface models in predicting progressive
failure.
To start with the question arises how to
interpret servo-controlled experimental observations in the
post-peak response regime and how to extract objective material
properties from novel image correlation systems. On the theoretical
side recent fracture energy-based softening models are contrasted in
the context of continuum plasticity and cohesive interface
formulations. The constitutive arguments of softening plasticity and
damage lead to failure diagnostics which distinguish among
continuous and discontinuous processes in the form of localization.
This leads to the concomitant argument how to assure positive energy
dissipation and the formulation of well-posed IBVP. Partial
remedies are regularization techniques which involve nonlocal and/or
multiscale aspects. For definiteness we examine the format of higher
grade material in the form of `micromorphic' and `micropolar'
continuum models which
introduce a natural length scale at the material level.
Aside from the constitutive aspects the concomitant numerical issues
need to deal with highly nonlinear and discontinuous degradation
processes. To this end we revisit the elementary model problems of
quasi-brittle materials in order to explain the difference of
snap-back in direct tension and compression in cohesive-frictional
materials. In tension snap-back develops during softening due to
unloading of the elastic domain in a serial system, while
compression mobilizes structural adaptation of the localized failure
processes in the realm of parallel systems. A number of
computational examples will help to illustrate these issues in 2D
and 3D applications involving reinforced concrete and infill masonry
structures, both are composites which exhibit large
differences of stiffness, strength, and ductility/toughness.
The constitutive and computational questions culminate in open
issues which came to the forefront during the recent NIST
investigation of the collapse of the WTC Twin Towers. In this
context the structural engineering community faced hard questions in
addressing impact/fracture and subsequent thermal collapse. In
fact, there is an increasing gap between the structural engineering
community and the academic research focus on micro- and nano-
investigation at atomistic and molecular levels. Current attempts to
bridge this increasing gap are laudatory showing modest promise.
Peter Wriggers
Material Characterization by Multi-Scale Simulations
Keywords: concrete, cement paste, friction, granular material, micro-scale, finite elements
Multi-scale models can be extremely helpful in the understanding of complex materials
used in engineering practice. In the presentation the basic theoretical strategy is developed.
Possible finite element methods to solve such problems are explained in detail and discussed.
These are based on homogenization techniques but also on true multi-scale solutions.
The developed methodology is then applied to a specific engineering materials like concrete or granular soil. Concrete has to be investigated on three different scales, the
hardened cement paste, the mortar and finally the concrete. Here a successive two-stage approach is followed in which first the multi-scale model of the cement paste and mortar is applied. The resulting homogenization can then used in a multi-scale mortar-concrete model. For the granular material homogenization is computed based on a three-dimensional discrete element model accounting for the frictional interface forces between the particles.
The model for the hardened cement paste is based on a three-dimensional computer tomography at the micrometer length scale. For this a finite element model is developed
with different constitutive equations for the three parts unhydrated residual clinker, pores
and hydrated products.
The constitutive equations at the micro-scale contains inelastic parameters, which cannot
be obtained through experimental testings. Therefore, one has to solve an inverse problem
which yields the identification of these properties. For computational efficiency and robustness, a combination of the stochastic genetic algorithm and the deterministic Levenberg-
Marquardt method is used. In order to speed-up the computation time significantly, a
client-server based system is used. Hence, all calculations are distributed automatically
within a network environment.
The resulting constitutive parameters on the micro-scale are then used in the homogenized
constitutive model for the mortar. But also in the multi-scale model for the mortar. Both
results are compared with each other but also with experimental data.
Further interesting applications occur for coupled problems where the interaction of freezing water and material has to be considered at micro-scale. The expansion of the ice leads to damage in
the micro-structure which yields an inelastic material behavior on the macro-scale. If such
a calculation is performed for different moistures and temperatures, a correlation between
moisture, temperature and the inelastic material behavior is obtained. Numerical examples
show, that the developed approach reproduces the material behavior realistically.
