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The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the contact-stabilized Newmark method recently suggested by Deuflhard et al., which is compared to the classical Newmark method and an improved energy dissipative version due to Kane et al. In the absence of contact, any such variant is equivalent to the Störmer-Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
We present a hierarchical a~posteriori error analysis for the
minimum value of the energy functional in symmetric obstacle
problems. The main result is that the energy of the exact solution
is, up to data oscillation, equivalent to an appropriate
hierarchical estimator. The proof of the main result does not
invoke any saturation assumption. Moreover, we prove an a
posteriori error estimate indicating that the estimator from
\cite{RHWHoppe_RKornhuber_1994a} is asymptotically reliable and we
give sufficient conditions for the validity of a saturation
assumption. Finally, we corroborate and complement our theoretical
results with numerical experiments.
We introduce geodesic finite elements as a new way to discretize
the nonlinear configuration space of a geometrically exact Cosserat rod.
These geodesic finite elements naturally generalize standard one-dimensional
finite elements to spaces of functions with values in a Riemannian manifold.
For the special orthogonal group, our approach reproduces the
interpolation formulas of [Crisfield/Jelenic:1999].
Geodesic finite elements are
conforming and lead to objective and path-independent problem formulations.
We introduce geodesic finite elements for general Riemannian manifolds,
discuss the relationship between geodesic finite elements and
coefficient vectors, and estimate the interpolation error.
Then we use them to find static equilibria of hyperelastic Cosserat rods.
Using the Riemannian trust-region algorithm of [Absil/Mahony/Sepulchre:2008]
we show numerically that the discretization error depends optimally on
the mesh size.
The identification of metastable conformations of molecules plays an
important role in computational drug design. One main difficulty is the
fact that the underlying dynamic processes take place in high dimensional
spaces. Although the restriction of degrees of freedom to a few dihedral
angles significantly reduces the complexity of the problem, the existing
algorithms are time-consuming. They are mainly based on the approximation
of a transfer operator by an extensive sampling of states according
to the Boltzmann distribution and short-time Hamiltonian dynamics simulations.
We present a method which can identify metastable conformations
without sampling the complete distribution. Our algorithm is based
on local transition rates and uses only pointwise information about the
potential energy surface. In order to apply the cluster algorithm PCCA+,
we compute a few eigenvectors of the rate matrix by the Jacobi-Davidson
method. Interpolation techniques are applied to approximate the thermodynamical
weights of the clusters. The concluding example illustrates
our approach for epigallocatechine, a molecule which can be described by
seven dihedral angles.
In order to compute the thermodynamic weights of the different metastable conformations
of a molecule, we want to approximate the molecule’s Boltzmann distribution in a reasonable
time. This is an essential issue in computational drug design. The energy landscape of active
biomolecules is generally very rough with a lot of high barriers and low regions. Many of the
algorithms that perform such samplings (e.g. the hybrid Monte Carlo method) have difficulties
with such landscapes. They are trapped in low-energy regions for a very long time and cannot
overcome high barriers. Moving from one low-energy region to another is a very rare event. For
these reasons, the distribution of the generated sampling points converges very slowly against
the thermodynamically correct distribution of the molecule.
The idea of ConfJump is to use a priori knowledge of the localization of low-energy regions
to enhance the sampling with artificial jumps between these low-energy regions. The artificial
jumps are combined with the hybrid Monte Carlo method. This allows the computation of
some dynamical properties of the molecule. In ConfJump, the detailed balance condition is
satisfied and the mathematically correct molecular distribution is sampled.
Biochemical interactions are determined by the 3D-structure of the involved components –
thus the identification of conformations is a key for many applications in rational drug design.
ConFlow is a new multilevel approach to conformational analysis with main focus on
completeness in investigation of conformational space.
In contrast to known conformational analysis, the starting point for design is a space-based
description of conformational areas. A tight integration of sampling and analysis leads to an
identification of conformational areas simultaneously during sampling. An incremental
decomposition of high-dimensional conformational space is used to guide the analysis. A new
concept for the description of conformations and their path connected components based on
convex hulls and Hypercubes is developed. The first results of the ConFlow application
constitute a ‘proof of concept’ and are further more highly encouraging. In comparison to
conventional industrial applications, ConFlow achieves higher accuracy and a specified
degree of completeness with comparable effort.
In this paper, we investigate the interconversion processes of the major flame retardant - 1,2,5,6,9,10-hexabromocyclododecane (HBCD) - by the means of statistical thermodynamics based on classical force-fields. Three ideas will be presented. First, the application of classical hybrid Monte-Carlo simulations for quantum mechanical processes will be justified. Second, the problem of insufficient convergence properties of hybrid Monte-Carlo methods for the generation of low temperature canonical ensembles will be solved by an interpolation approach. Furthermore, it will be shown how free energy differences can be used for a rate matrix computation. The results of our numerical simulations will be compared to experimental results.
Keywords: Markov process, molecular dynamics, rate matrix
This article deals with an efficient sampling of the stationary distribution
of dynamical systems in the presence of metastabilities. For such
systems, standard sampling schemes suffer from trapping problems and
critical slowing down. Starting multiple trajectories in different regions of
the sampling space is a promising way out. The different samplings represent
the stationary distribution locally very well, but are still far away
from ergodicity or from the global stationary distribution. We will show
how these samplings can be joined together in order to get one global
sampling of the stationary distribution.
For the treatment of equilibrated molecular systems in a heat bath we
propose a transition state theory that is based on conformation dynamics.
In general, a set-based discretization of a Markov operator P does not
preserve the Markov property. In this article, we propose a discretization
method which is based on a Galerkin approach. This discretization
method preserves the Markov property of the operator and can be interpreted
as a decomposition of the state space into (fuzzy) sets. The
conformation-based transition state theory presented here can be seen as
a first step in conformation dynamics towards the computation of essential
dynamical properties of molecular systems without time-consuming
molecular dynamics simulations.