296 research outputs found
Numerical Computation of Rank-One Convex Envelopes
We describe an algorithm for the numerical computation of the rank-one convex envelope of a function f:\MM^{m\times n}\rightarrow\RR. We prove its convergence and an error estimate in Lâ
Better Answers to Real Questions
We consider existential problems over the reals. Extended quantifier
elimination generalizes the concept of regular quantifier elimination by
providing in addition answers, which are descriptions of possible assignments
for the quantified variables. Implementations of extended quantifier
elimination via virtual substitution have been successfully applied to various
problems in science and engineering. So far, the answers produced by these
implementations included infinitesimal and infinite numbers, which are hard to
interpret in practice. We introduce here a post-processing procedure to
convert, for fixed parameters, all answers into standard real numbers. The
relevance of our procedure is demonstrated by application of our implementation
to various examples from the literature, where it significantly improves the
quality of the results
Regularity of Minimizers in Nonlinear Elasticity â the Case of a One-Well Problem in Nonlinear Elasticity
In this note sufficient conditions for bounds on the deformation gradient of a minimizer of a variational problem in nonlinear elasticity are reviewed. As a specific model class, energy densities which are the relaxation of the squared distance function to compact sets are considered and estimates in the space of functions with bounded oscillation are presented. An explicit example related to a one-well problem shows that assumptions of convexity are essential for uniform bounds on the deformation gradient. As an application of the relaxation of the energy in this special case it is indicated how general relaxation formulas for energies with p-growth can be obtained if the relaxation with quadratic growth satisfies natural assumptions
Stress-modulated growth in the presence of nutrients -- existence and uniqueness in one spatial dimension
Existence and uniqueness of solutions for a class of models for
stress-modulated growth is proven in one spatial dimension. The model features
the multiplicative decomposition of the deformation gradient into an
elastic part and a growth-related part . After the transformation due
to the growth process, governed by , an elastic deformation described by
is applied in order to restore the Dirichlet boundary conditions and
therefore the current configuration might be stressed with a stress tensor .
The growth of the material at each point in the reference configuration is
given by an ordinary differential equation for which the right-hand side may
depend on the stress and the pull-back of a nutrient concentration in the
current configuration, leading to a coupled system of ordinary differential
equations
Adapting Real Quantifier Elimination Methods for Conflict Set Computation
The satisfiability problem in real closed fields is decidable. In the context
of satisfiability modulo theories, the problem restricted to conjunctive sets
of literals, that is, sets of polynomial constraints, is of particular
importance. One of the central problems is the computation of good explanations
of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the
input constraints whose conjunction is already unsatisfiable. We adapt two
commonly used real quantifier elimination methods, cylindrical algebraic
decomposition and virtual substitution, to provide such conflict sets and
demonstrate the performance of our method in practice
Algorithmic strategies for applicable real quantifier elimination
One of the most important algorithms for real quantifier elimination is the quantifier elimination by virtual substitution introduced by Weispfenning in 1988. In this thesis we present numerous algorithmic approaches for optimizing this quantifier elimination algorithm. Optimization goals are the actual running time of the implementation of the algorithm and the size of the output formula. Strategies for obtaining these goals include simplification of first-order formulas,reduction of the size of the computed elimination set, and condensing a new replacement for the virtual substitution. Local quantifier elimination computes formulas that are equivalent to the input formula only nearby a given point. We can make use of this restriction for further optimizing the quantifier elimination by virtual substitution. Finally we discuss how to solve a large class of scheduling problems by real quantifier elimination. To optimize our algorithm for solving scheduling problems we make use of the special form of the input formula and of additional information given by the description of the scheduling problemEines der bedeutendsten Verfahren zur reellen Quantorenelimination ist die Quantorenelimination mittels virtueller Substitution, die von Weispfenning 1988 eingefĂŒhrt wurde. In der vorliegenden Arbeit werden zahlreiche algorithmische Strategien zur Optimierung dieses Verfahrens prĂ€sentiert. Optimierungsziele der Arbeit waren dabei die tatsĂ€chliche Laufzeit der Implementierung des Algorithmus sowie die GröĂe der Ausgabeformel. Zur Optimierung werden dabei die Simplifikation vonFormeln erster Stufe, die Reduktion der GröĂe der Eliminationsmenge sowie das Condensing, ein Ersatz fĂŒr die virtuelle Substitution,untersucht. Lokale Quantorenelimination berechnet Formeln, die nur inder NĂ€he eines gegebenen Punktes Ă€quivalent zur Eingabeformel ist. Diese EinschrĂ€nkung erlaubt es, das Verfahren weiter zu verbessern.Als Anwendung des Eliminationsverfahren diskutieren wir abschlieĂend, wie man eine groĂe Klasse von Schedulingproblemen mittels reeller Quantorenelimination lösen kann. In diesem Fall benutzen wir die spezielle Struktur der Eingabeformel und zusĂ€tzliche Informationen ĂŒber das Schedulingproblem, um die Quantorenelimination mittels virtueller Substitution problemspezifisch zu optimieren
- âŠ