Page 528 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 528
NONSMOOTH VARIATIONAL METHODS FOR PDES AND APPLICATIONS IN
MECHANICS (MS-8)
Analysis and Numerical Experiments of a Variance Reduction Technique
for Effective Energies of Random Atomic Lattices
Michael Kniely, michael.kniely@tu-dortmund.de
TU Dortmund University, Faculty of Mathematics, Germany
Coauthor: Julian Fischer
We discuss the calculation of effective energies of random materials modeled by the Thomas–
Fermi–von Weizsäcker (TFW) equations in the framework of the method of representative vol-
ume elements (RVEs). The TFW equations constitute a coupled system of nonlinear elliptic
equations and describe the distribution of electrons in the presence of a prescribed nuclear
charge density. The representative volume approximation is subject to a systematic error (due
to the restriction to finite material samples) and a random error (due to material differences in
different RVEs). Our emphasis lies on the reduction of the variance of the energy when evalu-
ated for the RVE, as the systematic error decreases exponentially as a function of the diameter
of the RVE. This variance reduction can be achieved by selecting the RVE in such a way that
it represents the statistical properties of the underlying material particularly well, an approach
proposed by Le Bris, Legoll, and Minvielle in the numerical homogenization of linear elliptic
equations. A rigorous analysis of this strategy has been carried out recently by Fischer for linear
elliptic PDEs.
For establishing the variance reduction in the case of the nonlinear TFW equations, we
need a locality result which ensures that perturbations of the nuclear density inside a bounded
region result in a change of the electronic density decaying exponentially away from this region.
We prove the required locality by extending a recent result by Nazar and Ortner for smeared
nuclear charges to the case of point nuclei represented by Dirac measures. We finally illustrate
the performance of the proposed selection method for RVEs compared to the standard RVE
approach by calculating the energy per atom of random AlTi lattices on RVEs of different size.
Shape differentiability of semilinear equilibrium-constrained optimization
Victor A. Kovtunenko, victor.kovtunenko@uni-graz.at
University of Graz, Austria
Coauthor: Karl Kunisch
A class of semilinear optimization problems linked to variational inequalities is studied with
respect to its shape differentiability. One typical example stemming from quasi-brittle fracture
describes an elastic body with a Barenblatt cohesive crack under the inequality condition of
non-penetration at the crack faces. The other conceptual model is described by a generalized
Stokes-Brinkman-Forchheimer’s equation under divergence-free and mixed boundary condi-
tions. Based on the Lagrange multiplier approach and using suitable regularization, an analyti-
cal formula for the shape derivative is derived from the Delfour-Zolesio theorem. The explicit
expression contains both primal and adjoint states and is useful for finding descent direction of
a gradient algorithm to identify an optimal shape, e.g., from boundary measurement data.
The authors thank the European Research Council (ERC) under the European Union’s Hori-
zon 2020 Research and Innovation Programme (advanced grant No. 668998 OCLOC) for par-
tial support.
526
MECHANICS (MS-8)
Analysis and Numerical Experiments of a Variance Reduction Technique
for Effective Energies of Random Atomic Lattices
Michael Kniely, michael.kniely@tu-dortmund.de
TU Dortmund University, Faculty of Mathematics, Germany
Coauthor: Julian Fischer
We discuss the calculation of effective energies of random materials modeled by the Thomas–
Fermi–von Weizsäcker (TFW) equations in the framework of the method of representative vol-
ume elements (RVEs). The TFW equations constitute a coupled system of nonlinear elliptic
equations and describe the distribution of electrons in the presence of a prescribed nuclear
charge density. The representative volume approximation is subject to a systematic error (due
to the restriction to finite material samples) and a random error (due to material differences in
different RVEs). Our emphasis lies on the reduction of the variance of the energy when evalu-
ated for the RVE, as the systematic error decreases exponentially as a function of the diameter
of the RVE. This variance reduction can be achieved by selecting the RVE in such a way that
it represents the statistical properties of the underlying material particularly well, an approach
proposed by Le Bris, Legoll, and Minvielle in the numerical homogenization of linear elliptic
equations. A rigorous analysis of this strategy has been carried out recently by Fischer for linear
elliptic PDEs.
For establishing the variance reduction in the case of the nonlinear TFW equations, we
need a locality result which ensures that perturbations of the nuclear density inside a bounded
region result in a change of the electronic density decaying exponentially away from this region.
We prove the required locality by extending a recent result by Nazar and Ortner for smeared
nuclear charges to the case of point nuclei represented by Dirac measures. We finally illustrate
the performance of the proposed selection method for RVEs compared to the standard RVE
approach by calculating the energy per atom of random AlTi lattices on RVEs of different size.
Shape differentiability of semilinear equilibrium-constrained optimization
Victor A. Kovtunenko, victor.kovtunenko@uni-graz.at
University of Graz, Austria
Coauthor: Karl Kunisch
A class of semilinear optimization problems linked to variational inequalities is studied with
respect to its shape differentiability. One typical example stemming from quasi-brittle fracture
describes an elastic body with a Barenblatt cohesive crack under the inequality condition of
non-penetration at the crack faces. The other conceptual model is described by a generalized
Stokes-Brinkman-Forchheimer’s equation under divergence-free and mixed boundary condi-
tions. Based on the Lagrange multiplier approach and using suitable regularization, an analyti-
cal formula for the shape derivative is derived from the Delfour-Zolesio theorem. The explicit
expression contains both primal and adjoint states and is useful for finding descent direction of
a gradient algorithm to identify an optimal shape, e.g., from boundary measurement data.
The authors thank the European Research Council (ERC) under the European Union’s Hori-
zon 2020 Research and Innovation Programme (advanced grant No. 668998 OCLOC) for par-
tial support.
526
