1,680 research outputs found
Shape optimisation for a class of semilinear variational inequalities with applications to damage models
The present contribution investigates shape optimisation problems for a class
of semilinear elliptic variational inequalities with Neumann boundary
conditions. Sensitivity estimates and material derivatives are firstly derived
in an abstract operator setting where the operators are defined on polyhedral
subsets of reflexive Banach spaces. The results are then refined for
variational inequalities arising from minimisation problems for certain convex
energy functionals considered over upper obstacle sets in . One
particularity is that we allow for dynamic obstacle functions which may arise
from another optimisation problems. We prove a strong convergence property for
the material derivative and establish state-shape derivatives under regularity
assumptions. Finally, as a concrete application from continuum mechanics, we
show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We
derive a necessary optimality system for optimal shapes whose state variables
approximate desired damage patterns and/or displacement fields
Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes
In this paper we study the sensitivity of nonlinear stochastic differential
equations of McKean-Vlasov type generated by stable-like processes. By using
the method of stochastic characteristics, we transfer these equations to the
non-stochastic equations with random coefficients thus making it possible to
use the results obtained for nonlinear PDE of McKean-Vlasov type generated by
stable-like processes in the previous works. The motivation for studying
sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis
of the mean-field games with common noise.Comment: arXiv admin note: text overlap with arXiv:1710.1060
- …