93 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
Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising
Regularity of elastic fields in composites
It is well known that high stress concentrations can occur in elastic
composites in particular due to the interaction of geometrical singularities
like corners,
edges and cracks and structural singularities like jumping material
parameters.
In the project C5 "Stress concentrations in heterogeneous materials" of
the SFB 404 "Multifield Problems in Solid and Fluid Mechanics"
it was mathematically analyzed where and which kind of stress
singularities in coupled linear and nonlinear elastic structures occur. In the
linear case asymptotic expansions near the geometrical and structural
peculiarities are derived, formulae for generalized stress intensity factors
included. In the nonlinear case such expansions are unknown in general and
regularity results are proved for elastic materials with power-law
constitutive equations with the help of the difference quotient technique
combined with a quasi-monotone covering condition for the subdomains and the
energy densities. Furthermore, some applications of the regularity results to
shape and structure optimization and the Griffith fracture criterion in linear
and nonlinear elastic structures are discussed. Numerical examples
illustrate the results
Applications of Asymptotic Analysis
This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods
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