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Shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elasticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the...
Shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elasticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the...
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
We examine an elliptic optimal control problem with control and state constraints in
ℝ3. An improved error estimate of
𝒪(hs)
with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation
involving piecewise constant functions for the control and piecewise linear for the state.
The derived order of convergence is illustrated by a numerical example.
We examine an elliptic optimal control problem with control and state constraints in
ℝ3. An improved error estimate of
𝒪(hs)
with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation
involving piecewise constant functions for the control and piecewise linear for the state.
The derived order of convergence is illustrated by a numerical example.
In this paper we prove a regularity
result for local minimizers of functionals of the Calculus of Variations of the
typewhere Ω is a bounded open set in , u∈(Ω; ), p> 1, n≥ 2 and N≥ 1.
We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give
a bound on the Hausdorff dimension of the singular set of minimizers.
Si prova resistenza locale della soluzione di una equazione di Riccati che si incontra in un problema di controllo ottimale. In ipotesi di regolarità per il costo si prova resistenza globale. Il problema astratto considerato è il modello di alcuni problemi di controllo ottimale governati da equazioni paraboliche con controllo sulla frontiera.
Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.
Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated.
The one- and multi-dimensional cases are treated separately.
Numerical examples illustrate the approach and as well as structural features of the solution.
The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical
variational approach without and with a regularization term in order to
smooth the contours of the classified image. Then we present the general
topological asymptotic analysis, and we finally introduce its application to
the grey-level image classification problem.
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound...
We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator...
We present an a posteriori error analysis of adaptive finite
element approximations of distributed control problems for second
order elliptic boundary value problems under bound constraints on
the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element
residuals. Since we do not assume any regularity of the data of
the problem, the error analysis further invokes data oscillations.
We prove reliability and efficiency of the error estimator...
The weak lower semicontinuity of the functional
is a classical topic that was studied thoroughly. It was shown that if the function is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on . However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.
We establish the existence of an optimal ``state-control'' pair for an optimal control problem of Lagrange type, monitored by a nonlinear elliptic partial equation involving nonmonotone nonlinearities.
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