92 research outputs found
Computing multiple solutions of topology optimization problems
Topology optimization problems often support multiple local minima due to a
lack of convexity. Typically, gradient-based techniques combined with
continuation in model parameters are used to promote convergence to more
optimal solutions; however, these methods can fail even in the simplest cases.
In this paper, we present an algorithm to perform a systematic exploratory
search for the solutions of the optimization problem via second-order methods
without a good initial guess. The algorithm combines the techniques of
deflation, barrier methods and primal-dual active set solvers in a novel way.
We demonstrate this approach on several numerical examples, observe
mesh-independence in certain cases and show that multiple distinct local minima
can be recovered
Preconditioned iterative methods for Navier-Stokes control problems
PDE-constrained optimization problems are a class of problems which have attracted much recent attention in scientific computing and applied science. In this paper, we discuss preconditioned iterative methods for a class of Navier-Stokes control problems, one of the main problems of this type in the field of fluid dynamics. Having detailed the Oseen-type iteration we use to solve the problems and derived the structure of the matrix system to be solved at each step, we utilize the theory of saddle point systems to develop efficient preconditioned iterative solution techniques for these problems. We also require theory of solving convection-diffusion control problems, as well as a commutator argument to justify one of the components of the preconditioner
Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems
This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop
A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow
We propose a semismooth Newton method for non-Newtonian models of
incompressible flow where the constitutive relation between the shear stress
and the symmetric velocity gradient is given implicitly; this class of
constitutive relations captures for instance the models of Bingham and
Herschel-Bulkley. The proposed method avoids the use of variational
inequalities and is based on a particularly simple regularisation for which the
(weak) convergence of the approximate stresses is known to hold. The system is
analysed at the function space level and results in mesh-independent behaviour
of the nonlinear iterations.Comment: 25 page
Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms
PDE-constrained optimization problems with control or state constraints are
challenging from an analytical as well as numerical perspective. The
combination of these constraints with a sparsity-promoting term
within the objective function requires sophisticated optimization methods. We
propose the use of an Interior Point scheme applied to a smoothed reformulation
of the discretized problem, and illustrate that such a scheme exhibits robust
performance with respect to parameter changes. To increase the potency of this
method we introduce fast and efficient preconditioners which enable us to solve
problems from a number of PDE applications in low iteration numbers and CPU
times, even when the parameters involved are altered dramatically
Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods
AbstractThis paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P1)–Q0 elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method
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