191 research outputs found
Towards a full higher order AD-based continuation and bifurcation framework
International audienceSome of the theoretical aspects of continuation and bifurcation methods devoted to the solution for nonlinear parametric systems are presented in a higher-order automatic differentiation (HOAD) framework. Besides benefits in terms of generality and ease of use, HOAD is used to assess fold and simple bifurcations points. In particular, the formation of a geometric series in successive Taylor coefficients allows for the implementation of an efficient detection and branch switching method at simple bifurcation points. Some comparisons with the Auto and MatCont continuation software are proposed. Strengths are then exemplified on a classical case study in structural mechanics
Nonlinear Systems: Asymptotic Methods, Stability, Chaos, Control, And Optimization
[No abstract available]201
An adaptive preconditioner for steady incompressible flows
This paper describes an adaptive preconditioner for numerical continuation of
incompressible Navier--Stokes flows. The preconditioner maps the identity (no
preconditioner) to the Stokes preconditioner (preconditioning by Laplacian)
through a continuous parameter and is built on a first order Euler
time-discretization scheme. The preconditioner is tested onto two fluid
configurations: three-dimensional doubly diffusive convection and a reduced
model of shear flows. In the former case, Stokes preconditioning works but a
mixed preconditioner is preferred. In the latter case, the system of equation
is split and solved simultaneously using two different preconditioners, one of
which is parameter dependent. Due to the nature of these applications, this
preconditioner is expected to help a wide range of studies
A Semismooth Predictor Corrector Method for Real-Time Constrained Parametric Optimization with Applications in Model Predictive Control
Real-time optimization problems are ubiquitous in control and estimation, and
are typically parameterized by incoming measurement data and/or operator
commands. This paper proposes solving parameterized constrained nonlinear
programs using a semismooth predictor-corrector (SSPC) method. Nonlinear
complementarity functions are used to reformulate the first order necessary
conditions of the optimization problem into a parameterized non-smooth
root-finding problem. Starting from an approximate solution, a semismooth
Euler-Newton algorithm is proposed for tracking the trajectory of the
primal-dual solution as the parameter varies over time. Active set changes are
naturally handled by the SSPC method, which only requires the solution of
linear systems of equations. The paper establishes conditions under which the
solution trajectories of the root-finding problem are well behaved and provides
sufficient conditions for ensuring boundedness of the tracking error. Numerical
case studies featuring the application of the SSPC method to nonlinear model
predictive control are reported and demonstrate the advantages of the proposed
method
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