Integrator | EPFL Graph Search

An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an important part of many engineering and scientific applications. Mechanical integrators are the oldest type and are still used for metering water flow or electrical power. Electronic analogue integrators are the basis of analog computers and charge amplifiers. Integration can also be performed by algorithms in digital computers. In signal processing circuits :See also Integrator at op amp applications and op amp integrator An electronic integrator is a form of first-order low-pass filter, which can be performed in the continuous-time (analog) domain or approximated (simulated) in the discrete-time (digital) domain. An integrator will have a low pass filtering effect but when given an offset it will accumulate a value building it until it r

Geometric, variational discretization of continuum theories

Mathieu Desbrun

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

Many physical and chemical reactions are driven by nonadiabatic processes, which imply the breakdown of the celebrated Born-Oppenheimer approximation. To understand these processes, experimentalists employ spectroscopic techniques. However, the obtained results are difficult to decipher, and accurate molecular quantum dynamics simulations are used to interpret the results.The second-order split-operator algorithm is one of the most popular numerical methods for simulating the nonadiabatic quantum dynamics because it is explicit, easy to implement, and it preserves many geometric properties of the exact solution. However, the second-order accuracy of this algorithm makes it unaffordable if very accurate results are needed, as tiny time steps are required. To remedy this lack of efficiency, we use composition methods to generate higher-order split-operator algorithms.Although compositions methods increase the accuracy of the standard split-operator algorithm to arbitrary even orders of convergence, the efficiency of the obtained algorithms is still questioned because the computational cost per time step increases drastically with the order of convergence. Therefore, using one- and three-dimensional models of NaI and pyrazine, respectively, we investigate the convergence, efficiency, and geometric properties of these high-order integrators and find that they are, for accurate simulations, more efficient than the standard split-operator algorithm while still preserving the same geometric properties. Besides employing these integrators for simulating the nonadiabatic quantum dynamics, we also explore quantum control and, more specifically, local control theory. This technique uses the instantaneous dynamics of the system to compute an electric field, which interacts with the system in order to drive the state in a desired direction. Because the electric field is obtained from the state itself, we demonstrate that this technique translates into a nonlinear time-dependent Schrödinger equation. Although it is geometric and second-order accurate for simple nonlinearities, the standard split-operator algorithm loses its time-reversal symmetry and second-order accuracy when employed for more complicated nonlinear time-dependent Schrödinger equations. One example of the latter is the one appearing in local control theory.We demonstrate that this lack of generality of the standard split-operator algorithm occurs due to its explicit nature. Thus, we propose two strategies to overcome this issue: First, we completely abandon the split-operator algorithm and present a numerical method based on the implicit midpoint method instead. Second, we make the standard split-operator algorithm implicit, which avoids abandoning the split-operator algorithm altogether. The accuracy and geometric properties of both strategies are then numerically verified using a two-dimensional model of retinal, a molecule whose photochemistry triggers the first event in the biological process of vision. The results demonstrate that both approaches yield second-order methods that preserve all the geometric properties of the exact solution. Because the developed integrators are symmetric, we further improve their accuracy and efficiency using composition methods.

EPFL2022

A multisymplectic integrator for elastodynamic frictionless impact problems

Alejandro Marcos Aragon, François Marie Alain Demoures, Mathieu Desbrun, François Gay-Balmaz, Tudor Ratiu

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational description of nonsmooth Lagrangian continuum mechanics, combined with generalized Lagrange multipliers to handle inequality constraints. We test the resulting explicit integrator for the longitudinal impact of two elastic linear bar models, and for the collision of a nonlinear geometrically exact beam model with a rigid plane. Numerical simulations for various physical parameters are presented to illustrate the behavior and performance of our approach. (C) 2016 Elsevier B.V. All rights reserved.

Elsevier2017

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

EPFL2022