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

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/88080/1/3224_ftp.pd

Towards Microscopic Understanding of the Phonon Bottleneck

The problem of the phonon bottleneck in the relaxation of two-level systems (spins) to a narrow group of resonant phonons via emission-absorption processes is investigated from the first principles. It is shown that the kinetic approach based on the Pauli master equation is invalid because of the narrow distribution of the phonons exchanging their energy with the spins. This results in a long-memory effect that can be best taken into account by introducing an additional dynamical variable corresponding to the nondiagonal matrix elements responsible for spin-phonon correlation. The resulting system of dynamical equations describes the phonon-bottleneck plateau in the spin excitation, as well as a gap in the spin-phonon spectrum for any finite concentration of spins. On the other hand, it does not accurately render the lineshape of emitted phonons and still needs improving.Comment: 13 Phys. Rev. pages, 5 figure captions (7 figures

R-adaptive multisymplectic and variational integrators

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure

Improvement and Application of Smoothed Particle Hydrodynamics in Elastodynamics

This thesis explores the mesh-free numerical method, Smooth Particle Hydrodynamics (SPH), presents improvements to the algorithm and studies its application in solid mechanics problems. The basic concept of the SPH method is introduced and the governing equations are discretised using the SPH method to simulate the elastic solid problems. Special treatments are discussed to improve the stability of the method, such as the treatment for boundary problems, artificial viscosity and tensile instability. In order to improve the stability and efficiency, (i) the classical SPH method has been combined with the Runge-Kutta Chebyshev scheme and (ii) a new time-space Adaptive Smooth Particle Hydrodynamics (ASPH) algorithm has been developed in this thesis. The SPH method employs a purely meshless Lagrangian numerical technique for spatial discretisation of the domain and it avoids many numerical difficulties related to re-meshing in mesh-based methods such as the finite element method. The explicit Runge-Kutta Chebyshev (RKC) scheme is developed to accurately capture the dynamics in elastic materials for the SPH method in the study. Numerical results are presented for several test examples applied by the RKC-SPH method compared with other different time stepping scheme. It is found that the proposed RKC scheme offers a robust and accurate approach for solving elastodynamics using SPH techniques. The new time-space ASPH algorithm which is combining the previous ASPH method and the RKC schemes can achieve not only the adaptivity of the particle distribution during the simulation, but also the adaptivity of the number of stage in one fixed time step. Numerical results are presented for a shock wave propagation problem using the time-space ASPH method compared with the analytical solution and the results of standard SPH. It is found that using the dynamic adaptive particle refinement procedure with adequate refinement criterion, instead of adopting a fine discretisation for the whole domain, can achieve a substantial reduction in memory and computational time, and similar accuracy is achieved

Adaptive time-step control for modal methods to integrate the neutron diffusion equation

[EN] The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P and financed with the help of a Primeros Proyectos de Investigacion (PAID-06-18) from Vicerrectorado de Investigacion, Innovacion y Transferencia of the Universitat Politecnica de Valencia.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Adaptive time-step control for modal methods to integrate the neutron diffusion equation. Nuclear Engineering and Technology. 53(2):399-413. https://doi.org/10.1016/j.net.2020.07.004S39941353