Time-dependent Spatial Domains Research Articles

Applications of a space-time FOSLS formulation for parabolic PDEs

Abstract In this work, we show that the space-time first-order system least-squares formulation (Führer, T. & Karkulik, M. (2021) Space–time least-squares finite elements for parabolic equations. Comput. Math. Appl.92, 27–36) for the heat equation and its recent generalization (Gantner, G. & Stevenson, R. (2021) Further results on a space-time FOSLS formulation of parabolic PDEs. ESAIM Math. Model. Numer. Anal.55, 283–299) to arbitrary second-order parabolic partial differential equations can be used to efficiently solve parameter-dependent problems, optimal control problems and problems on time-dependent spatial domains.

Controlling chaos by the system size

Despite the ubiquity of physical systems evolving on time-dependent spatial domains, understanding their regular and chaotic dynamics is still in a rudimentary state. While chaos implies that the system’s behavior can be altered by small perturbations, this sensitivity proves to be useful for control purposes. Here we report on the experimental discovery of a novel mechanism to control chaos by time-variation of the system (spatial domain) size: depending upon the rate of the latter, the chaotic state may be completely prevented. Our experimental observations are disentangled with theoretical insights and numerical modeling, which also reveals the ability to control spatio-temporal chaos, thus making the findings relevant to a wide range of natural phenomena.

Pattern formation on time-dependent domains

In the quest to understand the dynamics of distributed systems on time-dependent spatial domains, we study experimentally the response to domain deformations by Faraday wave patterns – standing waves formed on the free surface of a liquid layer due to its vertical vibration – chosen as a paradigm owing to their historical use in testing new theories and ideas. In our experimental set-up of a vibrating water container with controlled positions of lateral walls and liquid layer depth, the characteristics of the patterns are measured using the Fourier transform profilometry technique, which allows us to reconstruct an accurate time history of the pattern three-dimensional landscape and reveal how it reacts to the domain dynamics on various length and time scales. Analysis of Faraday waves on growing, shrinking and oscillating domains leads to a number of intriguing results. First, the observation of a transverse instability – namely, when a two-dimensional pattern experiences an instability in the direction orthogonal to the direction of the domain deformation – provides a new facet to the stability picture compared to the one-dimensional systems in which the longitudinal (Eckhaus) instability accounts for pattern transformation on time-varying domains. Second, the domain evolution rate is found to be a key factor dictating the patterns observed on the path between the initial and final domain aspect ratios. Its effects range from allowing the formation of complex sequences of patterns to impeding the appearance of any new pattern on the path. Third, the shrinkage–growth process turns out to be generally irreversible on a horizontally evolving domain, but becomes reversible in the case of a time-dependent liquid layer depth, i.e. when the dilution and convective effects of the domain flow are absent. These experimentally observed enigmatic effects of the domain size variations in time are complemented here with appropriate theoretical insights elucidating the dynamics of two-dimensional pattern evolution, which proves to be more intricate compared to one-dimensional systems.

Mobile observation for distributed parameter system with moving boundary over mobile sensor networks

This paper considers the state observation for a class of distributed parameter systems (DPSs) with moving boundaries modelled by parabolic partial differential equations (PDEs). The method of mobile observation is presented to improve the observation performance of the systems and to eliminate the influence of moving boundaries on system observation with the aids of mobile sensor networks (MSNs). The MSNs which can move throughout the time-dependent spatial domain are used to provide the spatially averaged observations of the DPSs. By using the abstract evolution equation theory of parabolic PDEs and the Lyapunov stability arguments, a centralised observer is designed and a mobile control scheme for each of the mobile sensors is presented while taking account of the dynamics of the MSNs. A numerical example is finally presented to illustrate the effectiveness and the advantages of the proposed approach.

A dual consistent summation-by-parts formulation for the linearized incompressible Navier–Stokes equations posed on deforming domains

In this article, well-posedness and dual consistency of the linearized constant coefficient incompressible Navier–Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem and improve the accuracy of gradients, the second order formulation is transformed to first order form. Boundary conditions that simultaneously lead to boundedness of the primal and dual problems are derived.Fully discrete finite difference schemes on summation-by-parts form, in combination with the simultaneous approximation technique, are constructed. We prove energy stability and discrete dual consistency and show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain. As a result of the aforementioned formulations, stability and discrete dual consistency follow simultaneously.The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

Modeling of hydraulic fracturing and designing of online pumping schedules to achieve uniform proppant concentration in conventional oil reservoirs

We present a novel control framework for the closed-loop operation of a hydraulic fracturing process. Initially, we focus on the development of a first-principle model of a hydraulic fracturing process. Second, a novel numerical scheme is developed to efficiently solve the coupled partial differential equations defined over a time-dependent spatial domain. Third, a reduced-order model is constructed, which is used to design a Kalman filter to accurately estimate unmeasurable states. Lastly, model predictive control theory is applied for the design of a feedback control system to achieve uniform proppant concentration across the fracture at the end of pumping by explicitly taking into account the desired fracture geometry, total amount of proppant injected, actuator limitations, and safety considerations. We demonstrate that the proposed control scheme is able to generate a spatial concentration profile which is uniform and close to the target concentration compared to that of the benchmark, Nolte's pumping schedule.

Improved control for distributed parameter systems with time-dependent spatial domains utilizing mobile sensor–actuator networks**Project supported by the National Natural Science Foundation of China (Grant Nos. 61174021 and 61473136).

A guidance policy for controller performance enhancement utilizing mobile sensor–actuator networks (MSANs) is proposed for a class of distributed parameter systems (DPSs), which are governed by diffusion partial differential equations (PDEs) with time-dependent spatial domains. Several sufficient conditions for controller performance enhancement are presented. First, the infinite dimensional operator theory is used to derive an abstract evolution equation of the systems under some rational assumptions on the operators, and a static output feedback controller is designed to control the spatial process. Then, based on Lyapunov stability arguments, guidance policies for collocated and non-collocated MSANs are provided to enhance the performance of the proposed controller, which show that the time-dependent characteristic of the spatial domains can significantly affect the design of the mobile scheme. Finally, a simulation example illustrates the effectiveness of the proposed policy.

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.

Stability on time-dependent domains: convective and dilution effects

We explore near-critical behavior of spatially extended systems on time-dependent spatial domains with convective and dilution effects due to domain flow. As a paradigm, we use the Swift–Hohenberg equation, which is the simplest nonlinear model with a non-zero critical wavenumber, to study dynamic pattern formation on time-dependent domains. A universal amplitude equation governing weakly nonlinear evolution of patterns on time-dependent domains is derived and proves to be a generalization of the standard Ginzburg–Landau equation. Its key solutions identified here demonstrate a substantial variety–spatially periodic states with a time-dependent wavenumber, steady spatially non-periodic states, and pulse-train solutions–in contrast to extended systems on time-fixed domains. The effects of domain flow, such as bifurcation delay due to domain growth and destabilization due to oscillatory domain flow, on the Eckhaus instability responsible for phase slips in spatially periodic states are analyzed with the help of both local and global stability analyses. A nonlinear phase equation describing the approach to a phase-slip event is derived. Detailed analysis of a phase slip using multiple time scale methods demonstrates different mechanisms governing the wavelength changing process at different stages.

Backstepping output-feedback control of moving boundary parabolic PDEs

This paper extends the backstepping-based observer design to the state estimation of parabolic PDEs with time-dependent spatial domain. The design is developed for the stabilization of a collocated boundary measurement and actuation of an unstable 1D heat equation with the application to the temperature distribution regulation in Czochralski crystal growth process. The PDE system that describes the estimation error dynamics is transformed to an exponentially stable target system through invertible transformations to obtain the time-varying kernel PDE defined on the time-varying triangular-shape domain. The exponential stability of the closed-loop system with an observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system.

An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels.

Blood flow in arterial systems can be described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the elasticity of the arterial walls. In this article, blood is treated as an incompressible Newtonian fluid that flows through compliant vessels of general cross section. A three-dimensional semi-implicit finite difference and finite volume model is derived so that numerical stability is obtained at a low computational cost on a staggered grid. The key idea of the method consists in a splitting of the pressure into a hydrostatic and a non-hydrostatic part, where first a small quasi-one-dimensional nonlinear system is solved for the hydrostatic pressure and only in a second step the fully three-dimensional non-hydrostatic pressure is computed from a three-dimensional nonlinear system as a correction to the hydrostatic one. The resulting algorithm is robust, efficient, locally and globally mass conservative, and applies to hydrostatic and non-hydrostatic flows in one, two and three space dimensions. These features are illustrated on nontrivial test cases for flows in tubes with circular or elliptical cross section where the exact analytical solution is known. Test cases of steady and pulsatile flows in uniformly curved rigid and elastic tubes are presented. Wherever possible, axial velocity development and secondary flows are shown and compared with previously published results.

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.

Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems.

Blood flow in arterial systems is described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the viscoelasticity of the arterial walls. These equations are simplified by assuming cylindrical geometry, axially symmetric flow, and hydrostatic equilibrium in the radial direction. In this paper, an efficient semi-implicit method is formulated in such a fashion that numerical stability is obtained at a minimal computational cost. The resulting computer model is relatively simple, robust, accurate, and extremely efficient. These features are illustrated on nontrivial test cases where the exact analytical solution is known and by an example of a realistic flow through a complex arterial system.

Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth

This paper considers the optimal control of convection–diffusion systems modeled by parabolic partial differential equations (PDEs) with time-dependent spatial domains for application to the crystal temperature regulation problem in the Czochralski (CZ) crystal growth process. The parabolic PDE model describing the temperature dynamics in the crystal region arising from the first principles continuum mechanics is defined on the time-varying spatial domain. The dynamics of the domain boundary evolution, which is determined by the mechanical subsystem pulling the crystal from the melt, are described by an ordinary differential equation for rigid body mechanics and unidirectionally coupled to the convection–diffusion process described by the PDE system. The representation of the PDE as an evolution system on an appropriate infinite-dimensional space is developed and the analytic expression and properties of the associated two-parameter semigroup generated by the nonautonomous operator are provided. The LQR control synthesis in terms of the two-parameter semigroup is considered. The optimal control problem setup for the PDE coupled with the finite-dimensional subsystem is presented and numerical results demonstrate the regulation of the two-dimensional crystal temperature distribution in the time-varying spatial domain.

Optimal boundary control of a diffusion–convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process

In this paper the optimal boundary control problem for diffusion–convection-reaction processes modeled by partial differential equations (PDEs) defined on time-dependent spatial domains is considered. The model of the transport system with time-varying domain arises in the context of high energy consuming Czochralski crystal growth process in which the crystal temperature regulation must successfully account for the change in the crystal spatial domain due to the crystal growth process realized by the pulling crystal out of melt. Starting from the first principles of continuum mechanics and transport theorem the time-varying parabolic PDE describing temperature evolution is derived and represented as a nonautonomous parabolic evolution system on an appropriately defined function space which is exactly transformed in the infinite-dimensional boundary control problem for which a boundary linear quadratic regulator is proposed. Properties of the solution of the time-varying parabolic PDEs given by the two-parameter evolutionary system are utilized in the synthesis of the optimal boundary regulator, and the control law is applied to the model given by a two-dimensional partial differential equation in the cylindrical coordinates representing the Czochralski crystal growth process with one-dimensional growth direction. Finally, numerical results demonstrate optimal stabilization of the two-dimensional temperature distribution in the crystal.

A linear stability theory on time-invariant and time-dependent spatial domains with symmetry: the drop splash problem

This work is a combination of physical and analytical considerations of lin- ear stability pictures on time-invariant and time-dependent spatial domains with symmetry. The discussion is offered in the context of the Rayleigh-Taylor instability (of a fluid inter- face accelerated in the direction of a heavier phase) applied to the drop splash problem, which provides a natural ground for developing stability theory on time-dependent spatial domains with O(2) symmetry. The peculiarity of the underlying linear model common in a number of other interfacial instabilities - linear oscillator ftt + a(k)f = 0 in the wavenumber k-space - allows one to establish a direct correspondence between stability pictures on time-invariant and time-dependent spatial domains. The stability analysis also leads to a notion of frustration in (linear) stability patterns.

Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes

3R23. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. - PD Christofides (Dept of Chem Eng, UCLA, Los Angeles CA 90095-1592). Birkhauser Boston, Cambridge MA. 2001. 248 pp. ISBN 0-8176-4156-4. $69.95.Reviewed by J Chow (Adv Tech Center, Org L9-24, Lockheed Martin, 3251 Hanover St, Palo Alto CA 94304-1191).The author has written a book that seeks to present practical, general nonlinear, and robust control methods for hyperbolic and parabolic PDE systems, and to illustrate their application to transport-reaction processes that are found in the chemical industry. There is also an attempt to compare their effectiveness with respect to traditional control methods for PDE systems. This book is written for process control engineers, researchers, and students at the graduate level.The field of chemical engineering contains many different examples of hyperbolic and parabolic PDE systems. As the author rightly states in his preface, “The interest in control of nonlinear partial differential equation (PDE) systems has been triggered by the need to achieve tight distributed control of transport-reaction processes that exhibit highly nonlinear behavior and strong spatial variations.” As a result, there is a requirement to provide the chemical process community with an overview of the most recent advances in nonlinear PDE control theory. In this book, general and practical methods for synthesizing nonlinear controllers for hyperbolic and parabolic PDE systems are systematically developed, and then extended to include robustness. Geometric and Lyapunov-based control techniques are used to synthesize nonlinear and robust controllers that use a finite number of measurement sensors and control actuators to achieve a stable closed-loop system in the face of model uncertainty. All the PDE systems considered in this text are assumed to have unique solutions that are sufficiently smooth. Readers are assumed to have a basic knowledge about PDE systems and control theories. The first two chapters focus first on quasi-linear, first-order hyperbolic PDE systems, and then on the same type of hyperbolic PDE systems, but those which include time-varying uncertain variables and unmodeled dynamics. These control methods are based on geometric control concepts and are applied to nonisothermal plug-flow reactor examples. The next two chapters focus on developing general methods for quasi-linear parabolic PDE systems using Galerkin’s method and approximate inertial manifolds, both with and without time-varying uncertain variables. Examples of the methodology’s successful application to the control of temperature profiles for catalytic rods and nonisothermal reactors are presented for illustration. The following chapter deals with the nonlinear and robust control of parabolic PDE systems with moving boundaries, ie, time-dependent spatial domains. General methods for synthesizing nonlinear and robust time-varying output feedback controllers are presented, again using a combination of Galerkin’s method and the approximate inertial manifold. The final chapter presents case studies in which the control methods derived in earlier chapters for parabolic PDE systems are applied. In particular, the control of a rapid thermal chemical vapor deposition process and a Czochralski crystal growth process are illustrated and simulated. The book ends with an extensive set of mathematical proofs to support the results developed in the preceding chapters. It also has an extensive bibliography and is sprinkled throughout with clear figures and examples. Nonlinear and Robust Control of PDE Systems is a well-written book that succeeds in its objectives and is possibly the first one to do so. The control theories and synthesis methodologies are described in exhaustive mathematical detail. This book can also be recommended to researchers and engineers in other fields who are faced with the task of developing nonlinear feedback and robust controllers for hyperbolic and parabolic PDE systems. They will find this book to be very useful because it provides the reader with the general framework for nonlinear feedback control based on detailed mathematical models.

Generalized parcel method for dispersed spray and message passing strategy on unstructured meshes

We consider a parallel computation on a message passing MPP of a dispersed spray in a gas such as the flow produced when fuel is injected at a high pressure into a combustion chamber of a piston engine. We suppose that collision effects and liquid breakup are negligible and we use the Stokes and d-square laws, respectively, for drag and vaporization mechanisms. The spray behavior is described in a Lagrangian formulation by a Fokker–Planck type equation. On the other hand the gas phase is described with a Eulerian formulation of the Navier–Stokes equations for a time-dependent spatial domain. In the context of unstructured moving grids, we propose a parallel approach for the particle–fluid numerical model, based on domain decomposition. Numerical experiments on the Cray T3E are performed and the effect of the spray imbalance on the global load is analyzed.

Computation of empirical eigenfunctions and order reduction for nonlinear parabolic PDE systems with time-dependent spatial domains

Computation of empirical eigenfunctions and order reduction for nonlinear parabolic PDE systems with time-dependent spatial domains

Crystal temperature control in the Czochralski crystal growth process

AbstractThis work proposes a control configuration and a nonlinear multivariable model‐based feedback controller for the reduction of thermal gradients inside the crystal in the Czochralski crystal growth process after the crystal radius has reached its final value. Initially, a mathematical model which describes the evolution of the temperature inside the crystal in the radial and axial directions and accounts for radiative heat exchange between the crystal and its surroundings and motion of the crystal boundary is derived from first principles. This model is numericully solved using Galerkin's method and the behaviour of the crystal temperature is studied to obtain valuable insights which lead to the precise formulation of the control problem, the design of a new control configuration for the reduction of thermal gradients inside the crystal and the derivation of a simplified 1‐D in a space dynamic model. Then, a model reduction procedure for partial differential equation systems with time‐dependent spatial domains (Armaou and Christofides, 1999) based on a combination of Galerkin's method with approximate inertial manifolds is used to construct a fourth‐order model that describes the dominant thermal dynamics of the Czochralski process. This low‐order model is employed for the synthesis of a fourth‐order nonlinear multivariable controller that can be readily implemented in practice. The proposed control scheme is successfully implemented on a Czochralski process used to produce a 0.7 m long silicon crystal with a radius of 0.05 m and is shown to significantly reduce the axial and radial thermal gradients inside the crystal. The robustness of the proposed controller with respect to model uncertainty is demonstrated through simulations.