Time-varying Domain Research Articles

Neumann boundary control for degenerate wave equations in time-varying domains

Degenerate wave equation with Neumann boundary control in time-varying domains is considered. The fixed endpoint has a Dirichlet-type boundary condition, a Neumann boundary control applied to the moving end. By using the Hilbert Uniqueness Method, we establish the exact boundary controllability for the degenerate wave equation and obtain estimation for the control time.

Review of Recent Developments in Hydrologic Forecast Merging Techniques

Accurate forecasting in hydrologic modeling is crucial for sustainable water resource management across various sectors, where predicting extreme flow phases holds particular significance due to their severe impact on the territory. Due to the inherent uncertainties in hydrologic forecasting, relying solely on a single rainfall–runoff model may not provide reliable predictions. To address this challenge, over the years, researchers have developed and applied hydrologic forecast merging (HFM) techniques that combine multiple models or ensembles to enhance forecast accuracy, reduce uncertainty, and increase confidence in the forecast. This review summarizes the progress in HFM techniques since the early 1990s and covers developments and applications in flow simulation, uncertainty analysis, monthly and seasonal streamflow predictions, ensemble forecasts, flood forecasting, and climate change analysis. The findings indicate that while HFM techniques outperform individual models regarding forecasting efficiency, their performance across applications is not uniform. Among the different methods, Bayesian model averaging (BMA) is the most popular due to its ability to reduce uncertainty and provide accurate and reliable forecasts in deterministic and probabilistic simulations. With their application simplicity, regression techniques are also robust and efficient as they perform competitively well across different model-merging applications. While specific techniques, such as model-dependent weighted averaging and neural network methods, effectively reduce forecast uncertainty, there is still room for improving forecast accuracy across different lead times. Future research can focus on advanced HFM techniques for estimating optimal weights in time-varying domains and overcoming limitations, like simulating low flows in seasonally dry catchments.

Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f(cdot ) to satisfy int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty for all tin mathbb{R}, we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H^{1} topology and the usual (L^{2},L^{2}) pullback mathscr{D}_{lambda}-attractor indeed can attract in the H^{1}-norm, provided that int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{H^{-1}(mathcal{O}_{s})},ds< infty and fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s})).

A multiscale asymptotic expansion for combustion system with composite materials

This paper discusses the multiscale asymptotic expansion for combustion system with composite materials or porous media in thermal protection system. The combustion system consists of two nonlinear equations with rapidly oscillating coefficients, which are posed on a time-varying domain and computationally expensive to solve with the finite element method. To efficiently solve this system, we propose the standard/novel multiscale asymptotic expansions by combining them with boundary layer solutions. Numerical simulation is performed to validate the accuracy and efficiency of the newly proposed methods, which show that the novel multiscale asymptotic expansion is more accurate than the standard one.

Boundary stabilisation of an unstable parabolic PDE with a time-varying domain and the external disturbance

This paper considers the stability of the one-dimensional parabolic system, where one end changes over time and the other is the control end with the external disturbance. Firstly, by the boundary immobilisation method, the displacement change of the system boundary is transferred into the equation so that the original system is transformed into a system with fixed boundaries. Secondly, by combining the backstepping transformation and the sliding mode control method, the feedback control is proposed to compensate the instability of the system itself and reject the matched disturbance. Then, the resulting closed-loop system will be in the form of , where generates a semigroup, and are bounded and unbounded operators respectively, and is the external input. The existence of the generalised solution to the closed-loop system is proved by using the eigenfunction expansion of the system solution. By the Lyapunov method, the closed-loop system is shown to be exponentially stable. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed controller.

Control of the Schrödinger equation by slow deformations of the domain

The aim of this work is to study the controllability of the Schrödinger equation i\partial_t u(t)=-\Delta u(t) on \Omega(t) with Dirichlet boundary conditions, where \Omega(t)\subset \mathbb{R}^N is a time-varying domain. We prove the global approximate controllability of the equation in L^2(\Omega) , via an adiabatic deformation \Omega(t)\subset\mathbb{R}^N ( t\in[0,T] ) such that \Omega(0)=\Omega(T)=\Omega . This control is strongly based on the Hamiltonian structure of the equation provided by Duca and Joly [Ann. Henri Poincaré 22 (2021), 2029–2063], which enables the use of adiabatic motions.We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.

Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.

Adaptive Control of Hyperbolic PDEs Coupled With a Disturbed and Highly Uncertain ODE

In this article, we address adaptive output-feedback boundary control of coupled hyperbolic partial differential equations (PDEs) with spatially varying coefficients and on a time-varying domain, whose uncontrolled boundary is coupled with a disturbed ordinary differential equation (ODE), where multiple parameters in the state matrix and the amplitudes of the harmonic disturbance are unknown. The asymptotic convergence to zero of the ODE state and the boundedness of the PDE states are ensured. This article is motivated by lateral vibration suppression of a mining cable elevator, where the interaction dynamics between the cage and the flexible guide is approximated as a viscoelastic system, including spring and damping, with unknown stiffness and damping coefficients. The performance of the proposed controller is tested in the application of the mining cable elevator by numerical simulation.

A high order moving boundary treatment for convection-diffusion equations

In this paper, a new type of inverse Lax-Wendroff boundary treatment is designed for high order finite difference schemes for solving general convection-diffusion equations on time-varying domain. This new method can achieve high order accuracy on Dirichlet boundary conditions with moving boundary. To ensure stability of the boundary treatment, we give a convex combination of the boundary treatments for the diffusion-dominated and the convection-dominated cases. A group convex combination of weights is carefully designed to avoid zero denominator, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. In order to match the time levels when constructing values of ghost points in the two intermediate stages of the third order Runge-Kutta method, we propose a new approximation to the mixed derivatives at the boundaries to ensure high order accuracy and to improve computational efficiency. In particular, we extend the boundary treatment to the compressible Navier-Stokes equations, which satisfies the isothermal no-slip wall boundary condition at any Reynolds number. We provide numerical tests for one- and two-dimensional problems involving both scalar equations and systems, demonstrating that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.

A Fourth-Order Unfitted Characteristic Finite Element Method for Solving the Advection-Diffusion Equation on Time-Varying Domains

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method.

Modeling of Three-Dimensional Potential Fluid Flow in a Time-Varying Domain

Modeling of Three-Dimensional Potential Fluid Flow in a Time-Varying Domain

Pullback attractors for 2D MHD equations on time-varying domains

&lt;p style='text-indent:20px;'&gt;In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.&lt;/p&gt;

L^{p}-regularity results for 2m-th order parabolic equations in time-varying domains

L^{p}-regularity results for 2m-th order parabolic equations in time-varying domains

Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Accretion and ablation, i.e., the addition and removal of mass at the surface, are important in a wide range of physical processes, including solidification, growth of biological tissues, environmental processes, and additive manufacturing. The description of accretion requires the addition of new continuum particles to the body, and is therefore challenging for standard continuum formulations for solids that require a reference configuration. Recent work has proposed an Eulerian approach to this problem, enabling side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid, which typically requires the deformation gradient that is not immediately available in the Eulerian formulation. To resolve this, the approach introduced the elastic deformation as an additional kinematic descriptor of the added material, and its evolution has been shown to be governed by a transport equation. In this work, the method of characteristics is applied to solve concrete simplified problems motivated by biomechanics and manufacturing. Specifically, (1) for a problem with both ablation and accretion in a fixed domain and (2) for a problem with a time-varying domain, the closed-form solution is obtained in the Eulerian framework using the method of characteristics without explicit construction of the reference configuration.

Extended Hamilton’s principle applied to geometrically exact Kirchhoff sliding rods

This article addresses the dynamic modeling of geometrically exact sliding Cosserat rods. Such systems need to consider non-material time-varying domains to which the Lagrangian view point of solid mechanics is inappropriate. In the article here presented, we use the geometrically exact model of inextensible Kirchhoff rods along a non-material domain whose time variations are not necessarily imposed but are governed by the dynamics, i.e. depend on the configuration of the rod. To progress through derivation, we use the variational calculus on Lie group introduced by Poincaré, and apply it to an extension of Hamilton’s principle holding for open rod systems, which is derived in the article. This extended variational principle uses a moving non-material tube across which the material rod slides. The resulting closed formulation of sliding rods dynamics takes the form of a set of non-material Cosserat–Poincaré’s partial differential equations governing the time-evolution of the cross-section pause of the non-material tube, coupled with an ordinary Lagrange’s differential equation for the sliding motion of the rod across the tube. While emphasize is on the dynamic formulations, the modeling approach is numerically illustrated on a few examples related to the so called sliding spaghetti problem.

An efficient high-order meshless method for advection-diffusion equations on time-varying irregular domains

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters and has O(Nlog⁡N) time complexity. We demonstrate high-orders of convergence for advection-diffusion equations on time-varying 2D and 3D domains for both small and large Peclet numbers. We also present timings that verify our complexity estimates. Finally, we utilize our method to solve a coupled 3D problem motivated by models of platelet aggregation and coagulation, once again demonstrating high-order convergence rates on a moving domain.

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in Rn+1, n≥2, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems:1.Let Ω1 and Ω2 be disjoint domains in Rn+1, n≥2, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set E⊂∂Ω1∩∂Ω2 of mutual absolute continuity of the associated caloric measures ωi with poles at p¯i=(pi,ti)∈Ωi, i=1,2. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of ω1|E is n+1 and the tangent measures of ω1 at ω1-a.e. point of E are equal to a constant multiple of the parabolic (n+1)-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis.2.If, additionally, ω1|E and ω2|E are doubling, log⁡dω2|Edω1|E∈VMO(ω1|E), and E is relatively open in the support of ω1, then their tangent measures at every point of E are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that if Ω1 is a δ-Reifenberg flat domain for δ small enough and Ω2=Rn+1∖Ω‾1, and log⁡dω2dω1∈VMO(ω1), then Ω1∩{t<t2} is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian.3.Finally, we establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure. Assuming that Ωi, 1≤i≤3, are quasi-regular domains for both the heat and the adjoint heat equations, the set of points on ∩i=13∂Ωi, where the three caloric measures are mutually absolutely continuous has null caloric measure. In the course of proving our main theorems, we obtain new results on heat potential theory, parabolic geometric measure theory, nodal sets of caloric functions, and Optimal Transport, that may be of independent interest.

Adaptive event-triggered PDE control for load-moving cable systems

Motivated by lateral vibration suppression of a mining cable elevator, which is a load-moving cable system, where the load moves along a viscoelastic guideway whose stiffness and damping coefficients are unknown, we present event-triggered adaptive output-feedback boundary control design of a hyperbolic PDE–ODE coupled system using the measurements at the PDE actuated boundary and the ODE, where the PDE subsystem is a class of 2 × 2 coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, and a high uncertainty exist in the system matrix of the ODE subsystem at the uncontrolled boundary of the PDE. A continuous-in-time observer-based adaptive backstepping control law is designed where the control gains can be self-tuned to adjust the system matrix of the ODE into a given target system matrix, based on which an observer-based dynamic event-triggering mechanism is built and the existence of a minimal dwell-time is proved. The asymptotic stability of the overall adaptive event-based output-feedback closed-loop system is proved via Lyapunov analysis. In numerical simulation, the performance of the proposed controller is verified in lateral vibration suppression of a mining cable elevator.

Causal regression for online estimation of highly nonlinear parametrically varying models

This paper presents an online estimation method for highly nonlinear systems. This method fits the data to a nonlinear in the parameters regression model, where each parameter is an unknown function of known variables that we call causal regressors. We define this model as an Autoregressive with eXogenous inputs and Causal-regressors-Dependent Parameters (ARX-CDP) model. We are interested on modeling problems where the causal regressors can vary rapidly, as well as the parameters, to the point that the latter cannot be estimated by standard methods. Since the parameters causally depend on the causal regressor but do not depend directly on time, we propose to model each parameter evolution as a random walk process in its causal regressors domain, rather than in the time variable domain, as usual. Therefore, the parameter prediction depends on some past estimates at instants when the causal regressors are similar to the present one. The absolute difference between the current and past causal regressors is used to measure this similarity. We call this prediction method as causal prediction, which, together with a standard recursive least square algorithm, used to update the model, makes up the online causal estimator. Finally, a finite impulse response and a system with multi-dependent parameters are modeled using this proposal. We show experimentally that the causal regression method has a similar performance with other offline methods from literature, with the important advantage of operating in online mode.

A Numerical Model for Steel Continuous Casting Problem in a Time-variable Domain

A mathematical model and numerical method for simulation of the continuous casting process in a variable in time domain are presented. The variable geometry of the slab is caused by the change in time of the width of the mould. The mathematical model of the process is a Stefan problem with prescribed convection and non-linear Robin boundary condition. Considered differential equation is approximated by a finite difference scheme, which is constructed in several steps. First, a semi-discrete problem is constructed using the method of characteristics with respect to the time variable. Then, at each time level, the current elliptic problem in a curvilinear domain is replaced by a problem in the parallelepiped domain using the fictitious domain method. Finally, the boundary-value problem in the parallelepiped domain is approximated by a finite-difference scheme. The constructed fully discrete problem in algebraic form is a system of nonlinear equations containing a diagonal monotone operator and a linear part with a symmetric and positive definite $$M$$ -matrix. To solve the resulting system of nonlinear algebraic equations, well-known iterative solution methods can be applied.