Time-dependent Domain Research Articles

Time-Dependent Reachable Domain and Its Application to Impulsive Orbital Pursuit–Evasion Analysis

Orbital pursuit–evasion (OPE) has garnered significant attention due to its potential applications in space rendezvous and proximity operations. Conventional methods using differential games do not fully address the impulsive OPE problem, which involves a discontinuous differential system. This paper presents the concept of a time-dependent reachable domain (TDRD) for analyzing the impulsive OPE problem. The range and time constraints for determining the TDRD are first given. These constraints can be solved to identify two different TDRDs: the RD at a given time and the RD over a duration. The numerical results obtained from the proposed method exhibit good agreement with those from Monte Carlo simulations, validating the practicality of using TDRDs for OPE analysis. Four different qualitatively OPE outcomes are defined based on the geometrical relationship between the spacecraft’s TDRDs. A series of application cases are comprehensively examined to prove that TDRDs directly predict the sufficient conditions leading to a specific OPE outcome. These findings demonstrate that the TDRD concept is a potent and efficacious instrument for studying the impulsive OPE problem.

The viscoelastic paradox in a nonlinear Kelvin–Voigt type model of dynamic fracture

In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox.

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.

An Implicitly Extended Crank–Nicolson Scheme for the Heat Equation on a Time-Dependent Domain

We consider a time-stepping scheme of Crank–Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld and Olskanskii (ESAIM: M2AN 53(2):585–614, 2019), we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps.

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equationwith periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

In this paper, we consider the heat-conducting compressible self-gravitating fluids in time-dependent domains, which typically describe the motion of viscous gaseous stars. The flow is governed by the 3D Navier–Stokes–Fourier–Poisson equations where the velocity is supposed to fulfill the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, the concept of ballistic energy is utilized in this work.

The existence of a weak solution for a compressible multicomponent fluid structure interaction problem

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modeled by a system resembling compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. The shell possesses a non-linear, non-convex Koiter energy. Considering that the densities are comparable initially we prove the existence of a weak solution until the degeneracy of the energy or the self-intersection of the structure occurs for two cases. In the first case the adiabatic exponents are assumed to satisfy max⁡{γ,β}>2, min⁡{γ,β}>0, and the structure involved is assumed to be non-dissipative. For the second case we assume the critical case max⁡{γ,β}≥2 and min⁡{γ,β}>0 and the dissipativity of the structure. The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy and the pressure, the almost compactness argument, added structural dissipation and suitable limit passages depending on uniform estimates.

An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain

Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of mathbb {R}^2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

On the global wellposedness of free boundary problem for the Navier-Stokes system with surface tension

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ωt⊂R3, t>0, to a problem in the lower half-space R−3. We then establish some time-weighted estimate of solutions, in an Lp-in-time and Lq-in-space setting, for the linearized problem around the trivial steady state with the help of Lr-Ls time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R−3 admits a global-in-time solution in the Lp-Lq setting and that the solution decays polynomially as time t tends to infinity under the assumption that p, q satisfy the conditions: 2<p<∞, 3<q<16/5, and (2/p)+(3/q)<1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R−3 to prove our main results mentioned above for the original problem in Ωt. Here we want to emphasize that it is not allowed to take p=q in the above assumption about p, q, which means that the different exponents p, q of Lp-Lq setting play an essential role in our approach.

Shock capturing for a high-order ALE discontinuous Galerkin method with applications to fluid flows in time-dependent domains

In recent decades, the arbitrary Lagrangian–Eulerian (ALE) approach has been one of the most popular choices to deal with the fluid flows having moving boundaries. The ALE finite volume (FV) method is well-known for its capability of shock capturing. Several ALE discontinuous Galerkin (DG) methods are developed to fully explore the advantages of high-order methods, especially in complex flows. The DG scheme is highly efficient and has high resolution in smooth parts of the flow while the FV scheme has advantage on the robustness against shocks. In this paper, we present a novel algorithm to combine both the ALE FV and ALE DG methods into one stable and efficient hybrid approach. The main challenge for a mixed ALE FV method and ALE DG method is to reduce the inconsistency between both discretizations. Of particular concern are the same mesh velocity distribution between DG and FV elements and a continuous mesh velocity at the coupling interfaces. Since the computational efficiency of the proposed algorithm is of major concern, performance aspects will be considered during the construction of the scheme. To investigate the accuracy of the new scheme, several benchmark test cases with prescribed movements are included. Afterwards, the algorithm is applied to more general and complex scenarios to show its ability to cope with complex setups. In our paper, the scheme is implemented into a loosely-coupled fluid–structure interaction (FSI) framework. With a shock-driven cylinder movement in a channel flow and a transonic flow-induced airfoil vibration, we demonstrate our method for FSI applications.

From Transience to Recurrence for Cox–Ingersoll–Ross Model When b &lt; 0

We consider the Cox–Ingersoll–Ross (CIR) model in time-dependent domains, that is, the CIR process in time-dependent domains reflected at the time-dependent boundary. This is a very meaningful question, as the CIR model is commonly used to describe interest rate models, and interest rates are often artificially set within a time-dependent domain by policy makers. We consider the most fundamental question of recurrence versus transience for normally reflected CIR process with time-dependent domains, and we examine some precise conditions for recurrence versus transience in terms of the growth rates of the boundary. The drift terms and the diffusion terms of the CIR processes in time-dependent domains are carefully provided. In the transience case, we also investigate the last passage time, while in the case of recurrence, we also consider the positive recurrence of the CIR processes in time-dependent domains.

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with quasi-δ boundary conditions. This is a particular class of self-adjoint boundary conditions compatible with the graph structure. We prove that global approximate controllability can be achieved using two physically distinct protocols: either using the boundary conditions as controls, or using time-dependent magnetic fields. Both cases have time-dependent domains for the Hamiltonians.

Axion electrodynamics: Green’s functions, zero-point energy and optical activity

Starting from the theory of Axion Electrodynamics, we work out the axionic modifications to the electromagnetic Casimir energy using the Green’s function method, both when the axion field is initially assumed purely time-dependent and when the axion field configuration is a static domain wall, so purely space-dependent. For the first case it means that the oscillating axion background is taken to resemble an axion fluid at rest in a conventional Casimir setup with two infinite parallel conducting plates, while in the second case we evaluate the radiation pressure acting on an axion domain wall. We extend previous theories in order to include finite temperatures. Various applications are discussed. (i) We review the theory of Axion Electrodynamics and particularly the energy–momentum conservation in a linear dielectric and magnetic material. We treat this last aspect by extending former results by Brevik and Chaichian (2022) and Patkos (2022). (ii) Adopting the model of the oscillating axion background we discuss the axion-induced modifications to the Casimir force between two parallel plates by using the Green’s function method. (iii) We calculate the radiation pressure acting on an axion domain wall at finite temperature T. Our results for an oscillating axion field and a domain wall are also useful for condensed matter physics, where some topological materials, “axionic topological insulators”, interact with the electromagnetic field with a Chern–Simons interaction, like the one in Axion Electrodynamics, and there are experimental systems analogous to time-dependent axion fields and domain walls as the ones showed e.g. by Jiang and Wilczek (2019) and Fukushima et al. (2019). (iv) We compare our results, where we assume time-dependent or space-dependent axion configurations, with the discussion of the optical activity of Axion Electrodynamics by Sikivie (2021) and Carrol et al. (1990). We also make comparisons with the properties of known materials, such as optically active, chiral media and the Faraday effect.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

An embedded strategy for large scale incompressible flow simulations in moving domains

In this work we describe a methodology to approximate the incompressible Navier-Stokes equations in time dependent domains. To deal with the motion of the domain, we employ a fixed mesh method that we call fixed-mesh ALE. It consists of writing the equations in a moving ALE reference system but then projecting them onto a fixed background mesh. This implies that the boundaries of the elements do not necessarily coincide with the physical boundaries, and thus there is the possibility of badly cut elements. We use a Nitsche's type formulation to prescribe the boundary conditions and stabilise the bad cuts by introducing a term that penalises the gradient of the unknown orthogonal to the finite element space in a patch that contains the badly cut element. The flow formulation is a stabilised finite element method that allows one to treat convection dominated flows and to use equal velocity–pressure interpolation. Furthermore, this formulation can be shown to behave as an implicit large eddy simulation approach. A key issue is that the sub-grid scales on which the formulation depends are allowed to be time dependent; this fact has proved to be crucial for the robustness of the approach. The calculation of the velocity and the pressure is segregated by using a fractional step scheme designed at the pure algebraic level, and the conditioning of the pressure equation is improved by using an artificial compressibility technique. Finally, an adaptive mesh refinement strategy is described. All the algorithms are implemented in a parallel environment. The strategy described can be applied to complex flow problems, and in particular here we show the simulation of the air flow generated by a train moving inside a tunnel.

Construction of a Right Inverse for the Divergence in Non-cylindrical Time Dependent Domains

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. The most precise estimates are given in weighted spaces, where the weight depends on the distance to the boundary. This allows for the deficit to be captured precisely in the vicinity of irregularities of the boundary. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier–Stokes equations in time dependent domains.

Global solutions to the free boundary value problem of a chemotaxis-Navier–Stokes system

In this paper, we investigate the global solvability of the chemotaxis-Navier–Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with physical boundary conditions on the free surface that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state , where is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-Navier–Stokes system on a time-dependent domain.

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. &amp; 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. &amp; 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.

Observability of a 1D Schrödinger Equation with Time-Varying Boundaries

We discuss the observability of a one-dimensional Schrödinger equation on certain time-dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we provide exact boundary observability when the curve satisfies some certain conditions. By duality theory, we establish the controllability of adjoint system.