Time-dependent Boundary Value Problem Research Articles

Well posedness of fluid-solid mixture models for biofilm spread

Two phase solid-fluid mixture models are ubiquitous in biological applications. For instance, models for growth of tissues and biofilms combine time dependent and quasi-stationary boundary value problems set in domains whose boundary moves in response to variations in the mechano-chemical variables. For a model of biofilm spread, we show how to improve its stability properties by characterizing the time derivatives of relevant quasi-stationary magnitudes in terms of additional boundary value problems. We also give conditions for well posedness of time dependent submodels set in moving domains depending on the motion of the boundary. After constructing solutions for transport, diffusion and elliptic submodels for volume fractions, displacements, velocities, pressures and concentrations with the required regularity, we are able to handle the full model of biofilm spread in moving domains assuming we know the dynamics of the boundary. These techniques are general and can be applied in models with a similar structure arising in biological applications.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion.

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Adaptive LOOCV-based kernel methods for solving time-dependent BVPs

In this article we propose an adaptive algorithm for the solution of time-dependent boundary value problems (BVPs). To solve numerically these problems, we consider the kernel-based method of lines that allows us to split the spatial and time derivatives, dealing with each separately. This adaptive scheme is based on a leave-one-out cross validation (LOOCV) procedure, which is employed as an error indicator. By this technique, we can first detect the domain areas where the error is estimated to be too large – generally due to steep variations or quick changes in the solution – and then accordingly enhance the numerical solution by applying a two-point refinement strategy. Numerical experiments show the efficacy and performance of our adaptive refinement method.

English

In this paper, we consider the direct and inverse problem for time-fractional diffusion in a domain with an impenetrable subregion. Here we assume that on the boundary of the subregion the solution satisfies a generalized impedance boundary condition. This boundary condition is given by a second order spacial differential operator imposed on the boundary. A generalized impedance boundary condition can be used to model corrosion and delimitation. The well-posedness for the direct problem is established where the Laplace Transform is used to study the time dependent boundary value problem. The inverse impedance problem of determining the parameters from the Cauchy data is also studied provided the boundary of the subregion is known. The uniqueness of recovering the boundary parameters from the Neumann to Dirichlet mapping is proven.

Dynamics of unsteady reactive flow of viscous nanomaterial subject to Ohmic heating, heat source and viscous dissipation

Owing to the improved thermophysical properties of nano-materials, the nanofluid convey novel applications in industries, engineering as well as bio-medicals. With high efficiency performances, the nanoparticles include applications in various cooling and heating systems, energy production, aerospace engineering, thermal extrusion, bio-medical applications like treatment of diseases, brain tumour, surgical applications and many more. Moreover, the magneto-nanofluids have the characteristics in the flow of blood in the human artery. The current analysis deals with the free convective unsteady flow of viscous nanofluid subject to the magnetic force over a vertical plate in presence of porous space. The heat transfer phenomenon is accessed by utilizing the additional features like Joule heating, viscous dissipation and absorption coefficients. In addition to the volume fraction of the nanofluid, the effect of the chemical reaction is also introduced. The solution procedure for the modified time-dependent boundary value problem is performed by using the code in-build MATLAB namely Built-in-Shooting method. The change in velocity, temperature and concentration of nano-materials is examined through various graphs. The physical consequences of flow parameters like porosity parameter K p 1 ⩽ K p ⩽ 3 , thermal Grashoff number Gr 1 ⩽ G r ⩽ 3 , solutal Grashoff number Gc 1 ⩽ G c ⩽ 3 , magnetic field parameter M 1 ⩽ M ⩽ 3 , thermal radiation parameter Nr 0.1 ⩽ N r ⩽ 1 , radiation absorption parameter Q 0 ⩽ Q ⩽ 0.5 , Prandtl number Pr 2 ⩽ Pr ⩽ 21 , Eckert number Ec 0 ⩽ E c ⩽ 0.4 , heat source S - 0.2 ⩽ S ⩽ 0.2 , chemical reaction parameter Kc 1 ⩽ K c ⩽ 4 , nanoparticles volume fraction ϕ 0.0 ⩽ ϕ ⩽ 0.1 , Lewis number Le 1 ⩽ L e ⩽ 10 on velocity, temperature and concentration distributions is graphically analyzed. The results show that skin friction coefficients decline with buoyant forces. A lower numerical variation in Nusselt number is resulted with heat source parameter. Moreover, a sharp fall in nanofluid concentration is noticed with increment of chemical reactants and Lewis number.

Effect of porous layer on the Faraday instability in viscous liquid

A linear stability analysis of a viscous liquid on a vertically oscillating porous plane is performed for infinitesimal disturbances of arbitrary wavenumbers. A time-dependent boundary value problem is derived and solved based on the Floquet theory along with the complex Fourier series expansion. Numerical results show that the Faraday instability is dominated by the subharmonic solution at high forcing frequency, but it responds harmonically at low forcing frequency. The unstable regions corresponding to both subharmonic and harmonic solutions enhance with the increasing value of permeability and yields a destabilizing effect on the Faraday instability. Further, the presence of porous layer makes faster the transition process from subharmonic instability to harmonic instability in the wavenumber regime. In addition, the first harmonic solution shrinks gradually and becomes an unstable island, and ultimately disappears from the neutral curve if the porous layer thickness is increased. In contrast, the first and second subharmonic solutions coalesce, and the onset of Faraday instability is dominated by the subharmonic solution. In a special case, the study of Faraday instability of a viscous liquid on a porous substrate can be replaced by a study of Faraday instability of a viscous liquid on a slippery substrate when the permeability of the porous substrate is very low. Further, the Faraday instability can be destabilized by introducing a slip effect at the bottom plane.

Quantum Spectral Methods for Differential Equations

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear equations or linear differential equations with complexity $\mathrm{poly}(\log d)$. While several of these algorithms approximate the solution to within $\epsilon$ with complexity $\mathrm{poly}(\log(1/\epsilon))$, no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity $\mathrm{poly}(\log d, \log(1/\epsilon))$.

The generalized finite difference method for an inverse time-dependent source problem associated with three-dimensional heat equation

This paper presents a meshless numerical scheme for recovering the time-dependent heat source in general three-dimensional (3D) heat conduction problems. The problem considered is ill-posed and the determination of the unknown heat source is achieved here by using the boundary condition, initial condition and the extra measured data from a fixed point placed inside the domain. The extra measured data are used to guarantee the uniqueness of the solution. The generalized finite difference method (GFDM), a recently-developed meshless method, is then adopted to solve the resulting time-dependent boundary-value problem. In our computations, the second-order Crank–Nicolson scheme is employed for the temporal discretization and the proposed GFDM for the spatial discretization. Several benchmark test problems with both smooth and piecewise smooth geometries have been studied to verify the accuracy and efficiency of the proposed method. No need to apply any well-known regularization strategy, the accurate and stable solution could be obtained with a comparatively large level of noise.

Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients

ABSTRACTFor two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast periodic coefficients of a small period , we derive a two-scale limit resolvent problem under a single generic decomposition assumption for the ‘stiff’ part. We show that this key assumption does hold for a large number of examples with a high contrast, both studied before and some recent ones, including those in linear elasticity and electromagnetism. Following ideas of V. V. Zhikov, under very mild restrictions on the regularity of the domain we prove that the limit resolvent problem is well-posed and turns out to be a pseudo-resolvent problem for a well-defined non-negative self-adjoint two-scale limit operator. A key novel technical ingredient here is a proof that the linear span of product test functions in the functional spaces corresponding to the degeneracies is dense in associated two-scale energy space for a general coupling between the scales. As a result, we establish (both weak and strong) two-scale resolvent convergence, as well as some of its further implications for the spectral convergence and for convergence of parabolic and hyperbolic semigroups and of associated time-dependent initial boundary value problems. Some of the results of this work were announced in Kamotski IV, Smyshlyaev VP. Two-scale homogenization for a class of partially degenerating PDE systems. arXiv:1309.4579v1. 2013 (https://arxiv.org/abs/1309.4579v1).

Meridional dynamics of grounded abyssal water masses on a sloping bottom in a mid-latitude -plane

Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods

It is well known the order reduction phenomenon which arises when exponential methods are used to integrate time-dependent initial boundary value problems, so that the classical order of these methods is reduced. In particular, this subject has been recently studied for Lie-Trotter and Strang exponential splitting methods, and the order observed in practice has been exactly calculated. In this article, a technique is suggested to avoid that order reduction.We deal directly with nonhomogeneous time-dependent boundary conditions, without having to reduce the problem to the homogeneous ones. We give a thorough error analysis of the full discretization and justify why the computational cost of the technique is negligible in comparison with the rest of the calculations of the method. Some numerical results for dimension splittings are shown, which corroborate that much more accuracy is achieved.

A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy

This work presents a simple finite element implementation of a geometrically exact and fully nonlinear Kirchhoff---Love shell model. Thus, the kinematics are based on a deformation gradient written in terms of the first- and second-order derivatives of the displacements. The resulting finite element formulation provides $$C^1$$C1-continuity using a penalty approach, which penalizes the kinking at the edges of neighboring elements. This approach enables the application of well-known $$C^0$$C0-continuous interpolations for the displacements, which leads to a simple finite element formulation, where the only unknowns are the nodal displacements. On the basis of polyconvex strain energy functions, the numerical framework for the simulation of isotropic and anisotropic thin shells is presented. A consistent plane stress condition is incorporated at the constitutive level of the model. A triangular finite element, with a quadratic interpolation for the displacements and a one-point integration for the enforcement of the $$C^1$$C1-continuity at the element interfaces leads to a robust shell element. Due to the simple nature of the element, even complex geometries can be meshed easily, which include folded and branched shells. The reliability and flexibility of the element formulation is shown in a couple of numerical examples, including also time dependent boundary value problems. A plane reference configuration is assumed for the shell mid-surface, but initially curved shells can be accomplished if one regards the initial configuration as a stress-free deformed state from the plane position, as done in previous works.

Application of generalized finite difference method to propagation of nonlinear water waves in numerical wave flume

In this paper, a numerical wave flume is formed by combining the generalized finite difference method (GFDM), the Runge–Kutta method, the semi-Lagrangian technique, the ramping function and the sponge layer to efficiently and accurately analyze the propagation of nonlinear water waves. On the basis of potential flow, the mathematical description of wave propagation is a time-dependent boundary value problem, governed by a Laplace equation for velocity potential and two nonlinear free-surface boundary conditions. The incident waves are introduced by imposing horizontal velocity along upstream boundary, as a sponge layer is placed at the end of flume to absorb wave energy and avoid any reflection of waves. The GFDM, a newly-developed meshless numerical method, and the second-order Runge–Kutta method were, respectively, adopted for spatial and temporal discretizations of the moving-boundary problems. The GFDM, which is truly free from mesh generation and numerical quadrature, is easy-to-program, straightforward and efficient, especially for moving-boundary problems. Four numerical examples are adopted in this paper to validate the stability, the efficiency and the accuracy of the proposed meshless numerical wave flume. The GFDM results were compared with other numerical solutions and experimental data to verify the merits and robustness of the proposed meshless numerical model.

Simulation of two-dimensional sloshing phenomenon by generalized finite difference method

In this paper, a meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to efficiently and accurately simulate the sloshing phenomenon in a two-dimensional numerical wave tank. When a numerical wave tank is excited horizontally or vertically, the disturbance on the free surface and the flow field in the tank is called sloshing. Based on the theorem of ideal fluid, the mathematical description of the sloshing problem is a time-dependent boundary value problem, governed by a second-order partial differential equation and two non-linear free-surface boundary conditions. In this paper, the GFDM and the explicit Euler method are adopted, respectively, for spatial and temporal discretizations of this moving-boundary problem. After the discretization by the explicit Euler method, the elevation of free surface is updated and a boundary value problem is yielded at every time step. Since the GFDM, a newly-developed domain-type meshless method, can truly get rid of time-consuming meshing generation and numerical quadrature, we adopted the GFDM to efficiently analyze this boundary value problem at every time step. To use the moving-least squares method of the GFDM can express the derivatives as linear combinations of nearby function values, such that the numerical procedures of the GFDM are very simple and efficient. We provided four numerical examples to verify the simplicity and the accuracy of the proposed meshless scheme. In addition, some factors of the proposed numerical scheme are systematically investigated via a series of numerical experiments.

Fast multipole accelerated dual reciprocity boundary element method for diffusion problems

A fast multipole boundary element method (FMBEM) is presented for diffusion problems based on a dual reciprocity formulation. In the dual reciprocity formulation, domain integrals that arise from solving the time-dependent boundary value problems are transformed into boundary integrals by constructing particular solutions. The time-derivatives in the governing differential equation are approximated with a first-order finite difference time-stepping scheme. Discontinuous linear elements, which are known to give more accurate results than constant or linear elements, are used in the implementation to discretize the boundary integral equations, in combination with the fast multipole method for speeding up the solution. Three numerical examples of diffusion are presented. The performance of the developed FMBEM is compared with that of a conventional BEM and a commercial finite element program. The results show that the developed FMBEM can be a reliable and efficient tool for solving diffusion problems.

The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King & Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King & Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

A thermodynamically compatible model for describing asphalt binders: solutions of problems

In this sequel to the first paper (Málek et al., 2014. International Journal of Pavement Engineering), in which we identified a generalisation of the model due to Burgers which was corroborated against two sets of experiments, including a challenging one showing distinctly different relaxation times for shear and normal stresses, we solve several time-dependent boundary value problems wherein the boundary of the material is deforming, that have relevance to applications involving asphalt. Problems wherein the boundary is subject to time-varying compressive loads such as those due to moving automobiles and the attendant rutting, and the compaction due to rollers are considered in additions to other problems.

Time-dependent capillary-gravity waves in the presence of current

Combined effects of current and surface tension on time-dependent transient boundary value problems (BVPs) associated with capillary-gravity wave motion in a two-layer fluid are analyzed under the assumption of small amplitude water wave theory in three dimensions. Plane wave solutions are discussed by analyzing the roots of the dispersion relation and phase velocity. Using the Laplace transform technique and Green’s function method, the time-dependent initial boundary value problem (IBVP) for the capillary-gravity wave motion are analyzed in both the cases of finite and infinite water depths. Using the method of stationary phase, the time-dependent Green’s function and surface and interface profiles are computed and analyzed for various values of physical parameters in two different cases. The time harmonic solutions are obtained as one of the special cases of the time-dependent IBVP in both the cases of finite and infinite water depths. It is observed that the direction of current and wave changes the phase velocity of the capillary-gravity wave motion along with the free surface and interface elevations. The position of the source significantly changes the amplitude of the source potentials in surface and internal modes. Further, the location of the interface has a significant influence on the pattern of surface and interface elevations.

Poroelastic relaxation indentation of thin layers of gels

We develop a method of poroelastic relaxation indentation (PRI) to characterize thin layers of gels. The solution to the time-dependent boundary-value problem is obtained in a remarkably simple form, so that the force-relaxation curve obtained by indenting a gel readily determines all the poroelastic constants of the gel—the shear modulus, Poisson’s ratio, and the effective diffusivity. The method is demonstrated with a layer of polydimethylsiloxane immersed in heptane.

Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.