Non-standard Boundary Conditions Research Articles

Neural Networks-Based Computational Modeling of Bilinear Control Systems for Conservation Laws: Application to the Control of Cogeneration

This paper considers the computational modeling of the class of bilinear control systems for hyperbolic conservation laws with nonstandard boundary conditions. These systems arise from (control) engineering applications of systems displaying propagation phenomena, i.e., integrating steam, water, and gas pipes. The aim of this paper is achieved by means of a systematic computational procedure previously introduced and adapted here for the class of systems under consideration. The procedure, based on a convergent Method of Lines ensures the convergence of the approximate numerical solution and also the preservation of the basic properties of the “true” solution as well as its Lyapunov stability. Thus, the approximate computational model allows numerical quantitative and qualitative analysis relevant to a specific problem. The computational efficiency of the procedure is ensured by its implementation based on some, possibly massively, parallel-structured devices belonging to the Artificial Intelligence field—the cell-based recurrent neural networks. As a case study, we consider a control system occurring in the cogeneration process (combined heat and electricity generation). A comparison between the results of the qualitative analysis and those of the numerical simulations demonstrates the correctness and effectiveness of the computational procedure for the dynamics and transients analysis. The paper ends with some conclusions and a list of open problems.

Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph

The aim of this work is to demonstrate the effectiveness of the extension theory of symmetric operators in the investigation of the stability of standing waves for the nonlinear Schrödinger equations with two types of non-linearities (power and logarithmic) and two types of point interactions ($\delta$- and $\delta'$-) on a star graph. Our approach allows us to overcome the use of variational techniques in the investigation of the Morse index for self-adjoint operators with non-standard boundary conditions which appear in the stability study. We also demonstrate how our method simplifies the proof of the stability results known for the NLS equation with point interactions on the line.

A two-phase model of early fibrous cap formation in atherosclerosis

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) are recruited from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and dangerously vulnerable to rupture.We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using data from a range of in vitro and in vivo studies, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development.

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

This paper deals with fast simulations of the haemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic haemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

A source identification problem in a time-fractional wave equation with a dynamical boundary condition

We focus on a inverse source problem in a partial differential equation containing a fractional derivative in time of the order 1<β<2. The equation is accompanied with a non-standard boundary condition which consists of the classic flux term and the dynamical time-fractional derivative term on the one part of the boundary. To determine both the solution of the equation and the source term, a measurement in a form of a integral over space domain is considered. Using a time-discretization and Rothe’s method, we prove an existence of the strong solution in the suitable functional spaces, and the error estimate is established; moreover, the uniqueness is addressed. The results are supported by some numerical experiments.

Hybrid Discontinuous Galerkin Discretisation and Domain Decomposition Preconditioners for the Stokes Problem

Abstract Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

Flexible heat pump or chiller with hybrid water/LiBr absorption/compression cycle

The concept and general considerations for the design of a hybrid absorption/compression cycle are presented. An electrically driven high-speed turbo compressor which is directly integrated in a single-stage water/LiBr absorption heat pump cycle provides a pressure lift for the refrigerant vapor transmitted from the evaporator to the absorber or from the desorber to the condenser, resulting in increased flexibility of the cycle for operation under non-standard boundary conditions. By detailed steady-state thermodynamic process simulation, three main operation modes are investigated concerning the required temperature level of the cooling water and the driving heat source as well as the increase of the chilled water capacity. Characteristic data for compression ratios between 1 and 3 are presented. Depending on the specific electrical input a significant change of the operating limits is observed. In order to assess the influence on the overall system efficiency, selected key figures such as driving power of the compressor, temperature lift, driving heat temperature level, as well as thermal and mechanical COP, are determined and discussed.As preparation for implementation of the concept with maximum flexibility, the required characteristic of the turbo compressor — governed by refrigerant vapor flow and pressure ratio — is deduced from the targeted operating envelope.Based on a fundamental physical model a preliminary design of the turbo compressor is found. This design has been further elaborated by a detailed fluid-dynamic and structural modeling of the compressor, allowing for a precise prediction of its performance and technical feasibility.

On the stabilization of a system of neutral type occurring in co-generation

There is considered a system of conservation laws with non-standard boundary conditions. At a certain level of linearization, a bilinear controlled system of functional equations is associated by integrating the Riemann invariants of the system along its characteristics. For this associated system the basic theory (existence, uniqueness and smooth data dependence) is developed. Then some invariant set accounting for the positiveness of the variables with physical significance is obtained. Further, its equilibria are shown to be stable but not asymptotically stable as suggests the Stability Postulate of N. G. Četaev. Feedback asymptotic stabilization is obtained by using a suitably designed Lyapunov functional. Using the representation formulae for the solutions, all properties and results thus obtained are projected back on the boundary value problem with bilinear control at the boundaries. This shows a way to obtain stability by the first approximation for the linearized conservation laws with nonlinear boundary conditions.

A Cantilevered Extensible Beam in Axial Flow: Semigroup Well-posedness and Postflutter Regimes

We consider a cantilevered (clamped-free) beam in an axial potential flow. Certain flow velocities may bring about a bounded-response instability in the structure, termed flutter. As a preliminary analysis, we employ the theory of large deflections and utilize a piston-theoretic approximation of the flow for appropriate parameters, yielding a nonlinear (Berger/Woinowsky-Krieger) beam equation with a nondissipative right-hand side. As we obtain this structural model via a simplification, we arrive at a nonstandard nonlinear boundary condition that necessitates careful well-posedness analysis. We account for beam rotational inertia effects and discuss technical issues that necessitate this feature. We demonstrate nonlinear semigroup well-posedness of the model with the rotational inertia terms. For the case with no rotational inertia, we utilize a Galerkin approach to establish the existence of weak, possibly nonunique, solutions. For the former, inertial model, we prove that the associated nongradient dynamical system has a compact global attractor. Finally, we study stability regimes and postflutter dynamics (nonstationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.

One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects

ABSTRACTIn the present work, the close similarity that exists between Mindlin’s strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and nonstandard boundary conditions in both the cases. C0-based finite element methods (FEMs) are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long-range interactions. In this case, the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom is required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher-order approximations are compared. Lagrangian, Bèzier and B-Spline approximations are considered for the analysis. It has been shown that such higher-order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion, both the models predict size effect for one-dimensional example. Further, the higher-continuous approximation results in less computational effort for nonlocal-differential model.

Existence and approximation for Navier–Stokes system with Tresca’s friction at the boundary and heat transfer governed by Cattaneo’s law

We consider an unsteady non-isothermal incompressible fluid flow. We model heat conduction with Cattaneo’s law instead of the commonly used Fourier’s law, in order to overcome the physical paradox of infinite propagation speed. We assume that the fluid viscosity depends on the temperature, while the thermal capacity depends on the velocity field. The problem is thus described by a Navier–Stokes system coupled with the hyperbolic heat equation. Furthermore, we consider non-standard boundary conditions with Tresca’s friction law on a part of the boundary. By using a time-splitting technique, we construct a sequence of decoupled approximate problems and we prove the convergence of the corresponding approximate solutions, leading to an existence theorem for the coupled fluid flow/heat transfer problem. Finally, we present some numerical results.

Structural dynamics of a dowelled-joist timber floor in the low-frequency range modelled using finite element simulation

This paper concerns the development and validation of Finite Element Methods (FEM) to simulate the dynamic response of a dowelled-joist timber floor. This is a solid floor comprised of timber joists connected using timber dowels with individual assemblies connected using inclined metal screws. The focus is on the structural dynamics in the low-frequency range up to 200Hz which is the relevant range for impact sound insulation and vibration serviceability. Dowel connections between the joists that formed each assembly were modelled using either rigid or spring connectors in the FEM models. The validation against experimental modal analysis showed that both approaches were valid in terms of the eigenfrequencies, Modal Assurance Criterion (MAC) and the spatial-average velocity with point excitation. Whilst the FEM model with spring connectors had a higher number of correlated modes in the MAC analysis, this required removal of many spurious modes before predicting the response. The validated models were used to demonstrate the potential in predicting assessment parameters for vibration serviceability that are contained in EN 1995-1-1 (Eurocode 5). This predictive approach to the evaluation of vibration serviceability has the advantage in that it can be used for non-standard timber floors with non-standard boundary conditions or floor plans.

Curvature-induced instability of a Stokes-like problem with non-standard boundary conditions

We present an analysis of a set of parametrized boundary conditions for a Stokes–Brinkman model in two space dimensions, discretized by finite elements. We particularly point out an instability which arises when these boundary conditions are posed on a curved line, which then leads to unphysical oscillations. In contrast to a Navier-slip condition, which is prone to Babuška's paradox, this instability can be traced back to the continuous level. We claim that the stability in these cases depend on the amount of curvature at the boundary, which is shown in a reduced setting in cylinder coordinates. The transition to a two dimensional Cartesian case is then based on numerical studies, which further substantiate the claim. Lastly, stabilization techniques are motivated that enhance the continuous FEM setting and are conveniently able to deal with arising oscillations.

Existence of global attractor for one-dimensional weakly damped nonlinear Schr\xf6dinger equation with Dirac interaction and artificial boundary condition in half-line

We consider a nonlinear Schrödinger equation with Dirac interaction defect. Moreover, non-standard boundary conditions are introduced in connection to the behavior of the solutions. First, we prove that this kind of Schrödinger equation can be characterized by an autonomous dynamical system. Then, based on this result, we show that such an equation possesses a maximal compact attractor in the weak topology of mathbf{H}^{mathbf{1}}.

Аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе стержней круглого сечения

In applications, it is of great importance to study the heat transfer process in periodic media containing vacuum interlayers or cavities through which the heat transfer is realized by radiation. A numerical solution of such problems requires considerable computational efforts and becomes, in fact, impossible for a large number of heat transferring elements, especially in the case of two-dimensional and three-dimensional structures. Therefore, it is important to construct effective approximation methods. This article continues a series of papers devoted to the construction and substantiation of special semi-discrete and asymptotic approximations of radiation-conductive heat transfer problems in periodic systems of heat-conducting elements separated by a vacuum. In this paper, we consider a stationary problem of radiation-conductive heat transfer in a periodic system consisting of absolutely black heatconducting rods of circular cross section with radius e packed in a square box. We constract two new approximate methods based on special discrete and asymptotic approximations of the original problem. The results of computational experiments are given, which make it possible to draw conclusions about the efficiency of methods and the nature of the dependence of their relative errors on the coefficient of thermal conductivity λ and the radius of the rods e. The seeking value is the absolute temperature. This function is the solution of the boundary-value problem for the stationary heat equation with nonlinear nonlocal intregral conditions describing the heat exchange by a radiation between the rods. Assuming that the temperature on each of the rods is approximately equal to the mean value over the cross section, we comstruct a discrete approximation of the original problem, which is a system of linear algebraic equations with respect to the fourth power of a temperature. Since the matrix of the system is symmetric and positive definite, the method of conjugate gradients can be used to solve it numerically. By its structure, the discrete problem is such that it can be considered as a difference approximation of the homogenized boundary-value problem for the Poisson equation with non-standard boundary conditions. This problem, which we consider as an asymptotic approximation of the original problem, is linear with respect to the fourth power of temperature. A series of computational experiments is carried out, which confirms the operability of the proposed methods for materials with a large coefficient of thermal conductivity. The relative errors of the methods tend to zero as the parameter tends to zero. For a fixed value of radius e, the relative errors of the methods decrease with increasing values of the thermal conductivity, reaching values in the tenths of a percent. As expected, both methods are practically unsuitable for poorly conductive materials and require substantial modification.

A theory of strain-gradient plasticity with effect of internal microforce

This paper develops a theory of strain gradient plasticity for isotropic bodies undergoing small deformation in the absence of plastic spin. The proposed theory is based on a system of microstresses which include a microstress vector consistent with microforce balance; the mechanical form of the second law of thermodynamics which includes work performed by the microstresses during plastic flow; and a constitutive theory that allows the free energy to depend on the elastic strain E??, divergence of plastic strain div E?? and the Burgers tensor G. Substitution of the constitutive relations into the microforce balance leads to a nonlinear partial differential equation in the plastic strain known as flow rule which captures the presence of an additional energetic length scale arising from the accounting of microstress vector. In addition to the flow rule, nonstandard boundary conditions are obtained, and as an aid to finite element solution a variational formulation of the flow rule is deduced. Finite element solution is obtained of one-dimensional problem of viscoplastic simple shearing under gravity force, where it is shown that for a fixed dissipative length scale, increase in the energetic length scales will result in decrease in the plastic strain.

Calibration and parameter variation using a finite element model for death time estimation: The influence of the substrate.

The most established method for temperature based death time estimation is based on an empirical double exponential model. New physically based approaches using numerical simulation techniques are subject of current research. A major advantage of such models is the possibility to incorporate non-standard boundary conditions. The aim of this study was to investigate the influence of the substrate on the cooling rate of a body in the early postmortem phase. A finite element model was used for parameter variation in terms of different substrate materials. Simulation results showed a considerable influence of substrate material on the postmortem cooling rate of a body. From a thermodynamical point of view, comparability between measurements on a steel trolley and real cooling scenarios with common substrates like normal floors, asphalt or soil remains questionable. It could be shown that not only the type of substrate but also its composition can have a considerable influence on the postmortem body cooling rate.

Mathematical analysis of a clonal evolution model of tumour cell proliferation

We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the clonal evolution model of cancer cell growth. This model has a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. We study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population extinguishes. With telomere restoration, exponential growth occurs in the linear model. We further characterise the specific growth behaviour of the cell population for special cases. We also study the effects of crowding induced mortality on the qualitative behaviour, and the existence and stability of steady states of a nonlinear model incorporating crowding effect. We present examples and extensive numerical simulations, which illustrate the rich dynamic behaviour of the linear and nonlinear models.

A new instability of the topological black hole

We investigate the stability of massless topological black holes in when minimally coupled to a scalar field of negative mass-squared. In many cases such black holes are unstable even though the field is above the BF bound and the geometry is locally AdS. The instability depends on the choice of boundary conditions for the scalars: scalars with non-standard (Neumann) boundary conditions tend to be more unstable, though scalars with standard (Dirichlet) boundary conditions can be unstable as well. This leads to an apparent mismatch between boundary and bulk results in the Vasiliev/vector-like matter duality.

Vector variational problem with knitting boundary conditions

We consider a variational problem with a polyconvex integrand and nonstandard boundary conditions that can be treated as minimization of the strain energy during the suturing process in plastic surgery. Existence of minimizers is proved and necessary optimality conditions are discussed.