Discrete Solution Research Articles

Delta-functions on recurrent random walks

If a random walk on a countable infinite state space is reversible, there are known necessary and sufficient conditions for the walk to be recurrent. When the condition of reversibility is dropped, by using discrete Dirichlet solutions and balayage (concepts familiar in potential theory) one could partially retrieve some of the above results concerning the recurrence and the transience of the random walk.

GPU-accelerated transient thermo-hydraulic simulation of weakly compressible restart flow of a non-Newtonian fluid in a long-buried hot oil pipeline

For the complex transient thermo-hydraulic restart process of a long-buried hot oil pipeline, it takes a long time to solve the restart simulation model, which cannot meet the requirements of rapid restart prediction and assessment. The modern GPU is specialized for compute-intensive and massively parallel computation, and it has great potential to accelerate transient thermo-hydraulic restart simulation. According to the computational characteristics of restart model solutions, four types of GPU parallel strategies, from the mapping of grids to the convergence judgment of the discrete equation solution, are combined to accelerate the whole process of a transient thermo-hydraulic restart simulation. The results indicate that the total computation time of the transient thermo-hydraulic simulation is reduced from 185.4 min to 8.8 min, corresponding to an acceleration rate of 21.1, which exhibits a good acceleration effect for a single GPU and is beneficial to the rapid assessment of restart schemes. Based on the GPU-accelerated numerical results, the complex transient thermo-hydraulic restart process with the changing soil temperature field, the non-Newtonian thixotropic behavior of crude oil and flow pattern evolution, is investigated. The analysis reveals that the dominant influencing factors of the restart process vary with the restart time, and the flow pattern evolution plays an important role in the thermo-hydraulic characteristics of the restart process.

Dynamic Phase‐Field Fracture in Viscoelastic Materials using a First‐Order Formulation

AbstractIn this contribution we present analytical results on a model for dynamic fracture in viscoelastic materials at small strains that have been obtained in full depth in [1]. In the model, the sharp crack interface is regularized with a phase‐field approximation, and for the phase‐field variable a viscous evolution with a quadratic dissipation potential is employed. A non‐smooth penalization prevents material healing. The viscoelastic momentum balance is formulated as a first order system and coupled in a nonlinear way to the non‐smooth evolution equation of the phase field. We give a full discretization in time and space using a discontinuous Galerkin method for the first‐order system. We discuss the existence of discrete solutions and, with the step size in space and time tending to zero, their convergence to a suitable notion of weak solution of the system. Eventually, we provide a numerical benchmark and compare it with simulation results found in [2].

An Online Hashing Algorithm for Image Retrieval Based on Optical-Sensor Network.

Online hashing is a valid storage and online retrieval scheme, which is meeting the rapid increase in data in the optical-sensor network and the real-time processing needs of users in the era of big data. Existing online-hashing algorithms rely on data tags excessively to construct the hash function, and ignore the mining of the structural features of the data itself, resulting in a serious loss of the image-streaming features and the reduction in retrieval accuracy. In this paper, an online hashing model that fuses global and local dual semantics is proposed. First, to preserve the local features of the streaming data, an anchor hash model, which is based on the idea of manifold learning, is constructed. Second, a global similarity matrix, which is used to constrain hash codes is built by the balanced similarity between the newly arrived data and previous data, which makes hash codes retain global data features as much as possible. Then, under a unified framework, an online hash model that integrates global and local dual semantics is learned, and an effective discrete binary-optimization solution is proposed. A large number of experiments on three datasets, including CIFAR10, MNIST and Places205, show that our proposed algorithm improves the efficiency of image retrieval effectively, compared with several existing advanced online-hashing algorithms.

Guidelines for RBF-FD Discretization: Numerical Experiments on the Interplay of a Multitude of Parameter Choices

There exist several discretization techniques for the numerical solution of partial differential equations. In addition to classical finite difference, finite element and finite volume techniques, a more recent approach employs radial basis functions to generate differentiation stencils on unstructured point sets. This approach, abbreviated by RBF-FD (radial basis function-finite difference), has gained in popularity since it enjoys several advantages: It is (relatively) straightforward, does not require a mesh and generalizes easily to higher spatial dimensions. However, its application is not quite as blackbox as it may appear at first sight. The computed solution might suffer severely from various sources of errors if RBF-FD parameters are not selected carefully. Through comprehensive numerical experiments, we study the influence of several of these parameters on the condition numbers of intermediate (local) weight matrices, on the condition number of the resulting (global) stiffness matrix and ultimately on the approximation error of the computed discrete solution to the partial differential equation. The parameters of investigation include the type of RBF (and its shape or other parameters if applicable), the degree of polynomial augmentation, the discretization stencil size, the underlying type of point set (structured/unstructured), and the total number of (interior and boundary) points to discretize the PDE, here chosen as a three-dimensional Poisson’s problem with Dirichlet boundary conditions. Numerical tests on a sphere as well as tests for the convection-diffusion equation are included in a supplement and demonstrate that the results obtained for the Laplace problem on a cube generalize to wider problem classes. The purpose of this paper is to provide a comprehensive survey on the various components of the basic algorithms for RBF-FD discretization and steer away from potential pitfalls such as computationally more expensive setups which not always lead to more accurate numerical solutions. We guide toward a compatible selection of the multitude of RBF-FD parameters in the basic version of RBF-FD. For many of its components we refer to the literature for more advanced versions.

A stabilized finite element method for the Stokes–Temperature coupled problem

In this work, we introduce and analyze a new stabilized finite element scheme for the Stokes–Temperature coupled problem. This new scheme allows equal order of interpolation to approximate the quantities of interest, i.e. velocity, pressure, temperature, and stress. We analyze an equivalent variational formulation of the coupled problem inspired by the ideas proposed in [3]. The existence of the discrete solution is proved, decoupling the proposed stabilized scheme and using the help of continuous dependence results and Brouwer's theorem under the standard assumption of sufficiently small data. Optimal convergence is proved under classic regularity assumptions of the solution. Finally, we present some numerical examples to show the quality of our scheme, in particular, we compare our results with those coming from a standard reference in geosciences described in [38].

Block-by-block method for solving non-linear Volterra integral equation of the first kind

In this paper, we focus on the numerical solution of the non-linear Volterra integral equation of the first kind. We start by converting its form to a second-kind Volterra equation and we construct some assumptions to give a sufficient condition to ensure the solution’s existence and uniqueness, this is on the one hand. On the other hand, to use it in the numerical and analytical analysis computation which put the two analyses compatible. Then, by applying block-by-block method, we transform our integral equation to a non-linear system. This system gives us a discrete numerical solution converging to our analytical solution. The approximate solution is computed using a Newton’s method and the numerical examples clearly demonstrate the efficiency of our proposed method compared with the Nystöm method.

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.

Discrete breathers in a mechanical metamaterial.

We consider a previously experimentally realized discrete model that describes a mechanical metamaterial consisting of a chain of pairs of rigid units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions with frequencies inside the gap between optical and acoustic dispersion bands. We compute numerically exact solutions of this type for several different parameter regimes and investigate their properties and stability. Our findings demonstrate that upon appropriate parameter tuning within experimentally tractable ranges, the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling and symmetry-breaking bifurcations, in addition to other mechanisms of stability change such as saddle-center and Hamiltonian Hopf bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system and paving the way for potential further experimental exploration of this rich nonlinear dynamical lattice setting.

Uniqueness' failure for the finite element Cauchy-Poisson's problem

We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn–Vogelius artifice and the condensation Steklov–Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called “weak spurious modes”, of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of “spurious modes”. We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in Azaïez et al. (2011) [4], and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material.

Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with free end and arbitrary initial temperature profile. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. By comparison of the discrete and continuum descriptions of kinetic temperature field, we reveal some restrictions to the latter. Specifically, the far-field kinetic temperature is well described by the continuum solution, which, however, deviates near and at the free end (boundary). We show analytically that, after thermal wave reflects from the boundary, the discrete solution for the kinetic temperature undergoes a jump near the free end. A comparison of the descriptions of heat propagation in the semi-infinite and infinite Hooke chains is presented. Results of the current paper are expected to provide insight into non-stationary heat transport in the semi-infinite lattices.

Measurement quantization

We present principles of Measurement Quantization (MQ) and approaches to measurement that support the discreteness of measure. Several claims are addressed. Notably, that measure is discrete with respect to the internal frame, non-discrete with respect to the system frame and that length is contracted due to the discreteness of measure. We address the relation of angular measure to momentum, the physical significance of count bounds and that the fundamental measures — more precise expressions for Planck’s units — are an emergent property of the internal frame. Quantum experiments by Shwartz, et. al and CODATA provide physical support. We predict and derive values for elementary charge and the gravitational, Hubble, reduced Planck, electric, magnetic, Coulomb, and fine structure constants. We then correlate gravity with electromagnetism (unification). We present expressions for galactic rotation, dark matter, dark energy, and accelerating expansion. MQ advances over Loop Quantum Gravity with two frames, the difference which leads to the physical constants and the laws of nature. We correlate the quantum and cosmological, describing an inflation free quantum epoch, why it ceases and expansion. Therein are solutions to the horizon problem and homogenous, isotropic properties of the universe. Predictions include length contraction unrelated to special or general relativity (SR/GR), 13-digit measures of the gravitational constant, the Planck momentum, and universal mass accretion. A calculation of CMB age, quantity, present-day density and temperature provides additional support. Also offered, discrete solutions to the size and age of the universe, ground state orbital, SR, GR, and equivalence.

Two-dimensional analysis on the magnetic field adjusted electrical behaviors in composite semiconductor structures

To explore the exact electric properties and reveal the intrinsic interaction mechanisms among multi-fields in composite magneto-electric-semiconductor structures, two-dimensional analyses are performed based on the coupled field theory. Expanding the basic physical quantities to Fourier series along the length, the governing equations are simplified. And then, the discrete solutions are derived by utilizing differential quadrature method. As applications, three composite structures are designed via considering different poling directions. Respectively, the extension, piecewise deformation and bending are realized. Before analysis, the convergence and correctness of adopted method are discussed systematically. In numerical calculation, the effect of material proportion on the perturbation carrier density is investigated. It is found the variation rule of perturbation carrier density is controlled by material proportion and deformation form simultaneously. Importantly, there are optimal material proportion ranges producing the carriers, nevertheless, the ranges are different for three structures. Additionally, two-dimensional distributions of perturbation carrier density, electric field, electric displacement and polarization are discussed. Along the thickness, the field quantities are symmetric for extension and piecewise deformation, but is antisymmetric for bending. Specially, in the structure with piecewise deformation, the potential barriers are realized. This work could be the guidance designing magneto-electric devices.

A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions

Abstract The Cahn–Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for this equation have been proposed. Recently, a family of models using Cahn–Hilliard-type equations on the boundary of the domain to describe adsorption processes was analysed (cf. Knopf, P., Lam, K. F., Liu, C. & Metzger, S. (2021) Phase-field dynamics with transfer of materials: the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. ESAIM: Math. Model. Numer. Anal., 55, 229–282). This family of models includes the case of instantaneous adsorption processes studied by Goldstein, Miranville and Schimperna (2011, A Cahn–Hilliard model in a domain with non-permeable walls. Phys. D, 240, 754–766) as well as the case of vanishing adsorption rates, which was investigated by Liu and Wu (2019, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal., 233, 167–247). In this paper, we are interested in the numerical treatment of these models and propose an unconditionally stable, linear, fully discrete finite element scheme based on the scalar auxiliary variable approach. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. Thereby, when passing to the limit, we are able to remove the auxiliary variables introduced in the discrete setting completely. Finally, we present simulations based on the proposed linear scheme and compare them to results obtained using a stable, nonlinear scheme to underline the practicality of our scheme.

Simulation of one-dimensional dam-break flood routing based on HEC-RAS

Dam-break is a serious disaster resulting in severe damage to downstream communities. Therefore, analyzing the affected range and the evolution process of dam-break floods in advance is valuable. However, the difficulties and challenges lie in the complexity of the breaking process of earth-rock dams, the uncertainty in the evolution of dam-break floods, and the geographical variability. Given this, the objective of this study is to analyze the characteristics of the dam-break flood evolution. The study chooses Chengbi River Reservoir as the research object, HEC-RAS as the simulation software, unsteady flow differential equations and one-dimensional Saint-Venant equations as the control equations, and it uses four-point implicit finite difference method for discrete solution. In this paper, the dam-break flood evolution is simulated under three boundary conditions (full breach, 1/2 breach, 1/3 breach), and the main results are as follows. From the dam site section to the Tianzhou hy-drological station section, the peak discharge decay rates of the three schemes are 78%, 77%, and 67%, respectively. The water level decay rates of the three schemes are 47%, 36%, and 30%, respectively. A 1 m increase in the bursting water level elevation increases the peak flow by ap-proximately 7%, and the highest water level in front of the dam by 1 m, and delays the peak time by 1.5 h on average. In addition, the preliminary inundation extent for the Baise City is obtained. The analysis results can provide a fundamental basis for flood control as well as a reference for flood disaster management.

Constraining Continuous Topology Optimizations to Discrete Solutions for Photonic Applications.

Photonic topology optimization is a technique used to find the permittivity distribution of a device that optimizes an electromagnetic figure-of-merit. Two common versions are used: continuous density-based optimizations that optimize a gray scale permittivity defined over a grid, and discrete level-set optimizations that optimize the shape of the material boundary of a device. In this work we present a method for constraining a continuous optimization such that it is guaranteed to converge to a discrete solution. This is done by inserting a constrained suboptimization with low computational overhead cost at each iteration of an overall gradient-based optimization. The technique adds only one hyperparameter with straightforward behavior to control the aggressiveness of binarization. Computational examples are provided to analyze the hyperparameter behavior, show this technique can be used in conjunction with projection filters, show the benefits of using this technique to provide a nearly discrete starting point for subsequent level-set optimization, and show that an additional hyperparameter can be introduced to control the overall material/void fraction. This method excels for problems where the electromagnetic figure-of-merit is majorly affected by the binarization requirement and situations where identifying suitable hyperparameter values becomes challenging with existing methods.

Identity-Based Unidirectional Collusion-Resistant Proxy Re-Encryption from U-LWE

Identity-based proxy re-encryption (IB-PRE) converts the ciphertext encrypted under the delegator’s identity to the one encrypted under the delegatee’s identity through a semitrusted proxy without leaking delegator’s private key and the underlying plaintext. At present, the security of most IB-PRE schemes relies on the hardness of the discrete logarithm solution or large integer decomposition and cannot resist attacks of the quantum algorithms. The majority of the IB-PRE schemes over lattice are secure only in the random oracle model. Aiming at such problems, the paper constructs a secure IB-PRE scheme over lattice in the standard model. In the scheme, the underlying encryption scheme proposed by Gentry et al. in EUROCRYPT 2010 is adopted to reduce the storage space of ciphertext. The proposed scheme is unidirectional collusion-resistant multihop and anonymous, and it is semantically secure against selective identity and chosen plaintext attack based on Decisional Learning With Errors with uniformly distributed errors (D-U-LWE) hard problem in the standard model.

Optimal Generator Policy for Hybrid Fuel UAV under Airspace Noise Restrictions

Here we study an optimal control problem involving energy management of a hybrid-fuel Unmanned Aerial Vehicle (UAV). The planning problem for a hybrid-fuel platform involves determining the path while managing the energy resources, which includes a policy for power modality switching whenever applicable. The hybrid-fuel platform considered here involves a generator and battery pack combined in a series fashion as energy sources on-board a UAV. Also included in the problem are the noise restrictions, which place constraints on generator operation depending on the airspace location. These emulate possible restrictions on UAV noise that occur in military surveillance missions or in urban path planning, where the collective noise of many UAVs, some with combustion engines, may be restricted in certain areas or times of the day. We present a hybrid methodology which starts from an initial path and generator pattern obtained from a mixed integer linear program (MILP) solution. The generator pattern from the discrete solution is then refined in an optimal control framework with an objective to minimize fuel usage, while considering the nonlinear battery and generator dynamics and noise-restriction constraints. Optimal control problem is solved with a nonlinear program solver, IPOPT. Numerical results are presented and analyzed with varying path lengths and scenarios. This work aims to serve as an initial study of this hybrid-fuel UAV problem within an optimal control framework, which can be extended to refinement of both the generator pattern and the trajectory in tandem, while considering vehicle and power dynamics that are often ignored in discrete path planning solutions.

Analysis of an HDG method for the Navier–Stokes equations with Dirac measures

In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier–Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and H(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, a posteriori error estimator, that provides an upper bound for the L2-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient a posteriori error estimator for the W1,q-seminorm in the velocity and Lq-norm in the pressure is also proved. Finally, a Banach’s fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained a posteriori error estimators.

Историчность внешней политики ЕС: историко-культурные условия наднациональной субъектности

Importance. Unlike national subjectivity, European foreign policy is quite historical, that is, it is not mythologized and has a chronologically distinct beginning. How then is the continuity inherent in subjectivity achieved if supranational foreign policy can be represented as a series of discrete solutions? Based on the fact that any foreign policy is a locus of cultural relations, the historical and cultural conditions of European subjectivity were examined on the example of European cinema. Materials and methods. A cultural method of historical research was formulated; materials were selected that seamlessly relate to the European tradition of cinema. Results and Discussion. Some conditions of supranational subjectivity were highlighted – first of all, it is a transformative experience associated with war. Conclusion. Cinema shows what European foreign policy is made of in addition to discrete solutions – ideas and traditions cannot show this, whereas modern cultural forms allow it to reveal the location of supranational subjectivity in the historical and cultural dynamics of the 20th century.