Discrete Schrodinger Operators Research Articles

Nonlocal hcp kernel functions based on ab initio calculations: Pertinent dislocation problems revisited

Eringen's nonlocal theory and an accurate determination of the nonlocal kernel functions for hexagonal close-packed (hcp) crystals are of interest. The kernel functions are closely related to the anisotropy as well as any crystalline symmetries. To this end, five new distinct nonlocal kernel functions which have the characteristics of discrete atomistic Green's functions in the stress space are obtained through consideration of the nonlocal dispersion relations associated with certain directions combined with ab initio Density Functional Perturbation Theory (DFPT) calculations of the pertinent phonon frequencies. This is the first work which provides the nonlocal hcp kernel functions pertinent to each component of the elastic moduli tensor of Eringen's nonlocal continuum theory based on quantum mechanical considerations. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameters associated with the hcp crystals of Mg, Cd, Ti, and Zn have been computed using ab initio Density Functional Theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a straight screw dislocation and a straight edge dislocation inside an infinitely extended hcp medium have been examined for the case where the dislocation lies in the basal plane. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. In contrast to the classical anisotropy theory that predicts singular shear stress on the slip plane at the core of the dislocation, the corresponding nonlocal shear stress component predicted by the present theory is zero at the dislocation core. It is noteworthy to mention that the normalized nonlocal shear stress distribution and its maximum value are greatly influenced by the anisotropy of the crystalline solids.

A series representation of the discrete fractional Laplace operator of arbitrary order

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Scalable Heat Transfer Characterization on Film Cooled Geometries Based on Discrete Green’s Functions

Abstract The aerothermal characterization of film-cooled geometries is traditionally performed at reduced temperature conditions, which then requires a debatable procedure to scale the convective heat transfer performance to engine conditions. This paper describes an alternative engine-scalable approach, based on Discrete Green’s Functions (DGF) to evaluate the convective heat flux along film-cooled geometries. The DGF method relies on the determination of a sensitivity matrix that accounts for the convective heat transfer propagation across the different elements in the domain. To characterize a given test article, the surface is discretized in multiple elements that are independently exposed to perturbations in heat flux to retrieve the sensitivity of adjacent elements, exploiting the linearized superposition. The local heat transfer augmentation on each segment of the domain is normalized by the exposed thermal conditions and the given heat input. The resulting DGF matrix becomes independent from the thermal boundary conditions, and the heat flux measurements can be scaled to any conditions given that Reynolds number, Mach number, and temperature ratios are maintained. The procedure is applied to two different geometries, a cantilever flat plate and a film-cooled flat plate with a 30 degree 0.125 in. cylindrical injection orifice with length-to-diameter ratio of 6. First, a numerical procedure is applied based on conjugate 3D unsteady Reynolds-averaged Navier–Stokes (URANS) simulations to assess the applicability and accuracy of this approach. Finally, experiments performed on a flat plate geometry are described to validate the method and its applicability. Wall-mounted thermocouples are used to monitor the surface temperature evolution, while a 10 kHz burst-mode laser is used to generate heat flux addition on each of the discretized elements of the DGF sensitivity matrix.

English

A special type of Jacobi matrices, discrete Schrodinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrodinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrodinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrodinger operators when their smallest and largest eigenvalues attain the extrema under certain constraints by use of the notion of generalized directional derivative and the method of Lagrange multiplier.

Conjugate Heat Transfer Analysis Using the Discrete Green's Function

Abstract Conjugate heat transfer problems generally require a coupled solution of the temperature fields in the fluid and solid domains. Implementing the boundary condition at the surface of the solid using a discrete Green's function (DGF) decouples the solutions. A DGF is determined first considering only the fluid domain with prescribed thermal boundary conditions, then the temperature distribution in the solid is calculated using standard numerical methods. The only compatibility requirement is that the DGF must be specified with the same discretization as the surface of the solid. The method is demonstrated for both steady-state and transient heating of a thin plate with laminar boundary layers flowing over both sides. The resulting set of linear algebraic equations for the steady-state problem or linear ordinary differential equations for the transient problem are easily solved using conventional scientific programming packages. The method converges with nearly second-order accuracy as the discretization resolution is increased.

Dataset Denoising Based on Manifold Assumption

Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the “true” geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem.

Schrödinger Operators Generated by Locally Constant Functions on the Fibonacci Subshift

We investigate the spectral properties of discrete one-dimensional Schrodinger operators whose potentials are generated by sampling along the elements of the Fibonacci subshift with a locally constant function. The fundamental trace map formalism for this model is presented and related to its spectral features via an extension of a multitude of works on the classical model, where the sampling function only depends on a single entry of the sequence.

An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

An efficient computational method to approximate the solution of a general class of nonlinear reaction-diffusion systems in Cartesian grids is presented. The proposed scheme uses the Local Discontinuous Galerkin (LDG) method as spatial discretization and the symmetric Strang operator splitting as time marching scheme. As a result, not only diffusion and reaction are decoupled; but the reaction sub-step reduces to a set of completely independent small local nonlinear systems of ordinary differential equations. To reduce their computational cost a Lagrange interpolatory technique is used. Diffusion sub-steps are approximated by a directional splitting method derived from an algebraic decomposition of the discrete Laplacian operator, which amounts to solve sets of completely independent small block tridiagonal linear systems; one set per direction. Stability of the diffusive step is analyzed. Finally the efficiency and accuracy of the proposed scheme is numerically investigated on well known Turing pattern formation models in 2D/3D.

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

AbstractA simple and efficient fourth‐order kernel‐free boundary integral method was recently proposed by Xie and Ying for constant coefficients elliptic PDEs on complex domains. This method is constructed by a compact finite difference scheme and works efficiently with fourth‐order accuracy in the maximum norm. But it is challenging to present the sharp error analysis of the resulting approach since the local truncation errors, at the irregular grid nodes near the interface, are only in the order of O(h3). The aim of this paper is to establish rigorous sharp error analysis. We prove that both the numerical solution and its gradient have fourth‐order accuracy in the discrete ℓ2‐norm, and the scheme has fourth‐order accuracy in the maximum norm based on the properties of discrete Green functions. Numerical examples are also provided to verify the error analysis.

Numerical study of anomalous diffusion of light in semicrystalline polymer structures

From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and energetics. Here we present a numerical study of anomalous diffusion of light through a semi-crystalline polymer structure where transport is guided by random disorder and nonlocal interactions. The numerical technique examines diffusion properties in one-dimensional (1D) space via the spectrum of an Anderson-type Hamiltonian with a discrete fractional Laplacian operator (-{\Delta})^s, 0<s<2 and a random distribution of disorder. The results show enhanced transport for s<1 (super-diffusion) and enhanced localization for s>1 (sub-diffusion) for most examined cases. An important finding of the present study is that transport can be enhanced at key spatial scales in the sub-diffusive case, where all states are normally expected to be localized for a (1D) disordered system.

Discrete Schrödinger Operator on a Tree, Angelesco Potentials, and Their Perturbations

We consider a class of discrete Schrodinger operators on an infinite homogeneous rooted tree. Potentials for these operators are given by the coefficients of recurrence relations satisfied on a multidimensional lattice by multiple orthogonal polynomials. For operators on a binary tree with potentials generated by multiple orthogonal polynomials with respect to systems of measures supported on disjoint intervals (Angelesco systems) and for compact perturbations of such operators, we show that the essential spectrum is equal to the union of the intervals supporting the orthogonality measures.

Solutions for discrete Schrödinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrödinger equations m(∑k∈Z(|Δuk−1|2+Vk|uk|2))(−Δ2uk−1+Vkuk)−ωuk=fk(uk)k∈Z,where m is a continuous function and V={Vk} is a positive potential, ω∈R, Δuk−1=uk−uk−1 and Δ2=Δ(Δ) is the one dimensional discrete Laplacian operator. Under some suitable assumptions on fk, we prove the existence of nontrivial solutions for this nonlocal problems by variation methods.

Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

We introduce the tensor numerical method for solution of the d-dimensional optimal control problems (d=2,3) with spectral fractional Laplacian type operators in constraints discretized on large n⊗d tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the spectral fractional d-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor grid are diagonalized by using the fast Fourier transform (FFT). Then the low rank approximation of the d-dimensional tensors obtained by folding of the corresponding large diagonal matrices of eigenvalues are computed, which allows to solve the governing equation for the control function in a tensor-structured format. The existence of low rank canonical approximation to the class of matrix valued functions involved is justified by using the sinc quadrature approximation method applied to the Laplace transform of the generating function. The linear system of equations for the control function is solved by the PCG iterative method with the rank truncation at each iteration step, where the low Kronecker rank preconditioner is pre-computed. The right-hand side, the solution vector, and the governing system matrix are maintained in the rank-structured tensor format which beneficially reduces the numerical cost to O(nlog⁡n), outperforming the standard FFT based methods of complexity O(n3log⁡n) for 3D case. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size n.

Discrete Laplace operator of 3-cochains

In this paper, we use the Nelson lemma to give a new proof for the essential self-adjointness of the discrete Laplace operator acting on 3-cochains, which we are defined in our previous paper [A. Baalal and K. Hatim, The discrete Laplacian of a 3-simplicial complex (2019), https://hal.archives-ouvertes.fr/hal-02105789 ]. Moreover, we establish on the infimum of the essential spectrum an upper bound.

A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space

Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

Open-Source Coprocessor for Integer Multiple Precision Arithmetic

This paper presents an open-source digital circuit of the coprocessor for an integer multiple-precision arithmetic (MPA). The purpose of this coprocessor is to support a central processing unit (CPU) by offloading computations requiring integer precision higher than 32/64 bits. The coprocessor is developed using the very high speed integrated circuit hardware description language (VHDL) as an intellectual property (IP) core. Therefore, it can be implemented within field programmable gate arrays (FPGAs) at various scales, e.g., within a system on chip (SoC), combining CPU cores and FPGA within a single chip as well as FPGA acceleration cards. The coprocessor handles integer numbers with precisions in the range 64 bits–32 kbits, with the limb size set to 64 bits. In our solution, the sign-magnitude representation is used to increase the efficiency of the multiplication operation as well as to provide compatibility with existing software libraries for MPA. The coprocessor is benchmarked in factorial ( n ! ), exponentiation ( n n ) and discrete Green’s function (DGF) computations on Xilinx Zynq-7000 SoC on TySOM-1 board from Aldec. In all benchmarks, the coprocessor demonstrates better runtimes than a CPU core (ARM Cortex A9) executing the same computations using a software MPA library. For sufficiently large input parameters, our coprocessor is up to three times faster when implemented in FPGA on SoC, rising to a factor of ten in DGF computations. The open-source coprocessor code is licensed under the Mozilla Public License.

The Discrete Green's Function for Convective Heat Transfer—Part 2: Semi-Analytical Estimates of Boundary Layer Discrete Green's Function

Abstract This is the second paper in a set that defines the discrete Green's function (DGF). This paper focuses first on the turbulent boundary layer and presents two different methods to estimate the DGF. The long-element formulation defines the DGF with just two simple algebraic equations, but it is not quantitatively accurate for short element lengths. A short element correction is derived, but must be recalculated for each selection of flow parameters and element lengths. A similarity solution is derived that allows accurate estimates of the DGF diagonal elements for laminar boundary layers and for turbulent boundary layers discretized with short element lengths. To illustrate other methods to derive DGFs in more complex flows, a low-resolution DGF for laminar stagnation line boundary layers is determined using the skin-friction formulation combined with similarity solutions for two different thermal boundary conditions. Stagnation line flow is shown to be highly sensitive to the thermal boundary condition, and this can be analyzed effectively using the DGF.

The Discrete Green's Function for Convective Heat Transfer—Part 1: Definition and Physical Understanding

Abstract The discrete Green's function (DGF) is a superposition-based descriptor of the relationship between the surface temperature and the convective heat transfer from a surface. The surface is discretized into a finite number of elements and the DGF matrix elements relate the heat transfer out of any element i to the temperature rise on every element j of the surface. For a given flow situation, the DGF is insensitive to the thermal boundary condition so it allows direct calculation of the heat transfer for any temperature distribution and noniterative solution of conjugate heat transfer problems. The diagonal elements of the matrix are determined solely by the local velocity field while the off-diagonals are determined by the spread of the thermal wake downstream of a heated element. An analytical DGF for the laminar flat-plate boundary layers is included as an example.

The existence and location of eigenvalues of the one-particle discrete Schrodinger operator on a lattice

The existence and location of eigenvalues of the one-particle discrete Schrodinger operator on a lattice

Threshold Analysis of the Three Dimensional Lattice Schrödinger Operator with Non-Local Potential

We consider a family of the discrete Schrodinger operators $$H_{\lambda\mu}$$ , depending on parameters, in the $$3$$ -dimensional lattice, $$\mathbb{Z}^{3}$$ with a non-local potential constructed via the Dirac delta function and the shift operator. The existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator are explicitly derived. The threshold eigenvalue is proven to be absorbed into the essential spectrum and it turns into an embedded eigenvalue at the left intercept of a particular parabola, and the threshold resonance at the other points of the parabola.