Discrete Solution Research Articles

Buckling of simply-supported rectangular Double–Double laminates

The family of Double–Double (DD) laminates is in focus of the present study. Well established buckling-cases for rectangular plates are examined in this article from a DD perspective. It is shown that the DD conventions allow for beneficial reformulation of the available equations, which leads to drastic simplification. It is demonstrated that the lightest DD laminate can directly be determined for a specified buckling load. Permutation discussions and the corresponding evaluation of thousands of discrete solutions, known for conventional laminates, become obsolete. The developed DD-buckling equations are further examined from an invariant-based perspective using IQ=Q11+Q22+Q66+Q12 which reveals the important role of the 1/IQ3 term for minimum-laminate-thickness calculations.

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

Discrete Breathers of Nonlinear Dimer Lattices: Bridging the Anti-continuous and Continuous Limits

In this work, we study the dynamics of an infinite array of nonlinear dimer oscillators which are linearly coupled as in the classical model of Su, Schrieffer and Heeger (SSH). The ratio of in-cell and out-of-cell couplings of the SSH model defines distinct $\textit{phases}$: topologically trivial and topologically non-trivial. We first consider the case of weak out-of-cell coupling, corresponding to the topologically trivial regime for linear SSH; for any prescribed isolated dimer frequency, $\omega_b$, which satisfies non-resonance and non-degeneracy assumptions, we prove that there are discrete breather solutions for sufficiently small values of the out-of-cell coupling parameter. These states are $2\pi/\omega_b$- periodic in time and exponentially localized in space. We then study the global continuation with respect to this coupling parameter. We first consider the case where $\omega_b$, the seeding discrete breather frequency, is in the (coupling dependent) phonon gap of the underlying linear infinite array. As the coupling is increased, the phonon gap decreases in width and tends to a point (at which the topological transition for linear SSH occurs). In this limit, the spatial scale of the discrete breather grows and its amplitude decreases, indicating the weakly nonlinear long wave regime. Asymptotic analysis shows that in this regime the discrete breather envelope is determined by a vector gap soliton of the limiting envelope equations. We use the envelope theory to describe discrete breathers for SSH- coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry, topologically nontrivial regimes, when the spectral gap is small. Our asymptotic theory shows excellent agreement with extensive numerical simulations over a wide range of parameters.

Efficient Secure Computation from SM Series Cryptography

The wireless network suffers from many security problems, and computation in a wireless network environment may fail to preserve privacy as well as correctness when the adversaries conduct attacks through backdoors, steganography, kleptography, etc. Secure computation ensures the execution security in such an environment, and compared with computation on the plaintext, the performance of secure computation is bounded by the underlying cryptographic algorithms and the network environment between the involved parties. Besides, the Chinese cryptography laws require the cryptographic algorithms that appeared in the commercial market to be authorized. In this work, we show how to implement oblivious transfer (OT), an important primitive in secure multiparty computation (MPC), using the Chinese government-approved SM2 and SM3 algorithms. The SM2 algorithm is based on the elliptic curve cryptography and is much faster than the discrete logarithm-based solutions. Moreover, by adopting the standard OT extension technique, we can extend the number of OTs efficiently with one more round of communication and invocations to the SM3 and SM4 algorithms. The OT primitive can be used in the Beaver multiplication triple generation and other MPC protocols, e.g., private set intersection. Therefore, we can utilize the SM series cryptography, specifically, the SM2, SM3, and SM4 algorithms, to build highly efficient secure computation frameworks which are suitable for the wireless network environment and for commercial applications in China. The experimental evaluation results show that our protocols have comparable performance to existing protocols; specifically, our protocols are quite suitable for bad network environments.

Analysis of a fully-discrete, non-conforming approximation of evolution equations and applications

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady [Formula: see text]-Navier–Stokes problem. The results of some numerical experiments are reported in the final section.

Integrable aspects, analytic solutions and their asymptotic analysis for a discrete relativistic Toda lattice system

In this paper, we investigate a discrete relativistic Toda lattice (mathrm{dRTL}_{+}(alpha )) system, which may describe particle vibrations in lattices with an exponential interaction force. First, we construct its discrete generalized (m,2N-m)-fold Darboux transformation, from which we can explicitly give its analytic solutions, such as discrete multi-soliton solutions, position controllable rational and semi-rational solutions and their hyperbolic-and-rational mixed solutions, whose properties and dynamics are analyzed and shown graphically. Second, the asymptotic behaviors of diverse exact solutions are analyzed, which shows that the interactions among different solutions are always elastic. In particular, the position of controllable rational solutions and asymptotic state analysis of discrete hyperbolic-and-rational mixed solutions are obtained and discussed for the first time. Finally, we study some integrable properties of this system, such as the integrable hierarchy and relevant Hamiltonian structures and conservation laws from a discrete spectral problem. These results may be helpful for understanding nonlinear lattice dynamics.

Virtual element method for linear elastic clamped plate model

In this paper, we provide the conforming virtual element method for a linear elastic clamped plate model for the first time. Based on the different regularities of the three components of the displacement vector field, we use H1-conforming virtual element to approximate the first two displacement components, and H2-conforming virtual element to approximate the third displacement component. Therefore, we establish the virtual element discrete scheme for the two-dimensional linear elastic clamped plate problem. Then, the existence and uniqueness of the discrete solution are proved and that the discrete solution can reach any order of convergence. Finally, the numerical experiments of circular plate and rectangular plate under different mesh division prove that the numerical method is convergent and stable.

Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains

We consider the problem of domain approximation in finite element methods for Maxwell equations on general curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest and an exact parametrization of the surface may not be readily available. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact mesh. We deduce conditions on the quality of these approximations that ensure rates of error convergence between discrete solutions—in the approximate domains—to the continuous one in the original domain. Moreover, we present numerical results validating our claims.

Divergence-Conforming Velocity and Vorticity Approximations for Incompressible Fluids Obtained with Minimal Facet Coupling

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

A hybridizable discontinuous Galerkin method for the fully coupled time-dependent Stokes/Darcy-transport problem

We present a high-order hybridized discontinuous Galerkin (HDG) method for the fully coupled time-dependent Stokes–Darcy-transport problem where the fluid viscosity and source/sink terms depend on the concentration and the dispersion/diffusion tensor depends on the fluid velocity. This HDG method is such that the discrete flow equations are compatible with the discrete transport equation. Furthermore, the HDG method guarantees strong mass conservation in the Hdiv sense and naturally treats the interface conditions between the Stokes and Darcy regions via facet variables. We employ a linearizing decoupling strategy where the Stokes/Darcy and the transport equations are solved sequentially by time-lagging the concentration. We prove well-posedness and optimal a priori error estimates for the velocity and the concentration in the energy norm. We present numerical examples that respect compatibility of the flow and transport discretizations and demonstrate that the discrete solution is robust with respect to the problem parameters.

Analysis study of the bee algorithms as a mechanism for solving combinatorial problems

Combinatorial optimization problems are problems that have a large number of discrete solutions and a cost function for evaluating those solutions in comparison to one another. With the vital need of solving the combinatorial problem, several research efforts have been concentrated on the biological entities behaviors to utilize such behaviors in population-based metaheuristic. This paper presents bee colony algorithms which is one of the sophisticated biological nature life. A brief detail of the nature of bee life has been presented with further classification of its behaviors. Furthermore, an illustration of the algorithms that have been derived from bee colony which are bee colony optimization, and artificial bee colony. Finally, a comparative analysis has been conducted between these algorithms according to the results of the traveling salesman problem solution. Where the bee colony optimization (BCO) rendered the best performance in terms of computing time and results.

Efficient time-stepping schemes for the Navier-Stokes-Nernst-Planck-Poisson equations

We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable approach. By introducing a dynamic equation for the auxiliary variable and reformulating the original equations into an equivalent system, we construct first- and second-order semi-implicit linearized schemes for the underlying problem. The main contribution of the paper includes: (1) the schemes are unconditionally stable in the sense that a discrete energy keeps decay during the time stepping; (2) the concentration components of the discrete solution preserve positivity and mass conservation; (3) the delicate implementation shows that the proposed schemes can be very efficiently realized, with computational complexity close to a semi-implicit scheme. As far as the best we know, it is the first time for constructing a second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method.

Adaptive partitioning-based discrete unified gas kinetic scheme for flows in all flow regimes

To improve the efficiency of the discrete unified gas kinetic scheme (DUGKS) in capturing cross-scale flow physics, an adaptive partitioning-based discrete unified gas kinetic scheme (ADUGKS) is developed in this work. The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation, which contains the initial distribution function and the local equilibrium state. The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes. When the contribution of the initial distribution function is negative, the local flow field can be regarded as the continuous flow and the Navier–Stokes (N-S) equations can be used to obtain the solution directly. Otherwise, the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefaction effect. Given this, in the ADUGKS, the computational domain is divided into the DUGKS cell and the N-S cell based on the contribution of the initial distribution function to the calculation of free streaming fluxes. In the N-S cell, the local flow field is evolved by solving the N-S equations, while in the DUGKS cell, both the discrete velocity Boltzmann equation and the corresponding macroscopic governing equations are solved by a modified DUGKS. Since more and more cells turn into the N-S cell with the decrease of the Knudsen number, a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.

Airfoil trailing-edge noise prediction based on CFD/CAA coupled boundary method

Flow-induced noise hybrid prediction method based on sound wave boundary condition is an effective numerical method to deal with a flow-induced noise problems of complex configuration. This method can not only finely simulate the sound source, but also analyze the noise propagation characteristics. In this paper, numerical methods such as space-time discrete scheme and boundary conditions involved in discrete solution of acoustic perturbation equations(APE) are developed, and strategies such as adding buffer layer are proposed to improve divergence caused by numerical discontinuity at the junction of sound source region and propagation region, so as to establish a high-precision flow-induced noise hybrid method. In order to study the rationality of flow-induced noise hybrid method, the numerical simulation of acoustic radiation at the trailing edge of NACA0012 airfoil in BANC experiment was carried out. Compared with the results of sound field solved by large eddy simulation method, the analysis shows that the high-precision flow-induced noise hybrid method based on sound wave boundary conditions can predict the spatial sound field distribution with high precision and directly reflect the interaction between sound waves. Compared with the results of large eddy simulation, the error of sound pressure level is less than 2 dB, and the noise directional distribution trend is consistent.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods.

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

AbstractThe Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

A comparative analysis of continuum plasticity, viscoplasticity and phase-field models for earthquake sequence modeling

This paper discusses continuum models for simulating earthquake sequences on faults governed by rate-and-state dependent friction. Through detailed numerical analysis of a conventional strike-slip fault, new observations regarding the use of various continuum earthquake models are presented. We update a recently proposed plasticity-based model using a consistently linearized formulation, show its agreement with discrete fault models for fault thicknesses of hundreds of meters, and demonstrate mesh objectivity for slip-related variables. To obtain a fully regularized fault width description with an internal length scale, we study the performance and mesh convergence of a plasticity-based model complemented by a Kelvin viscosity term and the phase-field approach to cohesive fracture. The Kelvin viscoplasticity-based model can introduce an internal length scale and a mesh-objective response. However, on grid sizes down to meters, this only holds for very high Kelvin viscosities that inhibit seismic slip rates, which renders this approach impractical for simulating earthquake sequences. On the other hand, our phase-field implementation for earthquake sequences provides a numerically robust framework that agrees with a discrete reference solution, is mesh objective, and reaches seismic slip rates. The model, unsurprisingly, requires highly refined grids around the fault zones to reproduce results close to a discrete model. Following this line, the effect of an internal length scale parameter on the phase-field predictions and mesh convergence are discussed.

Well‐posedness for a diffusion–reaction compartmental model simulating the spread of COVID‐19

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.

Numerical Analysis of a Swelling Poro-Thermoelastic Problem with Second Sound

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter.

The existence of discrete solitons for the discrete coupled nonlinear Schrödinger system

In this paper, we investigate the nonlinear coupled discrete Schrödinger equations with unbounded potentials. We find simple sufficient conditions for the existence of discrete soliton solution by using the Nehari manifold approach and the compact embedding theorem. Furthermore, by comparing the value of the action functional at the discrete soliton solution with those at nonzero solutions of one component zero, we demonstrate that both components of the discrete soliton solution are nontrivial.