Numerical Solution Scheme Research Articles

Algorithm for Construction of Quadrature Formulas with Exponential Convergence for Linear Operators Acting on Periodic Functions

We propose an algorithm for construction of quadrature formulas for linear operators acting on periodic functions. For analytic functions, the order of accuracy of quadrature formulas grows unboundedly with the growth of the number of the grid nodes. Within rather general restrictions on the kernels of linear operators, an exponential estimate of a quadrature formula is proved. To exemplify, we find quadrature formulas for the calculation of integral operators with logarithmic singularities which are used in the boundary element method to obtain overconvergent numerical schemes for solution of boundary value problems for harmonic and biharmonic equations on the plane.

A numerical method for solving hyperbolic partial differential equations with piecewise constant arguments and variable coefficients

This article deals with hyperbolic partial differential equations with piecewise constant arguments and variable coefficients. This study, therefore, with the aid of the finite difference technique, aims at presenting a numerical solution scheme for solving such types of equations. The stability, consistency, convergence, and convergence rate of our proposed numerical method are investigated. Moreover, the process of the computation of the analytical solution is studied. In order to support and confirm our theoretical results, some numerical examples are also presented. The figures of the numerical and analytical solutions and also the tables of errors are provided to demonstrate the validity of our proposed scheme.

Active Contour Model Using Fast Fourier Transformation for Salient Object Detection

The active contour model is a comprehensive research technique used for salient object detection. Most active contour models of saliency detection are developed in the context of natural scenes, and their role with synthetic and medical images is not well investigated. Existing active contour models perform efficiently in many complexities but facing challenges on synthetic and medical images due to the limited time like, precise automatic fitted contour and expensive initialization computational cost. Our intention is detecting automatic boundary of the object without re-initialization which further in evolution drive to extract salient object. For this, we propose a simple novel derivative of a numerical solution scheme, using fast Fourier transformation (FFT) in active contour (Snake) differential equations that has two major enhancements, namely it completely avoids the approximation of expansive spatial derivatives finite differences, and the regularization scheme can be generally extended more. Second, FFT is significantly faster compared to the traditional solution in spatial domain. Finally, this model practiced Fourier-force function to fit curves naturally and extract salient objects from the background. Compared with the state-of-the-art methods, the proposed method achieves at least a 3% increase of accuracy on three diverse set of images. Moreover, it runs very fast, and the average running time of the proposed methods is about one twelfth of the baseline.

Gauss optimization method for the dynamics of unilateral contact of rigid multibody systems

The discontinuous dynamical problem of multi-point contact and collision in multi-body system has always been a hot and difficult issue in this field. Based on the Gauss’ principle of least constraint, a unified optimization model for multibody system dynamics with multi-point contact and collision is established. The paper presents the study of the numerical solution scheme, in which particle swarm optimization method is used to deal with the corresponding optimization model. The article also presents the comparison of the Gauss optimization method (GOM) and the hybrid linear complementarity method (i.e. combining differential algebraic equations (DAEs) and linear complementarity problems (LCP)), commonly used to solve the dynamic contact problem of multibody systems with bilateral constraints. The results illustrate that, the GOM has the same advantage of dynamical modelling with LCP and when the redundant constraint exists, the GOM always has a unique solution and so no additional processing is needed, whereas the corresponding DAE-LCP method may have singular cases with multiple solutions or no solutions. Using numerical examples, the GOM is verified to effectively solve the dynamics of multibody systems with redundant unilateral and bilateral constraints without additional redundancy processing. The GOM can also be applied to the optimal control of systems in the future and combined with the parameter optimization of systems to handle dynamic problems. The work given provides the dynamics and control of the complex system with a new train of thought and method. The paper establishes a unified optimization model and the corresponding numerical solution scheme for multibody system dynamics with multi-point contact and collision based on Gauss’ principle of least constraint. Comparisons between Gauss optimization method (GOM) and the hybrid linear complementarity method (HLCM) are carried out through theoretical analysis and numerical examples, which show that GOM has the same advantage of dynamical modelling as HLCM and when redundant constraints exist, GOM has unique solutions without additional processes, whereas HLCM may have singular cases with multiple or no solutions.

Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish $\Gamma$-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.

Students' understanding of molar concentration

This article presents and discusses the results of a study that investigates university students’ comprehension of the concept of molar concentration, following teaching and evaluation of the subject. The specific problems underlying learning of this concept have not yet been focused on in sufficient detail or depth. A “Reasoning with molarity” questionnaire, which explores the logical–mathematical relationships between the variables n, V and M, was administered to 303 Argentine first-year university students. Results obtained from think-aloud interviews related to this questionnaire, which were performed with 18 students, are also analysed. These results reveal that approximately half the first-year university students have no clear conceptual understanding of molarity. The main difficulty arises in inverse proportionality tasks where the number of moles (extensive) and M (intensive) should be related qualitatively to determine which solution occupies the smallest volume. It was also established that much conceptual confusion, such as the lack of differentiation between n and M, lies behind the algorithmic, numerical solution strategy carried out exclusively by many students.

Mathematical modeling and dynamic characteristic analysis of a novel parallel tracking mechanism for inter-satellite link antenna

Inter-satellite link antenna tracking mechanism is an important enabling component of relay satellite for continuous real-time tracking of medium and low orbit target spacecraft. Facing the major demand of advanced technology development in aerospace engineering, this paper mainly tackles the problem on the rigid-flexible coupling dynamic modeling and dynamic characteristics analysis of a novel parallel tracking mechanism. Firstly, adopting a novel shape function of three-dimensional isoparametric beam element with curvature, the elasto-dynamic model is derived with floating coordinate method and Lagrangian equation. Then the natural frequencies of the system are investigated and ANSYS software is utilized to verify the model. An efficient numerical solution strategy consisting of the implicit integration algorithm and Baumgarte stabilization technology is employed to precisely evaluate the dynamic response of the system. Meanwhile, trajectory planning is carried out for high tracking precision. Furthermore, the validity of the numerical results of dynamic response is illustrated by simulations with the ADAMS software. Finally, the natural frequency sensitivity analysis is implemented by direct differential method with a novel sensitivity index to identify the structure weakness of the mechanism. A novel design idea for the cross-sectional parameters of each flexible beam is proposed with the sensitivity index. Above mentioned research results enrich the basic theoretical researches of parallel mechanism and lay a theoretical foundation for the practical application of tracking mechanism.

Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach

In this paper, a numerical solution strategy is proposed for studying the large deformations of rectangular plates made of hyperelastic materials in the compressible and nearly incompressible regimes. The plate is considered to be Mindlin-type, and material nonlinearities are captured based on the Neo-Hookean model. Based on the Euler–Lagrange description, the governing equations are derived using the minimum total potential energy principle. The tensor form of equations is replaced by a novel matrix–vector format for the computational aims. In the solution strategy, based on the variational differential quadrature technique, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing differential and integral matrix operators. Fast convergence rate, computational efficiency and simple implementation are advantages of this approach. The results are first validated with available data in the literature. Selected numerical results are then presented to investigate the nonlinear bending behavior of hyperelastic plates under various types of boundary conditions in the compressible and nearly incompressible regimes. The results reveal that the developed approach has a good performance to address the large deformation problem of hyperelastic plates in both regimes.

Numerical solution of stiff random ordinary differential equations via averaged schemes

Random ordinary differential equations (RODEs) are important differential equations that contain a stochastic process in their vector field functions. Examples of such stochastic processes are Brownian motion (i.e., Wiener process) and fractional Brownian motion. Taking a path from the stochastic process, these equations can be considered as ordinary differential equations, but numerical schemes for solving ordinary differential equations don't necessarily preserve their convergence order for these equations. In this paper, an implicit ϑ‐averaged scheme for numerical solution of RODEs is introduced. The proposed method involves averaging the noise inside the vector field and leads to integrals of the noise over discretization subintervals. Pathwise convergence and B‐stability of the above scheme are proved. Eventually, the desired averaged scheme is implemented on a medical model.

Towards real-time prediction of residual stresses induced by peripheral milling of Ti–6Al–4V

Residual stresses introduced into components in the course of manufacturing processes may considerably impair fatigue life and therefore the operational reliability and safety of the final product. Particularly in critical applications in the aerospace industry, where peripheral milling is a common surface finishing operation for components made from the titanium alloy Ti–6Al–4V, it is desirable to control the introduced residual stresses in the process while accounting for disturbance quantities such as tool wear. To this end, we propose a numerical scheme for the prediction of milling induced residual stresses that provides sufficient efficiency for real-time application. The scheme is based on a two-dimensional model where semi-analytical approaches from contact theory and thermoelasticity are combined with an approximate elasto-plastic solution technique based on an algorithm established in rolling contact mechanics to achieve the required performance in the plastic domain. Following the derivation of the numerical solution strategy, we turn our attention to the peripheral milling process under consideration and present predictions for the induced residual stresses along with a discussion of the general modeling approach, the major influencing factors on the predictions as well as efficiency aspects.

Comprehensive adaptive modelling of 1-D unsteady pipe network hydraulics

Multiple types of 1-D flow models simulate unsteady pipe network hydraulics. Water hammer models capture highly transient hydraulics, whereas incompressible flow formulations are more computationally efficient. Recent efforts to balance the accuracy–efficiency tension have been limited by numerical instability, model applicability, and inefficient solution schemes. Here, a novel adaptive hybrid transient model (AHTM) is presented that efficiently simulates the full range of unsteady flow conditions. The methodology combines the comprehensive global gradient algorithm (CGGA), a unique solver, with unsteady flow characterization indices and an adaptive scheme. Water hammer, rigid water column, and quasi-steady models are generalized to the CGGA within a unified framework; together with an adaptive scheme, the AHTM automatically adjusts the CGGA. Accordingly, dynamic effects are only simulated when present using an appropriately small time step. From two examples, the AHTM is shown to permit accurate and efficient simulations. Moreover, the framework is flexible and can be tailored to individual analyses.

A new numerical scheme based on Haar wavelets for the numerical solution of the Chen–Lee–Liu equation

This work based on a new numerical scheme for the numerical solutions of the Chen–Lee–Liu equation using Haar wavelet collocation method. In this work, we combined the backward Euler difference (BED) formula with the Haar wavelet collocation method (HWCM). The BED formula approximates the time derivative of the Chen–Lee–Liu equation, while the HWCM approximates the space derivatives of the Chen–Lee–Liu equation, consequently converting it into a finite system of algebraic equations. Furthermore, we validate the accuracy of the proposed method numerically and graphically with the help of several examples. Moreover, we have also presented an error analysis to prove the convergence of the proposed method.

Reliable Computer Simulation Methods for Electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation

Abstract The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it. The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems.

ψ-Haar wavelets method for numerically solving fractional differential equations

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.Design/methodology/approachAn operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems.FindingsThe error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples.Research limitations/implicationsThe method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity.Originality/valueEngineers and applied scientists may use the present method for solving fractional models appearing in applications.

Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure

We investigate how the observable relaxation behavior of an isolated quantum many-body system is modified in response to weak-to-moderate perturbations within a nonperturbative typicality framework. A key role is played by the so-called perturbation profile, which characterizes the dependence of the perturbation matrix elements in the eigenbasis of the unperturbed Hamiltonian on the difference of the corresponding energy eigenvalues. In particular, a banded matrix structure is quantitatively captured by a perturbation profile which approaches zero for large energy differences. The temporal modification of the relaxation is linked to the perturbation profile via a nonlinear integral equation, which admits approximate analytical solutions for sufficiently weak and strong perturbations, and for which we work out a numerical solution scheme in the general case. As an example, we consider a spin lattice model with a pronounced banded matrix structure, and we find very good agreement of the numerics with our analytical predictions without any free fit parameter.

Fully-coupled hydro-mechanical cracking using XFEM in 3D for application to complex flow in discontinuities including drainage system

The presence of uplift pressures in cracked plain concrete hydraulic structures is a major concern for their durability, serviceability and stability. To assess the performance of cracked structures several mechanical and hydraulic response parameters must be computed. This paper presents the development, implementation and application of a new nonlinear combined segregated fully-coupled hydromechanical model for application to non-planar 3D hydraulic fractures where complex flow is occurring. The eXtended Finite Element Method (XFEM) formulation is used, as it facilitates the computation of the crack aperture as well as the application of water pressure on crack surfaces for the simulation of hydraulic fracture initiation and propagation. To take into consideration the effects of drainage in numerical modeling and simulation, two coupled (multi-physics) finite element models are used; (1) one for the uplift pressures where a hydraulic mesh with refinement around drains is used, and (2) one for the mechanical response coupled with the computation of the unsaturated interstitial seepage problem. The two subproblems are solved using a partitioned procedure, as they have different resolution requirements. Finally, the mechanical mesh and the hydraulic mesh have non-matching discrete interfaces that must be coupled while respecting the equilibrium of the applied loads. Applications of the proposed hydromechanical constitutive model and numerical solution strategy to different problems are presented on a wedge splitting test and a full scale gravity dam including multiple drainage configurations. A case study adapted from a real dam is also given with a complex 3D non-planar discontinuity surface.

Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.

An advanced sampling technique for possibilistic uncertainty propagation

The ongoing demand for non-deterministic investigations of increasingly complex systems using possibility theory, and the closely connected fuzzy set theory, has led to the development of new numerical solution schemes. In this contribution, we present a sequential weighted sampling technique for the numerical solution of Zadeh’s extension principle which shows many advantages over conventional solution schemes in terms of flexibility and efficiency. The introduced solution strategy is applied to several test examples, among others from the field of structural mechanics, and its performance is compared with that of conventional sampling techniques.

A numerical methodology for simulation of non-Newtonian viscoelastic flows

The non-Newtonian fluids presenting viscoelastic flow behavior are found in many engineering applications. The development of a new numerical scheme for solution of this class of problems is the main goal of the present work. The proposed methodology adopts a second-order fully implicit finite difference approximation to discretize the convection and diffusion terms in the governing equations. Besides, the discretization is accomplished in a collocated mesh arrangement being used an Euler implicit pseudo-transient march in time aiming at steady-state solutions. Finally, it is worth mentioning that under-relaxation parameters are not needed, and the odd-even decoupling problem is avoided using artificial dissipations terms that are externally controlled by the user. The examples illustrating the application of the present method are: the non-Newtonian flows of viscoelastic materials in a plane channel and in a lid-driven cavity. The validation/verification performed indicates that the results are truly encouraging.

Simulation and Prediction of Storm Surges and Waves Using a Fully Integrated Process Model and a Parametric Cyclonic Wind Model

AbstractThis paper presents a fully integrated coastal process model and a simple parametric cyclonic wind‐pressure model for simulation of wind, storm surges, waves, tidal currents, and river flows. By sharing one computational grid within all those process modules and no need for switching executable codes from one module to another, this full‐coupling feature eliminates possible errors and loss of information due to interpolation and extrapolation of variables between different grids. To generate better cyclonic wind speed and barometric pressure, this parametric wind model includes nonlinear decay effect on wind intensity after hurricane's landfall. By implementing a new wind energy source term, the wave action model is capable of computing wave growth, propagation, and deformation through a regional‐scale domain from deepwater to shallow waters. Model validation and model skill assessment were performed by hindcasting wind, storm surges, waves, and river flows during Hurricane Gustav (2008) by using a high‐resolution grid covering the northern Gulf Coast. With improved wind fields estimated by the new parametric wind model, this fully integrated process model produced high‐quality wavefields in deep and shallow waters and storm tidal levels in the northern Gulf Coasts. Because of computing efficiency provided by seamless integration of process modules and optimized numerical solution schemes, faster‐than‐real‐time predictions of storm surges for the advisories during Hurricane Isaac (2012) were achieved by running the validated model in a desktop computer.