Accurate Numerical Solutions Research Articles

Dispersion Analysis of CIP-FEM for the Helmholtz Equation

When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number increases, which is known as “pollution effect” and which is directly related to the phase difference between the exact and numerical solutions, caused by the numerical dispersion. In this paper, we propose a dispersion analysis for the continuous interior penalty finite element method (CIP-FEM) and derive an explicit formula of the penalty parameter for the th order CIP-FEM on tensor product (Cartesian) meshes, with which the phase difference is reduced from to . Extensive numerical tests show that the pollution error of the CIP-FE solution is also reduced by two orders in with the same penalty parameter.

Convergence behaviour of inverse differential quadrature method for analysis of beam and plate structures

Modelling of structures like beams and plates for advanced engineering applications is a tedious task considering the complexity of the kinematical assumptions governing their behaviour. A fundamental requirement in understanding the behaviour of these engineering structures is providing accurate numerical solutions to the high-order system of partial differential equations derived from the kinematic assumptions. To circumvent error accumulation arising from the numerical differentiation of high-order systems, the inverse differential quadrature method (iDQM) was recently proposed, and it is increasingly gaining attention due to its associated spectral convergence and numerical stability. To fully exploit the merits of iDQM for wider applications, this study establishes an understanding of sensitivity of static, buckling, and free vibration iDQM-based solutions of beam and plate structures to the choice of basis functions and grid distributions. Moreover, the computational complexity imposed by the full preconditioning required to transform the inherently underdetermined iDQM system to a square system suitable for eigenvalue analyses is addressed by proposing two novel subsystem preconditioning methods. From the ensuing comprehensive convergence analysis, it is shown that iDQM can be generalised for a specific class of polynomials that exhibit spectral convergence and negligible sensitivity to grid distributions. For other classes of polynomials that offer ill-conditioned coefficient matrices, iDQM can be applied to problems in which the desired accuracy can be achieved with few grid points. Finally, the study highlights the excellent computational gains offered by subsystem preconditioning methods for obtaining numerically stable eigenvalue solutions with high-order accuracy.

Solving Stochastic Nonlinear Poisson-Boltzmann Equations Using a Collocation Method Based on RBFs

In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying the collocation method based on radial basis functions (RBFs) allows us to deal with the difficulties arising from the complexity of the domain. To indicate the accuracy of the RBF method, we present numerical results for two-dimensional models, we also study the stability of this method numerically. We examine our results with the RBF-reference value and the Chebyshev Spectral Collocation (CSC) method. Furthermore, we discuss finding the appropriate shape parameter to obtain an accurate numerical solution besides greatest stability. We have exerted the Newton–Raphson approach for solving the system of non-linear equations resulting from discretization by the RBF technique.

A modern analytic method to solve singular and non-singular linear and non-linear differential equations

This article circumvents the Laplace transform to provide an analytical solution in a power series form for singular, non-singular, linear, and non-linear ordinary differential equations. It introduces a new analytical approach, the Laplace residual power series, which provides a powerful tool for obtaining accurate analytical and numerical solutions to these equations. It demonstrates the new approach’s effectiveness, accuracy, and applicability in several ordinary differential equations problem. The proposed technique shows the possibility of finding exact solutions when a pattern to the series solution obtained exists; otherwise, only rough estimates can be given. To ensure the accuracy of the generated results, we use three types of errors: actual, relative, and residual error. We compare our results with exact solutions to the problems discussed. We conclude that the current method is simple, easy, and effective in solving non-linear differential equations, considering that the obtained approximate series solutions are in closed form for the actual results. Finally, we would like to point out that both symbolic and numerical quantities are calculated using Mathematica software.

Thermocapillarity in Cross Hybrid Nanofilm Flow Past an Unsteady Stretching Sheet

The present work is highly interested in examining the transport phenomena of the thin Cross hybrid nanofluid film flow over a continuously stretching surface. The proposed thin film flow study elucidates the film extrusion process, which is prominent in the packaging industry. With the intention of improvising the quality of the coating process, the thermocapillarity and injection effects have been probed in the present model. A suitable similarity transformation and the MATLAB software aid in producing accurate numerical solutions. The accumulated numerical results indicate that an increment in the hybrid nanofluid viscosity and surface tension intensity reduces the wall shear stress past the permeable stretching sheet and improves the heat transfer rate. Remarkably, negative film thickness has been identified when the unsteadiness parameter is greater than or equal to 0.9 while the thermocapillarity parameter falls within the range of 0 and 0.6.

Soil salinity simulation based on electromagnetic induction and deep learning

In pursuit of fine modeling and forecasting of water-salt transport in arid zones, this study uses the rapid electromagnetic induction survey as the main tool to establish an empirical relationship between soil volumetric water content, salinity and electrical conductivity based on on-site measurement data. By using deep learning to set up a surrogate model for electromagnetic induction inverse problem, the vertical distribution of soil salinity during salt leaching at non-growth season is revealed. HYDRUS-1D was used to perform inversions of soil hydraulic parameters to obtain accurate numerical solutions. The results show that the surrogate of electromagnetic induction inversion using a deep convolutional encoding-decoding network (NET) with a learning rate of 0.0006 and 1500 sets of training samples has high accuracy, with an R2 of 0.9930 and RMSE of 0.2585 mg/cm3. The HYDRUS inverse modeling was performed using synthetic ECa data, water content and soil salinity observation data, and the combination of ECa and shallow ground (10 cm deep) water content respectively. It was shown that although the accuracy of the water-salt transport model obtained by using ECa data is slightly lower than that of the inversion model using traditional observation data, it can still accurately simulate the water-salt distribution and has the incomparable advantages of being fast, convenient and low-cost, and the accuracy of the model is improved to some extent by incorporating shallow ground water content data into the inversion based on ECa data. Therefore, the water-salt transport simulation framework incorporating soil apparent electrical conductivity proposed in this study can accurately simulate salt transport and can provide strategies and methods for the simulation and prediction of soil water-salt transport using electromagnetic induction technology.

The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies

Finite element analysis of complex bodies is frequently used in design to determine the size of deformations. Successive iterations, with progressive refinement of mesh densities, are most often required to obtain a sufficiently accurate convergent numerical solution. This process is costly, time consuming, and requires superior hardware and software. The paper presents a quick and effortless way to determine a sufficiently accurate value of the numerical solution. The mentioned solution is obtained by amending the numerical solution resulting for a certain value of the mesh density of the studied body with an adequate proportionality coefficient determined following the deformation study of simple bodies differently subject to external forces. It is assumed that the elastic displacement of the various bodies has a similar evolution as the mesh density increases and that the values of the proportionality coefficients considered are approximately equal for identical mesh densities. Examples presented are related to the reference body of the mechanical press PAI 25.

Deep Learning Discrete Calculus (DLDC): a family of discrete numerical methods by universal approximation for STEM education to frontier research

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through the lens of applied mathematics. The DLDC methods aim to leverage the flexibility and ever-increasing resources of deep learning and rich literature on numerical analysis to formulate a general class of numerical methods that can directly use data with uncertainty to predict the behavior of an unknown system as well as elevate the speed and accuracy of numerical solution of the governing equations for known systems. The article is structured into two major sections. In the first section, the building blocks of the DLDC methods are presented and deep learning structures analogous to traditional numerical methods such as finite difference and finite element methods are constructed with a view to incorporate these techniques in Science, Technology, Engineering, Mathematics syllabus for K-12 students. The second section builds upon the building blocks of the previous discussion and proposes new solution schemes for differential and integral equations pertinent to multiscale mechanics. Each section is accompanied by a mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples.

An accurate and efficient numerical solution for the generalized Burgers–Huxley equation via Taylor wavelets method: Qualitative analyses and Applications

This paper aims to propose a highly accurate and simple algorithm based on the Taylor wavelet methods for obtaining the approximate solution of the generalized Burgers–Huxley equation. Additionally, various qualitative analyses including positivity-preservation, monotonicity-preservation, boundedness of the obtained solutions as well as convergence analysis of the proposed method have been provided. Furthermore, the applicability and validity of the proposed method are demonstrated on a benchmark equation. By comparing the approximate solutions with the exact solution, it is observed that the proposed method has recorded better results than the other methods in the literature. Although the generalized Burgers–Huxley equation has been studied throughout the study, it is worth emphasizing that the proposed method is a good solver for such nonlinear equations.

Residual error based adaptive method with an optimal variable scaling parameter for RBF interpolation

In infinitely smooth Radial Basis Function (RBF) based interpolation, the scaling parameter plays an important role to obtain an accurate and stable numerical solution. When this method is applied to interpolate a function with sharp gradients, then adaptive methods will also play a significant role in determining an optimal number of centers according to the user desired accuracy. In this article, we test an optimization algorithm developed using the nonlinear optimization to find a scaling parameter for RBF along with an adaptive residual subsampling method [1] RBF interpolation. In this process, at each stage of adoption, the available optimal shape parameters have been obtained by solving the system of non-linear equations.

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

AbstractOwing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

Finite-element method based solver for an evaporating sessile droplet on a heated surface

We report the development of a finite-element-based solver to compute transport of mass, momentum and energy during evaporation of a sessile droplet on a heated surface. The evaporation is assumed to be quasi-steady and diffusion-limited. The heat transfer between the droplet and substrate and mass transfer of liquid-vapor are solved using a two-way coupling. In particular, here, we develop and implement the formulation of fluid flow inside the droplet in the model. The continuity and Navier–Stokes equations are solved in axisymmetric, cylindrical coordinates. Jump velocity boundary condition is applied on the liquid–gas interface using the evaporation mass flux. The governing equations are discretized in the framework of the Galerkin weight residual approach. A mesh of finite triangular elements with six nodes is utilized, and quadratic shape functions are used to obtain the second-order accurate numerical solution. Two formulations, namely, penalty function and velocity pressure, are employed to obtain discretized equations. The numerical results are the same using both methods, and the latter is around 30–50% faster than the former for the cases of refined grid. Computed flow fields are in excellent agreement with published results. The solver’s capability is demonstrated by solving the internal flow field for a case of a heated substrate.

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.

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the interference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials.

Techniques for Mixed Multiple Shooting for solve Stiff Initial Value Problems

The object of the research is to develop a number of techniques on the subject of multiple shooting for solving stiff initial value problems, and these techniques increase the accuracy of numerical solutions of stiff problems, as well as reduce the accumulation of errors, In addition to reducing the algebraic equations to get to the solution. We used mixed-multiple shooting in the development of these techniques.

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

Cathode Shape Design for Steady-State Electrochemical Machining

The inverse or cathode shape design problem of electrochemical machining (ECM) deals with the computation of the shape of the tool cathode required for producing a workpiece anode of a desired shape. This work applied the complex variable method and the continuous adjoint-based shape optimization method to solve the steady-state cathode shape design problem with anode shapes of different smoothnesses. An exact solution to the cathode shape design problem is proven to exist only in cases when the function describing the anode shape is analytic. The solution’s physical realizability is shown to depend on the aspect ratio of features on the anode surface and the width of the standard equilibrium front gap. In cases where an exact and physically realizable cathode shape exists, the continuous adjoint-based shape optimization method is shown to produce accurate numerical solutions; otherwise, the method produces cathode shapes with singularities. For the latter cases, the work demonstrates how perimeter regularization can be applied to compute smooth approximate cathode shapes suitable for producing workpieces within the range of manufacturing tolerance.

Axisymmetric plumes due to fluid injection through a small source in a wet porous medium

A small spherical source discharges a fluid into a porous medium that is already fully saturated with another fluid. The injected fluid has higher density than the ambient fluid, and so it forms a plume that moves downward under the effects of gravity. We present a simple asymptotic analysis assuming the two fluids do not mix that gives the width of the plume far from the source as a function of the injected volume flux. A spectral method is then developed for solving the full nonlinear problem in Boussinesq theory. Accurate numerical solutions are presented, which show in detail the evolution of the plume of heavier injected fluid over time. Close agreement with the asymptotic plume shape far from the source is demonstrated at later times.

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

A numerical scheme is developed to solve a large time delay two‐parameter singularly perturbed one‐dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece‐wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second‐order convergent in the spatial direction and second‐order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

The present work purveys the steady two-dimensional viscous incompressible radial flow of Au-water and Ag-water nanofluids between plane walls which either converge or diverge in the presence of MHD effect. A uniform magnetic field is applied. The governing partial differential equations of the present physics and their appropriate boundary conditions are initially cast into the dimensionless forms to reduce into the ordinary differential equation. The resulting equation thus formed is then solved by adopting the spectral linearisation method (SLM) to get the accurate numerical solution. Solution errors and residual norms were analysed to elaborate the convergence and accuracy of the numerical solution. The present results are validated with favourable comparisons to previously published results due to the current investigations' unique cases. Parametric study of the governing parameters, namely the magnetic field strength parameter M, Reynolds number Re, angle of inclination α, and the volume fraction parameter ɸ on the non-dimensional velocity is conducted. The analysis discloses that the profiles of the flow are largely impacted by the physical parameters.