Newton Linearization Research Articles

H(div)-conforming and discontinuous Galerkin approach for Herschel–Bulkley flow with density-dependent viscosity and yield stress

This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

Developing computational methods of heat flow using bioheat equation enhancing skin thermal modeling efficiency

PurposeThe main objective of this paper is to investigate the response of human skin to an intense temperature drop at the surface. In addition, this paper aims to evaluate the efficiency of finite difference and finite volume methods in solving the highly nonlinear form of Pennes’ bioheat equation.Design/methodology/approachOne-dimensional linear and nonlinear forms of Pennes’ bioheat equation with uniform grids were used to study the behavior of human skin. The specific heat capacity, thermal conductivity and blood perfusion rate were assumed to be linear functions of temperature. The nonlinear form of the bioheat equation was solved using the Newton linearization method for the finite difference method and the Picard linearization method for the finite volume method. The algorithms were validated by comparing the results from both methods.FindingsThe study demonstrated the capacity of both finite difference and finite volume methods to solve the one-dimensional and highly nonlinear form of the bioheat equation. The investigation of human skin’s thermal behavior indicated that thermal conductivity and blood perfusion rate are the most effective properties in mitigating a surface temperature drop, while specific heat capacity has a lesser impact and can be considered constant.Originality/valueThis paper modeled the transient heat distribution within human skin in a one-dimensional manner, using temperate-dependent physical properties. The nonlinear equation was solved with two numerical methods to ensure the validity of the results, despite the complexity of the formulation. The findings of this study can help in understanding the behavior of human skin under extreme temperature conditions, which can be beneficial in various fields, including medical and engineering.

Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal–dual gap estimators

We consider nonsmooth partial differential equations associated with a minimization of an energy functional. We adaptively regularize the nonsmooth nonlinearity so as to be able to apply the usual Newton linearization, which is not always possible otherwise. We apply the finite element method as a discretization. We focus on the choice of the regularization parameter and adjust it on the basis of an a posteriori error estimate for the difference of energies of the exact and approximate solutions. Importantly, our estimates distinguish the different error components, namely those of regularization, linearization, and discretization. This leads to an algorithm that steers the overall procedure by adaptive stopping criteria with parameters for the regularization, linearization, and discretization levels. We prove guaranteed upper bounds for the energy difference and discuss the robustness of the estimates with respect to the magnitude of the nonlinearity when the stopping criteria are satisfied. Numerical results illustrate the theoretical developments.

Analysis of mixed convecto-magnetic Buongiorno nanofluid flow in a non-Darcy porous medium

Analysis of nanofluid flow phenomena through a stretching sheet in a non-Darcy porous medium has great importance in enhancing transport processes in energy systems like hybrid fuel cell technology, glass blowing, etc. The current study looks at two-dimensional flow over a convectively heated linear stretching sheet in presence of a magnetic field. The boundary layer flow caused by a sheet that is linearly stretched has been investigated numerically. Employing the appropriate similarity transformation, the governing partial differential equations, which define the flow regime, are converted into a set of ordinary differential equations. The mathematical calculations are carried out using a finite difference algorithm with the aid of Newton's linearization method, which allows us to handle non-linear terms very smoothly. The outcomes of this work have been demonstrated are based on eight parameters, such as Prandtl number ( Pr ) , Lewis number ( Le ) , Brownian motion parameter ( N b ) , thermophoresis parameter ( N t ) , magnetic parameter ( M ) , porosity parameter ( K ) , Frochemier number ( F r 1 ) , and convection Biot number ( Bi ) . The impacts of aforementioned parameters on thermal and concentration boundary layers are depicted graphically. It has been found that the heat transfer in the boundary layer rises as N t enhances, and deposits aggravating particle away from the fluid region and increases the volume percentage of nanoparticles. Moreover, it has been found that as the Le number rises, the concentration profiles become steeper and the species border layer becomes thinner. Furthermore, it has been encountered that the reduced Sherwoord number rises as Le number increases for N t < 0.1 but decreases as Le number increases for N t > 0.1 .

Nonsymmetric interior penalty Galerkin method for nonlinear time-fractional integro-partial differential equations

This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial–boundary-value problems. To tackle the non-linearity of the problem, we first use the Newton linearization process. Then we apply the non-symmetric interior penalty Galerkin method for the spatial variable, and obtain a semi-discrete problem in the time variable. Finally, to obtain the fully-discrete scheme, we use both L1-scheme and L2-scheme for time-fractional derivative, and trapezoidal rule for integral term in the semi-discrete problem. We derive L2-norm stability and error estimates. The validity of the error analysis is supported by numerical experiments.

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

AbstractThis article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history‐dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time‐independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non‐intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.

Superconvergence analysis of nonconforming finite element method for two-dimensional time-fractional Allen–Cahn equation

In this paper, we investigate the numerical solution of the time-fractional Allen–Cahn equation with variable diffusion coefficients. A fully discrete approximation is presented by the L2-1σ scheme on graded meshes and spatial high precision nonconforming finite element method (FEM). The nonlinear term is treated with the Newton linearization method. Based on the modified discrete fractional Grönwall inequality, we rigorously prove that the proposed scheme can achieve optimal convergence accuracy in both time and space directions. Furthermore, the H1-norm global superconvergence result is obtained through the interpolation postprocessing technique. Finally, numerical experiments confirm the correctness of our theoretical analysis.

Barycentric rational interpolation method for solving KPP equation

&lt;abstract&gt;&lt;p&gt;In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving the KPP equation is also proved. At last, two examples are given to prove the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation

&lt;abstract&gt;&lt;p&gt;The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Analytical and numerical solutions of time-fractional advection-diffusion-reaction equation

The analytical and numerical solutions of non-autonomous time-fractional partial advection-diffusion-reaction equations are the key topics of this article. Here, by using the Sumudu decomposition method and the maximum-minimum principle we establish the existence and uniqueness of the analytical solution to these problems. Further, we propose a computational scheme to obtain the numerical solution of the fractional diffusion equation. To obtain the numerical solution, first, we discretize the time domain by a graded mesh, and by using the L1−scheme we semi-discretize the fractional time derivative, and study the stability and convergence of the method. Later, by using the cubic spline method over a uniform mesh, we approximate the spatial derivative and obtain the fully discrete scheme. As a result of our convergence study, we have obtained second-order error estimates in the spatial variable and (2−β)-order with regard to the temporal variable. Also, we have applied the proposed method to solve semilinear problems after linearizing by the Newton linearization process. Numerical experiments are carried out to validate the proposed method, and the outcomes are compared with those obtained using earlier techniques described in the literature.

Robust and efficient primal-dual Newton-Krylov solvers for viscous-plastic sea-ice models

We present a Newton-Krylov solver for a viscous-plastic sea-ice model. This constitutive relation is commonly used in climate models to describe the material properties of sea ice. Due to the strong nonlinearity introduced by the material law in the momentum equation, the development of fast, robust and scalable solvers is still a substantial challenge. In this paper, we propose a novel primal-dual Newton linearization for the implicitly-in-time discretized momentum equation. Compared to existing methods, it converges faster and more robustly with respect to mesh refinement, and thus enables numerically converged sea-ice simulations at high resolutions. Combined with an algebraic multigrid-preconditioned Krylov method for the linearized systems, which contain strongly varying coefficients, the resulting solver scales well and can be used in parallel. We present experiments for two challenging test problems and study solver performance for problems with up to 8.4 million spatial unknowns.

Approximate analytical investigation of large-amplitude rolling motion of ships with the improved Galerkin method

In this study, the large-amplitude rolling motion of a ship was studied. First, the harmonic balance method was used to construct a reasonable initial approximate solution. Subsequently, the Newton linearization and Galerkin method were combined to obtain the analytical approximate solution of the ship's large-amplitude roll motion. By comparing with the numerical solution, the accuracy of the analytical approximation solution was verified. The results showed that the third-order analytical approximation had a higher accuracy for small and large amplitudes. Based on the analytical approximate solution, the influence of nonlinear quadratic damping and nonlinear restoring moment on the global stability of the ship under different amplitudes was analyzed. The theoretical results can guide the design and stability analysis of large rolling motions of ships.

Newton Linearization of the Curvature Operator in Structured Grid Generation with Sample Solutions

Elliptic grid generation equations based on the Laplacian operator have the well-known property of clustering the mesh near convex boundaries and declustering it near concave boundaries. In prior work, a new differential operator was derived and presented to address this issue. This new operator retains the strong smoothing properties of the Laplacian without the latter’s adverse curvature effects. However, the new operator exhibits slower convergence properties than the Laplacian, which can lead to increased turnaround times and in some cases preclude the achievement of convergence to machine accuracy. In the work presented here, a Newton linearization of the new operator is presented, with the objective of achieving more robust convergence properties. Sample solutions are presented by evaluating a number of solvers and preconditioners and assessing the convergence properties of the solution process. The efficiency of each solution method is demonstrated with applications to two-dimensional airfoil meshes.

Barycentric Rational Collocation Method for Nonlinear Heat Conduction Equation

Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.

The Mass-Lumped Midpoint Scheme for Computational Micromagnetics: Newton Linearization and Application to Magnetic Skyrmion Dynamics

Abstract We discuss a mass-lumped midpoint scheme for the numerical approximation of the Landau–Lifshitz–Gilbert equation, which models the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic field contributions, our setting covers the non-standard Dzyaloshinskii–Moriya interaction, which is the essential ingredient for the enucleation and stabilization of magnetic skyrmions. Our analysis also includes the inexact solution of the arising nonlinear systems, for which we discuss both a constraint-preserving fixed-point solver from the literature and a novel approach based on the Newton method. We numerically compare the two linearization techniques and show that the Newton solver leads to a considerably lower number of nonlinear iterations. Moreover, in a numerical study on magnetic skyrmions, we demonstrate that, for magnetization dynamics that are very sensitive to energy perturbations, the midpoint scheme, due to its conservation properties, is superior to the dissipative tangent plane schemes from the literature.

Adaptively switched time stepping scheme for direct aeroacoustic computations

The physical variables involved in a flow field are of very different scales to those of the aeroacoustic field within the flow. Thus, direct aeroacoustic computations (DACs) require extremely small time steps to capture the fluctuations in the aeroacoustic field and stabilize the numerical scheme. As a result, the computation time for DACs can be very long. This paper describes an implicit Adaptively Switched Time Stepping (ASTS) scheme that enables the use of larger time steps when solving DAC problems. ASTS is based on the adaptive selection of the time stepping scheme according to the value of the residual. Early in the iteration process, an artificial time term is used to afford larger time steps while ensuring that the numerical scheme remains stable. Once the residual is less than some preset reference value, the artificial time term is removed, thus eliminating the Newton linearization error. Simulation results show that ASTS can reduce the computational cost of DACs by a factor of more than eight while maintaining the required level of accuracy.

A SUBGRID STABILIZED METHOD FOR NAVIER-STOKES EQUATIONS WITH NONLINEAR SLIP BOUNDARY CONDITIONS

In this paper, we consider a subgrid stabilized Oseen iterative method for the Navier-Stokes equations with nonlinear slip boundary conditions and high Reynolds number. We provide one-level and two-level schemes based on this stability algorithm. The two-level schemes involve solving a subgrid stabilized nonlinear coarse mesh inequality system by applying m Oseen iterations, and a standard one-step Newton linearization problems without stabilization on the fine mesh. We analyze the stability of the proposed algorithm and provide error estimates and parameter scalings. Numerical examples are given to confirm our theoretical findings.

Keller-Box Scheme to Mixed Convection Flow Over a Solid Sphere with the Effect of MHD

Mixed convection is the combination of a free convection caused by the buoyancy forces due to the different density and a forced convection due to external forces that increase the heat exchange rate. This means that, in free convection, the effect of external forces is significant besides buoyancy forces. In this study the fluid type with viscoelastic effect is non-Newtonian. The viscoelastic fluids that pass over a surface of a sphere form a thin layer, which due to their dominant viscosity is called by the border layer. The obtained limiting layer is analyzed with the thickness of the boundary layer- near the lower stagnating point, then obtained dimensional boundary layer equations, continuity, momentum, and energy equations. These dimensional boundary layer equations are then transformed into non-dimensional boundary layer equations by using non-dimensional variables. Further, the non-dimensional boundary layer equations are transformed into ordinary differential equations by using stream function, so that obtained the non-similar boundary layer equations. These non-similar boundary layer equations are solved numerically by using finite difference method of Keller-Box. The discretization results are non-linear and it should be linearized using newton linearization technique. The numerical solutions are analyzed the effect of Prandtl number, viscoelastic, mixed convection, and MHD parameters towards velocity profile, temperature profile, and wall temperature.

Thermal radiation effect on unsteady mixed convection boundary layer flow and heat transfer of nanofluid over permeable stretching surface through porous medium in the presence of heat generation.

Here, we study the effect of mixed convection and thermal radiation on unsteady boundary layer of heat transfer and nanofluid flow over permeable moving surface through a porous medium. The effect of heat generation is also discussed. The equations governing the system are the continuity equation, momentum equation and the heat transfer equation. These governing equations transformed into a system of nondimensional equations contain many physical parameters that describe the study. The transformed equations are solved numerically using an implicit finite difference technique with Newton's linearization method. The thermo-physical parameters describe the study are the mixed convection parameter α, , the Radiation parameter Rd, , porous medium parameter k, , the nanoparticles volume ,, the suction or injection parameter fw, , the unsteadiness parameter At, and the heat source parameter λ = 0.5 .The influence of the thermo-physical parameters is obtained analytically and displayed graphically. Comparisons of some special cases of the present study are performed with previously published studies and a good agreement is obtained.

Multi-level mixed finite element algorithms for the stationary incompressible magneto-hydrodynamics equations

In this article, three multi-level mixed finite element algorithms for the stationary incompressible magneto-hydrodynamics (MHD) equations are proposed and analyzed. Combining multi-level technique with mixed finite element method, three kinds of multi-level mixed finite element algorithms based on the Stokes linearization, Oseen linearization and Newton linearization are established. Stability and optimal convergence rate for the multi-level mixed finite element algorithms are derived. Finally, performance of the three designed algorithms is investigated by some numerical tests.