Stable Numerical Solution Research Articles

Two high-precision compact schemes for the dissipative symmetric regular long wave (SRLW) equation by multiple varying bounds integral method

This paper mainly focuses on the numerical study of fourth-order nonlinear dissipative symmetric regular long wave equation. We propose two new methods: the Multiple Varying Bounds Integral (MVBI) method and Taylor Function Fitted (TFF) method. With the multiple varying bounds integral method, all the derivatives in the space direction of the differential equation can be eliminated and we can get different numerical formats by adjusting the integral bound parameters. According to the physical properties of the original differential equation, we can choose an appropriate format from them. Meanwhile, with the Taylor function fitted method, the derivatives of the function at one point, such as first-order and second-order, can be approximated by the original function value at the points around it. Hence, with the MVBI method and TFF method, we can establish two compact and high-precision numerical schemes. In addition, we prove that these numerical schemes are consistent with the original equation on the energy property. Next, the convergence and stability of numerical solution U and P̃ are both proved. Finally, numerical experiments are carried out to verify the effectiveness of numerical schemes.

Modeling Brittle Failure in Rock Slopes Using Semi‐Lagrangian Nonlocal General Particle Dynamics

ABSTRACTThe nonlocal general particle dynamics (NGPD) has been successfully developed to model crack propagation and large deformation problems. In this paper, the semi‐Lagrangian nonlocal general particle dynamics (SL‐NGPD) is proposed to solve brittle failure in rock slopes. In SL‐NGPD, the interaction between particles due to deformation is calculated in the initial configuration, while the friction contact interaction from discontinuities is calculated in the current configuration. The Van der Waals force model is utilized for friction contact. The bond‐level energy‐based failure criterion is developed to predict tensile/compressive‐shear mix‐mode cracks. The artificial viscosity and damage correction are used to enhance the numerical stability and accuracy when modeling brittle failure. The SL‐NGPD paradigm is numerically implemented through adaptive dynamic relaxation and predictor–corrector schemes for stable numerical solutions. The stability and accuracy of SL‐NGPD are verified by simulating compression tests. Thereafter, the crack coalescence patterns of double‐flaw specimens are investigated to understand the triggering failure mechanism of jointed rock slopes. Finally, the progressive failure process of the rock slope with step‐path joints is simulated to demonstrate its validity and robustness in modeling brittle failure in rockslides. The numerical results illustrate that the proposed SL‐NGPD is promising and performant for analyzing brittle failure problems in geotechnical engineering.

Regularized kernel function methods for the backward heat conduction problem

In this letter, we are interested in the regularized numerical techniques for backward heat conduction problems, which aim at the detection of the previous status of physical field from its present information. It is well known that backward heat conduction problems are ill-posed. The small perturbation in the final status may result in large change in the previous status of physical field. It is difficult to get the stable numerical solutions of backward heat conduction problems. Based on the reproducing kernel theory, the letter aims at finding a new regularized kernel function methods for backward heat conduction problems.

Numerical and Experimental Determination of Selected Performance Indicators of the Liquid Flat-Plate Solar Collector under Outdoor Conditions

The paper proposes applying an in-house mathematical model of a liquid flat-plate solar collector to calculate the collector time constant. The described model, proposed for the first time in an earlier study, is a one-dimensional distributed parameter model enabling simulations of the collector operation under arbitrarily variable boundary conditions. The model is based on the solution of energy balance equations for all collector components. The formulated differential equations are solved iteratively using an implicit difference scheme. To obtain a stable numerical solution, it is necessary to use appropriate steps of time and spatial division. These were found by comparing the results obtained from the model with the results of the analytical solution available in the literature for the transient state, which constitutes the novelty of the present study. The accuracy of the results obtained from the model was verified experimentally by comparing the measured and calculated history of the fluid temperature at the outlet of the collector. The calculation of the collector time constant is proposed in the paper as an example of the model’s practical application. The results of the time constant calculation were compared with the values obtained experimentally on the test stand. This is another novelty of the presented research. The analysed collector instantaneous efficiency was then calculated for selected outdoor conditions. The presented mathematical model can also be used to verify the correctness of the collector operation. By comparing, on an ongoing basis, the measured and calculated values of the fluid temperature at the collector outlet, conclusions can be drawn about the process of solar glass fouling or glycol gelling. The simplicity of the model and the low computational demands enable such comparisons in an online mode.

A local meshless method for the numerical solution of multi-term time-fractional generalized Oldroyd-B fluid model

This work presents an accurate and efficient method, for solving a two dimensional time-fractional Oldroyd-B fluid model. The proposed method couples the Laplace transform (LT) with a radial basis functions based local meshless method (LRBFM). The suggested numerical scheme first uses the LT which transform the given equation to an elliptic equation in LT space, and then it utilizes the LRBFM to solve transformed equation in LT space, and then the solution is converted back into the time domain via the improved Talbot's scheme. The local meshless methods are widely recognized for scattered data interpolation and for solving PDEs in complex shaped domains. The adaptability, simplicity, and ease of use are features that have led to the popularity of local meshless methods. The local meshless methods are easy and straightforward, they only requires to solve linear system of equations. The main objective of using the LT is to avoid the computation of costly convolution integral in time-fractional derivative and the effect of time stepping on accuracy and stability of numerical solution. The stability and the convergence of the proposed numerical scheme are discussed. Further, the Ulam-Hyers (UH) stability of the proposed model is discussed. The accuracy and efficiency of the suggested numerical approach have been demonstrated using numerical experiments on five different domains with regular nodes distribution.

Phase field approach to predict mixed-mode delamination and delamination migration in composites

In this work, we present a mixed-mode phase field approach to model delamination progression and its migration. The proposed model includes the volumetric and deviatoric effects that occur during the mixed-mode delamination and migration. The volumetric–deviatoric split is combined with a power law criterion of delamination to capture the mixed-mode effects. An anisotropic tensor is used in the crack surface density function for capturing the anisotropic fracture. Two history parameters are introduced to ensure the irreversibility of the damage field throughout the evolution process of delamination and to consider the maximum value of the strain energy. A staggered approach is implemented for solving the equilibrium and evolution equations. The algorithm enables to obtain stable numerical solutions with faster convergence. Several examples of mode I and mode II as per standard tests have been incorporated to demonstrate the working of the proposed method. The numerical examples are validated by comparison with experimental results from the literature. It is observed that the proposed model can predict crack paths and angles closer to the experimental results. Examples of multiple delamination and delamination migration are also studied to understand the kinking in the delamination scenario. It is observed that the delamination migration is governed by the ratio of notch length a to the distance of the load from the edge L, and by the stress state in front of the crack tip. In the case of laminates with multiple interphases, the event of delamination impinging and kinking at interphase depends on the fracture toughness ratio of the bulk and the interphase. The example of the single-edge notched composite lamina and open-hole tension test with different fiber orientations has been validated with the literature and experiments.

Stability Analysis and Simulation of Diffusive Vaccinated Models

This paper begins by analyzing the key mathematical properties of diffusive vaccinated models, including existence, uniqueness, positivity, and boundedness. Equilibria are identified, and the basic reproductive number is calculated. The Banach contraction mapping principle is applied to rigorously establish the solution existence and uniqueness. In order to understand the disease’s time transmission, it is important to examine the global stability of the equilibrium points. Disease‐free equilibrium and endemic equilibrium are the two equilibria in this model. Here, we demonstrate that the endemic equilibrium is worldwide asymptotic stable when the basic reproductive number is greater than 1, and the disease‐free equilibrium is globally asymptotic stable whenever the basic reproductive number is less than 1. Moreover, based on the Caputo fractional derivative of order and the implicit Euler’s approximation, we offered an unconditionally stable numerical solution for the resultant system. This work explores the solution of some significant population models of noninteger order using an approach known as the iterative Laplace transform. The proposed methodology is developed by effectively combining Laplace transformation with an iterative procedure. A series form solution that exhibits some convergent behavior towards the precise solution can be attained. It is noted that there is a close contact between the obtained and precise solutions. Moreover, the suggested method can handle a variety of fractional order derivative problems because it involves minimal computations. This information will be helpful in further studies to determine the ideal strategy of action for preventing or stopping the spread disease transmission.

Fractional order study of magnetohydrodynamical time-dependent flow of Prandtl fluid

Although the thermal and flow analysis of complex mechanisms of electrochemistry, dispersion and viscoelasticity is significantly addressed in the literature but none of the studies are made through the concept of fractional differential operators with computationally stable numerical solutions. The current work is a first attempt for the fractional classification and computational analysis of the Prandtl fluid. The physical model is formulated to examine the thermal, flow and concentration transference for a naturally convective, time-dependent, incompressible, and stagnation point flow of Prandtl fluid over an infinite plate under the impacts of magnetic, chemical reaction, Soret and Dufour. The mathematical expression of the model is expressed in terms of partial differential equation coupled with Caputo's fractional temporal differential operator. The computational code based on finite difference scheme is formulated to simulate the model. The detailed analysis of the flow pattern, thermal management and concentration of the diffusive species are made computationally. It is noted that involvement of the fractional parameter is useful to control the flow motion, heat transport and concentration of the diffusive species according to the required physical system. The temperature of the system increases 64 % against the variation of radiation parameter when fractional order parameter α=0.4 then the integer order. The study will be a useful addition to non-Newtonian computational mechanics and numerical treatment of such complex models.

A Compact Split-step Finite Difference Method for Solving the Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation arises from quantum mechanics and is extensively used in many fields of science and engineering. Thus, it is important to construct the high-order and stable numerical scheme of the Schrödinger equation. To solve the high-order and stable numerical solution of the nonlinear Schrödinger equation, the compact split-step finite difference method and the local one-dimensional method are combined in this paper. To attain high-order accuracy in time and space, the 4-order compact finite difference in space discretization is combined with the L-stable Simpson method in time discretization. Therefore, a scheme with 4-order accuracy in space and 3-order accuracy in time is obtained, and the stability of the scheme is analyzed. Finally, numerical results manifest that the devised scheme can supply accurate and stable results to the nonlinear Schrödinger equation.

Structure-preserving discretization of fractional vector calculus using discrete exterior calculus

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curlαgradα=0 and divαcurlα=0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.

Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme.

Refined Schemes for Computing the Dynamics of Elastoviscoplastic Media

For a stable numerical solution of the system of equations governing an elastoviscoplastic continuous medium model, a second-order explicit-implicit scheme is proposed. An explicit approximation is used for the equations of motion, and an implicit approximation, for the constitutive relations containing a small relaxation time parameter in the denominator of the nonlinear free terms. A second-order implicit approximation for isotropic and anisotropic elastoviscoplastic models is constructed to match the orders of approximation of the explicit elastic and implicit adjustment steps. Refined formulas for correcting the stress deviators after the elastic step are derived for various viscosity function representations. The resulting solutions of the second-order implicit approximation for the stress deviators of the elastoviscoplastic equations admit passage to the limit as the relaxation time tends to zero. The correcting formulas obtained via this passage to the limit can be treated as regularizers of the numerical solutions to the elastoplastic systems.

The moving finite element method with streamline-upwind Petrov–Galerkin for magnetohydrodynamic flows problems at high Hartmann numbers

It is widely acknowledged that both the streamline-upwind Petrov–Galerkin (SUPG) method and moving mesh method can enhance the stability of finite element method in solving convection-dominated convection–diffusion problems. However, in the case of strong convection-dominated conditions, they still exhibit numerical oscillations and the solutions will not even converge. In order to address similar phenomena observed in solving magnetohydrodynamic (MHD) flow problems at high Hartmann numbers with strong convective dominance, the MHD equation was first decoupled into two convection–diffusion equations. Subsequently, a new approach called the moving mesh streamline-upwind Petrov–Galerkin method (MM-SUPG) was proposed by coupling the moving mesh technique with SUPG method. The MM-SUPG method contains two independent part: the finite element method with SUPG for solving the decoupled MHD equation and a mesh-redistribution algorithm. To verify its effectiveness for solving the MHD equation at high Hartmann number, a classical MHD fluid flow problem was numerically tested using both the MM-SUPG method and the moving finite element method (MFEM). Compared to the MFEM, the proposed method not only yields stable numerical solutions under strong convective dominance while reducing unnecessary false oscillations, but also improves calculation accuracy to some extent using the same mesh.

Optimal error estimates of a SAV–FEM for the Cahn–Hilliard–Navier–Stokes model

We carry out the optimal error estimates of a scalar auxiliary variable (SAV) based the Euler semi-implicit finite element method for the Cahn–Hilliard–Navier–Stokes (CHNS) system in this paper. A SAV is introduced to reformulate the CHNS system into an equivalent system. We apply the Euler semi-implicit method and finite element method for the temporal and spatial discretizations to obtain a linear and unconditional energy stable scheme. The stability of numerical solutions under different norms and the optimal error estimates for our proposed numerical schemes are also presented. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method, and the discrete energy is dissipated in numerical simulations.

Generalized thermal properties of hybrid NANOLIQUID composed of aluminum oxide (Al2O3) and silver (Ag) nanoparticles with water (H2O) as base liquid

AbstractHeat transfer rate is numerically analyzed in convective flow of Al2O3‐Ag/ H2O hybrid nanoliquid through a stretching sheet by incorporating induced magnetic field. Results of entropy generation in system are evaluated as well. Considered physical factors associated with heat transfer are heat generation parameter and viscous dissipation. The system of nonlinear partial differential equations is modeled and dimensionally simplified by implementing boundary layer approximation assumption and proper similarity transformations. Adam's Bashforth method is applied to get highly accurate and stable numerical solutions. Numerical results of flow variables, entropy generation number and physical quantities are interpreted by way of graphs and bar charts to perceive the extensive significance of the problem. It is visualized that rise in numeric values of mixed convection parameter λ1 leads to enhance velocity; entropy generation number and Nusselt number while suppress temperature. High magnitude of heat generation parameter δ augments velocity and temperature but reverse behavior is observed for Nusselt number and entropy generation number. Moreover, the factor of viscous dissipation significantly modifies rate of flow and heat transfer under the effect of no‐slip condition on sheet. The present study is useful in different fields of industries, technological processes, mechanical processes, and electrical processes due to the applications of magnetic hybrid nanofluid with improved heat and mass transfer.

3D Sensitivity Kernels With Full Attenuation Computed by a Combination of the Strong Stability Preserving Runge‐Kutta Method and the Scattering‐Integral Method

AbstractTo date developments of seismic attenuation models of the Earth have lagged those of velocity models. This is partly due to difficulties in isolating waveform perturbations caused by attenuation and velocity heterogeneities and partly due to different theories (e.g., ray theory, finite frequency theory) used in most previous and current studies to approximate sensitivity kernels and invert for the attenuation structure. We present in this paper a new method for computing the 3D waveform Fréchet kernels that account for full physical‐dispersion and dissipation attenuation. For solving the 3D isotropic anelastic wave equation described by the generalized Maxwell body model, we extend our previously proposed full waveform modeling method in Cartesian coordinates to that in spherical coordinates, which can provide stable numerical solutions even in the presence of strong attenuation. Then, we apply the scattering‐integral method for calculating 3D travel time and amplitude sensitivity kernels with respect to velocity and attenuation structures. We demonstrate the accuracy of our forward method and the effectiveness of the implementation of absorbing and free surface boundary conditions through numerical tests. Moreover, by choosing the Northwestern United States region as a realistic example, we verify the accuracy of the computed 3D sensitivity kernels through comparing the waveform measurements with predictions from the kernels. Finally, we discuss the importance of calculating full anelastic sensitivity kernels including both effects of physical dispersion and dissipation, where we specially explore the effect of scattering due to random velocity and attenuation heterogeneities on waveform measurements.

Nonlocal geometric fracture theory: A nonlocal‐local asymptotically compatible framework for continuous–discontinuous deformation of solid materials

SummaryOne of the fundamental but unsolved problems in fracture mechanics of solid materials is how to describe the continuous and discontinuous deformation of continuum in a unified manner. This paper aims to develop a novel nonlocal geometric fracture theory attempting to address this essential issue. Using the concepts of nonlocal calculus, a new nonlocal equation for the deformation of continuum is obtained by minimizing a nonlocal Lagrangian. This is done to ensure that the proposed nonlocal model can be used to describe the continuous and discontinuous deformation of a continuum in a unified geometric manner, as well as to be asymptotically compatible with the classical continuum theory. Inspired by the phase field fracture theory, a nonlocal geometric crack surface functional is proposed to approximate the Griffith fracture energy in continuum. Note, however, that a distinguished feature of the proposed functional is that it enables strong discontinuities appear in the crack phase fields, which is difficult for the phase field fracture theory. After obtaining the nonlocal fracture energy, a nonlocal elastic energy is phenomenologically decomposed into volumetric and deviatoric parts, and then a system of nonlocal geometric fracture equations is derived to describe the damage nucleation, cracks initiation and propagation, which covers the whole physical process of fracture in solid materials. A staggered discontinuous Galerkin method is developed to reach an accurate and stable numerical solution. Numerical examples demonstrate the proposed theory can well capture the transition from the continuous to discontinuous deformation of the continuum and accurately describe the topological evolution of cracks in a unified framework.

A Physically‐Based Reverse‐Stage Routing Model Considering Lateral Flow for Establishing Normal Rating Curves at Ungauged Upstream River Sections

AbstractMany of the world‐rivers are often ungauged or poorly gauged due to inadequate streamflow monitoring networks. Even with the limited available monitoring stations, only a few of them monitor both stage and discharge data. But when there is a need for estimation of discharges at a gauging station, where only stage data is monitored, one may employ a reverse routing technique using the stage data monitored at a nearby downstream gauging station. This study develops a novel single‐parameter reverse‐stage routing (RSR) model based on the second‐order approximate water surface gradient governing equation to estimate stage and discharge at any scantily‐gauged upstream river section using the known stage information available at the downstream section. A novel criterion is also developed for checking the applicability of the RSR model. Reverse routing experiments, carried out by the RSR model using different hypothetical downstream stage hydrographs in a number of hypothetical trapezoidal and rectangular channel reaches to reproduce the benchmark upstream stage and discharge hydrographs, demonstrate the good performance of the RSR model with the Nash–Sutcliffe Efficiency >98%, absolute volume conservation error <1.5%, and an absolute error in peak ≤1%. Subsequently, the RSR model was tested for three real‐river case studies in India and Italy with good reproduction performances, and along with the development of the corresponding normal rating curves at the upstream river sections. The study results reveal that the parsimonious RSR model has good potential for solving reverse routing problems with stable numerical solutions for rivers under ungauged and scantily‐gauged scenarios.

On the design of stable, consistent, and conservative high-order methods for multi-material hydrodynamics

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.

Control of Tooth Form Deformation in Heat Treatment of Spiral Bevel Gears Based on Reverse Adjustment of Cutting Parameters.

The tooth surface structure of spiral bevel gear is complex and requires high machining accuracy. In order to reduce the tooth form deformation of heat treatment, this paper proposes a reverse adjustment correction model of tooth cutting for heat treatment tooth form deformation of spiral bevel gear. Based on the Levenberg-Marquardat method, a stable and accurate numerical solution for the reverse adjustment amount of the cutting parameters is solved. Firstly, a mathematical model of the tooth surface of spiral bevel gears was established based on the cutting parameters. Secondly, the effect law of each cutting parameter on tooth form was studied by using the method of small variable perturbation. Finally, based on the tooth form error sensitivity coefficient matrix, a reverse adjustment correction model of tooth cutting is established to compensate the heat treatment tooth form deformation by reserving the tooth cutting allowance in the tooth cutting stage. The effectiveness of the reverse adjustment correction model of tooth cutting was verified through experiments on reverse adjustment of tooth cutting processing. The experimental results show that the accumulative tooth form error of the spiral bevel gear after heat treatment is 199.8 μm, which is reduced by 67.71%, and the maximum tooth form error is 8.7 μm, which is reduced by 74.75%, after reverse adjustment of cutting parameters. This research can provide technical support and a theoretical reference for heat treatment tooth form deformation control and high-precision tooth cutting processing of spiral bevel gears.