Numerical Convergence Research Articles

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

Convergence results for controlled contractions and applications to a system of Volterra integral equations

This work presents the concept of controlled dislocated metric spaces, and the Banach contraction principle in these spaces is examined. Next, we provide extensive examples of numerical convergence of fixed points to demonstrate the practical implications of such spaces. Along with specific instances, it presents the idea of a rational contraction of the ϝ − type to determine common fixed points in such spaces. Moreover, numerical analysis and graphical comparisons are used to further assess the existence of common fixed points based on the provided notions. Lastly, using these ideas to solve a Volterra integral equation shows how useful they are for finding common solutions, improving our comprehension of fixed point theory in these spaces, and creating new avenues for applications.

Sensitivity of the future evolution of the Wilkes Subglacial Basin ice sheet to grounding-line melt parameterizations

Abstract. Projections of Antarctic Ice Sheet mass loss and therefore global sea level rise are hugely uncertain, partly due to how mass loss of the ice sheet occurs at the grounding line. The Wilkes Subglacial Basin (WSB), a vast region of the East Antarctic Ice Sheet, is thought to be particularly vulnerable to deglaciation under future climate warming scenarios. However, future projections of ice loss, driven by grounding-line migration, are known to be sensitive to the parameterization of ocean-induced basal melt of the floating ice shelves and, specifically, to the adjacent grounding line – termed grounding-line melt parameterizations (GLMPs). This study investigates future ice sheet dynamics in the WSB with respect to four GLMPs under both the upper and lower bounds of climate warming scenarios from the present to 2500, with different model resolutions, ice shelf melt parameterizations (ISMPs) and choices of sliding relationships. The variation in these GLMPs determines the distribution and the amount of melt applied in the finite-element assembly procedure on partially grounded elements (i.e. elements containing the grounding line). Our findings indicate that the GLMPs significantly affect both the trigger timings of tipping points and the overall magnitude of ice mass loss. We conclude that applying full melting to the partially grounded elements, which causes melting on the grounded side of the grounding line, should be avoided under all circumstances due to its poor numerical convergence and substantial overestimation of ice mass loss. We recommend preferring options that depend on the specific model context, by either (1) not applying any melt immediately adjacent to the grounding line or (2) employing a sub-element parameterization.

Numerical Investigations on the Enhancement of Convective Heat Transfer in Fast-Firing Brick Kilns

In order to reduce CO2 emissions in the brick manufacturing process, the effectiveness of the energy-intensive firing process needs to be improved. This can be achieved by enhancing the heat transfer in order to reduce firing times. As a result, current development of tunnel kilns is oriented toward fast firing as a long-term goal. However, a struggling building sector and complicated challenges, such as different requirements for product quality, have impeded developments in this direction. This creates potential for the further development of oven designs, such as improved airflow through the kiln. In this article, numerical flow simulations are used to investigate two different reconstruction measures and compare them to the initial setup. In the first measure, the kiln height is reduced, while in the second measure, the kiln cars are adjusted to alternate the height of the bricks so that every other pair of bricks is elevated, creating a staggered arrangement. Both measures are investigated to determine the effect on the heating rate compared to the initial configuration. A transient grid independence study is performed, ensuring numerical convergence and the setup is validated by experimental results from measurements on the initial kiln configuration. The simulations show that lowering the kiln height improves the heat transfer rate by 40%, while the staggered arrangement of the bricks triples it. This leads to an average brick temperature after two hours which is around 130 °C higher compared to the initial kiln configuration. Therefore, the firing time can be significantly reduced. However, the average pressure loss coefficient rises by 70% to 90%, respectively, in the staggered configuration.

Mind the Kinematics Simulation of Planet–Disk Interactions: Time Evolution and Numerical Resolution

Planet–disk interactions can produce kinematic signatures in protoplanetary disks. While recent observations have detected non-Keplerian gas motions in disks, their origins are still being debated. To explore this, we conduct 3D hydrodynamic simulations using the code FARGO3D to study nonaxisymmetric kinematic perturbations at two scale heights induced by Jovian planets in protoplanetary disks, followed by examinations of detectable signals in synthetic CO emission line observations at millimeter wavelengths. We advocate for using residual velocity or channel maps, generated by subtracting an azimuthally averaged background of the disk, to identify planet-induced kinematic perturbations. We investigate the effects of two basic simulation parameters, simulation duration and numerical resolution, on the simulation results. Our findings suggest that a short simulation (e.g., 100 orbits) is insufficient to establish a steady velocity pattern given our chosen viscosity (α = 10−3) and displays plenty of fluctuations on an orbital timescale. Such transient features could be detected in observations. By contrast, a long simulation (e.g., 1000 orbits) is required to reach steady state in kinematic structures. At 1000 orbits, the strongest detectable velocity structures are found in the spiral wakes close to the planet. Through numerical convergence tests, we find hydrodynamics results converge in spiral regions at a resolution of 14 cells per disk scale height or higher. Meanwhile, synthetic observations produced from hydrodynamic simulations at different resolutions are indistinguishable with 0.″1 angular resolution and 10 hr of integration time on Atacama Large Millimeter/submillimeter Array.

VENICE: A multi-scale operator-splitting algorithm for multi-physics simulations

Context. We present VENICE, an operator-splitting algorithm to integrate a numerical model on a hierarchy of timescales. Aims. VENICE allows a wide variety of different physical processes operating on different scales to be coupled on individual and adaptive time-steps. It therewith mediates the development of complex multi-scale and multi-physics simulation environments with a wide variety of independent components. Methods. The coupling between various physical models and scales is dynamic, and realised through (Strang) operators splitting using adaptive time-steps. Results. We demonstrate the functionality and performance of this algorithm using astrophysical models of a stellar cluster, first coupling gravitational dynamics and stellar evolution, then coupling internal gravitational dynamics with dynamics within a galactic background potential, and finally combining these models while also introducing dwarf galaxy-like perturbers. These tests show numerical convergence for decreasing coupling timescales, demonstrate how VENICE can improve the performance of a simulation by shortening coupling timescales when appropriate, and provide a case study of how VENICE can be used to gradually build up and tune a complex multi-physics model. Although the examples provided here couple dedicated numerical models, VENICE can also be used to efficiently solve systems of stiff differential equations.

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

ABSTRACTUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

Characterization of the angular coefficient method on 2D and 3D piecewise smooth boundaries

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.

A fast H3N3‐2σ$_\sigma$‐based compact ADI difference method for time fractional wave equations

AbstractIn the present paper, we derive a fast H3N3‐2‐based compact alternating direction implicit (ADI) scheme for the time fractional wave equation in two‐dimensional spatial domains. The time fractional derivative involved in the equation is discretized by the H3N3‐2 formula combined with sum‐of‐exponential technique. The former was first proposed by Du et al., while the latter has become a widely used technique to reduce the computational cost in the processing of convolution integrals. The spatial discretization adopts the standard compact ADI method, which can ensure that the space can reach the fourth‐order accuracy without increasing the amount of computation. The numerical stability and convergence of the difference scheme are analyzed rigorously. As an auxiliary illustration, a numerical example is given to verify the validity of the theoretical findings.

Designing a Fully‐Tunable and Versatile TKE‐l Turbulence Parameterization for the Simulation of Stable Boundary Layers

AbstractThis study presents the development of a so‐called Turbulent Kinetic Energy (TKE)‐l, or TKE‐l, parameterization of the diffusion coefficients for the representation of turbulent diffusion in neutral and stable conditions in large‐scale atmospheric models. The parameterization has been carefully designed to be completely tunable in the sense that all adjustable parameters have been clearly identified and the number of parameters has been minimized as much as possible to help the calibration and to thoroughly assess the parametric sensitivity. We choose a mixing length formulation that depends on both static stability and wind shear to cover the different regimes of stable boundary layers. We follow a heuristic approach for expressing the stability functions and turbulent Prandlt number in order to guarantee the versatility of the scheme and its applicability for planetary atmospheres composed of an ideal and perfect gas such as that of Earth and Mars. Particular attention has been paid to the numerical stability and convergence of the TKE equation at large time steps, an essential prerequisite for capturing stable boundary layers in General Circulation Models (GCMs). Tests, parametric sensitivity assessments and preliminary tuning are performed on single‐column idealized simulations of the weakly stable boundary layer. The robustness and versatility of the scheme are assessed through its implementation in the Laboratoire de Météorologie Dynamique Zoom GCM and the Mars Planetary Climate Model and by running simulations of the Antarctic and Martian nocturnal boundary layers.

An alternative method to the Takagi-Taupin equations for studying dark-field X-ray microscopy of deformed crystals.

This study introduces an alternative method to the Takagi-Taupin equations for investigating the dark-field X-ray microscopy (DFXM) of deformed crystals. In scenarios where dynamical diffraction cannot be disregarded, it is essential to assess the potential inaccuracies of data interpretation based on the kinematic diffraction theory. Unlike the Takagi-Taupin equations, this new method utilizes an exact dispersion relation, and a previously developed finite difference scheme with minor modifications is used for the numerical implementation. The numerical implementation has been validated by calculating the diffraction of a diamond crystal with three components, wherein dynamical diffraction is applicable to the first component and kinematic diffraction pertains to the remaining two. The numerical convergence is tested using diffraction intensities. In addition, the DFXM image of a diamond crystal containing a stacking fault is calculated using the new method and compared with the experimental result. The new method is also applied to calculate the DFXM image of a twisted diamond crystal, which clearly shows a result different from those obtained using the Takagi-Taupin equations.

A new space–time localized meshless method based on coupling radial and polynomial basis functions for solving singularly perturbed nonlinear Burgers’ equation

In this paper, the singularly perturbed nonlinear Burgers’ problem (SPBP) with small kinematic viscosity 0<ϵ≪1 is solved using a new Space–Time Localized collocation method based on coupling Polynomial and Radial Basis Functions (STLPRBF). To our best knowledge, it is the first time that the solution of SPBP is accurately approximated using the space–time meshless method without applying any adaptive refinement technique. The method is based on solving the problem without distinguishing between space and time variables, which eliminates the need for time discretization schemes. To address the inherent non-linearity of the problem, the method employs an iterative algorithm based on quasilinearization technique. The efficiency and accuracy of the proposed method are demonstrated by solving different examples of one- and two-dimensional SPBP with very small ϵ up to 10−10. Additionally, the numerical convergence of the method with respect to ϵ and also to the number of collocation points has been investigated. The comparison of the STLPRBF results with other published ones is presented.

Dynamics for rigid–flexible coupled solar panel multibody system composed of composite laminated plates

Composite laminates are widely employed in the fabrication of solar panels in modern spacecraft engineering because of their superior durability, mechanical qualities, and cost-effectiveness. This research is dedicated to multibody system dynamics modeling of composite laminate solar panels, specifically the coupling system between the rigid main body and the flexible solar panels. In this study, the rigid body is modeled using the natural coordinate formulation (NCF), and for the flexible solar panel, the absolute nodal coordinate formulation (ANCF) with a 36-DOF reduced-order composite thin shell element is used to accurately capture the coupling effect of the solar panel under large motion and deformation. Compared to a 48-DOF fully parameterized thick plate element, this method efficiently avoids shear locking and improves numerical solution convergence efficiency. Based on the Kirchhoff theory and the constitutive relationship of composite laminates, a dynamic model of the multibody system is developed, taking into account the rigid-flexible coupling effect and the kinematic constraints. This work thoroughly discusses the dynamic behavior of solar panel deployment using a comparative analysis of numerical simulation and virtual prototype simulation, and the accuracy is verified. The findings indicate that the flexible characteristics and fiber orientation of composite laminates have a significant impact on the dynamic response of solar panels during deployment. Furthermore, the bang-bang attitude controller is designed, and the difference in dynamic response between the two modeling methods in controlled and uncontrolled states is studied, providing an important theoretical foundation and technical support for solar panel design and analysis.

Numerical modeling of transient water table in shallow unconfined aquifers: A hyperbolic theory and well-balanced finite volume scheme

We present a new methodology capable of modeling transient motion of shallow phreatic surface of groundwater in unconfined aquifers. This methodology is founded on a new and comprehensive theory for water table motion and a corresponding efficient numerical scheme. In the theoretical aspect, we derived a new set of governing equations constituted by a depth-averaged continuity equation and momentum equations based on unsteady Darcy’s law. The derived governing equations are of the hyperbolic type and possess stiff terms in the momentum equations due to the inertia motion in a characteristic time scale that is relatively shorter than the time scale of seepage motion. To effectively solve the derived hyperbolic system with stiff terms, in the numerical aspect, we utilize f-wave propagation algorithm, an explicit finite volume method, that can ensure numerical convergence and well-balancing solutions when momentum is rapidly relaxing to an equilibrium of steady state. Verification is successfully performed by comparing the results with analytic solutions to the classic problem of multidimensional spreading of a groundwater mound. This study demonstrates that the proposed methodology can accurately and satisfactorily simulate the spatiotemporal distribution of shallow water table and its wetting front in unconfined aquifers.

학생의 퀀텀점프 성장을 위한 대학의 마스터키 융합교육 전략

Objectives The purpose of this study is to establish a convergence education plan to quantum jump students from local private universities whose academic capabilities are gradually declining, to nurture them into talented individuals who can take on various challenges after graduation, and to establish detailed strategies to implement them. Methods Reflecting the internal needs of students for activating convergence education obtained through the survey results of previous research and external factors such as the situation of local private universities, the need for master key education, and the trend of introduction of minor degree in domestic universities, specific convergence education strategies that can be effectively applied in universities are presented step by step. Results The strategic steps of master-key convergence education proposed in this study are: 1. Using the convergence university as a convergence education platform. 2. Strengthening the basic academic capabilities of freshmen by reviewing middle and high school courses. 3. Exploring various majors by easily teaching non-majors the core contents of each department. 4. Opening a small-scale curriculum called ‘convergence module’ by combining subjects with common educational goals from one or more departments, and awarding nano-degree to students who complete it. 5. Universities, students and/or local companies create numerous convergence majors by combining various convergence modules according to their individual needs. Conclusions Through the master-key convergence education strategies proposed in this study, universities aiming to newly introduce convergence education or make significant changes in existing curriculum can establish effective convergence education plans that can be immediately implemented with minimal reorganization of the current curriculum.

An Innovative and Efficient Implementation of Matrix-Free Newton Krylov Method for Neutronics/Thermal-hydraulics Coupling Simulation

The core physical behavior of reactors is essentially the result of multi-physical fields coupling feedback. High-fidelity neutronics/thermal-hydraulics (N/TH) analysis can simulate and predict nuclear reactor core phenomena realistically, providing advanced and reliable technical means during the design and safety analysis of nuclear reactor. In this work, an efficient and robustness coupling method using power density as the coupling parameter, Matrix-Free Newton Krylov (MFNK) method, is successfully developed and innovatively implemented in HNET for high-fidelity N/TH coupling simulation. To enhance the efficiency and stability, the multi-level generalized equivalence theory-based CMFD (ML-gCMFD) iterative acceleration method and ML-gCMFD coupling acceleration method are proposed. In addition, the nonlinear preconditioning and hybrid perturbation size formula are implemented to further improve the convergence. Finally, to evaluate the numerical accuracy, convergence, efficiency and stability of MFNK method, a series of representative problems, including a three-dimensional (3D) single fuel pin problem, VERA Benchmark Problem 6, and VERA Benchmark Problem 7, are analyzed by comparing with the current N/TH coupling methods. Numerical results indicate that MFNK method can obtain strong stability, high convergence performance, and relatively high computational efficiency while ensuring high accuracy. It demonstrates that MFNK method has significant performance advantages and potential for high-fidelity N/TH coupling simulation.

Convergent stochastic algorithm for estimation in general multivariate correlated frailty models using integrated partial likelihood

The Cox model with unspecified baseline hazard is often used to model survival data. In the case of correlated event times, this model can be extended by introducing random effects, also called frailty terms, leading to the frailty model. Few methods have been put forward to estimate parameters of such frailty models, and they often consider only a particular distribution for the frailty terms and specific correlation structures. In this paper, a new efficient method is introduced to perform parameter estimation by maximizing the integrated partial likelihood. The proposed stochastic estimation procedure can deal with frailty models with a broad choice of distributions for the frailty terms and with any kind of correlation structure between the frailty components, also allowing random interaction terms between the covariates and the frailty components. The almost sure convergence of the stochastic estimation algorithm towards a critical point of the integrated partial likelihood is proved. Numerical convergence properties are evaluated through simulation studies and comparison with existing methods is performed. In particular, the robustness of the proposed method with respect to different parametric baseline hazards and misspecified frailty distributions is demonstrated through simulation. Finally, the method is applied to a mastitis and a bladder cancer dataset.

High-Accuracy Numerical Methods and Convergence Analysis for Schrödinger Equation with Incommensurate Potentials

High-Accuracy Numerical Methods and Convergence Analysis for Schrödinger Equation with Incommensurate Potentials

Model-free chemomechanical interfaces: History-dependent damage under transient mass diffusion

This paper presents a data-driven framework based on distance functional for chemo-mechanical cohesive interfaces to capture transient diffusion and resulting interfacial damage evolution. The framework eliminates reliance on complex constitutive models and phenomenological assumptions, especially avoiding the coupling tangent terms with a given material database. The interfacial chemical potential and its jumps are used to expand the phase space for describing the chemical states of interfaces. Subsequently, a novel chemo-mechanical distance norm, including traction-separation pair, interfacial chemical potential-concentration pair and corresponding chemical potential jump-flux pair, is presented. Momentum conservation law for interfacial tractions and mass conservation law for interfacial concentration are enforced via Lagrange multipliers. For tracking the history-dependent state of the interface damage, an internal variable, termed interface integrity, is introduced to condition the interfacial database and manage the subsets mapping strategies, following physically motivated evolution constraints. Numerical examples are conducted to investigate the efficiency and capability of the present framework. A monotonic loading test validates a good numerical convergence relative to the number of data points. Subsequent cyclic loading simulations, compared with the reference solutions, show that the algorithm is well suitable to predict the history-dependent interface damage evolution under complex loading paths. Finally, the chemo-mechanically coupled examples capture phenomena such as interfacial swelling and interface degradation induced by transient diffusion and stress-driven diffusion. The current work provides a promising tool for understanding chemo-mechanical behaviors of interfaces within heterogeneous composites.