Explicit Scheme Research Articles

Study on the influence of perforation parameters on hydraulic fracture initiation and propagation based on CDEM

Perforation hydraulic fracturing plays a crucial role in the development of oil and gas reservoirs, and perforation layout can significantly affect the initiation and propagation of complex reservoir rock fractures. In this paper, we used a mixed finite-discrete element method with an explicit iterative scheme called the continuous–discontinuous element method (CDEM) to study the process of hydraulic fracture propagation through perforation. The coupling effects of fluid flow and reservoir mechanics are fully considered, and the accuracy of the model in simulating the directional propagation of perforation hydraulic fractures is validated by comparison with physical experimental results. We also discussed in detail the impact of perforation pattern, density, depth, and aperture on crack initiation, propagation, and complexity, and used the new index of fracture degree to quantitatively evaluate the fracturing effects under the influence of multiple perforation layout parameters. The results indicate that: (i) hydraulic fractures near the wellbore compete for initiation, and the length distribution of cracks is uneven and intricate; (ii) linear perforation patterns produce the maximum initiation pressure, which is negatively correlated with the perforation density and positively correlated with the perforation depth and aperture; (iii) the fracture degree is mainly affected by perforation density and depth, and is less affected by aperture. Perforation can effectively control the fracture morphology and initiation pressure, and has a substantial influence on the crack propagation layout near the wellbore. This study provides a basis for the rational selection and optimization of perforation parameters, which can effectively improve the fracturing effects of tight oil and gas reservoirs.

On the enriched mixed Galerkin method with θ scheme for the elastic wave equation

In this paper, we propose the application of an enriched Galerkin (EG) method to solve the elastic wave equation in the “displacement pseudo-pressure” mixed form, which is known for being locking-free. Unlike the continuous Galerkin (CG) method and the discontinuous Galerkin (DG) method, the EG method is based on the weak form of the DG method, and the enriched space combines a continuous piecewise linear vector-valued function space with some discontinuous piecewise linear functions. Specifically, we employ the enriched space to discretize the displacement, while the pseudo-pressure is discretized using the P0 element space. As a result, compared to the DG method, the EG method significantly reduces the number of degrees of freedom. Furthermore, we adopt a more general θ-scheme for time discretization, which behaves as an implicit scheme for 14≤θ≤1 and as an explicit scheme for 0≤θ<14. We establish error estimates for both the semi-discrete and fully-discrete schemes. The accuracy of the proposed method is validated through numerical examples, which not only confirm the theoretical analysis but also provide simulations of elastic wave propagation.

A yarn-scale woven fabric model including significant slippage formulated within the Arbitrary Lagrangian Eulerian framework

A numerical model is proposed to simulate significant yarn slippage and unweaving in woven fabrics using the Arbitrary Lagrangian Eulerian (ALE) method. Woven fabrics were modeled at the yarn scale using 1D ALE elements with additional slip displacement degrees of freedom (DOF). The relatively weak bending stiffness of the yarn was included using a rotation-free approach. A weave kinematic constraint between crossing yarns allowed an efficient description of the deformation modes due to the yarn assembly such as in-plane shear. In particular, normal compaction at weave points was taken into account using an additional crimp amplitude DOF and couplings of the fabric’s cohesion with the normal and lateral compaction were considered. To deal with yarn unweaving, a specific void-ALE 1D element was developed in order to avoid remeshing, time-step reduction (explicit scheme) and early deactivation of the weave kinematic constraint. To demonstrate the capabilities of this numerical model, common characterization tests such as in-plane and out-of-plane pullouts, which induced unweaving were simulated. Finally, a first forming simulation on a hemisphere was performed to verify whether the cohesion of the fabric remained well represented in situations without loss of cohesion before simulating forming on a prismatic punch involving significant yarn slippage.

The mediating role of capital on deposit insurance and financial stability: evidence from Islamic and conventional banks

PurposeThis study aims to evaluate the interaction between bank capital and explicit deposit insurance scheme (DIS) on the financial stability of Islamic and conventional banks.Design/methodology/approachThe author's sample covers 52 Islamic and 108 conventional banks operating in 12 countries over the period 2000–2021 using the random-effects generalized least squares (RE-GLS) regression technique.FindingsThe author's results reveal that bank capital negatively mediates the relationship between explicit DIS and the financial stability of both Islamic and conventional banks. Additionally, explicit DIS has a positive impact on the financial stability of conventional banks. However, the results are mixed for Islamic banks, as the effect of explicit DIS is positive for the Middle East and North Africa (MENA) region but negative for the South and Southeast Asia (SSA) region. Finally, the interaction between explicit DIS and the COVID-19 pandemic has a negative effect on conventional banks operating in the MENA region, while it has a positive effect on Islamic banks operating in the SSA region.Research limitations/implicationsThe findings of this paper have important implications for regulators in evaluating DIS policies and in anticipating any potential adverse consequences that might arise for both Islamic and conventional banks in normal and crisis times. Policymakers should strive to preserve the benefits of DIS while mitigating the destabilizing effects of its interaction with capital ratios.Originality/valueThis study introduces a novel aspect by examining the mediating role of capital in the relationship between explicit DIS and the financial stability of Islamic and conventional banks.

Accuracy Analysis for Explicit-Implicit Finite Volume Schemes on Cut Cell Meshes

The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. In May and Berger (J Sci Comput 71: 919–943, 2017), the mixed explicit-implicit approach in general and MUSCL-Trap (monotonic upwind scheme for conservation laws and trapezoidal scheme) in particular have been introduced to solve this problem by using implicit time stepping on the cut cells. Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping. In this contribution, we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme. As this result is unlikely to hold in two dimensions, we also introduce two new versions of mixed explicit-implicit schemes based on exchanging the explicit scheme. We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.

An efficient and fast adaptive numerical method for a novel phase-field model of crystal growth

In this study, we present an efficient, fast, and fully explicit adaptive numerical scheme for solving a recently developed novel phase-field model of crystal growth in both two-dimensional (2D) and three-dimensional (3D) spaces. The novel phase-field model employed in this research incorporates a term specifically designed to eliminate the artificial curvature effect, thereby facilitating accelerated evolution when compared to traditional models. This enhancement significantly enhances the efficiency and speed of the simulation, leading to more expedited results. The crystal growth model employed in this study incorporates a phase-field equation to accurately represent the crystal interface, in addition to a heat equation that effectively models the distribution of temperature. To effectively solve the phase-field equation, we employ an adaptive numerical algorithm that optimizes the computational process. Our numerical scheme, specifically tailored for simulating dendritic growth, incorporates an adaptive narrow-band domain approach to accurately resolve the interfacial transition layer of the phase field. Furthermore, we enhance computational efficiency by implementing a double-sized grid for the temperature distribution, further improving the overall efficiency of the model. By combining these strategies, we achieve accurate and efficient solutions for the dendritic growth model. To validate the accuracy and efficiency of our proposed adaptive numerical method for solving the phase-field equation of dendritic growth, we conduct a series of numerical experiments in both 2D and 3D spaces. In these experiments, we assess the performance of our algorithm, analyzing its ability to accurately capture the intricate dynamics of dendritic growth while maintaining computational efficiency. By thoroughly evaluating the results obtained from these experiments, we provide strong evidence supporting the reliability and effectiveness of our adaptive numerical algorithm.

Advancing wave equation analysis in dual-continuum systems: A partial learning approach with discrete empirical interpolation and deep neural networks

In this work, we propose a partial learning approach using partially explicit discretization for solving the wave equations. The considered mathematical model involves a dual-continuum system of hyperbolic second-order partial differential equations. We employ a finite element method for spatial approximation, while for temporal approximation, we utilize a partially explicit scheme. Specifically, we adopt the explicit scheme for the low-conductive continuum and the implicit scheme for the high-conductive continuum.Our proposed method’s key idea lies in employing a partial learning approach to remove the necessity of finding the difficult-to-compute (implicit) part of the solution. Instead of attempting a full training of the numerical solution, we focus on training only a portion of it that requires the most computational cost. We use a proper orthogonal decomposition with a discrete empirical interpolation method to facilitate machine learning. By employing these methods, we can train a deep neural network to predict the values of the implicit part of the solution at specific observed locations and then restore the field.We consider a two-dimensional model problem in heterogeneous media to test the effectiveness of the proposed approach. The numerical results demonstrate that our partial learning approach yields a satisfactory approximation. By combining the reduced-order modeling techniques, the partially explicit time scheme, and the deep neural network, we removed the necessity of finding the difficult-to-compute part of the solution while maintaining high accuracy.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

&lt;abstract&gt;&lt;p&gt;In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.&lt;/p&gt;&lt;/abstract&gt;

Study on coupled problems of thermoelasticity in Strains

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.

Calculation of the wave field in a layered soil foundation using the grid method

The article contains a solution to the problem of propagation of waves in a soil massif arising from the movement of subway trains. This problem often arises when predicting vibration levels in buildings located near metro lines. The problem is considered in a two-dimensional formulation, taking into account the actual location of soil layers with different physical and mechanical properties. The soil is modeled using the assumption of its linear deformability, which, to a first approximation, is acceptable due to small vibration displacements. To solve the problem, the finite difference method is used, which allows taking into account the various physical properties of individual soil layers and the structure of subway tunnels and at the same time is one of the simplest numerical methods. To obtain a solution, the direct integration method is used using an explicit scheme, which requires setting sufficiently small time steps. The results are presented as graphs of vibration displacements obtained at various points in time. The graphs illustrate the different nature of the wave field at different load frequencies. The dynamic load is specified in the form of vibrations of the walls of the subway tunnel.

The Significance of the Melt-Pond Scheme in a CMIP6 Global Climate Model

Abstract The impact of melt ponds on sea ice albedo has been observed and documented. In general circulation models, ponds are now accounted for through indirect diagnostic treatments (“implicit” schemes) or prognostic melt-pond parameterizations (“explicit” schemes). However, there has been a lack of studies showing the impacts of these schemes on simulated Arctic climate. We focus here on rectifying this using the general circulation model HadGEM3, one of the few models with a detailed explicit pond scheme. We identify the impact of melt ponds on the sea ice and climate, and associated ice–ocean–atmosphere interactions. We run a set of constant forcing simulations for three different periods and show, for the first time, that using mechanistically different pond schemes can lead to very significantly different sea ice and climate states. Under near-future conditions, an implicit scheme never yields an ice-free summer Arctic, while an explicit scheme yields an ice-free Arctic in 35% of years and raises autumn Arctic air temperatures by 5° to 8°C. We find that impacts on climate and sea ice depend on the ice state: under near-future and last-interglacial conditions, the thin sea ice is very sensitive to pond formation and parameterization, whereas during the preindustrial period the thicker sea ice is less sensitive to the pond scheme choice. Both of these two commonly used parameterizations of sea ice albedo yield similar results under preindustrial conditions but in warmer climates lead to very different Arctic sea ice and ocean and atmospheric temperatures. Thus, changes to physical parameterizations in the sea ice model can have large impacts on simulated sea ice, ocean, and atmosphere. Significance Statement This study investigates the impacts of melt ponds on Arctic sea ice under different climate conditions, using the HadGEM3-GC3.1-LL general circulation model (GCM). Additionally, we study the impact of changing the type of pond scheme used. We find that changing the pond scheme causes large differences to how a GCM simulates Arctic sea ice, the ocean, and the atmosphere, for both near-future and warmer paleoclimate conditions. These large differences have not been found previously, because this is one of the first GCM studies of this type. Our results demonstrate the importance of melt ponds, and their wider impacts on ocean and atmosphere. Furthermore, they suggest that better evaluation of the representation of sea ice processes is vital for the robust projection of future climate change.

Coupled Problem on Thermo-Elasticity in Strains for an Isotropic Parallelepiped

This work is devoted to mathematical and numerical modeling of the coupled dynamic problem of thermoelasticity in deformations. A numerically related boundary value problem of thermoelasticity in deformations for a parallelepiped with the corresponding initial and boundary conditions is formulated and solved. Grid equations are constructed using the finite-difference method in the form of explicit and implicit schemes. In this case, the solution of the explicit scheme is reduced to recurrent relations with respect to deformations and temperature. In the case of implicit difference schemes, the equations are solved by sequential application of the sweep method. The validity of the formulated boundary value problems is justified by comparing the numerical results obtained with different methods based on two different models.

COVID-19 pandemic and explicit deposit insurance scheme: an analytical study

COVID-19 pandemic and explicit deposit insurance scheme: an analytical study

A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations

In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more difficult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using both the obtained solution of the conformable time-derivative heat equation and the inverse Cole-Hopf transformation. This approach shows its efficiency to deal with this class of fractional nonlinear problems. Some numerical experiments are displayed to consolidate our approach

On the arbitrary precision and the explicit time integration scheme for semi-discretized finite element method based using wave kernel function

On the arbitrary precision and the explicit time integration scheme for semi-discretized finite element method based using wave kernel function

Passive-guaranteed modeling and simulation of a finite element nonlinear string model

This paper proposes to solve the dynamics of the Kirchhoff-Carrier nonlinear string model using the finite elements method. In order to ensure the power balance of the resulting finite dimensional model it is rewritten in the Port-Hamiltonian System (PHS) formalism. Using a discrete gradient and a quadratization of the Hamiltonian, an explicit power-preserving numerical scheme is proposed. Results of simulation are presented.

Stability analysis of an explicit numerical scheme for the Allen-Cahn equation with high-order polynomial potentials

&lt;abstract&gt;&lt;p&gt;The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle.&lt;/p&gt;&lt;/abstract&gt;

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

An Explicit Scheme for Energy-Stable Simulation of Mass-Barrier Collisions with Contact Damping and Dry Friction

Collisions feature in the acoustics of many musical instruments, and as such need to be included in physical models that aim to capture the resulting dynamics. Previous studies have established various ways of constructing energy-stable schemes for numerical modelling of collisions, usually in implicit form. Recent methods use energy quadratisation in the derivation of explicit update forms, which are particularly useful in a real-time synthesis context. This approach is extended here by including contact damping and sliding friction in a 2D one-mass impact oscillator model, which serves as an initial testbed problem. The scheme’s guaranteed passivity is formally shown, and brief simulation and convergence results are included.

Сравнительный анализ проекционного и диффузионного методов обеспечения соленоидальности расчетного магнитного поля

Numerical modeling of current-carrying plasma dynamics is based, as a rule, on multiphysics models, which include equations describing MHD waves, transport and and dissipative processes accompanying the exchange of momentum and energy with the electromagnetic field. To solve the equations describing the evolution of the magnetic field, including as a result of magnetic diffusion, the grid system of equations of the discrete MHD model is not always constructed in such a way that the magnetic field divergence constraint is satisfied “automatically”. As a result, numerical errors can accumulate, creating the effect of the appearance of non-physical “magnetic charges” and plasma flows caused by these charges that do not correspond to the true physical situation. To maintain the solenoidal condition, calibration of the calculated magnetic field is used. In this work, in computational experiments with the MHD MARPLE code (M.V. Keldysh Institute of Applied Mathematics RAS), the practical accuracy of calibration methods such as projection and diffusion is assessed. The magnetic field correction was calculated using a stabilized explicit scheme for solving parabolic equations. It has been shown via numerical experiments that the diffusion method is more accurate and is not inferior to the projection method in terms of efficiency.