Dissipation Law Research Articles

Second-Order, Energy-Stable and Maximum Bound Principle Preserving Schemes for Two-Phase Incompressible Flow

In this paper, we propose several linear fully discrete schemes for the mass-conserved Allen-Cahn-Navier–Stokes equation, based on the generalized stabilized exponential scalar auxiliary variable approach in time and the marker and cell (MAC) scheme in space. It is quite remarkable that our schemes can guarantee second-order accuracy in space provided the maximum bound principle (MBP) is satisfied, whereas most previous work can only possess first-order accuracy in space. We rigorously show that the constructed schemes satisfy the unconditional energy dissipation law and preserve the MBP. Finally, various numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed schemes.

Experimental study on the splitting tensile failure of a carbon nanotube-modified fly ash foamed concrete filler

To study the enhancement effect of carbon nanotubes (CNTs) on the splitting tensile properties of foamed concrete backfill in which cement and fly ash were used as the cementitious materials and natural sand was used as the aggregate, specimens of CNT-modified foamed concrete backfill were prepared. Brazilian splitting tests were used to investigate the splitting tensile strength of the CNT-modified foamed concrete backfill, and the digital speckle correlation method was used to analyze the stress field characteristics and crack expansion law of the specimens during splitting tensile testing. The stress–strain characteristics and energy dissipation laws of the backfill were studied at various static loading rates, and a relationship between the splitting tensile strength, ultimate strain, and loading rate was established. The results showed that at the optimum CNT content of 0.05%, the peak strength and ultimate strain of the modified foamed concrete backfill increased by an average of 67.2% and 21.7%, respectively. Moreover, after modification with CNTs, the foamed concrete backfill was less likely to develop strain concentration areas before reaching peak strength. The triangular stable loadbearing structure formed by the modified foamed concrete backfill after splitting caused the end of the stress–strain curve to exhibit varying degrees of “backlash”. For the CNT-modified foamed concrete backfill, the peak strength correlated logarithmically with the loading rate, while the ultimate strain correlated as a power function of the loading rate. At a low loading rate, the CNT-modified foamed concrete backfill dissipated less energy, and the reverse was true for higher rates.

Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

On the maximum bound principle and energy dissipation of exponential time differencing methods for the matrix-valued Allen–Cahn equation

Abstract This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen–Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen–Cahn equation—both being linear schemes—unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Furthermore, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen–Cahn equation, and their convergence analysis is also provided. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields.

Energy Dissipation Law of the Temporal Variable-Step Fractional BDF2 Scheme for Space–Time-Fractional Cahn–Hilliard Equation

A high-order variable-step numerical scheme is formulated for the space–time-fractional Cahn–Hilliard equation, employing the variable-step fractional BDF2 formula. The unique solvability and mass conservation at the discretization setting are established. Subject to the constraint of time-step ratios, i.e., 0.4159≤rk≤4.660, a careful analysis based on the discrete gradient structure of the fractional BDF2 formula reveals that the proposed scheme adheres to the energy dissipation law. Remarkably, the modified energy exhibits asymptotic compatibility with that of the classical Cahn–Hilliard equation. Moreover, the modified energy dissipation law of the resulting scheme for the space–time-fractional Cahn–Hilliard equation aligns asymptotically with that of the variable-step BDF2 scheme for its classical counterpart. Finally, a few numerical experiments combined with the adaptive method are presented, which confirm the accuracy and efficiency of the proposed scheme.

Mechanical Failure Characteristics and Energy Dissipation Laws of the Coal–Concrete Combination under Impact Rates

Mechanical Failure Characteristics and Energy Dissipation Laws of the Coal–Concrete Combination under Impact Rates

Decoupled and energy stable schemes for phase-field surfactant model based on mobility operator splitting technique

In this paper, we investigate numerical methods for the phase-field surfactant (PFS) model, which is a gradient flow system consisting of two nonlinearly coupled Cahn-Hilliard type equations. The main challenge in developing high-order efficient energy stable methods for this system results from the nonlinearity and the strong coupling in the two variables in the free energy functional. We propose two fully decoupled, linear and energy stable schemes based on a linear stabilization approach and an operator splitting technique. We rigorously prove that both schemes can preserve the original energy dissipation law. The techniques employed in these schemes are then summarized into an innovative approach, which we call the mobility operator splitting (MOS), to design high-order decoupled energy stable schemes for a wide class of gradient flow systems. As a particular case, MOS allows different time steps for updating respective variables, leading to a multiple time-stepping strategy for fast-slow dynamics and thus serious improvement of computational efficiency. Various numerical experiments are presented to validate the accuracy, efficiency and other desired properties of the proposed schemes. In particular, detailed phenomena in thin-film pinch-off dynamics can be clearly captured by using the proposed schemes.

General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs

In this paper, we propose a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws, offering a unique approach from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent form, where the free energy of the transformed system takes a quadratic form in terms of the state variables. This transformation is known as energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced-order models that continue to respect the energy dissipation law. Secondly, our proposed numerical approach automatically provides algorithms to discretize these reduced-order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. Our platform offers a distinctive approach for deriving structure-preserving reduced-order models for a wide range of PDE systems with underlying thermodynamic principles.

Effects of bedding angles on rockburst proneness of layered anisotropic phyllites

To examine the effect of bedding angle upon burst proneness in terms of energy, phyllites with seven various bedding angles are selected for conventional uniaxial compression and single-cyclic loading–unloading uniaxial compression tests. The ejection and failure during compression process of phyllites are monitored in real-time by high-speed camera system. The results demonstrate that the phyllites with different bedding angles all consistently follow the linear energy storage and dissipation (LESD) law during compression. The ultimate energy storage of phyllites with varying bedding angles can be calculated precisely via using the LESD law. Based on this, four kinds of energy-based rockburst indices are applied to quantitatively assess the burst proneness for phyllites. Combined with the recorded images of high-speed camera system, ejection distance, and mass of rock fragments and powder, the burst proneness for phyllites with various bedding angles is qualitatively evaluated adopting the far-field ejection mass ratio. Next, burst proneness of anisotropic phyllites is assessed quantitatively and qualitatively. It is found that phyllites with bedding angles of 0°, 15°, and 90° have a high burst proneness, and that with bedding angle of 30° has a medium burst proneness, whereas the ones with bedding angles of 45°, 60°, and 75° have a low burst proneness. Finally, the published experimental data of shale and sandstone specimens with different bedding angles are extracted, and it is preliminarily verified that the bedding angle does not change the LESD law of rocks.

Failure characteristics and pressure relief effectiveness of non-persistent jointed rock mass with holes

In tunnel engineering, the rock mass contains a significant number of irregularly distributed joints, and typically exhibits high energy accumulation, thereby posing a risk of rockburst occurrence. Therefore, it is of paramount importance to investigate the fracture propagation behavior in jointed rock masses and assess the impact of borehole pressure relief on mitigating rockburst occurrences for effective prevention and control measures. This paper focuses on failure characteristics and pressure relief effectiveness of non-persistent jointed rock mass with holes through laboratory testing and numerical simulation. In laboratory experiments, rock samples are prepared to include a range of crack dip angles and circular holes. Then, the crack propagation law of crack inclination and circular hole is studied by AE and DIC technology. The experimental results show that with the increase of fracture dip angle, the peak strength and energy change of the sample decrease first and then increase. Due to the existence of holes, the crack propagation direction of the original crack is changed. After drilling, the strain energy of the sample is obviously reduced, which shows that the drilling pressure relief effect is obvious, which can effectively reduce the energy accumulated inside the rock mass and reduce the risk of rockburst. Finally, the PFC numerical simulation software is used to analyze the micro-failure process and energy change law of the sample from three aspects: the relative position of cracks and holes, the diameter of boreholes and the spacing of boreholes. Further understanding of the energy dissipation law and mechanical behavior characteristics of jointed rock mass provides a reference for exploring the pressure relief effect of rock mass and preventing rockburst.

Stability for degenerate wave equations with drift under simultaneous degenerate damping

In this paper we study the stability of two different problems. The first one is a one-dimensional degenerate wave equation with degenerate damping, incorporating a drift term and a leading operator in non-divergence form. In the second problem we consider a system that couples degenerate and non-degenerate wave equations, connected through transmission, and subject to a single dissipation law at the boundary of the non-degenerate equation. In both scenarios, we derive exponential stability results.

Investigation of Seismic Behavior of Slope Supported by Pile-Anchor Structure with Dampers

ABSTRACT The pile-anchor structure has been widely studied by global researchers. However, there is still little research on seismic optimization measures considering energy dissipation. In this research, the seismic performance of the pile-anchor structure is optimized by setting viscous dampers on the anchor cables. Shaking table tests are conducted to compare the seismic response of this new structure with that of the ordinary structure. Meanwhile, an analysis of the seismic energy dissipation law under the support of both structures was conducted. Those results show that the seismic performance of the pile-anchor structure can be greatly improved by setting dampers.

Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows

The scalar auxiliary variable (SAV) approach was considered by Shen et al. in Shen et al. (2019) and has been widely used to simulate a series of gradient flows. However, the SAV-based schemes are known for the stability of a ‘modified’ energy. In this paper, we construct a series of modified SAV approaches with unconditional energy dissipation law based on several improvements to the classic SAV approach. Firstly, by introducing the three-step technique, we can reduce the number of constant coefficient linear equations that need to be solved at each time step, while retaining all of its other advantages. Secondly, the addition of energy-optimal technique and Lagrange multiplier technique can make the numerical schemes have the advantage of preserving the original energy dissipation. Thirdly, we use the first-order approximation of the energy balance equation in the GSAV approach, instead of discretizing the dynamic equation of the auxiliary variable, so that we can construct the high-order unconditional original energy stable numerical schemes. Finally, representative numerical examples show that the efficiency and accuracy of the proposed schemes are improved.

Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the non-preservation of steady-state solution and original energy dissipation law. To overcome these disadvantages, some new integrating factor methods are developed by using two classes of difference correction, including the telescopic correction and nonlinear-term translation correction, enforcing the preservation of steady-state solution. Then the original energy dissipation properties of the new methods are examined by using the associated differential forms and the differentiation matrices. As applications, some new integrating factor Runge-Kutta methods up to third-order maintaining the original energy dissipation law are constructed by applying the difference correction strategies to some popular explicit integrating factor methods in the literature. Extensive numerical experiments are presented to support our theory and to demonstrate the improved performance of new methods.

An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach

Abstract An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=\tau _{k}/\tau _{k-1}< r_{\max }\approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.

An artificial compressibility SAV finite element method for the time-dependent natural convection problem

In this article, we present, analyze, and test a first-order implicit-explicit type scheme based on the artificial compressibility (AC) method and the scalar auxiliary variable (SAV) approach for solving the time-dependent natural convection problem. The proposed method is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization, and explicit treatment for nonlinear terms. It effectively decouples the velocity and pressure via artificial compressibility, thereby reducing computational complexity and execution time. Moreover, we use the SAV approach for the convective terms, it leads to the scheme is linear, and only require solving a sequence of linear differential equation with constant coefficients at each time step. It is proved that the solution of this method is bounded and satisfying the law of energy dissipation. The rigorous error estimate of velocity, pressure, and temperature are derived. Finally, some numerical tests are implemented to verify the theoretical analysis and illustrate the efficiency of the presented method.

Energy-stable auxiliary variable viscosity splitting (AVVS) method for the incompressible Navier–Stokes equations and turbidity current system

In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model

AbstractWe study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

Analysis and numerical methods for nonlocal‐in‐time Allen‐Cahn equation

AbstractIn this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.