Finite-difference Calculations Research Articles

Study on progressive failure mode of surrounding rock of shallow buried bias tunnel considering strain-softening characteristics

Mountain tunnels portal often have to pass through slope terrain unavoidably, thus forming a shallow buried bias tunnel. During the construction of shallow buried bias tunnel, disasters such as slope sliding and tunnel collapse frequently occur. The failure mode of surrounding rock obtained by current research is based on the limit equilibrium theory, which cannot reflect the progressive failure characteristics of the surrounding rock of shallow buried bias tunnel. In order to reveal the failure mechanism of the gradual instability of surrounding rock of shallow buried bias tunnel, the problem of gradual failure of the surrounding rock is reduced to an elastic–plastic analysis problem for surrounding rock considering the strain-softening characteristics. Based on the elastic–plastic analysis of the failure process of shallow buried bias tunnel, MATLAB was used to compile a program to read the finite-difference calculation result file, extract the effective information such as shear strain and tensile strain at the center point of each unit, and establish the analysis method of the progressive failure mode of shallow buried bias tunnel. The reliability of the method proposed was verified by comparing the failure process of the model test with the development process of shear strain increment. Under the condition of no support, the formation mechanism of failure plane of surrounding rock on both sides of shallow buried bias tunnel is different. The shallow buried side is the shear failure plane formed by the collapse of surrounding rock, while the deep buried side of the tunnel is the shear failure plane formed by the collapse of surrounding rock and slope sliding. Under the conditions of excavation and support, the failure plane of the shallow buried bias tunnel can be divided into three parts according to the formation sequence and reasons. The part I is the failure plane, which is formed by active shear under the influence of tunnel excavation. The part II is the failure plane formed by tensile crack of slope top. The part III is the failure plane formed by passive shear under the push of the soil in the upper part of the slope.

Fracture prediction and application in carbonate reservoirs based on the rescaled range analysis- finite difference numerical analysis methods

For carbonate reservoirs, the study of fractures is crucial. In this paper, the identification and prediction of fractures in carbonate reservoirs are carried out by applying R/S-FD numerical analysis method by using core, thin section and imaging logging data. The analysis concludes that the AC, CAL, DEN, and GR curves are very sensitive to the response of fracture development segments as calculated by the rescaled range analysis (R/S) analysis method, but some fractures are still misidentified. After finite difference (FD) calculation on the basis of R/S calculation, high-angle structural fractures can be accurately identified, and the R/S-FD fracture identification results of different curves can be verified with the results of core and imaging logging, and the fracture identification accuracy can reach 71.69%. The number of fracture development segments is positively correlated with the cumulative oil production, the number of producing layers and the cumulative thickness of producing layers, and the wells with more fracture development segments have higher cumulative oil production, more producing layers and larger cumulative thickness of producing layers. This method improves the accuracy of fracture identification in carbonate reservoirs, which is of great significance for oil and gas exploration.

Theoretical Study on Diaphragm Wall and Surface Deformation Due to Foundation Excavation Based on Three-Parameter Kerr Model

To calculate the horizontal displacement of the diaphragm wall and surface settlement caused by foundation pit excavation, the three-parameter Kerr foundation model was applied to a diaphragm wall and derived the flexural differential equations of the diaphragm wall and calculated the horizontal displacement of the diaphragm wall using the finite difference calculation method. The boundary element method combined with the Mindlin displacement solution was then used to invert the additional horizontal stress near the diaphragm wall. Lastly, the Mindlin solution was used to calculate the surface settlement. The effectiveness of the proposed calculation method was verified by comparing the horizontal displacement of the diaphragm wall and the surface settlement between the theoretical calculation and the actual project. The theory proves that there is a certain connection between the horizontal displacement of the diaphragm wall and the surface settlement, and the horizontal displacement of the diaphragm wall is larger than the surface settlement. Using this theory to further analyze the foundation pit construction parameters, the greater the thickness and elasticity modulus of the diaphragm wall, and the greater the diameter and number of internal supports, the smaller the horizontal displacement of the diaphragm wall and the surface settlement. The theory can accurately predict the horizontal displacement of the diaphragm wall and surface settlement and provides guidance for the construction of foundation pit projects.

Calculating high-pressure vibrational frequencies analytically with the extended hydrostatic compression force field approach.

We report the implementation of the analytical Hessian for the mechanochemical extended hydrostatic compression force field method in the Q-Chem program package. To verify the implementation, the analytical Hessian was compared with finite difference calculations. In addition, we calculated the pressure dependency of the Raman active vibrational modes of methane, ethane, and hydrogen, as well as all IR and Raman active modes of Buckminsterfullerene, and compared the results with experimental and theoretical data. Our implementation paves the way for the analysis of geometric points on a pressure-deformed potential energy surface and provides a straightforward model to calculate the vibrational properties of molecules under high pressure.

Oceanic crust—seismic structure, lithology and the cause of the 2A Event at borehole 504B

SUMMARY This study focuses on the 3-D velocity structure and thickness of ∼7-Myr-old oceanic crust surrounding borehole 504B, located ∼235 km from the intermediate-spreading Costa Rica Rift (Panama Basin). It investigates how well seismic structure determined by 3-D tomography compares with actual lithology and, consequently, what the origin and cause might be of an amplitude anomaly, the 2A Event, that is observed in multichannel seismic data. Our P-wave model shows an ∼0.3-km-thick sediment layer of velocity between ∼1.6 and 1.9 km s−1 (gradient 1.0 s−1), bound at its base by a velocity step to 4.8 km s−1 at the top of oceanic crustal Layer 2. Layer 2 itself is subdivided into two main units (2A and 2B) by a vertical velocity gradient change at 4.5 km depth, with a gradient of 1.7 s−1 above (4.8–5.8 km s−1) and 0.7 s−1 below (5.8–6.5 km s−1). The base of Layer 2, in turn, is defined by a change in gradient at 5.6 km depth. Below this, Layer 3 has a velocity range of 6.5–7.5 km s−1 and a gradient of ∼0.3 s−1. Corresponding S-wave igneous layer velocities and gradients are: Layer 2A, 2.4–3.1 km s−1 and 1.0 s−1; Layer 2B, 3.1–3.7 km s−1 and 0.5 s−1; Layer 3, 3.7–4.0 km s−1 and 0.1 s−1. The 3-D tomographic models, coupled with gravity modelling, indicate that the crust is ∼6 km thick throughout the region, with a generally flat-lying Moho. Although the P- and S-wave models are smooth, their velocities and gradients are remarkably consistent with the main lithological layering subdivisions logged within 504B. Thus, using the change in velocity gradient as a proxy, Layer 2 is interpreted as ∼1.8 km thick and Layer 3 as ∼3.8 km thick, with little vertical variation throughout the 3-D volume. However, the strike of lateral gradient variation is not Costa Rica Rift-parallel, but instead follows the orientation of the present-day adjacent Ecuador Rift, suggesting a reorientation of the Costa Rica Rift spreading ridge axis. Having determined its consistency with lithological ground-truth, the resulting P-wave model is used as the basis of finite difference calculation of wave propagation to find the origin of the 2A Event. Our modelling shows that no distinct interface, or transition, is required to generate this event. Instead, it is caused by averaging of heterogeneous physical properties by the seismic wave as it propagates through Layer 2 and is scattered. Thus, we conclude that the 2A Event originates and propagates exclusively in the lower part of Layer 2A, above the mean depth to the top of the dykes of Layer 2B. From our synthetic data we conclude that using the 2A Event on seismic reflection profiles as a proxy to determine the Layer 2A/2B boundary's depth will result in an overestimate of up to several hundred metres, the degree of which being dependent on the specific velocity chosen for normal moveout correction prior to stacking.

Efficient physics-informed neural networks using hash encoding

Physics-informed neural networks (PINNs) have attracted much attention in scientific computing as their functional representation of partial differential equation (PDE) solutions offers flexibility and accuracy features. However, their training cost has limited their practical use as a real alternative to classic numerical methods. Thus, we propose to incorporate multi-resolution hash encoding into PINNs to improve the training efficiency, as such encoding offers a locally-aware (at multi-resolution) coordinate input to the neural network. Borrowed from the neural representation field community (NeRF), we investigate the robustness of calculating the derivatives of such hash-encoded neural networks with respect to the input coordinates, which is often needed by the PINN loss terms. We propose to replace the automatic differentiation with finite-difference calculations of the derivatives to address the discontinuous nature of such derivatives. We also share the appropriate ranges for the hash encoding hyperparameters to obtain robust derivatives. We test the proposed method on several benchmark problems, and the proposed method admits about a 10-fold improvement in efficiency over the vanilla PINN implementation.

Digital visualization in graduate transport phenomena

AbstractThe use of contour flood plots and color digital movies is investigated for teaching fluid mechanics, heat transfer, and mass transfer at the first‐year graduate student level. The color visualizations are intended to not only stimulate curiosity but also encourage enthusiasm and determination in learning difficult material. Although there is a significant amount of finite‐difference material, the focus is on analytical solutions. In particular, the mathematical problems involve error functions and Fourier series, which generally prove uninteresting and unpopular. Students are encouraged to produce the visualizations themselves, in addition to being provided with digital results in various forms. Excel can be used to generate both the analytical and finite difference calculations, but Python or another computer language can be employed. While software, such as Matplotlib with Python, Gnuplot, or Tecplot, can create excellent contour flood plots, results from Excel are also good, and Powerpoint will produce movies from individual frames. Student response is positive, although quantifying their reaction is difficult based on the small class sizes. Not surprisingly, the use of color appeals to students and adds to their interest. In addition, students claim increased understanding with contour plots and movies, but the visualizations should be used in conjunction with more traditional graphs, perhaps augmented with consistent use of color. However, the impact on motivation due to color presentation is harder to establish. Interestingly, both graduate and senior undergraduate students showed about equal preference for the rainbow and viridis color palettes.

Surface phase-field surfactant fluid model and its practical closest point type finite difference computation

To investigate the fluid surfactant material on three-dimensional objects, we herein develop a simple and practical model based on phase-field method and design its efficient algorithm. The binary fluids and surfactant are represented by two order parameters. The fluid evolution is governed by the incompressible Navier–Stokes equations. To achieve high-performance computation, a linear semi-implicit scheme with stabilizers is proposed for the phase-field type equations in time. The projection method with pressure correction is used to decouple the velocity and the pressure. Based on the closest-point type embedded narrow domain method and a simple pseudo-Neumann boundary treatment, the calculation on curved surface is extended into a narrow band region including the surface. The surface spatial operators are approximately replaced by the standard spatial operators. In the three-dimensional narrow band, the finite difference method is used to discretize the equations on staggered grids. The mass compensation and vector projection techniques are designed to obtain physically meaningful solutions. In each time iteration, the proposed algorithm is efficient to implement. Numerical results indicate that the proposed method is capable to simulate surfactant-laden phase separation, fluid instabilities, and coarsening dynamics with variable mobility on various surfaces.

Anisotropy distorts the spreading of a fixed volume porous gravity current

We consider the release and subsequent gravity-driven spreading of a dense finite volume of fluid in an anisotropic porous medium bounded by an impermeable substrate. When the permeability in the vertical direction is much smaller than in the horizontal direction, as is the case in many real geological reservoirs, this restricts the spread of the current to a very thin layer near the impermeable base. Using a combination of asymptotic analysis and finite difference computations of Darcy flow, we show that there exist two distinct flow regimes. At early times, the bulk of the current descends slowly and uniformly, injecting fluid into thin finger-like regions near the base. At much later times, the current transitions to the classical gravity-driven solution and continues to spread with a self-similar shape. One interesting consequence is that the swept volume of the current grows differently depending on the anisotropy of the medium. This has important consequences for managing contaminant spills, where it is important to minimize the contacted volume of the aquifer, or during geological CO 2 sequestration where a larger contacted volume results in more CO 2 being stored.

Significance of body acceleration and gold nanoparticles through blood flow in an uneven/composite inclined stenosis artery: A finite difference computation

Stenosis is a tiny plaque-like structure that builds up in the arterial wall owing to the sediment of cholesterol, fats, and pearly substances. Such inward proliferation in arteries significantly inhibits blood flow, which leads to a lack of nutrients and oxygen in the organs. Therefore, exploring the transport characteristics of blood fluid flow in stenosis arteries plays a prominent role in enhancing blood transportation. As a result, the present mathematical model is devoted to scrutinizing the flow of Sutterby gold blood nanofluid in two distinct stenosis arteries with periodic body acceleration. It is observed that the Sutterby rheology model is treated as blood, and the single-phase model is used for exposing the nanofluid behaviour. Dimensional non-linear PDEs of the current model are reduced to the set of dimensionless PDEs with the help of non-similar variables. A finite-difference approach is manipulated to compute the dimensionless PDEs. The physical features of governing flow parameters on the Sutterby nanofluid velocity, temperature, resistance impedance, flow rate, and wall shear stress are exposed through graphs. It is found that the composite stenosis has a lower wall shear stress than the irregular stenosis. Sutterby blood nanofluid velocity is elevated with the rising of nanoparticle volume fraction. When employing a gold nanofluid containing 5% volume fraction, the temperature in the irregular artery rises by 26.004% compared to the base fluid (blood). Similarly, in the composite artery, utilizing the same 5% volume fraction of the gold nanofluid leads to a temperature increase of 32.6207%. The blood flow pattern exhibits a 0.2340% higher in the irregular artery as compared to the composite artery.

Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation

The dispersion analysis of numerical schemes for finite difference computations based on 2D compressible linearized Navier–Stokes equation (LNSE) is presented here. The analysis presents a more holistic view of the applicability of the schemes while solving compressible NSE, rather than analyzing model 1D or 2D convection or convection–diffusion equations. A theoretical dispersion analysis of model 2D−LNSE is performed first which categorizes it into vortical, entropic and two acoustic modes. The variation of the theoretical dispersion relation for each of the modes with wavenumber, Mach number and Reynolds number is analyzed. It is noted that the entropic mode is the most diffusive among all the modes. Two acoustic modes are less diffusive than the vortical mode up to a certain absolute wavenumber Kb depending upon the Mach number and Reynolds number. It is also shown that while vortical and entropic modes are non-dispersive in nature, the two acoustic modes are dispersive only up to K=Kb. Next, numerical dispersion and diffusion corresponding to each of the modes are analyzed for second order central difference (CD2) and sixth order central compact difference scheme proposed in Lele (J. Comput. Phys., Vol-103(1), 1992) for discretization of convective and diffusive derivatives using fourth-order Runge–Kutta (RK4) time integration method. Results show the existence of numerical unstable zone and regions where spurious upstream propagating waves are noted for each of the modes in the spectral plane. From these and the relative size of the dispersion relation preserving zones for the modes, applicability of the numerical schemes can be assessed. Analysis of the variation of the numerical unstable zone with Mach number, orientation of the mean flow and the unit CFL number Nc=Δt/Δx for each of the modes are presented. It is noted that entropic mode displays the lowest critical CFL number of all the four modes. Lastly, linearized and nonlinear numerical simulations for the convection of acoustic and vortical pulse with varying spectral content are presented. The linearized simulations are also compared with corresponding exact solutions. Obtained results indicate a direct correspondence of the stability analysis performed based on 2D−LNSE even for nonlinear simulations which result in additional intra modal energy transfer and interactions.

Iron Permeable Reactive Barrier for Removal of Lead from Contaminated Groundwater

The possibility of using zero-valent iron as permeable reactive barrier in removing lead from a contaminated groundwater was investigated. In the batch tests, the effects of many parameters such as contact time between adsorbate and adsorbent (0-240 min), initial pH of the solution (4-8), sorbent dosage (1-12 g/100 mL), initial metal concentration (50-250 mg/L), and agitation speed (0-250 rpm) were studied. The results proved that the best values of these parameters achieve the maximum removal efficiency of Pb+2 (=97%) were 2 hr, 5, 5 g/100 mL, 50 mg/L and 200 rpm respectively. The sorption data of Pb+2 ions on the zero-valent iron have been performed well by Langmuir isotherm model in compared with Freundlich model under the studied conditions. Finite difference method and computer solutions (COMSOL) multiphysics 3.5a software based on finite element method were used to simulate the one-dimensional equilibrium transport of lead through sand aquifer with and without presence of barrier. The predicted and experimental results proved that the reactive barrier plays a potential role in the restriction of the contaminant plume migration and a reasonable agreement between these results was recognized.

Transition moments beyond the electric-dipole approximation: Visualization and basis set requirements.

In the simulation of x-ray absorption spectroscopy, the validity of the electric-dipole approximation comes into question. Three different schemes exist to go beyond this approximation: the first scheme is based on the full semi-classical light-matter interaction, whereas the latter two schemes, referred to as the generalized length and velocity representation, are based on truncated multipole expansions. Even though these schemes have been successfully implemented in several quantum chemistry codes, their basis set requirements remained largely unknown. Here, we assess basis set requirements of these three schemes. We have considered 1s1/2 and 7s1/2 → 7p1/2 transitions in the radium atom, representative of core and valence excitations, respectively, and carried out calculations with dyall.aeXz (X = 2, 3, 4) basis sets at the four-component relativistic TD-HF level of theory. Our basis set study was greatly facilitated by the generation and visualization of radial distributions of transition moment densities, allowing for a straightforward comparison with equivalent finite-difference calculations. Pertaining to the truncated interaction, we find that the length representation electric multipole is the easiest to converge, requiring the dyall.ae2z basis for low-order multipoles and the dyall.ae4z basis at higher orders. The magnetic multipole moments follow a similar trend although they are more difficult to converge. The velocity representation electric multipoles are the most difficult to converge: at high orders, the dyall.ae3z and dyall.ae4z basis sets introduce artificial peaks and oscillations, which increase the overall error. These artifacts are associated with linear dependence issues in the small component space of larger basis sets. The full interaction operator, however, does not suffer from these problems, and we therefore recommend its use in the simulation of x-ray spectroscopy.

Group refractive index via auto-differentiation and neural networks

In this article, using principles of automatic differentiation, we demonstrate a generic deep learning representation of group refractive index for photonic channel waveguides. It enables evaluation of group refractive indices in a split of second, without any traditional numerical calculations. Traditionally, the group refractive index is calculated by a repetition of the optical mode calculations via a parametric wavelength sweep of finite difference (or element) calculations. To the direct contrary, in this work, we show that the group refractive index can be quasi-instantaneously obtained from the auto-gradients of the neural networks that models the effective refractive index. We embed the wavelength dependence of the effective index in the deep learning model by applying the scaling property of the Maxwell’s equations and this eliminates the problems caused by the curse of dimensionality. This work portrays a very clear illustration on how physics-based derived optical quantities can be calculated instantly from the underlying deep learning models of the parent quantities using automatic differentiation.

Improved fast iterative method for higher calculation accuracy of traveltimes

Fast Iterative Method (FIM), a grid-based ray-tracing algorithm, has a computational efficiency far exceeding that of conventional fast marching method (FMM) in serial computing, because it adopts an iterative algorithm and can be calculated in parallel. However, the finite difference scheme of the FIM only employs the first-order Godunov upwind finite difference scheme, resulting in low calculation accuracy of arrival traveltimes. This paper proposes a new algorithm of FIM, which utilizes second-order and mixed-order difference schemes under the parallel computing structure, effectively improves the calculation accuracy of traveltimes, and basically maintains the original computational efficiency. Especially, improved finite difference calculation in the diagonal direction of square grids and the double-grid technique near the source are applied to further reduce overall calculation error. Numerical experiments show that for the given model, the calculation error can be reduced from 2 ms of the original FIM calculation to 0.05 ms, and the calculation accuracy is increased by 40 times, while the time consumption increases only 27%.

Machine Learning Predicting Optimal Preparation of Silica-Coated Gold Nanorods for Photothermal Tumor Ablation

Gold nanorods (GNRs) coated with silica shells are excellent photothermal agents with high surface functionality and biocompatibility. Understanding the correlation of the coating process with both structure and property of silica-coated GNRs is crucial to their optimizing preparation and performance, as well as tailoring potential applications. Herein, we report a machine learning (ML) prediction of coating silica on GNR with various preparation parameters. A total of 306 sets of silica-coated GNRs altogether were prepared via a sol-gel method, and their structures were characterized to extract a dataset available for eight ML algorithms. Among these algorithms, the eXtreme gradient boosting (XGboost) classification model affords the highest prediction accuracy of over 91%. The derived feature importance scores and relevant decision trees are employed to address the optimal process to prepare well-structured silica-coated GNRs. The high-throughput predictions have been adopted to identify optimal process parameters for the successful preparation of dumbbell-structured silica-coated GNRs, which possess a superior performance to a conventional cylindrical core-shell counterpart. The dumbbell silica-coated GNRs demonstrate an efficient enhanced photothermal performance in vivo and in vitro, validated by both experiments and time domain finite difference calculations. This study epitomizes the potential of ML algorithms combined with experiments in predicting, optimizing, and accelerating the preparation of core-shell inorganic materials and can be extended to other nanomaterial research.

Multiscale extensions for enhancing coarse grid computations

Many phenomena in Physics and Engineering are modeled by Partial Differential Equations. Typically, analytical solutions of such problems are unknown. Hence, numerical schemes are applied for approximating the exact solutions. In order to accurately approximate the exact solutions, very fine grids are required. However, the computations become too costly. In this work, a multiscale iterative approach is utilized for enhancing the accuracy of coarse grid computations which results from a high-order finite difference scheme. The main idea is to run the finite difference computations on a coarse grid until some intermediate time level. Then, the coarse grid results are interpolated and extended to a fine grid using Laplacian Pyramids, a multiscale iterative approach. Finally, the finite difference scheme is employed on the finer grid until a prescribed final time. Comparing the results to those obtained on a fine grid, we see that the convergence rates obtained from the two methods are comparable, while the computational time is significantly reduced.A modified multimodal Laplacian Pyramids algorithm for predicting future values of the solution is also suggested. The method approximates and extends a function based on two or more input modalities coded by a series of multiscale kernels, which are averaged as a convex combination. In this work, the modalities are the numerical model’s approximations of the solution of the differential equation and its derivative at previous time steps, and the goal is to predict the solution at a proceeding time step. It can be seen that, by adapting the convex combination of the kernels to local regions of the solution with stronger or weaker gradients, the predicted results are improved.

Rational Fabrication of Ag Nanocone Arrays Embedded with Ag NPs and Their Sensing Applications.

Colloidal lithography is used to design and construct a high-performance plasmonic sensor based on Ag nanocone arrays embedded with Ag NPs. The surface plasmon polariton (SPP) of the Ag nanocone array and the localized surface plasmon resonance (LSPR) of Ag NPs inside the nanocones can both couple incident photons. Sharp reflectance troughs are considerably enhanced by coupling the SPPs and LSPR, which is made possible by carefully tuning the nanocone sizes. To maximize the line shape and sensitivity, other geometric factors, such as the thickness of the silver layer and the size of the Ag NPs, are modified. Finite-difference time-domain computations confirm these hypotheses and experimental findings. We use well-researched solvents with various refractive indices as a model system to demonstrate good sensing performance as a proof of concept. The crystal used in this investigation has the ideal refractive index sensitivity, having 500 nm lattice constant, 350 nm nanocone height, and 350 nm base diameter (aspect ratio = 1). The Ag nanocone array embedded with Ag NPs is a good contender for a sensing platform due to its compact structure and efficient read-out apparatus.

Adjoint Equations for Beam-Wave Interaction and Optimization of TWT Design

By formulating and solving the adjoint equations governing the beam–wave interaction in a traveling-wave tube (TWT), we show the partial derivatives with respect to design parameters of various TWT figures of merit (FoMs) may be efficiently calculated. FoMs include average gain, gain flatness, and gain–bandwidth product, and design parameters include beam voltage and circuit geometry. We use these derivatives in an optimization algorithm that finds parameter values that minimize or maximize the desired FoM. We show that a 1-D large-signal simulation code, such as CHRISTINE, may be easily modified to compute the adjoint solutions. We further show that only three runs of the modified code suffice to compute the partial derivatives of the output power and phase at a specified frequency with respect to an arbitrary number of parameters, resulting in potentially large savings in computing time compared with direct, finite difference calculation of the partial derivatives. We illustrate the method by optimizing the beam voltage and gap spacing of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> -band folded-waveguide (FWG) TWT. The formulation given here applies only to TWT slow wave structures, such as coupled-cavity and FWGs, and to klystrons, composed of discrete gaps followed by drift spaces; it does not apply to helix structures, which may be the subject of a future paper.

Anti-reflection metallic anode-enhanced performance of organic solar cells via cross coupling between Fabry-Perot cavity modes and microcavity modes.

An effective anti-reflection metallic anode with the structure of glass/dielectric2 /Ag (D1D2M) is demonstrated both in small-molecule (SM) and conjugated polymer (CP) organic solar cells (OSCs). The anti-reflection mechanism is investigated by the finite-difference time-domain numerical calculation method and the experimental method. By tuning the refractive index and the thickness of the D2 layer, the reflection light is confined in the Fabry-Perot (F-P) cavity modes, which effectively enhances the transmittance of the D1D2M anode in the wavelength range of 420 nm-800 nm. Compared with the conventional glass/Ag (D1M) anode, the experimental transmittance of the D1D2M anode is enhanced by 33.24% at a wavelength of 550 nm. By replacing the D1M anode with the D1D2M anode in the OSCs, the F-P cavity modes cross couple with the microcavity modes in the active layers. As a result, the absorption intensity is obviously increasing in a wide angle range (0≤θ≤85∘) in the wavelength ranges of 475 nm-650 nm and 540 nm-720 nm for the SM and CP OSCs, respectively. The short circuit current density and power conversion efficiency of the SM OSC is increased by 25.07% and 27.23%, respectively.