Physics-informed neural network-based petroleum reservoir simulation with sparse data using domain decomposition

Recent advances in deep learning have expanded new possibilities for fluid flow simulation in petroleum reservoirs.However,the predominant approach in existing research is to train neural networks using high-fidelity numerical simulation data.This presents a significant challenge because the sole source of authentic wellbore production data for training is sparse.In response to this challenge,this work introduces a novel architecture called physics-informed neural network based on domain decomposition(PINN-DD),aiming to effectively utilize the sparse production data of wells for reservoir simulation with large-scale systems.To harness the capabilities of physics-informed neural networks(PINNs)in handling small-scale spatial-temporal domain while addressing the challenges of large-scale systems with sparse labeled data,the computational domain is divided into two distinct sub-domains:the well-containing and the well-free sub-domain.Moreover,the two sub-domains and the interface are rigorously constrained by the governing equations,data matching,and boundary conditions.The accuracy of the proposed method is evaluated on two problems,and its performance is compared against state-of-the-art PINNs through numerical analysis as a benchmark.The results demonstrate the superiority of PINN-DD in handling large-scale reservoir simulation with limited data and show its potential to outperform conventional PINNs in such scenarios.

A Parallel Global-Local Mixed Evolutionary Algorithm for Multimodal Function Optimization Based on Domain Decomposition

This paper presents a parallel two-level evolutionary algorithm based on domain decomposition for solving function optimization problem containing multiple solutions. By combining the characteristics of the global search and local search in each sub-domain, the former enables individual to draw closer to each optima and keeps the diversity of individuals, while the latter selects local optimal solutions known as latent solutions in sub-domain. In the end, by selecting the global optimal solutions from latent solutions in each sub-domain, we can discover all the optimal solutions easily and quickly.

A refined Frequency Domain Decomposition tool for structural modal monitoring in earthquake engineering

Output-only structural identification is developed by a refined Frequency Domain Decomposition（rFDD） approach, towards assessing current modal properties of heavy-damped buildings（in terms of identification challenge）, under strong ground motions. Structural responses from earthquake excitations are taken as input signals for the identification algorithm. A new dedicated computational procedure, based on coupled Chebyshev Type Ⅱ bandpass filters, is outlined for the effective estimation of natural frequencies, mode shapes and modal damping ratios. The identification technique is also coupled with a Gabor Wavelet Transform, resulting in an effective and self-contained time-frequency analysis framework. Simulated response signals generated by shear-type frames（with variable structural features） are used as a necessary validation condition. In this context use is made of a complete set of seismic records taken from the FEMA P695 database, i.e. all 44 ＂Far-Field＂（22 NS, 22 WE） earthquake signals. The modal estimates are statistically compared to their target values, proving the accuracy of the developed algorithm in providing prompt and accurate estimates of all current strong ground motion modal parameters. At this stage, such analysis tool may be employed for convenient application in the realm of Earthquake Engineering, towards potential Structural Health Monitoring and damage detection purposes.

Domain Decomposition for Wavelet Single Layer on Geometries with Patches

We focus on the single layer formulation which provides an integral equation of the first kind that is very badly conditioned. The condition number of the unpreconditioned system increases exponentially with the multiscale levels. A remedy utilizing overlapping domain decompositions applied to the Boundary Element Method by means of wavelets is examined. The width of the overlapping of the subdomains plays an important role in the estimation of the eigenvalues as well as the condition number of the additive domain decomposition operator. We examine the convergence analysis of the domain decomposition method which depends on the wavelet levels and on the size of the subdomain overlaps. Our theoretical results related to the additive Schwarz method are corroborated by numerical outputs.

A Fully Nonlinear HOBEM with the Domain Decomposition Method for Simulation of Wave Propagation and Diffraction

A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.

Meshless Method with Domain Decomposition for Submerged Porous Breakwaters in Waves

Based on the improved version of the meshless singular boundary method(ISBM)in multi domain(MD),a numerical method is proposed in this paper to study the interaction of submerged permeable breakwaters and regular waves at normal incidence.To account for fluid flow inside the porous breakwaters,the conventional model of Sollitt and Cross for porous media is adopted.Both single and dual trapezoidal breakwaters are examined.The physical problem is formulated in the context of the linear potential wave theory.The domain decomposition method(DDM)is employed,in which the full computational domain is decomposed into separate domains,that is,the fluid domain and the domains of the breakwaters.Respectively,appropriate mixed type boundary and continuity conditions are applied for each subdomain and at the interfaces between domains.The solution is approximated in each subdomain by the ISBM.The discretized algebraic equations are combined,resulting in an overdetermined full system that is solved using a least-square solution procedure.The numerical results are presented in terms of the hydrodynamic quantities of reflection,transmission,and wave-energy dissipation.The relevance of the results of the present numerical procedure is first validated against data of previous studies,and then selected computations are discussed for various structural conditions.The proposed method is demonstrated to be highly accurate and computationally efficient.

Modified domain decomposition method for Hamilton-Jacobi-Bellman equations

This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and in- compressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangu- lar sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method （CSM） is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neu- mann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet bound- ary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison be- tween the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

Parallel computing for finite element structural analysis using conjugategradient method based on domain decomposition

Parallel finite element method using domain decomposition technique is adapted to a distributed parallel environment of workstation cluster. The algorithm is presented for parallelization of the preconditioned conjugate gradient method based on domain decomposition. Using the developed code, a dam structural analysis problem is solved on workstation cluster and results are given. The parallel performance is analyzed.

Fourier analysis of Schwarz domain decomposition methods for the biharmonic equation

Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.

A DOMAIN DECOMPOSITION ALGORITHM WITH FINITE ELEMENT-BOUNDARY ELEMENT COUPLING

A domain decomposition algorithm coupling the finite element and the boundary element was presented. It essentially involves subdivision of the analyzed domain into sub-regions being independently modeled by two methods, i.e., the finite element method （FEM） and the boundary element method （BEM）. The original problem was restored with continuity and equilibrium conditions being satisfied on the interface of the two sub-regions using an iterative algorithm. To speed up the convergence rate of the iterative algorithm, a dynamically changing relaxation parameter during iteration was introduced. An advantage of the proposed algorithm is that the locations of the nodes on the interface of the two sub-domains can be inconsistent. The validity of the algorithm is demonstrated by the consistence of the results of a numerical example obtained by the proposed method and those by the FEM, the BEM and a present finite element-boundary element （FE-BE） coupling method.

DOMAIN DECOMPOSITION METHODS FOR SOLVING PDE's ON MULTI-PROCESSORS

In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.

Method for wheat ear counting based on frequency domain decomposition of MSVF-ISCT

Wheat ear counting is a prerequisite for the evaluation of wheat yield.A wheat ear counting method based on frequency domain decomposition is proposed in this study to improve the accuracy of wheat yield estimation.The frequency domain decomposition of wheat ear image is completed by multiscale support value filter(MSVF)combined with improved sampled contourlet transform(ISCT).Support Vector Machine(SVM)is the classic classification and regression algorithm of machine learning.MSVF based on this has strong frequency domain filtering and generalization ability,which can effectively remove the complex background,while the multi-direction characteristics of ISCT enable it to represent the contour and texture information of wheat ears.In order to improve the level of wheat yield prediction,MSVF-ISCT method is used to decompose the ear image in multiscale and multi direction in frequency domain,reduce the interference of irrelevant information,and generate the sub-band image with more abundant information components of ear feature information.Then,the ear feature is extracted by morphological operation and maximum entropy threshold segmentation,and the skeleton thinning and corner detection algorithms are used to count the results.The number of wheat ears in the image can be accurately counted.Experiments show that compared with the traditional algorithms based on spatial domain,this method significantly improves the accuracy of wheat ear counting,which can provide guidance and application for the field of agricultural precision yield estimation.

Deep Domain Decomposition Methods:Helmholtz Equation

This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.

CONSTRUCTION OF A PRECONDITIONER FOR DOMAIN DECOMPOSITION METHODS WITH POLYNOMIAL LAGRANGIAN MULTIPLIERS

In this paper we consider domain decomposition methods with polynomial Lagrangian multipliers to two-dimentional elliptic problems, and construct a kind of simple preconditioners for the corresponding interface equation. It will be shown that condition number of the resulting preconditioned interface matrix is almost optimal (namelys it has only logarithmic growth with dimension of the local interface space).

GLOBAL SUPERCONVERGENCES OF THE DOMAIN DECOMPOSITION METHODS WITH NONMATCHING GRIDS

Focuses on a study which determined the use of the global convergences of the domain decomposition methods with Lagrangian multiplier and nonmatching grids in solving the second order elliptic boundary value problems. Background on domain decomposition and global superconvergence; Correction scheme and estimates; Numerical examples.

A PARALLEL ITERATIVE DOMAIN DECOMPOSITION ALGORITHM FOR ELLIPTIC PROBLEMS

An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments arc conducted to show the effectiveness of the method.

Coupling potential and viscous flow models with domain decomposition for wave propagations

Either potential flow or viscous flow based model may be flawed for numerical wave simulations.The two-way coupling of potential and viscous flow models with the domain decomposition utilizing respective strengths has been a trending research topic.In contrast to existing literatures in which closed source potential models were used,the widely used open source OceanWave3D,OpenFOAM-v2012 are used in the present research.An innovative overlapping two-way coupling strategy is developed utilizing the ghost points in OceanWave3D.To guarantee computational stability,a relaxation zone used both for outlet damping and data transfer is built over the overlapping region in OceanWave3D.The free surface elevation in the relaxation zone is directly probed in OpenFOAM while the velocity potential is indirectly built upon its temporal variation which is calculated by the free surface boundary condition using the probed velocity.Strong coupling is achieved based on the fourth-order Runge-Kutta(RK)algorithm.Both two-and three-dimensional tests including linear,nonlinear,irregular,and multi-directional irregular waves,are conducted.The effectiveness of the coupling procedure in bidirectional data transfer is fully demonstrated,and the model is validated to be accurate and efficient,thus providing a competitive alternative for ocean wave simulations.

Extension of Near-Wall Domain Decomposition to Modeling Flows with Laminar-Turbulent Transition

The near-wall domain decomposition method(NDD)has proved to be very efficient for modeling near-wall fully turbulent flows.In this paper the NDD is extended to non-equilibrium regimeswith laminar-turbulent transition(LTT)for the first time.The LTT is identified with the use of the e^(N)-method which is applied to both incompressible and compressible flows.TheNDD ismodified to take into account LTT in an efficientway.In addition,implementation of the intermittency expands the capabilities of NDD to model non-equilibrium turbulent flows with transition.Performance of the modified NDD approach is demonstrated on various test problems of subsonic and supersonic flows past a flat plate,a supersonic flow over a compression corner and a planar shock wave impinging on a turbulent boundary layer.The results of modeling with and without decomposition are compared in terms of wall friction and show good agreement with each other while NDD significantly reducing computational resources needed.It turns out that the NDD can reduce the computational time as much as three times while retaining practically the same accuracy of prediction.

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy’s law.The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface.Some computational examples are presented and discussed.