Domain Decomposition Framework Research Articles

Time-resolved modeling of dropwise condensation patterns formed on a nanopillared substrate

Instantaneous condensation patterns formed over a vertical nanopillared copper surface are modeled by including drop level details including nucleation, growth, coalescence, droplet jumping and gravitational instability. The vapor phase is taken to be saturated and condensation is driven by a colder wall that imposes a given degree of subcooling. The mathematical model is setup in the form of a condensation cycle, starting from drop nucleation all the way to instability followed by fresh nucleation. In view of the condensing surface being pillared, a variety of droplet morphologies are possible, including Cassie, Wenzel, and partially-wetting that depend on the prevailing condensation conditions. The degree of subcooling is varied over 1 – 28 K, covering a wide range of condensation characteristics from Cassie jumping to Wenzel flooding with increasing temperature difference. Coalescence-based droplet jumping for Cassie and partially-wetting droplet configurations are incorporated as well. Gravitational instability of the largest drop leads to sweeping along the substrate followed by surface renewal and renucleation. With this overall approach, it is possible to explicitly determine the drop-size distribution as a function of time. The entire model is multiscale in space and time, making it a computationally demanding simulation. A domain decomposition framework is used and computations are carried out in a high-performance computing system with a parallel architecture employing message passing interface (MPI). The time-resolved model has been extensively tested against experimental data reported in the literature. The partially-wetting configuration for a subcooling of 8 K has a larger average droplet size and fewer jumping events generating similar heat fluxes as for Cassie droplets at a lower degree of subcooling (ΔT = 5 K) where the average droplet size is smaller and jumping events are frequent. In addition, the Cassie state yields an enhancement factor of around 2.5 relative to a flat surface while Wenzel droplets cause flooding and need subcooling levels as high as 24 K for similar heat flux values.

Field-split preconditioned active-set reduced-space algorithm for complex black oil reservoir simulation at large-scale

Accurate black oil reservoir simulation is important for the understanding of underground resources in hydrology and petroleum reservoir engineering. The success of fast prediction relies on a robust and scalable black oil simulator, which can handle the complex subsurface fluid dynamics with coupling several different physical processes, and meanwhile run efficiently on distributed memory parallel computers. In this work, we introduce a highly parallel simulator to accurately compute the reservoir flow coupling with wells, including some adaptive fully implicit schemes for the temporal discretization, the improved active-set reduced-space algorithm for the nonlinear algebraic system, and several types of field-split methods for preconditioning. This parallel simulator is applied on the domain decomposition framework, which is designed to overcome the computational issues of traditional simulators implemented for personal computers or small workstations. High-resolution reservoir simulation results on the Society of Petroleum Engineers (SPE) comparative solution project or real-field cases are obtained and analyzed, and the parallel performance of the algorithm is studied on a supercomputer with scaling up to 27440 processor cores. The numerical experiments indicate that the proposed black oil simulator is capable of accurately predicting the highly complex physical processes and interactions between wells and the reservoir for full field scale simulation.

Domain decomposition methods and acceleration techniques for the phase field fracture staggered solver

AbstractThe phase field modeling of fracture is able to simulate the nucleation and the propagation of complex crack patterns. However, the relatively small internal lengths that are required usually lead to very fine meshes and high computational costs, especially for three‐dimensional applications. In the present work, additional cost also comes from the implicit dynamics regularization of unstable crack propagations which potentially leads to a large variation of time steps when switching from quasi‐static to dynamic regimes. To reduce the time to solution in this context, this study proposes a domain decomposition framework and acceleration techniques for the phase field fracture staggered solver. The displacement subproblem and the phase field one are solved with parallel domain decomposition solvers. Dual domain decomposition methods provide low cost preconditioner well adapted to the phase field subproblem. For displacement subproblems undergoing unstable crack propagations, primal domain decomposition methods are preferred to be less sensitive to the treatment of floating substructures. Preconditioners performances are assessed and scalability studies over academic test cases, up to 324 subdomains, are presented. Finally, the robustness of the approach is illustrated on two semi‐industrial simulations.

Nonlinearly preconditioned semismooth Newton algorithms for nonlinear nonsmooth systems

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called ‘bad components’ causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods.

Domain Decomposition Framework for Maxwell Finite Element Solvers and Application to PIC

Domain Decomposition Framework for Maxwell Finite Element Solvers and Application to PIC

Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction

This work proposes a special type of Finite Element (FE) technology – the Empirical Interscale FE method – for modeling heterogeneous structures in the small strain regime, for both dynamic and static analyses. The method combines a domain decomposition framework, where interface conditions are established through “fictitious” frames, with dimensional hyperreduction at subdomain level. Similar to other multiscale FE methods, the structure is assumed to be partitioned into coarse-scale elements, each of these elements is equipped with a fine-scale subgrid, and the displacements of the boundaries of the coarse-scale elements are described by standard polynomial FE shape functions. The distinguishing feature of the proposed method is the employed “interscale” variational formulation, which directly relates coarse-scale nodal internal forces with fine-scale stresses, thereby avoiding the typical nested local/global problems that appear, in the nonlinear regime, in other multiscale methods. This distinctive feature, along with hyperreduction schemes for nodal internal and external body forces , greatly facilitate the implementation of the proposed formulation in existing FE codes for solid elements. Indeed, one only has to change the location and weights of the integration points, and to replace a few polynomial-based FE matrices with “empirical” operators, i.e., derived from the information obtained in appropriately chosen computational experiments. We demonstrate that the elements resulting from this formulation are not afflicted by volumetric locking when dealing with nearly-incompressible materials, and that they can handle non-matching fine-scale grids as well as curved structures. Last but not least, we show that, for periodic structures, this method converges upon mesh refinement to the solution delivered by classical first-order computational homogenization. Thus, although the method does not presuppose scale separation, it can represent solutions in this limiting case by taking sufficiently small coarse-scale elements.

An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method.

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective.

Toward Large-Scale Graph-Based Traffic Forecasting: A Data-Driven Network Partitioning Approach

Network partitioning is recognized as an effective auxiliary approach for solving transportation tasks on large-scale traffic networks in a domain-decomposition (DD) manner. Most of the existing related partitioning algorithms are explicitly designed to traffic management problems and merely focus on the implied topology of the networks. In this article, toward the practical problems that happened to traffic forecasting (TF) tasks, we propose a network-partitioning-based DD framework to improve graph convolutional network (GCN)-based predictors’ performance on large-scale transportation networks. Particularly, we devise a data-driven network-partitioning approach, namely, speed-matching-partitioning (SMP), which employs not only the topological features but also the traffic speed observations of traffic networks for partitioning. Additionally, we propose a data-parallel training strategy that feeds partitioned subnetworks into independent predictors for parallel training. The proposed approach is tested by comprehensive case studies on three real-world data sets to evaluate its effectiveness. The results indicate that the proposed approach can help improve GCN-based predictors’ accuracy and training efficiency on both small and relatively large traffic data sets. Furthermore, we investigate the model sensitivity to the selection of graph representations and framework parameters, and the learning efficiency of the data-parallel training strategy.

A STOCHASTIC DOMAIN DECOMPOSITION AND POST-PROCESSING ALGORITHM FOR EPISTEMIC UNCERTAINTY QUANTIFICATION

Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires computational modeling of the PDE in bounded/unbounded regions. Uncertainty in the computational model manifests through lack of precise knowledge of the input field or configuration. Uncertainty quantification (UQ) in the output physical process is typically carried out by modeling the uncertainty using a random field, governed by an appropriate covariance function. This leads to solving high-dimensional stochastic counterparts of the PDE computational models. Such UQ-PDE models require a large number of simulations of the PDE in conjunction with samples in the high-dimensional probability space, with probability distribution associated with the covariance function. Those UQ computational models having explicit knowledge of the covariance function are known as aleatoric UQ (AUQ) models. The lack of such explicit knowledge leads to epistemic UQ (EUQ) models, which typically require solution of a large number of AUQ models. In this article, using a surrogate, post-processing, and domain decomposition framework with coarse stochastic solution adaptation, we develop an offline/online algorithm for efficiently simulating a class of EUQ-PDE models. We demonstrate the algorithm for celebrated bounded and unbounded spatial region models, with high-dimensional uncertainties.

Extending the Capabilities of Data-Driven Reduced-Order Models to Make Predictions for Unseen Scenarios: Applied to Flow Around Buildings

We present a data-driven or non-intrusive reduced-order model (NIROM) which is capable of making predictions for a significantly larger domain than the one used to generate the snapshots or training data. This development relies on the combination of a novel way of sampling the training data (which frees the NIROM from its dependency on the original problem domain) and a domain decomposition approach (which partitions unseen geometries in a manner consistent with the sub-sampling approach). The method extends current capabilities of reduced-order models to generalise, i.e., to make predictions for unseen scenarios. The method is applied to a 2D test case which simulates the chaotic time-dependent flow of air past buildings at a moderate Reynolds number using a computational fluid dynamics (CFD) code. The procedure for 3D problems is similar, however, a 2D test case is considered sufficient here, as a proof-of-concept. The reduced-order model consists of a sampling technique to obtain the snapshots; a convolutional autoencoder for dimensionality reduction; an adversarial network for prediction; all set within a domain decomposition framework. The autoencoder is chosen for dimensionality reduction as it has been demonstrated in the literature that these networks can compress information more efficiently than traditional (linear) approaches based on singular value decomposition. In order to keep the predictions realistic, properties of adversarial networks are exploited. To demonstrate its ability to generalise, once trained, the method is applied to a larger domain which has a different arrangement of buildings. Statistical properties of the flows from the reduced-order model are compared with those from the CFD model in order to establish how realistic the predictions are.

An AI-based non-intrusive reduced-order model for extended domains applied to multiphase flow in pipes

The modeling of multiphase flow in a pipe presents a significant challenge for high-resolution computational fluid dynamics (CFD) models due to the high aspect ratio (length over diameter) of the domain. In subsea applications, the pipe length can be several hundreds of meters vs a pipe diameter of just a few inches. Approximating CFD models in a low-dimensional space, reduced-order models have been shown to produce accurate results with a speed-up of orders of magnitude. In this paper, we present a new AI-based non-intrusive reduced-order model within a domain decomposition framework (AI-DDNIROM), which is capable of making predictions for domains significantly larger than the domain used in training. This is achieved by (i) using a domain decomposition approach; (ii) using dimensionality reduction to obtain a low-dimensional space in which to approximate the CFD model; (iii) training a neural network to make predictions for a single subdomain; and (iv) using an iteration-by-subdomain technique to converge the solution over the whole domain. To find the low-dimensional space, we compare Proper Orthogonal Decomposition with several types of autoencoder networks, known for their ability to compress information accurately and compactly. The comparison is assessed with two advection-dominated problems: flow past a cylinder and slug flow in a pipe. To make predictions in time, we exploit an adversarial network, which aims to learn the distribution of the training data, in addition to learning the mapping between particular inputs and outputs. This type of network has shown the potential to produce visually realistic outputs. The whole framework is applied to multiphase slug flow in a horizontal pipe for which an AI-DDNIROM is trained on high-fidelity CFD simulations of a pipe of length 10 m with an aspect ratio of 13:1 and tested by simulating the flow for a pipe of length 98 m with an aspect ratio of almost 130:1. Inspection of the predicted liquid volume fractions shows a good match with the high fidelity model as shown in the results. Statistics of the flows obtained from the CFD simulations are compared to those of the AI-DDNIROM predictions to demonstrate the accuracy of our approach.

Massively Parallel Discontinuous Galerkin Surface Integral Equation Method for Solving Large-Scale Electromagnetic Scattering Problems

In this communication, we present flexible and efficient solutions of large-scale electromagnetic scattering problems using the parallel discontinuous Galerkin (DG) surface integral equation (SIE) method. A simplified formulation of the conventional DG is used by removing the interior penalty term, thereby avoiding computational burden of the contour integral in the conventional DG. It is demonstrated numerically that the simplification causes convergence deficiency and changes the accuracy convergence rate of the solution for objects with sharp corners. The former is overcome by constructing preconditioners using the near-field matrix of the whole region, and the latter is proved to have a neglectable influence on the accuracy of the final solutions as long as normal mesh density requirement is satisfied. The well-scaling ternary parallelization approach of the multilevel fast multipole algorithm (MLFMA) is incorporated into the DG domain decomposition framework to reduce the complexity of the solution and strength its capability for large-scale problems, with the distribution of the near-field-related calculation adjusted to adapt the nonuniform meshes for a better workload balance. Numerical results are included to validate the accuracy and demonstrate the scalability and versatility of the proposed method. In addition, we demonstrated the effectiveness of our algorithm by solving a high-definition complicated aircraft model with inlets and exhausts involving over one billion unknowns.

Parallel framework for dynamic domain decomposition of data assimilation problems: a case study on Kalman Filter algorithm

We focus on Partial Differential Equation (PDE) based Data Assimilatio problems (DA) solved by means of variational approaches and Kalman filter algorithm. Recently, we presented a Domain Decomposition framework (we call it DD-DA, for short) performing a decomposition of the whole physical domain along space and time directions, and joining the idea of Schwarz' methods and parallel in time approaches. For effective parallelization of DD-DA algorithms, the computational load assigned to subdomains must be equally distributed. Usually computational cost is proportional to the amount of data entities assigned to partitions. Good quality partitioning also requires the volume of communication during calculation to be kept at its minimum. In order to deal with DD-DA problems where the observations are nonuniformly distributed and general sparse, in the present work we employ a parallel load balancing algorithm based on adaptive and dynamic defining of boundaries of DD -- which is aimed to balance workload according to data location. We call it DyDD. As the numerical model underlying DA problems arising from the so-called discretize-then-optimize approach is the constrained least square model (CLS), we will use CLS as a reference state estimation problem and we validate DyDD on different scenarios.

An Anomalous Diffusion Approach for Speckle Noise Reduction in Medical Ultrasound Images

Medical ultrasound images are usually degraded by a specific type of noise, called speckle. The presence of speckle noise in medical ultrasound images will reduce the image quality and affect the effective information, which can potentially cause a misdiagnosis. Therefore, medical image enhancement processing has been extensively studied and several denoising approaches have been introduced and developed. In the current work, a robust fractional partial differential equation (FPDE) model based on the anomalous diffusion theory is proposed and used for medical ultrasound image enhancement. An efficient computational approach based on a combination of a time integration scheme and localized meshless method in a domain decomposition framework is performed to deal with the model. {In order to evaluate the performance of the proposed de-speckling approach, it is used for speckle noise reduction of a synthetic ultrasound image degraded by different levels of speckle noise. The results indicate the superiority of the proposed approach in comparison with classical anisotropic diffusion denoising model (Catt$acute{e}$'s pde model).}

A Three-field Based Optimization Formulation for Flow Simulations in Networks of Fractures on Nonconforming Meshes

A new numerical scheme is proposed for flow computation in complex discrete fracture networks. The method is based on a three-field domain decomposition framework in which independent variables are introduced at the interfaces generated in the process of decoupling the original problem on the whole network into a set of fracture-local problems. A PDE-constrained formulation is then used to enforce compatibility conditions at the interfaces. The combination of the three-field domain decomposition and of the optimization-based coupling strategy results in a novel method which can handle nonconforming meshes, independently built on each geometrical object of the computational domain, and ensures a local mass conservation property at fracture intersections, which is of paramount importance for hydrogeological applications. An iterative solver is devised for the method, suitable for parallel implementation on parallel computing architectures.

A novel domain decomposition framework for the ground state solution of Bose–Einstein condensates

In this study, we present a new type of domain decomposition algorithm to obtain the ground state solution for Bose–Einstein condensates. Using our proposed algorithm, instead of directly solving the nonlinear eigenvalue problem, one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method, and subsequently solve a small-scale nonlinear eigenvalue problem on an enriched space simultaneously. Because solving large-scale nonlinear eigenvalue problem directly is time intensive, our algorithm can obviously improve the efficiency of producing simulation for Bose–Einstein condensates. In addition, any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework, which makes the algorithm considerably flexible. Two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm.

Method of Distributions for Quantification of Geologic Uncertainty in Flow Simulations

AbstractProbabilistic models of subsurface flow and transport are required for risk assessment and reliable decision making under uncertainty. These applications require accurate estimates of confidence intervals, which generally cannot be ascertained with statistical moments such as mean (unbiased estimate) and variance (a measure of uncertainty) of a quantity of interest (QoI). The method of distributions provides this information by computing either the probability density function or the cumulative distribution function (CDF‐) of the QoI. The standard method can be orders of magnitude faster than Monte Carlo simulations (MCS) but is applicable to stationary, mildly to moderately heterogeneous porous media in which the coefficient of variation of input parameters (e.g., log‐conductivity) is below four. Our CDF‐random domain decomposition (RDD) framework alleviates these limitations by combining the method of distributions and RDD; it also accounts for uncertainty in the geologic makeup of a subsurface environment. For a given realization of the geological map, we derive a deterministic equation for the conditional CDF of hydraulic head of steady single‐phase flow. The solutions of this equation are then averaged over realizations of the geological map‐ to compute the hydraulic head CDF. Our numerical experiments reveal that the CDF‐RDD method remains accurate for two‐dimensional flow in a porous medium composed of two heterogeneous hydrofacies, a setting in which the original CDF method fails. For the same accuracy, the CDF‐RDD method is an order of magnitude faster than MCS.

An overlapping domain decomposition framework without dual formulation for variational imaging problems

In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel--Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel structure is constructed using a custom overlapping domain decomposition scheme with the notion of essential domains. As a solver for the constructed formulation, we propose a decoupled augmented Lagrangian method for untying the coupling of adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian method is provided. We present implementation details and numerical examples for various model problems including total variation minimizations and higher order models.

Hybrid FEM-DDM and BEM-BoR for the Analysis of Multiscale Composite Structures

We propose a novel scheme to accelerate the computation of the finite-element method-boundary element method (FEM-BEM) for 3-D electromagnetic scattering. It builds on the FEM-domain decomposition method (FEM-DDM) and the degenerated boundary element method (BEM) developed for the body of revolution (BoR), which is, thus, referred to as BEM-BoR. When coupled with FEM in the framework of domain decomposition, the infinite computational domain is truncated by a closed rotationally symmetric surface enclosing the 3-D scatterer, and BEM-BoR is then applied on this surface as a radiation boundary condition (RBC). The finite-domain interior to this surface is further decomposed into several subdomains, which are connected by the second-order transmission conditions (SOTCs) imposed on nonconformal meshes. We refer to the resultant solver as a hybrid FEM-DDM and BEM-BoR (FEM-BEM-BoR). Since BEM-BoR can be divided into several 2-D problems instead of the original 3-D one, the proposed FEM-BEM-BoR is more efficient than the original FEM-BEM. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the accuracy and efficiency improvement of FEM-BEM-BoR. In addition, its practical applicability is demonstrated by modeling large-scale objects.

Uncertainty propagation in dynamic sub-structuring by model reduction integrated domain decomposition

This paper addresses computational aspects in dynamic sub-structuring of built-up structures with uncertainty. Component mode synthesis (CMS), which is a model reduction technique, has been integrated within the framework of domain decomposition (DD), so that reduced models of individual sub-systems can be solved with smaller computational cost compared to solving the full (unreduced) system by DD. This is particularly relevant for structural dynamics applications where the overall system physics can be captured by a relatively low number of modes. The theoretical framework of the proposed methodology has been extended for application in stochastic dynamic systems. To limit the number of eigen-value analyses to be performed corresponding to the random realizations of input parameters, a locally refined high dimensional model representation model with stepwise least squares regression is presented. Effectively, a bi-level decomposition is proposed, one in the physical space and the other in the stochastic space. The geometric decomposition in the physical space by the proposed model reduction-based DD reduces the computational cost of a single analysis of the system and the functional decomposition in the stochastic space by the proposed meta-model lowers the number of simulations to be performed on the actual system. The results achieved by solving a finite-element model of an assembled beam structure and a 3D space frame illustrate good performance of the proposed methodology, highlighting its potential for complex problems.