Coarsening Strategy Research Articles

Adaptive finite element–discrete element analysis for stratal movement and microseismic behaviours induced by multistage propagation of three-dimensional multiple hydraulic fractures

PurposeOptimized three-dimensional (3D) fracture networks are crucial for multistage hydrofracturing. To better understand the mechanisms controlling potential disasters as well as to predict them in 3D multistage hydrofracturing, some governing factors, such as fluid injection-induced stratal movement, compression between multiple hydraulic fractures, fracturing fluid flow, fracturing-induced microseismic damaged and contact slip events, must be properly simulated via numerical models. This study aims to analyze the stratal movement and microseismic behaviours induced by multistage propagation of 3D multiple hydraulic fractures.Design/methodology/approachAdaptive finite element–discrete element method was used to overcome the limitations of conventional finite element methods in simulating 3D fracture propagation. This new approach uses a local remeshing and coarsening strategy to ensure the accuracy of solutions, reliability of fracture propagation path and computational efficiency. Engineering-scale numerical models were proposed that account for the hydro-mechanical coupling and fracturing fluid leak-off, to simulate multistage propagation of 3D multiple hydraulic fractures, by which the evolution of the displacement, porosity and fracture fields, as well as the fracturing-induced microseismic events were computed.FindingsStratal movement and compression between 3D multiple hydraulic fractures intensify with increasing proximity to the propagating fractures. When the perforation cluster spaces are very narrow, alternate fracturing can improve fracturing effects over those achieved via sequential or simultaneous fracturing. Furthermore, the number and magnitude of microseismic events are directly proportional to the stratal movement and compression induced by multistage propagation of fracturing fracture networks.Originality/valueMicroseismic events induced by multistage propagation of 3D multiple hydraulic fractures and perforation cluster spaces and fracturing scenarios that impact the deformation and compression among fractures in porous rock matrices are well predicted and analyzed.

Algebraic multigrid for the nonlinear powerflow equations

AbstractIn a recent article, one of the authors developed a multigrid technique for coarse‐graining dynamic powergrid models. A key component in this technique is a relaxation‐based coarsening of the graph Laplacian given by the powergrid network and its weighted graph, which is represented by the admittance matrix. In this article, we use this coarsening strategy to develop a multigrid method for solving a static system of nonlinear equations that arises through Ohm's law, the so‐called powerflow equations. These static equations are tightly knitted to the dynamic model in that the full powergrid model is an algebraic‐differential system with the powerflow equations describing the algebraic constraints. We assume that the dynamic model corresponds to a stable operating powergrid, and thus, the powerflow equations are associated with a physically stable system. This stability permits the coarsening of the powerflow equations to be based on an approximate graph Laplacian, which is embedded in the powerflow system. By algebraically constructing a hierarchy of approximate weighted graph Laplacians, a hierarchy of nonlinear powerflow equations immediately becomes apparent. This latter hierarchy can then be used in a full approximation scheme (FAS) framework that leads to a nonlinear solver with generally a larger basin of attraction than Newton's method. Given the algebraic multigrid (AMG) coarsening of the approximate Laplacians, the solver is an AMG‐FAS scheme. Alternatively, using the coarse‐grid nodes and interpolation operators generated for the hierarchy of approximate graph Laplacians, a multiplicative‐correction scheme can be derived. The derivation of both schemes will be presented and analyzed, and numerical examples to demonstrate the performance of these schemes will be given.

An adaptive Finite Element strategy for the numerical simulation of additive manufacturing processes

In this work an adaptive Finite Element strategy to deal with the numerical simulation of Additive Manufacturing (AM) processes is presented. The Selective Laser Melting (SLM) is chosen as the reference technology because of its great diffusion in the industrial manufacturing chain, although the proposed methodology can be applied to the numerical simulation of all types of AM. An octree-based mesh adaptivity approach is adopted allowing for the use of much finer meshes within the processing zone, the so called Thermo-Mechanically Affected Zone (TMAZ), if compared to the rest of the computational domain. Although the adaptive meshing is vital to keep controlled the computational resources through the entire simulation of the fabrication process, the accuracy of the results can be compromised by the coarsening strategy, and particularly when simulating the SLM process, where the mesh size can vary from microns (TMAZ) to centimetres (close to the build-plate). This loss of accuracy can spoil the original efforts in refining the mesh in the process zone. Therefore a strategy to compensate for information loss in the adaptive refinement simulation of additive manufacturing processes is developed. The main idea is to add two correction terms which compensate for the loss of accuracy in the coarsening process of the mesh in the already manufactured regions. The proposed correction terms can be interpreted as a Variational Multiscale enhancement on the adaptive mesh. This allows one to successfully simulate the additive manufacturing process by using an adaptively coarsened mesh, with results which have an accuracy very similar to the one of a uniformly refined mesh simulation, at a fraction of the computational cost. Numerical examples illustrate the performance of the proposed strategy.

Two-level difference scheme for the two-dimensional Fokker–Planck equation

In this paper, we propose a two-level difference scheme for solving the two-dimensional Fokker–Planck equation. This equation is a parabolic type equation which governs the time evolution of probability density function of the stochastic processes. In addition, these equations preserve positivity and conservation. The Chang–Cooper discretization scheme is used, which ensures second-order accuracy, positiveness, and satisfies the conservation of the total probability. In particular, we investigate a two-level scheme with factor-three coarsening strategy. With coarsening by a factor-of-three we obtained simplified inter-grid transfer operators and thus have a significant reduction in CPU time. Numerical experiments are performed to validate efficiency of the proposed Chang–Cooper two-level algorithms to stationary and time-dependent Fokker–Planck equations, respectively.

Formula omitted]-multigrid methods and their comparison to [formula omitted]-multigrid methods within Isogeometric Analysis

Over the years, Isogeometric Analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p-multigrid method, in which coarsening is applied in the spline degree p instead of the mesh width h, and compare it to h-multigrid methods. Since the use of classical smoothers (e.g. Gauss–Seidel) results in a p-multigrid/h-multigrid method with deteriorating performance for higher values of p, the use of an ILUT smoother is investigated as well. Numerical results and a spectral analysis indicate that the use of this smoother exhibits convergence rates essentially independent of h and p for both p-multigrid and h-multigrid methods. In particular, we compare both coarsening strategies (e.g. coarsening in h or p) adopting both smoothers for a variety of two and three dimensional benchmarks. Furthermore, the ILUT smoother is compared to a state-of-the-art smoother (Hofreither and Takacs 2017) using both coarsening strategies. Finally, the proposed p-multigrid method is used to solve linear systems resulting from THB-spline discretizations.

Constraint bubbles and affine regions

We propose to enhance the capability of standard free-surface flow simulators with efficient support for immersed bubbles through two new models: constraint-based bubbles and affine fluid regions. Unlike its predecessors, our constraint-based model entirely dispenses with the need for advection or projection inside zero-density bubbles, with extremely modest additional computational overhead that is proportional to the surface area of all bubbles. This surface-only approach is easy to implement, realistically captures many familiar bubble behaviors, and even allows two or more distinct liquid bodies to correctly interact across completely unsimulated air. We augment this model with a per-bubble volume-tracking and correction framework to minimize the cumulative effects of gradual volume drift. To support bubbles with non-zero densities, we propose a novel reduced model for an irregular fluid region with a single pointwise incompressible affine vector field. This model requires only 11 interior velocity degrees of freedom per affine fluid region in 3D, and correctly reproduces buoyant, stationary, and sinking behaviors of a secondary fluid phase with non-zero density immersed in water. Since the pressure projection step in both the above schemes is a slightly modified Poisson-style system, we propose novel Multigrid-based preconditioners for Conjugate Gradients for fast numerical solutions of our new discretizations. Furthermore, we observe that by enforcing an incompressible affine vector field over a coalesced set of grid cells, our reduced model is effectively an irregular coarse super-cell. This offers a convenient and flexible adaptive coarsening strategy that integrates readily with the standard staggered grid approach for fluid simulation, yet supports coarsened regions that are arbitrary voxelized shapes, and provides an analytically divergence-free interior. We demonstrate its effectiveness with a new adaptive liquid simulator whose interior regions are coarsened into a mix of tiles with regular and irregular shapes.

Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach

We introduce a general multi-material topology optimization framework for large deformation problems that effectively handles an arbitrary number of candidate hyperelastic materials and addresses three major associated challenges: material interpolation, excessive distortion of low-density elements, and computational efficiency. To account for many nonlinear elastic materials, we propose a material interpolation scheme that, instead of interpolating multiple material parameters (such as Young’s modulus), interpolates multiple nonlinear stored-energy functions. To circumvent convergence difficulties caused by excessive distortions of low-density elements under large deformations, an energy interpolation scheme is revisited to account for multiple candidate hyperelastic materials. Computational efficiency is addressed from both structural analysis and optimization perspectives. To solve the nonlinear state equations efficiently, we employ the lower-order Virtual Element Method in conjunction with tailored adaptive mesh refinement and coarsening strategies. To efficiently update the design variables of the multi-material system, we exploit the separable nature and improve the ZPR (Zhang–Paulino–Ramos) update scheme to account for positive sensitivities and update the design variables associated with each volume constraint in parallel. Four design examples with three types of nonlinear material models demonstrate the efficiency and effectiveness of the proposed framework.

Hybrid multigrid methods for high-order discontinuous Galerkin discretizations

The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems, which are, for example, a key ingredient of incompressible flow solvers in the field of computational fluid dynamics. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid problem, but also leads to a discretization most suitable for state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The relevant design choices regarding the selection of optimal multigrid coarsening strategies among the various possibilities are discussed with the metric of computational costs as the driving force for algorithmic selections. We find that a transfer to a continuous function space at highest polynomial degree (or on the finest mesh), followed by polynomial and geometric coarsening, shows the best overall performance. The success of this particular multigrid strategy is due to a significant reduction in iteration counts as compared to a transfer from discontinuous to continuous function spaces at lowest polynomial degree (or on the coarsest mesh). The coarsening strategy with transfer to a continuous function space on the finest level leads to a multigrid algorithm that is robust with respect to the penalty parameter of the symmetric interior penalty method. Detailed numerical investigations are conducted for a series of examples ranging from academic test cases to more complex, practically relevant geometries. Performance comparisons to state-of-the-art methods from the literature demonstrate the versatility and computational efficiency of the proposed multigrid algorithms.

Algorithm 1003

Partitioning graphs is a common and useful operation in many areas, from parallel computing to VLSI design to sparse matrix algorithms. In this article, we introduce Mongoose, a multilevel hybrid graph partitioning algorithm and library. Building on previous work in multilevel partitioning frameworks and combinatoric approaches, we introduce novel stall-reducing and stall-free coarsening strategies, as well as an efficient hybrid algorithm leveraging (1) traditional combinatoric methods and (2) continuous quadratic programming formulations. We demonstrate how this new hybrid algorithm outperforms either strategy in isolation, and we also compare Mongoose to METIS and demonstrate its effectiveness on large and social networking (power law) graphs.

A coarsening method for bipartite networks via weight-constrained label propagation

A multilevel method is a scalable strategy to solve optimization problems in large bipartite networks, which operates in three stages. Initially the input network is iteratively coarsened into a hierarchy of gradually smaller networks. Coarsening implies in collapsing vertices into so-called super-vertices which inherit properties of their originating vertices. An initial solution is obtained executing the target algorithm in the coarsest network. Finally, this solution is successively projected back over the inverse sequence of coarsened networks, up to the initial one, yielding an approximate final solution. Despite its potential applicability, the strategy faces several theoretical and practical limitations. Coarsening is usually attained following a user-defined policy to match vertices pairwise. However, the network reduction process is extremely slow and may yield degraded solutions due to propagation of poor matches. Additionally, proper parameterization of coarsening algorithms is difficult, as well as ensuring the super-vertices preserve the relevant properties. We address these issues with a near-linear complexity coarsening strategy based on weight-constrained label propagation. Our strategy collapses groups of vertices, rather than pairs, yielding faster and more extensive network reduction. Moreover, users may specify the desired size of the coarsest network and control super-vertex weights. The applicability of our solution is illustrated in multiple scenarios, namely: multilevel implementation of an existing high-cost community detection algorithm; as a direct community detection algorithm; finally, network visualization, in connection with force-directed graph drawing algorithms. Results provide empirical evidence on the potential of our proposal to foster novel applications of the multilevel method in bipartite networks.

Coarse graining Euler-Lagrange simulations of cohesive particle fluidization

Although Euler-Lagrange simulations can be performed with millions of particles, particle coarsening where particles are replaced by parcels is necessary for simulation of large particulate and fluid-particle flows. The present study examines coarsening strategies for cohesive particles, where cohesion arises through either van der Waals interaction or liquid bridges between particles. In the latter case, the dynamics of liquid transfer between particles is also taken into account. Strategies based on matching dimensionless overlap, stress and effective coefficient of restitution are shown to lead to same coarse graining rules, while that based on matching the Bond number yields a different set of rules. Test simulations involving fluidization of cohesive particles reveal that the stress-based coarse graining rules provide better approximation of the average slip velocity between the gas and the particles.

Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field.We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space.The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.

3D voxel homogenized limit analysis of single-leaf non-periodic masonry

This paper presents an extensive investigation on the out-of-plane collapse behavior of single-leaf non-periodic masonry walls. This is achieved by deriving out-of-plane homogenized failure surfaces from test-windows that are extracted from various location within the same non-periodic masonry wall. The concept of “test-window” is inspired by that of Statistically Equivalent Periodic Unit Cell (SEPUC), which is an evolution of that of Representative Element of Volume (REV), and is needed for successfully applying homogenization to non-periodic masonry. An innovative feature introduced in this paper is the automatic generation of a suitable 3D finite element mesh directly from the sketch of the considered masonry test-window, based on a so-called “voxel strategy” that converts each voxel (the 3D equivalent of a pixel) into a finite element. Moreover, this mesh generating procedure enables the correct representation of the transversal layout of the test-window, and it also contains a coarsening strategy that allows a reduction of the overall number of finite elements in the 3D mesh. For the derivation of the out-of-plane homogenized failure surfaces a linear programming problem is solved, which is based on the upper bound theorem of limit analysis coupled with a homogenization approach. The investigation is performed on six real case studies displaying different degrees of non-periodicity: for the four test-windows of each case study, two out-of-plane homogenized failure surfaces are extracted (flexural and torsional) as well as three relevant deformed modes at collapse. Eventually, the results are critically commented; comparisons are drawn among the case studies, highlighting the influence of each non-periodic masonry bond on the results.

Numerical verification of an Octree mesh coarsening strategy for simulating additive manufacturing processes

Thermo-mechanical finite element modeling of additive manufacturing processes, such as Directed Energy Deposition and Laser Powder Bed Fusion, has been widely applied for the prediction and mitigation of part distortion. However, as the size of modeled geometries gets larger, the number of nodes and elements required in the finite element mesh increases significantly. Because runtime will increase as more nodes are added, it is not practical to conduct full simulations of large builds using standard static meshes. Advanced meshing strategy is required to reduce the run time and to retain the accuracy of the prediction. In this work, a mesh coarsening strategy is evaluated for predicting temperature, distortion, and residual stress in additive manufacturing, aiming to achieve feasible run times with reasonable accuracy on large builds. Directed Energy Deposition of thin wall geometries built from Inconel® 625 and Ti6Al4V is used as a reference and models with 2 levels of Octree mesh coarsening are investigated. The thermal history, in situ distortion, residual stress, and run times are compared with previously experimentally validated static mesh results. Two levels of mesh coarsening is found to be the most effective case for both materials reducing the computational time by 75% while reporting less than 2.5% error for the peak distortion and negligible error for the thermal history difference as compared to the static mesh. Keeping two fine layers of elements underneath the heat source is found to be the most efficient in terms of prediction accuracy and run time.

Hypergraph Partitioning With Embeddings

Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations as hypergraph partitioning problems. A hypergraph is a generalization of a traditional graph wherein “hyperedges” may connect any number of nodes. As a result, hypergraph partitioning is an NP-Hard problem to both solve or approximate. State-of-the-art algorithms that solve this problem follow the multilevel paradigm, which begins by iteratively “coarsening” the input hypergraph to smaller problem instances that share key structural features. Once identifying an approximate problem that is small enough to be solved directly, that solution can be interpolated and refined to the original problem. While this strategy represents an excellent trade off between quality and running time, it is sensitive to coarsening strategy. In this work we propose using graph embeddings of the initial hypergraph in order to ensure that coarsened problem instances retrain key structural features. Our approach prioritizes coarsening within self-similar regions within the input graph, and leads to significantly improved solution quality across a range of considered hypergraphs. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Reproducibility:</i> All source code, plots and experimental data are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://sybrandt.com/2019/partition</uri> .

Sampling and multilevel coarsening algorithms for fast matrix approximations

SummaryThis paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider the column subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.

Marc: Multi-Granular Representation Learning for Networks Based on the 3-Clique

Network embedding is of paramount importance in many real applications, such as node classification, network visualization, and link prediction. Existing methods can effectively encode different structural properties into representations. Most of them are single-granular representation learning methods that do not enable the network to be easily analyzed at various scales. There are many kinds of hierarchical structures in reality. In this paper, we propose a novel algorithm multi-granular representation learning for networks based on the 3-clique named Marc. It makes the representations of the current network be formed by both considering the structure of this granular network and inheriting the coarser network representation results. Firstly, we propose the 3-clique coarsening strategy. A 3-clique is coarsened to be a supernode in the coarser network. These coarsened networks with different granularities, preserve the original network's main structure. Secondly, we use the single-granular optimization objective function to obtain the nodes' representations of each granular network. Finally, the refinement model learns the nodes' representations, starting from the coarsest network. The learned representations of supernodes serve as good initializations for embedding the corresponding coarsened nodes from the finer network. This process is repeated until we obtain an embedding for each node in the original network which preserves the relationship among multi-granular networks. The experimental results on five public datasets, Wiki, BlogCatalog, Cora, CiteSeer, and DBLP, demonstrate that Marc has a better Macro F1 value for classification tasks than the baseline methods.

Predictable Thread Coarsening

Thread coarsening on GPUs combines the work of several threads into one. We show how thread coarsening can be implemented as a fully automated compile-time optimisation that estimates the optimal coarsening factor based on a low-cost, approximate static analysis of cache line re-use and an occupancy prediction model. We evaluate two coarsening strategies on three different NVidia GPU architectures. For NVidia reduction kernels we achieve a maximum speedup of 5.08x, and for the Rodinia benchmarks we achieve a mean speedup of 1.30x over 8 of 19 kernels that were determined safe to coarsen.

Hydrography‐Driven Coarsening of Grid Digital Elevation Models

AbstractA new grid coarsening strategy, denoted as hydrography‐driven coarsening, is developed in the present study. The hydrography‐driven coarsening is designed to retain the essential hydrographic features of valleys and channels observed in high‐resolution digital elevation models: (1) depressions are filled in the considered high‐resolution digital elevation model, (2) the obtained topographic data are used to extract a reference grid network composed of all surface flow paths, (3) the Horton order is assigned to each link of the reference grid network, and (4) within each coarse grid cell, the elevation of the point lying along the highest‐order path of the reference grid network and displaying the minimum distance to the cell center is assigned to this coarse grid cell center. The capabilities of the hydrography‐driven coarsening to provide consistent surface flow paths with respect to those observed in high‐resolution digital elevation models are evaluated over a synthetic valley and two real drainage basins located in the Italian Alps and in the Italian Apennines. The hydrography‐driven coarsening is found to yield significantly more accurate valley and channel profiles than existing coarsening strategies. In absolute terms, the obtained valleys and channels compare favorably with those observed. In addition, the proposed coarsening strategy is found to reduce drastically the impact of depression‐filling in the obtained coarse digital elevation models. The hydrography‐driven coarsening strategy is therefore advocated for all those cases in which the relief of the extracted drainage network is an important hydrographic feature.

An efficient multilevel scheme for coarsening large scale social networks

The explosive growth of data raised from social networks, hinders researchers from analysing them in a good way. So, is it possible to rapidly “zoom-out” from this huge network while preserving its whole structure? In fact, this technique is named “graph’s reduction” and represents a significant task in social networks’ analysis. Thus, several methods have been developed to pull a smaller succinct version of the graph. Some of them belong to the category of “graph sampling” and risk losing key characteristics of communities. Others are part of “coarsening strategy” and designed to cope with the problem of community discovering, which is our desired purpose. In this paper, we propose a multi-level coarsening algorithm called MCCA (Multi-level Coarsening Compact Areas). The main strategy of this algorithm is to merge well connected zones in every level by updating edge and vertex weight until a stopping criterion is met. Using real-world social networks, we evaluate the quality and scalability of MCCA. Furthermore, we compared it with eight known proposals. We also show how our method can be used as a preliminary step for community detection without much loss of information.