Coarsening Strategy Research Articles

An isogeometric approach to coupled thermomechanics in 3D via hierarchical adaptivity

In the present work, a complete numerical scheme to perform adaptive refinement and coarsening in the context of isogeometric analysis applied to thermomechanics is introduced. In particular, truncated hierarchical B-splines (THB-splines) are considered as a basis for the analysis and suitably graded isogeometric meshes are constructed to guarantee admissible configurations at each refinement and coarsening iteration. The algorithms are developed and implemented in a full three-dimensional framework and applied to solve coupled thermoelastic problems. The proposed numerical framework is verified on two relevant problems with respect to overkill solutions, obtained either with a commercial finite element software or with uniform isogeometric meshes. Different refinement and coarsening strategies are considered and compared. The obtained results demonstrate the ability of the THB-spline-based adaptive scheme to deliver robust discretizations guaranteeing a suitable trade-off between computational accuracy and number of degrees of freedom.

Adaptive material evaluation by stabilized octree and sandwich coarsening in FFT‐based computational micromechanics

AbstractThe computational efficiency of FFT‐based computational micromechanics is deeply rooted in the underlying regular, that is, Cartesian, discretization. The bottleneck for most industrial applications is evaluating the typically rather expensive constitutive law on the regular grid. In the work at hand, we exploit coarsening strategies to evaluate the material law with the intention of speeding up the overall computation time while retaining the level of achieved accuracy. Inspired by wavelet‐compression techniques, we form aggregates of voxels where the local strain tensors are close, and compute the stresses on these coarsened elements. If done naively, such a strategy will lead to intrinsic instabilities whose origin is apparent from a mathematical perspective. As a remedy, we introduce a stabilization technique which is inspired by hourglass control well‐known for underintegrated finite elements. We introduce octree as well as sandwich coarsening, discuss the handling of internal variables, report on the efficient implementation of the concepts and demonstrate the effectiveness of the developed technology on simple as well as industrial examples.

Meshless Geometric Multigrid Method for Complex Geometries with Improved Cell Coarsening Algorithm

In this paper, we present an improved multicloud methodology to extend the multicloud convergence accelerator, which depends on meshless discretization on coarse level computation of the geometric multigrid method, to cell finite volume (CFV) on three-dimensional space. To achieve this, we first identify the primary challenge in the meshless coarsening application to CFV. Specifically, the challenge lies in the poorly coarsened grid, which results in a degradation of the convergence rate. The reason for this is that the coarsening stencils were coupled with the discretization stencil. To address the issue, we proposed a new coarsening strategy on CFV that adequately utilizes node information on the fine-level mesh. The proposed methodology shows superior coarsening rates compared to the basic meshless cell coarsening, regardless of the dimension and type of mesh elements, resulting in significant speedup in convergence. Various computational fluid dynamics (CFD) simulations with three-dimensional complex geometries are presented, and their results demonstrate approximately seven times faster convergence compared to the single-grid method. The overall results verify that the multicloud method with the proposed coarsening procedure provides a powerful tool for practical CFD simulations by reducing the computation time, and it enables researchers and engineers to simulate and optimize real-world systems more efficiently.

Neural Architecture Search for GNN-Based Graph Classification

Graph classification is an important problem with applications across many domains, for which graph neural networks (GNNs) have been state-of-the-art (SOTA) methods. In the literature, to adopt GNNs for the graph classification task, there are two groups of methods: global pooling and hierarchical pooling. The global pooling methods obtain the graph representation vectors by globally pooling all of the node embeddings together at the end of several GNN layers, whereas the hierarchical pooling methods provide one extra pooling operation between the GNN layers to extract hierarchical information and improve the graph representations. Both global and hierarchical pooling methods are effective in different scenarios. Due to highly diverse applications, it is challenging to design data-specific pooling methods with human expertise. To address this problem, we propose PAS (Pooling Architecture Search) to design adaptive pooling architectures by using the neural architecture search (NAS). To enable the search space design, we propose a unified pooling framework consisting of four modules: Aggregation, Pooling, Readout, and Merge. Two variants, PAS-G and PAS-NE, are provided to design the pooling operations in different scales. A set of candidate operations is designed in the search space using this framework. Then, existing human-designed pooling methods, including global and hierarchical ones, can be incorporated. To enable efficient search, a coarsening strategy is developed to continuously relax the search space, and then a differentiable search method can be adopted. We conduct extensive experiments on six real-world datasets, including the large-scale datasets MR and ogbg-molhiv. Experimental results in this article demonstrate the effectiveness and efficiency of the proposed PAS in designing the pooling architectures for graph classification. The Top-1 performance on two Open Graph Benchmark (OGB) datasets 1 further indicates the utility of PAS when facing diverse realistic data. The implementation of PAS is available at: https://github.com/AutoML-Research/PAS.

Learning node representation via Motif Coarsening

Motifs, as fundamental units of the graph, play a significant role in modeling complex systems in a variety of domains, including social networks, as well as biology and neuroscience. Motif preservation is a widely studied problem that provides new avenues for structure preservation. This paper is dedicated to exploring the significance of motifs with various patterns and effectively incorporating different motif patterns into node-level graph representation learning. We propose a novel node Representation learning framework via Motif Coarsening (RMC), which effectively incorporates different granularity structural information into node representation learning. RMC consists of two parallel components, the node representation learning aggregator and the motif-based node representation learning aggregator. In the node representation learning process, RMC directly encodes lower-order structures into node representation by a one-layer graph convolution network. For the motif-based node representation learning process, we propose a Motif Coarsening strategy for incorporating motif structure into the graph representation learning process. Furthermore, the MotifRe-Weighting strategy is proposed to biased convert motif representation into motif-based node representation. We verify the effectiveness of RMC by several node-related tasks on a series of widely used real-world datasets. Experimental results demonstrate that our proposed framework delivers superior promising representation performance to existing benchmarks. Ablation experiments proved that RMC has potential as an auxiliary framework, which indicates the excellent quality of Motif Coarsening and MotifRe-Weighting strategies over existing benchmarks from several evaluation metrics, involving mean classification accuracy, Micro-F1, and Macro-F1.

On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

Algebraic Multigrid Preconditioner for Statically Condensed Systems Arising from Lowest-Order Hybrid Discretizations

We address the numerical solution of linear systems arising from the hybrid discretizations of second-order elliptic partial differential equations. Such discretizations hinge on a hybrid set of degrees of freedom (DoFs), respectively, defined in cells and faces, which naturally gives rise to a global hybrid system of linear equations. Assuming that the cell unknowns are only locally coupled, they can be efficiently eliminated from the system, leaving only face unknowns in the resulting Schur complement, which is also called the statically condensed matrix. We propose in this work an algebraic multigrid (AMG) preconditioner specifically targeting condensed systems corresponding to lowest-order discretizations (piecewise constant). Like traditional AMG methods, we retrieve geometric information on the coupling of the DoFs from algebraic data. However, as the condensed matrix only gives information on the faces, we use the uncondensed version to reconstruct the connectivity graph between elements and faces. An aggregation-based coarsening strategy mimicking a geometric coarsening or semicoarsening can then be set up to build coarse levels. Numerical experiments are performed on diffusion problems discretized by the hybrid high-order method at the lowest order. Our approach uses a K-cycle to precondition an outer flexible Krylov method. The results demonstrate similar performances, in most cases, compared to a standard AMG method and a notable improvement on anisotropic problems with Cartesian meshes.

Residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems

AbstractIn this paper, we derive a residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems. The present approach does not involve the Helmholtz decomposition while analyzing the reliability of the estimator. The constants involved in the estimators are independent of the jump of the diffusion coefficient across the interface, and the quasi‐monotonocity assumption on the diffusion coefficient is relaxed. The reliability bound of the estimator consists of the element residual, the edge flux jump and the edge solution jump. The efficiency of the estimator is analyzed by employing a coarsening strategy introduced by Chen and Feng's study. We derive both global upper bound and a local lower bound for the error and an adaptive space–time algorithm is prescribed using the derived estimators. Numerical results illustrating the behavior of the estimators are provided.

A painless multi-level automatic goal-oriented [formula omitted]-adaptive coarsening strategy for elliptic and non-elliptic problems

This work extends an automatic energy-norm hp-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global h- and p-refinements with a coarsening step. Thus, at each unrefinement step, we eliminate the basis functions with the lowest contributions to the solution. When solving elliptic problems using energy-norm adaptivity, the removed basis functions are those with the lowest contributions to the energy of the solution. For non-elliptic problems or goal-oriented adaptivity, we propose an upper bound of the error representation expressed in terms of an inner product of the specific equation, leading to error indicators that deliver quasi-optimal hp-unrefinements. This unrefinement strategy removes unneeded unknowns possibly introduced during the pre-asymptotical regime. In addition, the grids over which we perform the unrefinements are arbitrary, and thus, we can limit their size and associated computational costs. We numerically analyze our algorithm for energy-norm and goal-oriented adaptivity. In particular, we solve two-dimensional (2D) Poisson, Helmholtz, convection-dominated equations, and a three-dimensional (3D) Helmholtz-like problem. In all cases, we observe exponential convergence rates. Our algorithm is robust and straightforward to implement; therefore, it can be easily adapted for industrial applications.

High‐order multigrid strategies for hybrid high‐order discretizations of elliptic equations

AbstractThis study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high‐order method. Combinations of ‐, ‐, and ‐coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non‐nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.

Multilevel Coarsening for Interactive Visualization of Large Bipartite Networks.

Bipartite networks are pervasive in modeling real-world phenomena and play a fundamental role in graph theory. Interactive exploratory visualization of such networks is an important problem, and particularly challenging when handling large networks. In this paper we present results from an investigation on using a general multilevel method for this purpose. Multilevel methods on networks have been introduced as a general approach to increase scalability of community detection and other complex optimization algorithms. They employ graph coarsening algorithms to create a hierarchy of increasingly coarser (reduced) approximations of an original network. Multilevel coarsening has been applied, e.g., to the problem of drawing simple (“unipartite”) networks. We build on previous work that extended multilevel coarsening to bipartite graphs to propose a visualization interface that uses multilevel coarsening to compute a multi-resolution hierarchical representation of an input bipartite network. From this hierarchy, interactive node-link drawings are displayed following a genuine route of the “overview first, zoom and filter, details on demand” visual information seeking mantra. Analysts may depart from the coarsest representation and select nodes or sub-graphs to be expanded and shown at greater detail. Besides intuitive navigation of large-scale networks, this solution affords great flexibility, as users are free to select different coarsening strategies in different scenarios. We illustrate its potential with case studies involving real networks on distinct domains. The experimental analysis shows our strategy is effective to reveal topological structures, such as communities and holes, that may remain hidden in a conventional node-link layout. It is also useful to highlight connectivity patterns across the bipartite layers, as illustrated in an example that emphasizes the correlation between diseases and genes in genetic disorders, and in a study of a scientific collaboration network of authors and papers.

An efficient and synchronous mesh refining–coarsening strategy in kinematic upper-bound limit analysis

This study presents an efficient and robust adaptive remeshing strategy in the kinematic upper-bound limit analysis in association with using six-node triangular elements and Second–Order Cone Programming (SOCP). In particular, a Synchronous Mesh Refining–Coarsening (SMRC) algorithm following a posterior plastic dissipation-based error indicator and the well-known Delaunay triangulation are implemented into the Upper Bound Finite Element Method (UBFEM). Detailed numerical implementation of the proposed method is presented, and its computational performance is demonstrated through several benchmark problems and comparisons with those solutions obtained using Pure Mesh Refinement (PMR) scheme. The proposed method, named UBFEM-SMRC, is proved to generate more rigorous solutions for all considered examples and significantly reduce the computational burdens of the UBFEM-PMR. Additionally, more clear adaptively refined mesh zones with intensive plastic dissipations, which can readily reproduce the potential failure mechanisms of geotechnical problems, are observed by successive performing of the proposed synchronous adaptive mesh refining–coarsening process.

A multi-criteria h-adaptive finite-element framework for industrial part-scale thermal analysis in additive manufacturing processes

This work presents an h-adaptive finite-element (FE) strategy to address the numerical simulation of additive manufacturing (AM) of large-scale parts. The wire-arc additive manufacturing is chosen as the demonstrative technology for its manufacturing capabilities suitable for industrial purposes. The scanning path and processing parameters of the simulation are provided via a RS-274 (GCode) file, being the same as the one delivered to the AM machine. The approach is suitable for industrial applications and can be applied to other AM processes. To identify the location in the FE mesh of the heat affected zone (HAZ), a collision detection algorithm based on the separating axis theorem is used. The mesh is continuously adapted to guarantee the necessary mesh resolution to capture the phenomena inside and outside the HAZ. To do so, a multi-criteria adaptive mesh refinement and coarsening (AMR) strategy is used. The AMR includes a geometrical criterion to guarantee the FE size within the HAZ, and a Zienkiewicz–Zhu-based a-posteriori error estimator to guarantee the solution accuracy elsewhere. Thus, the number of active FEs is controlled and mesh manipulation by the end-user is avoided. Numerical simulations comparing the h-adaptive strategy with the (reference) fixed fine meshes are performed to prove the computational cost efficiency and the solution accuracy.

Multi-Scale Modelling of Aggregation of TiO2 Nanoparticle Suspensions in Water.

Titanium dioxide nanoparticles have risen concerns about their possible toxicity and the European Food Safety Authority recently banned the use of TiO2 nano-additive in food products. Following the intent of relating nanomaterials atomic structure with their toxicity without having to conduct large-scale experiments on living organisms, we investigate the aggregation of titanium dioxide nanoparticles using a multi-scale technique: starting from ab initio Density Functional Theory to get an accurate determination of the energetics and electronic structure, we switch to classical Molecular Dynamics simulations to calculate the Potential of Mean Force for the connection of two identical nanoparticles in water; the fitting of the latter by a set of mathematical equations is the key for the upscale. Lastly, we perform Brownian Dynamics simulations where each nanoparticle is a spherical bead. This coarsening strategy allows studying the aggregation of a few thousand nanoparticles. Applying this novel procedure, we find three new molecular descriptors, namely, the aggregation free energy and two numerical parameters used to correct the observed deviation from the aggregation kinetics described by the Smoluchowski theory. Ultimately, molecular descriptors can be fed into QSAR models to predict the toxicity of a material knowing its physicochemical properties, enabling safe design strategies.

An efficient algebraic multigrid method for second‐order elliptic equation on polygonal domains

Based on a coarsening strategy of adjacency matrix, a new algebraic prolongation operator is developed for standard V‐cycle multigrid method to accelerate the whole process. An efficient algebraic multigrid (EAMG) method is proposed for solving large‐scale linear system, which arise from finite element (FE) discretization of second‐order elliptic boundary value problem. Numerical experiments on polygonal domains are conducted to demonstrate that the EAMG computation is more efficient than standard method.

Towards robust, fast solutions of elliptic equations on complex domains through hybrid high‐order discretizations and non‐nested multigrid methods

AbstractThe use of modern discretization technologies such as hybrid high‐order (HHO) methods, coupled with appropriate linear solvers, allow for the robust and fast solution of partial differential equations (PDEs). Although efficient linear solvers have recently been made available for simpler cases, complex geometries remain a challenge for large scale problems. To address this problem, we propose in this work a geometric multigrid algorithm for unstructured non‐nested meshes. The non‐nestedness is handled in the prolongation operator through the use of the ‐orthogonal projection from the coarse elements onto the fine ones. However, as the exact evaluation of this projection can be computationally expensive, we develop a cheaper approximate implementation that globally preserves the approximation properties of the ‐orthogonal projection. Additionally, as the multigrid method requires not only the coarsening of the elements, but also that of the faces, we leverage the geometric flexibility of polytopal elements to define an abstract non‐nested coarsening strategy based on element agglomeration and face collapsing. Finally, the multigrid method is tested on homogeneous and heterogeneous diffusion problems in two and three space dimensions. The solver exhibits near‐perfect asymptotic optimality for moderate degrees of approximation.

Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport

Hierarchical abstractions are a methodology for solving large-scale graph problems in various disciplines. Coarsening is one such approach: it generates a pyramid of graphs whereby the one in the next level is a structural summary of the prior one. With a long history in scientific computing, many coarsening strategies were developed based on mathematically driven heuristics. Recently, resurgent interests exist in deep learning to design hierarchical methods learnable through differentiable parameterization. These approaches are paired with downstream tasks for supervised learning. In practice, however, supervised signals (e.g., labels) are scarce and are often laborious to obtain. In this work, we propose an unsupervised approach, coined OTCoarsening, with the use of optimal transport. Both the coarsening matrix and the transport cost matrix are parameterized, so that an optimal coarsening strategy can be learned and tailored for a given set of graphs. We demonstrate that the proposed approach produces meaningful coarse graphs and yields competitive performance compared with supervised methods for graph classification and regression.

Algorithm 1016

In this article, we introduce the Python framework PyMGRIT, which implements the multigrid-reduction-in-time (MGRIT) algorithm for solving (non-)linear systems arising from the discretization of time-dependent problems. The MGRIT algorithm is a reduction-based iterative method that allows parallel-in-time simulations, i.e., calculating multiple time steps simultaneously in a simulation, using a time-grid hierarchy. The PyMGRIT framework includes many different variants of the MGRIT algorithm, ranging from different multigrid cycle types and relaxation schemes, various coarsening strategies, including time-only and space-time coarsening, and the ability to utilize different time integrators on different levels in the multigrid hierachy. The comprehensive documentation with tutorials and many examples and the fully documented code allow an easy start into the work with the package. The functionality of the code is ensured by automated serial and parallel tests using continuous integration. PyMGRIT supports serial runs suitable for prototyping and testing of new approaches, as well as parallel runs using the Message Passing Interface (MPI). In this manuscript, we describe the implementation of the MGRIT algorithm in PyMGRIT and present the usage from both a user and a developer point of view. Three examples illustrate different aspects of the package itself, especially running tests with pure time parallelism, as well as space-time parallelism through the coupling of PyMGRIT with PETSc or Firedrake.

Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise

Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme with three-factor coarsening to solve the corresponding discretized control problem is presented. On staggered grids, a three-factor coarsening strategy helps in simplifying the inter-grid transfer operators and reduction in computation (CPU time). For smoothing, a distributive Gauss–Seidel scheme with a line search strategy is employed. To validate the proposed multigrid staggered grid framework, numerical results are presented with white noise at the end.

Adaptive Finite Element–Discrete Element Analysis for the Stress Shadow Effects and Fracture Interaction Behaviours in Three-Dimensional Multistage Hydrofracturing Considering Varying Perforation Cluster Spaces and Fracturing Scenarios of Horizontal Wells

Optimization of complex fracture networks improves the fracturing effects and enhances production in multistage hydrofracturing technology. To understand the controlling mechanisms of multistage hydrofracturing in unconventional tight reservoirs, some governing issues, such as hydro-mechanical coupling, stress shadow effects, propagation interaction behaviours of three-dimensional (3D) multiple fractures, and 3D multistage hydrofracturing, should be addressed. However, the characterization of perforation cluster spaces and fracturing scenarios of horizontal wells, which significantly affect the evolution of the stress field and 3D morphology of the fracture network, is a challenge. In this study, to overcome the drawbacks of the traditional finite-element method in simulating 3D fracture propagation, the adaptive finite element–discrete element method is used. This method uses a local remeshing and coarsening strategy to ensure the accuracy of solutions, reliability of the fracture propagation path, and computational efficiency. The study proposes 3D engineering-scale numerical models, considering the crucial hydro-mechanical coupling and fracturing fluid leak-off, to simulate 3D multistage hydrofracturing and fracture interaction behaviours. The numerical results show that the stress shadow effects and fracture interaction behaviours become more intense once the spaces between different propagating fractures become thinner due to superposition and reduction effects in fracturing-induced shear stress variation areas. The alternate fracturing can reduce the stress shadow effects through adjusting the sequence of perforation clusters that are activated and injected with fracturing fluid. When the perforation cluster spaces become narrow, the alternate fracturing scenario can yield more fracturing fracture areas and improve the fracturing effects as compared to sequential and simultaneous fracturing.