Refinement Criteria Research Articles

Adaptive stabilized mixed formulation for phase field fracture modeling of nearly incompressible finite elasticity

The recently proposed mixed displacement-pressure-phase field formulation characterized by damage-mitigating incompressibility constraints excels in fracture modeling of nearly incompressible hyperelastic materials. However, such formulations generally call for mixed finite element (FE) allocations that satisfy the inf-sup condition, denying the popular lower equal-order linear FE pairs, like linear polynomial (P1) interpolation for all fields. To overcome this drawback, we develop a stabilized mixed formulation (P1/P1S) by resorting to a pressure projection technique. The lure of this formulation lies in its simplicity and versatility, allowing low-order P1 interpolation for all field variables. With this trait in mind, an efficient adaptive meshing strategy is designed, thereby speeding up computation by more than 15 times. For better coping with the adaptive scenarios involving crack nucleation, we also propose a novel energy-based mesh refinement criterion. The numerical treatment of the stabilized mixed FE formulation is elaborated, as well as the core operations of the adaptive meshing technique. The accuracy, efficiency, and robustness of the proposed formulation are validated by several representative numerical examples. Remarkably, a mixed I/II fracture test thereof is in good agreement with the experiment and thus promises to be a new benchmark.

Bioinformatics pipeline unveils genetic variability to synthetic vaccine design for Indian SARS-CoV-2 genomes

In the worrisome scenarios of various waves of SARS-CoV-2 pandemic, a comprehensive bioinformatics pipeline is essential to analyse the virus genomes in order to understand its evolution, thereby identifying mutations as signature SNPs, conserved regions and subsequently to design epitope based synthetic vaccine. We have thus performed multiple sequence alignment of 4996 Indian SARS-CoV-2 genomes as a case study using MAFFT followed by phylogenetic analysis using Nextstrain to identify virus clades. Furthermore, based on the entropy of each genomic coordinate of the aligned sequences, conserved regions are identified. After refinement of the conserved regions, based on its length, one conserved region is identified for which the primers and probes are reported for virus detection. The refined conserved regions are also used to identify T-cell and B-cell epitopes along with their immunogenic and antigenic scores. Such scores are used for selecting the most immunogenic and antigenic epitopes. By executing this pipeline, 40 unique signature SNPs are identified resulting in 23 non-synonymous signature SNPs which provide 28 amino acid changes in protein. On the other hand, 12 conserved regions are selected based on refinement criteria out of which one is selected as the potential target for virus detection. Additionally, 22 MHC-I and 21 MHC-II restricted T-cell epitopes with 10 unique HLA alleles each and 17 B-cell epitopes are obtained for 12 conserved regions. All the results are validated both quantitatively and qualitatively which show that from genetic variability to synthetic vaccine design, the proposed pipeline can be used effectively to combat SARS-CoV-2.

Adaptive fourth-order phase field method for rock fractures using novel refinement criteria and improved data transfer operators

High computing cost restricts the application of phase field models in geotechnical engineering (e.g., in blasting, oil and gas exploration, and rock landslides). To improve computational efficiency, this paper proposes an adaptive isogeometric method of the phase-field model for simulating rock fracture using a novel refinement criterion and an improved data transfer operator (HBFT). The proposed method is shown to decrease the calculation time and storage requirements by over 90% compared to the uniform refinement in most cases, and the computing time of incorporating non-equal order cells is 35.23% less than that of the equal order case. Notably: (1) the proposed refinement criterion is simple and efficient, and relies only on the 1D knot vector of the IGA to guarantee the hierarchical difference of adjacent cells.(2) the proposed HBFT only transfers the history variables in the local region to be refined, while keeping the variables in other regions unchanged; additionally, compared with the global and cell-by-cell versions in the traditional BFT, the proposed HBFT not only has the potential to avoid solving large-scale linear equations of the global version, but also alleviates, to a certain extent, the requirement of the cell-by-cell version for the full integration cell.

Machine learning based refinement strategies for polyhedral grids with applications to virtual element and polyhedral discontinuous Galerkin methods

We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the “shape” of an element so that “ad-hoc” refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity.

Wake modeling and simulation of an experimental wind turbine using large eddy simulation coupled with immersed boundary method alongside a dynamic adaptive mesh refinement

Accurate numerical models of the flow around wind turbines is highly relevant for optimizing the energy efficiency of wind farms. The main goal of this work is to apply large eddy simulation (LES) along with an immersed boundary (IB) method to generate spatial and temporal information of the velocity field and forces acting on a fully-resolved NREL S809 airfoil and NREL Phase VI wind turbine. The numerical framework used in the simulations performs LES under a Cartesian block-structured mesh that is dynamically refined via adaptive mesh refinement (AMR) to increase accuracy and reduce computational costs. The mesh refinement criteria is based on Lagrangian points and vorticity. Solid surfaces of the wind turbine are captured using the Lagrangian formulation of the IB method. The simulations are based on the NREL Phase VI wind turbine, where blade analysis of lift and drag coefficients were performed, followed by time-averaged streamwise velocity and vorticity evaluations. For different angles of attack, the wake centerline velocity analysis demonstrates high energy extraction in the near wake of the wind turbine blade. In terms of validation, the drag coefficient of the NREL S809 airfoil achieves better agreement with the experimental data than the lift coefficient. Validation from the NREL Phase VI shows that the power performance difference between the simulation and experiment is lower than 6%. Wake analysis of the wind turbine scenario are based on downstream profiles of the time-averaged streamwise velocity. The numerical simulation shows a higher loss of kinetic energy in the near wake region, mainly in the centerline, compared to benchmark data, but achieves very similar results to literature in the far-wake region.

Computation of interfacial flows using Continuous Surface Stress method with adaptive mesh refinement in a quad/octree grid structure

A new implementation of the Continuous Surface Stress (CSS) method into a quad/octree grid structure is presented, which realizes effective adaptive mesh refinement (AMR) for computation of interfacial flows with significant topology changes. The momentum conservation property of the CSS method is exactly retained. The extension of the CSS method to the AMR framework and a designed phase-based refinement criterion are described in detail. The new implementation works on 2D/3D and on axis-symmetric grids. The proposed method is shown to be accurate in the computation of the Laplace pressure. The spurious velocity is evaluated based on the benchmark cases of static droplet and translating droplet. The test case of multiple droplets colliding onto a liquid film illustrates the functionality and efficiency of the method in dealing with complex interfacial flows. Thanks to the high resolution realized by AMR, it is shown in computation of the process of binary droplet collisions that the method is capable of capturing finest interfacial structures subjected to complex topology changes.

Lessons for adaptive mesh refinement in numerical relativity

We demonstrate the flexibility and utility of the Berger–Rigoutsos adaptive mesh refinement (AMR) algorithm used in the open-source numerical relativity (NR) code GRChombo for generating gravitational waveforms from binary black-hole (BH) inspirals, and for studying other problems involving non-trivial matter configurations. We show that GRChombo can produce high quality binary BH waveforms through a code comparison with the established NR code Lean. We also discuss some of the technical challenges involved in making use of full AMR (as opposed to, e.g. moving box mesh refinement), including the numerical effects caused by using various refinement criteria when regridding. We suggest several ‘rules of thumb’ for when to use different tagging criteria for simulating a variety of physical phenomena. We demonstrate the use of these different criteria through example evolutions of a scalar field theory. Finally, we also review the current status and general capabilities of GRChombo.

A moving embedded boundary approach for the compressible Navier-Stokes equations in a block-structured adaptive refinement framework

A computational technique has been developed to perform compressible flow simulations involving moving boundaries using an embedded boundary approach within the block-structured adaptive mesh refinement (SAMR) framework of AMReX [1,91,92]. We leverage the SAMR capability to obtain quantitatively accurate results whilst using robust, second-order finite volume schemes. A conservative, unsplit, cut-cell approach is utilized and a ghost-cell approach is developed for computing the flux on the moving, embedded boundary faces. A third-order least-squares formulation has been developed to compute the wall velocity gradients, and was found to significantly improve the performance of the solver in terms of the quantitative comparison of surface quantities such as the skin friction coefficient. Various test cases are performed to validate the method, and compared with analytical, experimental, and other numerical results in literature. Inviscid and viscous test cases are performed that span a wide regime of flow speeds – acoustic (harmonically pulsating sphere), smooth flows (expansion fan created by a receding piston) and flows with shocks (shock-cylinder interaction, shock-wedge interaction, pitching NACA 0012 airfoil and shock-cone interaction). A closed system with moving boundaries – an oscillating piston in a cylinder, showed that the percentage error in mass within the system decreases with refinement, demonstrating that the numerical scheme is conservative with grid refinement, but is not discretely conservative. Viscous test cases involve that of a horizontally moving cylinder at Re=40, an inline oscillating cylinder at Re=100, and a transversely oscillating cylinder at Re=185. The judicious use of adaptive mesh refinement with appropriate refinement criteria to capture the regions of interest leads to well-resolved flow features, and good quantitative comparison is observed with the results available in literature.

Solving the Schrödinger-Poisson system using the coordinate adaptive moving mesh method

In this paper, we implement the Adaptive Moving Mesh method (AMM) to the solution of initial value problems involving the Schr\"odinger equation, and more specifically the Schr\"odinger-Poisson system of equations. This method is based on the solution of the problem on a discrete domain, whose resolution is coordinate and time-dependent, and allows to dynamically assign numerical resolution in terms of desired refinement criteria. We apply the method to solve various test problems involving stationary solutions of the SP system, and toy scenarios related to the disruption of subhalo s made of ultralight bosonic dark matter traveling on top of host galaxies.

Adaptive grid refinement for ship resistance computations

This paper studies the possibility of using adaptive grid refinement for routine, automated simulations of ship resistance in calm water. With the increase in maturity and reliability of mesh adaptation methods, the main remaining challenge is the creation of straightforward, universal user guidelines which allow these computations to be run correctly, without resorting to trial and error to set the parameters.The paper uses the mesh adaptation in the flow solver ISIS-CFD. For this solver, a simulation protocol for resistance computation is proposed, which specifies for example the choice of the refinement criterion and the global mesh size. To investigate the reliability and generality of this protocol, it is fine-tuned on one test case and then applied unchanged to three different cases.The tests show that the solutions have good behaviour and compare well with experiments. Furthermore, numerical uncertainty estimation works for these cases, which increases the trustworthiness of the solutions. Where this is tested, the mesh adaptation produces the same solutions as traditional meshing methods with reduced computational costs. As such, it is shown that mesh adaptation for resistance computations is possible today on a routine basis and that it is advantageous compared with other meshing techniques.

Efficient distributed approach for density-based topology optimization using coarsening and h-refinement

This work presents an efficient parallel implementation of density-based topology optimization using Adaptive Mesh Refinement (AMR) schemes to reduce the computational burden of the bottleneck of the process, the evaluation of the objective function using Finite Element Analysis (FEA). The objective is to obtain an equivalent design to the one generated on a uniformly fine mesh using distributed memory computing but at a much cheaper computational cost. We propose using a fine mesh for the optimization and a coarse mesh for the analysis using coarsening and refinement criteria based on the thresholding of design variables. We evaluate the functional on the coarse mesh using a distributed conjugate gradient solver preconditioned by an algebraic multigrid (AMG) method showing its computational advantages in some cases by comparing with geometric multigrid (GMG) and AMG methods in two- and three-dimensional problems. We use different computational resources with small regularization distances for such comparisons. We also evaluate the performance and scalability of the proposal using a different number of computing cores and distributed computing hosts. The numerical results show a significant increment of the computing performance for the overall computing time of the proposal combining dynamic coarsening, adaptive mesh refinement, and distributed memory computing architectures.

Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows

A parallel anisotropic block-based adaptive mesh refinement (AMR) algorithm is proposed for the prediction of physically complex flow problems having disparate spatial and temporal scales and exhibiting highly anisotropic features on three-dimensional multi-block body-fitted hexahedral meshes with non-uniform grid blocks. The proposed AMR scheme makes use of a binary tree hierarchical data structure to permit anisotropic refinement of grid blocks in a preferred coordinate direction as dictated by appropriately selected physics-based refinement criteria. The anisotropic coarsening of the grid blocks in a manner that is independent of the refinement history allows the mesh to rapidly re-adapt for evolving unsteady flow applications. Moreover, the proposed anisotropic AMR scheme adopts a non-uniform representation of the cells within each block by directly using the neighboring cells as the ghost cells, even at a grid resolution change. This affords a number of computational advantages especially related to evaluating the solution in the ghost cells as well as to ensuring flux conservation at block interfaces. A modified upwind-based finite-volume spatial discretization scheme is applied in conjunction with the AMR scheme to the solution of Euler and Navier-Stokes equations for inviscid and viscous compressible gaseous flows. Steady-state and time-varying flow problems are considered on anisotropic adapted meshes. The potential flexibility and efficiency of the proposed enhanced anisotropic AMR scheme are demonstrated for the simulation of a number of representative flows of varying complexity.

Simulation of free-surface flows using the Lattice Boltzmann method with the AMR technique

High resolution grids are generally preferred near the free surface to capture desired phenomena. In the present paper, an adaptive mesh refinement (AMR) framework based on the hash table data structure is introduced to the lattice Boltzmann method (LBM). The AMR refinement criterion is based on the movement of the free surface captured through interface nodes. The performance of the AMR-based LBM solver is analyzed through several benchmark cases. Results show that the proposed numerical framework can properly track the free surface with a spatial convergence rate between the first and second order. In addition, the AMR framework is of benefit to efficiently simulating free-surface flows since it requires much less nodes to achieve high resolution near the interface.

Stochastic stability in the large population and small mutation limits for coordination games

We consider a model of stochastic evolution in symmetric coordination games with $ K\ge 2 $ strategies played by myopic agents. Agents employ the best response with mutations choice rule and simultaneously revise strategies in each period. We form the dynamic process as a Markov chain with state space being the set of best responses in order to overcome difficulties that arise with the large population. We examine the long run equilibria for both orders of limits where the small noise limit and the large population limit are taken sequentially. We characterize an equilibrium refinement criterion that is common among both orders of limits.

Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the “shape” of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements' shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is independent of the differential model the overall computational costs are kept low.

Quasicontinuum method combined with microplane model

In this paper, we show how certain ideas of the microplane theory can be used in the quasicontinuum method for an irregular structural lattice with axial interactions representing a heterogeneous material. The paper first analyzes the microplane material model, and then the relation between the lattice model and the microplane model is described. Based on this analogy, a microplane-based summation rule for the quasicontinuum method is proposed. Five simplified approaches adopting various levels of simplification are introduced using interpolation, global or local microplane-based homogenization, and an isotropic or anisotropic microplane model. For an adaptive quasicontinuum method, a new refinement criterion based on strain at individual microplanes is proposed and investigated. All presented approaches have been implemented in OOFEM (Patzák, 2012) an open-source object-oriented code. Accuracy, efficiency and specific properties of all simplified models based on the quasicontinuum idea are evaluated by comparing the results with the fully resolved lattice model for a number of examples in 2D and 3D. The presented results show that using the proposed microplane-based quasicontinuum approaches, a significant simplification of the problem can be reached while keeping the error acceptable.

Multiscale modeling of fragmentation in an electrospray plume

We present a 3D-particle-in-cell (PIC) approach to modeling electrospray plumes typical of those formed by externally wetted emitter devices. Numerical grid-resolution techniques suitable for capturing strong electric fields in the emitter region were explored, and grid refinement criteria were quantified. The molecular dynamics simulations of the EMIM−BF4 ionic liquid system were modeled to determine the fragmentation mechanism in the presence of an electric field and dimer temperature as well as to provide fragmentation rates for the PIC simulations. An energy analysis of the molecular dynamics (MD) fragmentation demonstrated that the key mechanism for dimer fragmentation corresponds to a decrease in the Coulomb energy between the cation and anion in the system and that dimers of temperatures 300 and 600 K are extremely stable for electric fields smaller than 1.5 V/nm. Using probabilities of fragmentation consistent with the MD simulations, we implemented a dimer fragmentation model in our PIC simulations. The ion energy distribution functions obtained from the PIC simulations were used to predict retarding potential analysis (RPA) curves that were compared directly to measurements. The sensitivity of the RPA shape to the fragmentation probability was found to be significant. By comparing predicted and measured RPA curves for both negative and positive operation modes, and the fact that dimers do not fragment for electric fields less than 0.6 V/nm, we conclude that fragmentation of dimers occurs spontaneously due to their high thermal energies.

Average-based mesh adaptation for hybrid RANS/LES simulation of complex flows

Generating meshes with the right resolution is crucial for hybrid RANS/LES simulations of high Reynolds number flow with complex physical phenomena and geometries. This makes automatic mesh generation through adaptive refinement an interesting option. However, since the behavior of these turbulence models depends on the local grid size, mesh changes in time as a result of adaptive refinement may affect the production and destruction of turbulence. Therefore, grid adaptation with refinement criteria based on time-averaged quantities is proposed here to produce meshes that do not evolve rapidly in time and that are suitable for capturing the unsteady flow at each instant. Two averaging approaches, averaging over the instantaneous flow field, and averaging over the refinement criterion, are tested to simulate a turbulent flow behind a backward-facing step with a Detached Eddy Simulation type turbulence model. Compared to grid adaption based on instantaneous solutions, the computational cost is reduced, the accuracy of the solutions improves and an adapted mesh which has a generally static topology based on the main flow features is obtained. The investigation of this refinement process for two realistic test cases, a ship and a hybrid delta wing, confirms the reliability of the average-based adaptation and shows that automatic meshing for hybrid RANS/LES simulations of complex flows is possible.

Simpler and better: Supply chain contracting in the presence of contract unobservability and upstream competition

A retail platform or intermediary can purchase products from multiple upstream manufacturers through different types of contracts. Traditional studies assume that the contracts and their terms are common knowledge; that is, each party of the supply chain can observe the contracts reached. In practice, however, this is not necessarily the case. Taking the result under contract observability as a benchmark, this paper investigates the contracting of the supply chain comprised of one retail platform and two upstream manufacturers when contracts are unobservable. Two representative contracts are considered: the traditional and succinct wholesale price contract and the nonlinear, more complicated, two-part tariff contract. We apply the framework of perfect Bayesian equilibrium and also use passive beliefs as an equilibrium refinement criterion to analyze the model. Our study shows that the classical result – that the more complicated contract can increase channel profitability – may no longer be valid when contracts are unobservable to competing manufacturers. On the contrary, the manufacturers and the whole supply chain can benefit from adopting wholesale price contracts rather than two-part tariff contracts. To examine the robustness of our findings, we consider a commission fee contract and another classic demand function in an extended analysis, and the results show that the main findings and key insights remain valid. The intuition behind these findings is that the unobservability of contracts affects the opportunistic behavior of competing manufacturers. The practical value of these findings is mainly reflected in the evidence that supply chain members must carefully consider the impact of contract (un)observability when agreeing on contracts.

Implicit Space-Time Domain Decomposition Approach for Solving Multiphase Miscible Flow: Accuracy and Scalability

Summary In this paper, we propose a fully implicit space-time multiscale scheme to improve computational efficiency in solving nonlinear multiphase flow in porous media. Here, error estimators are used for adaptively changing the spatial-temporal mesh. This algorithm applied to the black-oil model is compared to a standard control volume approach using a fine time and spatial mesh. Error estimators are introduced to determine subdomains of the reservoir in which high nonlinearity hinders Newtonian convergence. This is followed by applying local fine timesteps to these marked regions, whereas the remaining regions retain the coarse time scale. Once a temporal discretization is determined for different parts of the reservoir, the spatial mesh is refined for treating saturation fronts. The nonmatching interfaces arising from different temporal and spatial scales are resolved by the enhanced velocity method, which enforces strongly the continuity of fluxes. This whole system is solved monolithically. Results from a three-phase black-oil model are described. The multiscale solution is compared to a uniformly fine spatial mesh and fine timestepping solution to confirm accuracy. Solutions from both Gaussian and channelized permeability fields are presented. In the multiscale solution, we observe temporal refinements being applied to the water and gas saturation fronts by reducing the timestep size to guarantee Newtonian convergence. We also observe that the extent of refinements is balanced between the two saturation fronts based on their respective nonlinearity. For Gaussian permeability fields, the algorithm only treats saturation fronts spatially, whereas for channelized permeability fields, the geological features are also considered. Production profiles for the three phases match well between the two solutions under specific refinement criteria. We also investigate the improvement in computational efficiency, as well as the algorithm scalability in regard to increasing problem sizes. We observe a speedup of approximately 10 for solving the linear system, and such speedup increases as the problem size expands. In reservoir simulation, schemes that attempt to decouple the diffusion and advection process become less robust as the physics becomes more complex. Our approach, which is focused on handling the high nonlinearities, can be treated fully implicitly and provide a reduction in simulation costs. This approach can also be applied to model order reduction in providing accurate snapshot solutions. NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.