Approximate Analysis Research Articles

Phase reduction of strongly coupled limit-cycle oscillators

The phase reduction method has played a key role in the theoretical and experimental analysis of synchronization dynamics of coupled oscillators, but its applicability is limited to the case of weak coupling. In this study, a phase equation that describes strongly coupled limit-cycle oscillators is derived, where the orbit of each oscillator can deform largely depending on the strong parametric coupling. We extend the phase reduction theory for strongly perturbed limit-cycle oscillators and develop an approximate self-consistency analysis to obtain a reduced description of the oscillator dynamics using only the phase variables of the oscillators. Numerical simulations show that the proposed phase equation improves the accuracy to predict the dynamics of strongly coupled oscillators over the conventional description.

Designing Cladded Components for High Temperature Nuclear Service: Part II—Design Rules

AbstractThe challenge of using existing ASME Section III, Division 5, class A metallic materials for the construction of structural components of advanced reactors with corrosive coolants could be mitigated by allowing designers to use cladding to protect the base material from corrosion. However, the existing Section III, Division 5 rules provide no guidance on the evaluation of strain accumulation and creep-fatigue damage in cladded components. The availability of design rules for cladded components that do not require long-term clad material testing could promote the application of the cladding approach to accelerate the deployment schedule of these advanced reactor systems. To avoid long-term properties for the clad materials Part I of this work proposes two approximate design analysis methods for two types of clad materials—soft clads that creep much faster and have lower yield stress than the class A base material, and hard clads that creep much slower and have higher yield stress than the class A base material. The proposed analysis methods approximate the response of a soft clad by treating it as perfectly compliant and of a hard clad by treating it as linear elastic. Based on these approximate design analysis strategies this Part II develops a complete set of design rules for class A components cladded with either soft or hard clad materials. Part II discusses the reasoning behind the proposed design rules and uses example finite element analyses of representative reactor components to illustrate the use of these design methods.

Designing Cladded Components for High Temperature Nuclear Service: Part I—Analysis Methods

AbstractUse of corrosion-resistant cladding can greatly extend the design life of structural components in many advanced reactor systems. However, there are currently no ASME design rules for cladded components to guard against creep-fatigue failure and ratcheting strain accumulation in elevated temperature nuclear service. This paper, presented in two parts, addresses this gap by proposing a design strategy for cladded components that do not require long-term testing of clad materials. The proposed approach relies on approximate design analysis methods for two types of clad materials—a soft clad that creeps much faster than and has lower yield stress than the class A base material and a hard clad that creeps much slower than and has higher yield stress than the class A base material. Part I discusses the approximate analysis strategies for the clad materials—treat a soft clad as perfectly compliant and a hard clad as linear elastic—and Part II develops a complete set of design rules for each of the two types of cladded components. Finite element analyses of representative high temperature reactor components show that the proposed design analysis methods can bound the design quantities in soft cladded components and approximate the design quantities in hard cladded components.

Nine-Point Iterated Rectangle Dichotomy for Finding All Local Minima of Unknown Bounded Surface

Abstract A method is suggested to find all local minima and the global minimum of an unknown two-variable function bounded on a given rectangle regardless of the rectangle area. The method has eight inputs: five inputs defined straightforwardly and three inputs, which are adjustable. The endpoints of the initial intervals constituting the rectangle and a formula for evaluating the two-variable function at any point of this rectangle are the straightforward inputs. The three adjustable inputs are a tolerance with the minimal and maximal numbers of subintervals along each dimension. The tolerance is the secondary adjustable input. Having broken the initial rectangle into a set of subrectangles, the nine-point iterated rectangle dichotomy “gropes” around every local minimum by successively cutting off 75 % of the subrectangle area or dividing the subrectangle in four. A range of subrectangle sets defined by the minimal and maximal numbers of subintervals along each dimension is covered by running the nine-point rectangle dichotomy on every set of subrectangles. As a set of values of currently found local minima points changes no more than by the tolerance, the set of local minimum points and the respective set of minimum values of the surface are returned. The presented approach is applicable to whichever task of finding local extrema is. If primarily the purpose is to find all local maxima or the global maximum of the two-variable function, the presented approach is applied to the function taken with the negative sign. The presented approach is a significant and important contribution to the field of numerical estimation and approximate analysis. Although the method does not assure obtaining all local minima (or maxima) for any two-variable function, setting appropriate minimal and maximal numbers of subintervals makes missing some minima (or maxima) very unlikely.

Efficient steady state analysis of the grid using electromagnetic transient models

• Time-domain steady-state (TDSS) methodology efficiently solves for an accurate steady-state of the grid using EMT models. • The paper presents a decoupling approach that iterates on a waveform to efficiently solve for the steady-state. • The simulation results highlight improvement in accuracy over power flow while solving in fewer iterations compared to EMT analysis. Modern grids contain an increasing number of non-linear grid devices that are accurately modeled only in the time domain. Performing an accurate steady-state analysis with such models requires a transient simulation to infinite (sufficiently long) time, which can be computationally prohibitive. Power flow provides an efficient, but approximate steady state analysis using models based on a single nominal frequency, which fails to represent the true steady-state. We present a new method for efficiently analyzing the steady state response using traditional transient models that is inspired by harmonic balance methods. Rather than solving the underlying ODE by discretizing time, our method iterates on the entire waveform to find an accurate time domain waveform that captures the steady-state dynamics for all three phases. This presents a novel, efficient methodology for accurately studying the steady-state of the power grid. We introduce the concept by studying the steady state of a 14-bus system under various scenarios and highlight the difference between the steady-states achieved through power-flow and our method that uses EMT models.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

Analysis and Optimization of Energy Variations in MMCs With Variable DC Voltages

The modular multilevel converter (MMC) containing full-bridge submodules (FBSMs) can increase the modulation index to reduce the energy storage requirement. The MMCs containing FBSMs can also adjust the dc voltage to fit special applications. However, limited research explores the energy storage characteristics of MMCs with variable dc voltages, and few effective approaches can ensure low-energy storage requirements. This study proposes an approximate analysis method for the energy storage requirement in MMC with variable dc voltages. Results reveal that the energy variation under low dc voltages restricts the reduction effects of the increased modulation index on the energy storage requirement. Then, on the basis of the fact that the low ac component of the arm current under low dc voltages provides a large current margin, an optimal circulating current (OCC) is proposed to optimize the energy variation under low dc voltages. In addition, a practical curve-fitting approach is proposed to facilitate the calculation of OCC in real-time controllers. With these approaches, the energy storage requirement with variable dc voltage becomes similar to that with constant rated dc voltage. Simulation and experimental results verify the analysis.

Operational signature-based symbolic hierarchical clustering for building energy, operation, and efficiency towards carbon neutrality

Buildings are considered the enormous source of untapped energy efficiency potential in the global carbon neutrality. It is necessary to ensure that buildings are energy-efficient using operational pattern analytics and diagnostics. Therefore, this study proposes a novel symbolic hierarchical clustering method (named HOS-SAX) to evaluate the building system operation, efficiency, and energy usage patterns. The proposed HOS-SAX method is intended to enhance the existing methods that focus only on the energy usage characteristics and thus offer limited insights on the building system and operational efficiency. The proposed method consists of: (1) Holistic Operational Signature (HOS) and (2) HOS-based symbolic aggregate approximation (SAX) analyses. A HOS analysis is conducted to derive the representative operational signatures for building operation and efficiency using system-, building-, and weather-level data. Then, SAX is performed with the operational signatures derived from the HOS to cluster the building operation patterns. In a case study for a district heating substation serving residential buildings, the HOS-SAX cluster analysis showed 15 sections in the cluster map that visualize the: (1) energy usage, (2) design efficiency, and (3) control efficiency. The cluster map revealed that the sections that operated inefficiently account for approximately 71% of the entire operation period. Moreover, it is expected that the supply temperature of 0.87 °C can be reduced in the most inefficient sections.

Analysis of sheet pile in dense and loose soil using finite element method

Sheet pile wall acts as a support system for soils in excavation or backfill structures of a wall. There is different loading condition on a sheet pile wall depending upon the different type of soil. The study is carried out to check the stability of sheet pile walls using finite element analysis software PLAXIS 2D. Different types of soil combinations are considered in this study, such as DD (Dense backfill and Dense foundation soil), LL (Loose backfill and Loose foundation soil), DL (Dense backfill and Loose foundation soil), and LD (Loose backfill and Dense foundation soil). The embedment depth of the sheet pile varies along with the thickness of the pile. The PZ35 sheet pile is used as the optimum sheet pile because of the reduction in horizontal displacement in all soil combinations and its lesser thickness compared to the PZ40 sheet pile. The approximate cost analysis of steel used in PZ27 and PZ35 sheet pile shows that the with PZ35 sheet pile cost is increased by 20 % with 50 % reduction in horizontal displacement. The cushion material (well-graded sand) below the building load causes a decrease in the horizontal deflection of the sheet pile.

Design and Robust Performance Analysis of Low-Order Approximation of Fractional PID Controller Based on an IABC Algorithm for an Automatic Voltage Regulator System

In this paper, a low-order approximation (LOA) of fractional order PID (FOPID) for an automatic voltage regulator (AVR) based on the modified artificial bee colony (ABC) is proposed. The improved artificial bee colony (IABC) high-order approximation (HOA)-based fractional order PID (IABC/HOA-FOPID) controller, which is distinguished by a significant order approximation and by an integer order transfer function, requires the use of a large number of parameters. To improve the AVR system’s performance in terms of transient and frequency response analysis, the memory capacity of the IABC/HOA-FOPID controller was lowered so that it could fit better in the corrective loop. The new robust controller is named the improved artificial bee colony (IABC) low-order approximation (LOA)-based fractional order PID (IABC/LOA-FOPID). The performance of the proposed IABC/LOA-FOPID controller was compared not only to the original ABC algorithm-tuned PID controller, but also to other controllers tuned by state-of-the-art meta-heuristic algorithms such as the improved whale optimization algorithm (IWOA), particle swarm optimization (PSO), cuckoo search (CS), many optimizing liaisons (MOL), genetic algorithm (GA), local unimodal sampling (LUS), and the tree seed algorithm (TSA). Step response, root locus, frequency response, robustness test, and disturbance rejection abilities are all compared. The simulation results and comparisons with the proposed IABC/LOA-FOPID controller and other existing controllers clearly show that the proposed IABC/LOA-FOPID controller outperforms the optimal PID controllers found by other algorithms in all the aforementioned performance tests.

Results on approximate controllability for a second-order semilinear nonlocal control system with monotonic nonlinearity

ABSTRACT In this paper, we propose the approximate controllability analysis for a second semilinear system under nonlocal conditions using monotone nonlinearity as a tool. Monotonicity is a significant feature in many communications applications that use digital-to-analog converter circuits. While non-monotonicity prevents such applications from working, nonlinearity makes them work. Therefore, it is of great interest to study a problem while assuming the nonlinear function is monotonic. Nonlocal conditions have a finer effect on the solution and are additional specifications for the physical measurements beyond the conventional ones. In order to determine the controllability of the considered system, we formulate the control function. In addition to this, we have broadened the scope of the results for the impulsive system. At last, we discuss two examples.

Mis-Splicing Due to Somatic U2AF2 Mutations in Myeloid Neoplasms

Background Recent genetic studies have identified frequent mutations affecting a number of genes encoding RNA splicing factors (SDs) in myelodysplastic syndromes (MDS) and related myeloid neoplasms (MNs). Among these, most frequently mutation are SF3B1, SRSF2, U2AF1, and ZRSR2, whose functions have been intensively studied. By contrast, U2AF2, another SF, is also recurrently mutated in MNs. U2AF2 is a component of the U2 auxiliary factor that forms a heterodimer with U2AF1 for the recognition of the 3’ splice site (3'SS). U2AF2 contains a sequence-specific RNA-binding region with two RNA recognition motifs and identifies polypyrimidine (Py) tract signals of nascent transcripts. However, compared to other splicing factor mutations, U2AF2 mutations are much rare, preventing the detailed analysis of their role in leukemogenesis. Methods To characterize the role of U2AF2 mutations, we systematically analyzed mutation spectrum in 6,369 with different MNs using targeted-capture sequencing. We also performed RNA sequencing of cKit(+) bone marrow cells from 6 U2AF2 mutated MDS cases along with 52 MDS patients without common splicing factor mutations and 25 healthy individuals. THP1 and HL60 leukemia cells with exogenous expression of wildtype or mutant U2AF2 were established to evaluate the splicing response. We also used clustered regularly interspaced short palindromic repeats /CRISPR-associated protein-9 nuclease (CRISPR/cas9) to introduce the p.190_195del mutation to U2AF2 in K562 and MOLM-13 leukemia cells, generating an isogenic model so that splicing alterations can be attributed solely to mutant U2AF2. Results In total, U2AF2 mutations were found in 34 (0.53%) of 6,369 MN cases, of which 29 had the recurrent p.190_195del mutation, the median age of U2AF2 mutations was 55 years (range 31-79) and 55% were diagnosed with MDS, 31% with CMML and 7% with AML. U2AF2-mutated patients showed a male predominance (90%). The most frequently co-mutated gene was ASXL1 with 52%, followed by SETBP1 (35%) STAG2 (21%),RUNX1 (21%), CSF3R, NRAS, DNMT3A, NF1, PTPN11, ETV6, and STAT3. To investigate the effects of the U2AF2 p.190_195del mutation on RNA splicing, we analyzed transcriptome sequencing, followed by the rMATS bioinformatics pipeline to determine alternative splicing (AS) events . Various types of AS events were identified, including skipped exons (SEs), alternative 5’ ss exons (A5SSs), alternative 3’ ss exons (A3SSs), retained introns (RIs), and mutually exclusive exons (MXEs). Among these, the skipping exons are the most frequent in both U2AF2-mutant primary MDS cases and cell lines. To explore these alternative splicing results in more detail, we next examined differentially skipped exons. Unsupervised uniform manifold approximation and projection (UMAP) analysis based on the standardized splicing ratio of skipping exons segregated U2AF2-mutated patients from other MDS cases without common splicing factor mutations and healthy individuals. To prioritize mutant U2AF2-induced alterative splicing events, we intersected significant skipping exons across 2 datasets: MDS patient samples with and without U2AF2 p.190_195del mutations, and MDS patient samples with U2AF2 p.190_195del mutations and healthy individuals. The differential skipping exon usages and differential gene expression profiles enables both consensus sequence analysis and pathway/gene-set enrichment analysis. Conclusions Our study revealed that the U2AF2 p.190_195del regulated aberrant alternative splicing facilitated MDS progression through perturbations in splice isoforms. A better understanding of the U2AF2 p.190_195del and its critical specific changes will provide novel insights into disease pathophysiology and potentially inform future treatment decisions.

Approximation and convergence analysis of optimal control for non-instantaneous impulsive fractional evolution hemivariational inequalities

In this work, a fractional evolution hemivariational inequalities driven by non-instantaneous impulses is studied. The solvability of the proposed system is obtained by fractional calculus, properties of generalized Clarke’s subdifferential and Dhage fixed point theorem. This article also presents the construction of the lower-dimensional approximation system for the proposed model, and its convergence of the mild solution is obtained. Besides, by considering suitable assumptions, the local approximation and uniform convergence results are established for the proposed system’s mild solution. Furthermore, sufficient conditions for the existence of Lagrange optimal control problem and convergence analysis of optimal control for the proposed model are formulated and proved. At last, an example is given for the illustration of invented new theoretical results.

A 23–31 GHz high‐gain GaN low‐noise amplifier using T‐type matching networks and multiple feedback techniques for broadband 5G millimeter‐wave applications

AbstractA 23–31 GHz three‐stage low‐noise amplifier (LNA) based on 100 nm gallium nitride‐on‐Si technology is presented with a total circuit size of 2.0 × 1.4 mm2. To achieve an excellent noise figure (NF) and gain, a special inductive dual‐source feedback network is used; it contributes to noise matching and input conjugate matching by resonating with the parasitic gate‐source capacitance of the transistor. T‐type networks with low‐quality factor are carefully employed for broadband matching circuits to extend the bandwidth. Meanwhile, a lossy RLC feedback network of the output stage can further obtain a wider frequency band and improve the stability through Miller approximation analysis of the input quality factor and gain of the closed‐loop system. The proposed LNA achieves a NF of 1.9–2.5 dB, a small signal gain of 24–27 dB, and an output 1 dB compression point (P1dB) of 16 dBm across the band.

ЗАВИСИМОСТЬ ЧАСТОТЫ ВРАЩЕНИЯ ЛОПАСТЕЙ МЕШАЛКИ МОКРОГО ПЫЛЕУЛОВИТЕЛЯ ОТ ПЛОТНОСТИ ЖИДКОСТИ

One of the undesirable factors when grinding grain with hammer crushers is the formation of flour dust, which is fire and explosive, as well as harmful to the health of service personnel. The applied fabric dust collectors ineffectively purify the air coming out of the pneumatic system of the crusher. One of the directions of development of dust collection equipment is the development of wet dust collectors or the improvement of existing ones. The proposed two-stage wet dust collector gives greater efficiency of air purification from dust, which is shown by laboratory studies. One of the factors affecting the quality of cleaning is the speed of rotation of the agitator blade, which varies depending on the concentration of dust in the liquid of the first stage. Not the entire dust-air mixture passes through the upgraded dust collector, but only the part that is redundant. This allows you to increase the life of the dust collector before the next service. The aim of the work is to study the dependence of the rotation speed of the blade of the agitator of the first stage of the dust collector on the density of the liquid. Methods of mathematical modeling of the agitator operation, approximation and regression analysis were used for the research, correlation analysis was performed. Theoretical studies have revealed a hyperbolic dependence of the rotation frequency of the paddle agitator on the density of the colloidal solution, however, experimental data provided other quantitative indicators. To determine the necessary parameters of the equation, the least squares method was used to approximate the experimental data. The calculated data obtained were compared with the experimental ones by the coefficient of determination, and the model was highly adequate, R2 = 0,9705. The presented method of selecting the regression equation shows good convergence with the experiment. The equations derived for a laboratory installation can be converted to an industrial design using the theory of similarity.

Numerical approximation and error analysis for Caputo–Hadamard fractional stochastic differential equations

Numerical approximation and error analysis for Caputo–Hadamard fractional stochastic differential equations

Abstract 14950: Single-Cell Transcriptomic Analysis Reveals Emergence of Inflammatory and Activated Arterial Endothelial Cells Promoted by Cross-Talk With Mural Cells During Development of Severe Pulmonary Arterial Hypertension

Introduction: Pulmonary arterial hypertension (PAH) is a lethal disease characterized by occlusive arterial remodelling thought to be triggered by endothelial cell (EC) injury; however, the pathobiological mechanisms are poorly understood. We employed single cell RNA sequencing (scRNAseq) to define transcriptomic changes in the SU5416/chronic hypoxia (SU/CH) rat model. Methods: Sprague-Dawley rats were injected with 20mg/kg SU subcutaneously and exposed to 3 weeks of CH (10% O 2 ). Right ventricular systolic pressure (RVSP) was measured at baseline, 1, 3, 5, and 8-weeks, then lungs were explanted, digested, and dissociated into single cells which were sequenced using the 10x genomics platform. Results: Elevation of RVSP plateaued at >100 mmHg by 5 weeks. Dimensionality reduction of the scRNAseq data was performed using Uniform Manifold Approximation and Projection (UMAP) analysis resulting in 24 lung clusters. Cell prioritization analysis performed using Augur identified vascular cells as the most affected by SU/CH, including arterial ECs, pericytes, and gCap ECs. After sub-clustering the EC populations, eight distinct clusters were identified representing all expected subtypes (capillary, arterial, venous, and lymphatic). As well, two novel EC clusters were detected. The first showed reduced expression of classical EC genes ( Cdh5 , Cldn5 ) and increased proinflammatory markers ( RT1-Da , Cd74 ), termed ‘inflammatory’ ECs. The second was an arterial cluster that exhibited reduced expression of Dll4 and increased Cxcl12 and Fn1 , termed ‘activated’ arterial ECs. Both populations emerging at 1-week post SU, persisting throughout PAH progression. While many EC populations showed marked differential gene expression (DGE) at 1 week, the activated arterial cluster was the only cluster showing progression of DGE throughout progression of PAH. Receptor-ligand analysis with NicheNet identified activated arterial ECs as a top receiver cell responding to multiple ligands, including Bmp4/5 , Col4a1 , and Vegfa/c, which were largely derived from mural fibroblasts. Conclusion: Emergence of inflammatory and activated arterial ECs likely contribute to vascular remodelling associated with PAH promoted by cross-talk with lung stromal cells.

Abstract 15203: Single Cell Transcriptional Analysis Implicates Acta2 Positive Pericytes in Arterial Muscularization and Imbalance in Bmp and Angpt Signaling During Development of Pulmonary Arterial Hypertension

Introduction: Pulmonary arterial hypertension (PAH) is a lethal disease characterized by marked vascular remodeling, inflammation and increased pulmonary vascular resistance. Pericytes normally play a key role in maintaining microvascular homeostasis; however, they can also contribute to increased muscularization of distal arterioles in PAH by giving rise to smooth cells (SMCs). Methods: Sprague-Dawley rats were injected with 20mg/kg SU subcutaneously and exposed to 3 weeks of chronic hypoxia (CH, 10% O2). Right ventricular systolic pressure (RVSP) was measured at baseline, 1, 3, 5, and 8-weeks, then lungs were explanted, digested, and dissociated into single cells which were sequenced using the 10x genomics platform. Results: RVSP was significantly elevated at 1 week after SU (47mmHg) reaching a plateau of >100 mmHg by 5 weeks. Dimensionality reduction of the scRNAseq data was performed using Uniform Manifold Approximation and Projection (UMAP) analysis resulting in 24 lung clusters and sub-clustering of stomal populations yielded seven clusters. In addition to fibroblasts, SMC/myofibroblasts and mesothelial cells, we identified 2 distinct pericyte clusters both expressing canonical pericyte markers such as Pdgfrb and Csgp4 ( Ng2 ) and differentiated by presence or absence of Acta2 . Acta2 + pericytes exhibited high expression of contractile markers (i.e., Myh11 and Tagln ), consistent with partial SMC differentiation, and underwent a substantial (>3-fold) expansion during PAH development. The Acta2 + pericytes also exhibited profound downregulation in the endothelial protective gene, Bmpr2 , with a reciprocal upregulation of Rbpms2 , a gene which inhibits BMP signaling. Similarly, reciprocal reduction in Angpt1 , an agonist of the endothelial Tie2 receptor and an increase in expression the partial Tie2 antagonist, Angpt2 , thereby favoring EC activation and inflammation. Conclusion: These results suggest expansion of Acta2 + pericytes in PAH may contribute to distal arteriolar muscularization as well as an imbalance in BMP/TGFß and ANGPT1/2 signaling, promoting EC apoptosis, activation, and inflammation in the SU/CH model.

A priori error analysis for an isogeometric discontinuous Galerkin approximation for convection problems on surfaces

In this paper, we identify and study the new isogeometric analysis penalty discontinuous Galerkin (DG) methods of convection problems on implicitly defined surfaces with optimal convergence properties. Like all other known discontinuous Galerkin methods on flat space or Euclidean space using polynomials of degree k≥0 for the unknown, the orders of convergence in L2 norm and DG norm are k+1 and k+12, respectively, which shows the resulting methods on surfaces can be implemented as efficiently as they are for the case of flat space or Euclidean space. The theoretical results are illustrated by two numerical experiments.

From ridges to manifolds: 3D characterization of the moving groups in the Milky Way disc

Context. The details of the effect of the bar and spiral arms on the disc dynamics of the Milky Way are still unknown. The stellar velocity distribution in the solar neighbourhood displays kinematic substructures, which are possibly signatures of these processes and of previous accretion events. With the Gaia mission, more details of these signatures, such as ridges in the Vϕ − R plane and thin arches in the Vϕ − VR plane, have been revealed. The positions of these kinematic substructures, or moving groups, can be thought of as continuous manifolds in the 6D phase space, and the ridges and arches as specific projections of these manifolds. Aims. Our aim is to detect and characterize the moving groups along the Milky Way disc, sampling the galactocentric radial and azimuthal velocities of the manifolds through the three dimensions of the disc: radial, azimuthal, and vertical. Method. We developed and applied a novel methodology to perform a blind search for substructure in the Gaia EDR3 6D data, which consists in the execution of the wavelet transform in independent small volumes of the Milky Way disc, and the grouping of these local solutions into global structures with a method based on the breadth-first search algorithm from graph theory. We applied the same methodology to simulations of barred galaxies to validate the method and for comparison with the data. Results. We reveal the skeleton of the velocity distribution, uncovering projections that were not possible before. We sample nine main moving groups along a large region of the disc in configuration space, covering up to 6 kpc, 60 deg, and 2 kpc in the radial, azimuthal, and vertical directions, respectively, extending significantly the range of previous analyses. In the radial direction we find that the groups deviate from the lines of constant angular momentum that one would naively expect from an epicyclic approximation analysis of the first-order effects of resonances. We reveal that the spatial evolution of the moving groups is complex and that the configuration of moving groups in the solar neighbourhood is not maintained along the disc. We also find that the azimuthal velocity of the moving groups that are mostly detected in the inner parts of the disc (Arcturus, Bobylev, and Hercules) is non-axisymmetric. For Hercules we measure an azimuthal gradient of −0.50 km s−1 deg−1 at R = 8 kpc. We detect a vertical asymmetry in the azimuthal velocity for the Coma Berenices moving group, which is not expected for structures originating from a resonance of the bar, supporting the previous hypothesis of the incomplete vertical phase mixing of the group. In our simulations we extract substructures corresponding to the outer Lindblad resonance and the 1:1 resonances and observe the same deviation from constant angular momentum lines and the non-axisymmetry of the azimuthal velocities of the moving groups in the inner part of the disc. Conclusions. This data-driven characterization is a starting point for a holistic understanding of the moving groups. It also allows for a quantitative comparison with models, providing a key tool to comprehend the dynamics of the Milky Way.