Geometric Structure Research Articles

Theoretical insights into enantioselective [2 + 1] cyclopropanation reactions of diazo compounds with electron-deficient olefins.

The cyclopropane skeleton plays a significant role in bioactive molecules due to its distinctive structural properties. This has sparked keen interest and in-depth exploration in the field of stereoselective synthesis of cyclopropane derivatives. In the present study, the mechanism and the origin of stereoselectivity of diastereodivergent synthesis of cyclopropane derivatives via the catalyst-free [2 + 1]-cyclopropanation reactions of 3-diazo-N-methylindole (R1) with two types of electron-deficient olefins (R2 and R3) in both aqueous and toluene media have been studied using the DFT calculations. The findings indicate that these [2 + 1] cycloaddition reactions proceed in two stages, where the first step is not only the rate-determining step but also critically dictates the stereoselectivity of the product. The calculated diastereomeric ratios are in agreement with the experimental results. Furthermore, by utilizing non-covalent interaction (NCI) analysis and energy decomposition analysis based on molecular force fields (EDA-FF), we elucidated that the electrostatic interactions between reactant fragments in the transition state TS1s for the first step are the predominant factors determining the stereoselectivity, as opposed to the experimentally hypothesized steric hindrance and π-π stacking interactions. The geometrical structures of all minima and transition states on the potential energy surface (PES) in solvents water and toluene were fully optimized using the DFT method at the M06-2X(D3)/SMD/6-31 + G(d,p) level of theory. Single-point energy calculations were carried out based on the optimized geometries in the solution at the M06-2X(D3)/6-311 + G(d,p) level. All the DFT calculations were performed using the Gaussian 09 software. The optimized molecular structures were visualized using CYLview software. NCI analysis was performed using the Multiwfn and VMD softwares. The Multiwfn program was also used for CDFT and EDA-FF analyses.

Intrinsic Chiroptical Activity and Excitonic Properties of Group II-VI Magic-Size Clusters.

Semiconductor magic-size clusters (MSCs), lying in the local minima of the potential landscape, are important intermediates that emerge during the synthesis of colloidal quantum dots. They have definite geometrical and electronic structures, thus serving as atomically precise building blocks for assembling supramolecular structures and devices with unprecedented functionalities. Here we report the intrinsic chiroptical activity in the magic-size cadmium and zinc chalcogenide clusters with magic numbers of 13, 33, and 34 possessing unique core-shell structures. They are responsive to circularly polarized light from the ultraviolet to visible region, with size-tunable energy gap, absorption wavelength, and excitonic characteristics. The origin of the chiroptical activity and the evolution of excitonic states with magic size are disclosed by time-dependent density functional theory calculations within a correlated electron-hole picture. This molecular-level understanding of the photophysical properties of group II-VI MSCs provides essential guidelines for utilizing them for chiral optoelectronics and photonics.

Boosting Electrochemical Urea Synthesis via Constructing Ordered Pd–Zn Active Pair

HighlightsOrdered intermetallic pallium-zinc (PdZn) electrocatalyst comprising a high density of PdZn pairs is synthesized for achieving co-adsorption and co-activation carbon dioxide and nitrate.Both operando measurements and theoretical calculations reveal that the PdZn pairs provide a dual-site geometric structure conducive to the key C–N coupling with a low kinetical barrier.The intermetallic PdZn displays excellent performance for the co-reduction of nitrate (NO3–) and carbon dioxide (CO2) toward urea, with a maximum FEurea of 62.78% at a small potential of – 0.4 V versus RHE.

Relative equilibria of mechanical systems with rotational symmetry

We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class which we call normal, and a non-generic abnormal class. The eigenvalues of the locked inertia tensor descend to shape-space and endow it with the geometric structure of a three-web with the property that any normal relative equilibrium occurs as a critical point of the potential restricted to a leaf from the web. To demonstrate the utility of this web structure we show how the spherical three-body problem gives rise to a web of Cayley cubics on the three-sphere, and use this to fully classify the relative equilibria for the case of equal masses.

Unveiling Atomic-Scaled Local Chemical Order of High-Entropy Intermetallic Catalyst for Alkyl-Substitution-Dependent Alkyne Semihydrogenation.

High-entropy intermetallic (HEI) nanocrystals, composed of multiple elements with an ordered structure, are of immense interest in heterogeneous catalysis due to their unique geometric and electronic structures and the cocktail effect. Despite tremendous efforts dedicated to regulating the metal composition and structures with advanced synthetic methodologies to improve the performance, the surface structure, and local chemical order of HEI and their correlation with activity at the atomic level remain obscure yet challenging. Herein, by determining the three-dimensional (3D) atomic structure of quinary PdFeCoNiCu (PdM) HEI using atomic-resolution electron tomography, we reveal that the local chemical order of HEI regulates the surface electronic structures, which further mediates the alkyl-substitution-dependent alkyne semihydrogenation. The 3D structures of HEI PdM nanocrystals feature an ordered (intermetallic) core enclosed by a disordered (solid-solution) shell rather than an ordered surface. The lattice mismatch between the core and shell results in apparent near-surface distortion. The chemical order of the intermetallic core increases with annealing temperature, driving the electron redistribution between Pd and M at the surface, but the surface geometrical (chemically disordered) configurations and compositions are essentially unchanged. We investigate the catalytic performance of HEI PdM with different local chemical orders toward semihydrogenation across a broad range of alkynes, finding that the electron density of surface Pd and the hindrance effect of alkyl substitutions on alkynes are two key factors regulating selective semihydrogenation. We anticipate that these findings on surface atomic structure will clarify the controversy regarding the geometric and/or electronic effects of HEI catalysts and inspire future studies on tuning local chemical order and surface engineering toward enhanced catalysts.

Multi-layer Bundling as a New Approach for Determining Multi-scale Correlations Within a High-Dimensional Dataset

The growing complexity of biological data has spurred the development of innovative computational techniques to extract meaningful information and uncover hidden patterns within vast datasets. Biological networks, such as gene regulatory networks and protein-protein interaction networks, hold critical insights into biological features’ connections and functions. Integrating and analyzing high-dimensional data, particularly in gene expression studies, stands prominent among the challenges in deciphering these networks. Clustering methods play a crucial role in addressing these challenges, with spectral clustering emerging as a potent unsupervised technique considering intrinsic geometric structures. However, spectral clustering’s user-defined cluster number can lead to inconsistent and sometimes orthogonal clustering regimes. We propose the Multi-layer Bundling (MLB) method to address this limitation, combining multiple prominent clustering regimes to offer a comprehensive data view. We call the outcome clusters “bundles”. This approach refines clustering outcomes, unravels hierarchical organization, and identifies bridge elements mediating communication between network components. By layering clustering results, MLB provides a global-to-local view of biological feature clusters enabling insights into intricate biological systems. Furthermore, the method enhances bundle network predictions by integrating the bundle co-cluster matrix with the affinity matrix. The versatility of MLB extends beyond biological networks, making it applicable to various domains where understanding complex relationships and patterns is needed.

Formation of Delocalized Linear M-B-M Covalent Bonds: A Combined Experimental and Theoretical Study of BM2(CO)8+ (M = Co, Rh, Ir) Complexes.

Investigations of transition-metal boride clusters not only lead to novel structures but also provide important information about the metal-boron bonds that are critical to understanding the properties of boride materials. The geometric structures and bonding features of heteronuclear boron-containing transition metal carbonyl cluster cations BM(CO)6+ and BM2(CO)8+ (M = Co, Rh, and Ir) are studied by a combination of the infrared photodissociation spectroscopy and density functional calculations at B3LYP/def2-TZVP level. The completely coordinated BM2(CO)8+ complexes are characterized as a sandwich structure composed of two staggered M(CO)4 fragments and a boron cation, featuring a D3d symmetry and 1Eg electronic ground state as well as metal-anchored carbonyls in an end-on manner. In conjunction with theoretical calculations, multifold metal-boron-metal bonding interactions in BM2(CO)8+ complexes involving the filled d orbitals of the metals and the empty p orbitals of the boron cation were unveiled, namely, one σ-type M-B-M bond and two π-type M-B-M bonds. Accordingly, the BM2(CO)8+ complexes can be described as a linear conjugated (OC)4M═B═M(CO)4 skeleton with a formal B-M bond index of 1.5. The three delocalized d-p-d covalent bonds render compensation for the electron deficiency of the cationic boron center and endow both metal centers with the favorable 18-electron structure, thus contributing much to the overall structural stability of the BM2(CO)8+ cations. As a comparison, the saturated BRh(CO)6+ and BIr(CO)6+ complexes are determined to be a doublet Cs-symmetry structure with an unbridged (OC)2B-M(CO)4 pattern, involving a two-center σ-type (OC)2B → M(CO)4+ dative single bond along with a weak covalent B-M half bond. This work offers important insight into the structure and bonding of late transition metal boride carbonyl cluster cations.

Formation of Delocalized Linear M-B-M Covalent Bonds: A Combined Experimental and Theoretical Study of BM2(CO)8+ (M = Co, Rh, Ir) Complexes.

Investigations of transition-metal boride clusters not only lead to novel structures but also provide important information about the metal-boron bonds that are critical to understanding the properties of boride materials. The geometric structures and bonding features of heteronuclear boron-containing transition metal carbonyl cluster cations BM(CO)6+ and BM2(CO)8+ (M = Co, Rh, and Ir) are studied by a combination of the infrared photodissociation spectroscopy and density functional calculations at B3LYP/def2-TZVP level. The completely coordinated BM2(CO)8+ complexes are characterized as a sandwich structure composed of two staggered M(CO)4 fragments and a boron cation, featuring a D3d symmetry and 1Eg electronic ground state as well as metal-anchored carbonyls in an end-on manner. In conjunction with theoretical calculations, multifold metal-boron-metal bonding interactions in BM2(CO)8+ complexes involving the filled d orbitals of the metals and the empty p orbitals of the boron cation were unveiled, namely, one σ-type M-B-M bond and two π-type M-B-M bonds. Accordingly, the BM2(CO)8+ complexes can be described as a linear conjugated (OC)4M═B═M(CO)4 skeleton with a formal B-M bond index of 1.5. The three delocalized d-p-d covalent bonds render compensation for the electron deficiency of the cationic boron center and endow both metal centers with the favorable 18-electron structure, thus contributing much to the overall structural stability of the BM2(CO)8+ cations. As a comparison, the saturated BRh(CO)6+ and BIr(CO)6+ complexes are determined to be a doublet Cs-symmetry structure with an unbridged (OC)2B-M(CO)4 pattern, involving a two-center σ-type (OC)2B → M(CO)4+ dative single bond along with a weak covalent B-M half bond. This work offers important insight into the structure and bonding of late transition metal boride carbonyl cluster cations.

Multi-scale analysis of the community structure of the Twitter discourse around the Italian general elections of September 2022

We perform a multi-scale analysis of the geometric structure of the network of X (Twitter at the time of data collection) interactions surrounding the Italian snap general elections of September 25th 2022. We identify within it the communities related to the major Italian political parties and after it we analyse both the large-scale structure of interactions between different parties, showing that it resembles the coalitions formed in the run-up to the elections and the internal structure of each community. We observe that some parties have a very centralised communication with the major leaders clearly occupying the central role, while others have a more horizontal communication strategy, with many accounts playing an important role. We observe that this can be characterized by checking whether the network in the community has a strongly connected giant component or not.

Hydrogen production catalysed by atomically precise metal clusters.

Atomically precise metal clusters that possess the exact atom number, definitive composition, and tunable geometric and electronic structures have emerged as ideal model catalysts for many important chemical processes. Recently, metal clusters have been widely used as excellent catalysts for hydrogen production to explore the relationship between the structure and catalytic properties at the atomic level. In this review, we systematically summarize the significant developments concerning metal clusters as electrocatalysts and photocatalysts for hydrogen generation. This review also puts forward the challenges and perspectives of atomically precise metal clusters in electrocatalysis and photocatalysis in the hope of providing a valuable reference for the rational design of high-performance catalysts for hydrogen production.

Hexa‐atom Pt Catalyst Fabricated by a Ligand Engineering Strategy for Efficient Hydrogen Oxidation Reaction

Atomically precise supported nanocluster catalysts (APSNCs), which feature exact atomic composition, well‐defined structures, and unique catalytic properties, offer an exceptional platform for understanding the structure‐performance relationship at the atomic level. However, fabricating APSNCs with precisely controlled and uniform metal atom numbers, as well as maintaining a stable structure, remains a significant challenge due to uncontrollable dispersion and easy aggregation during synthetic and catalytic processes. Herein, we developed an effective ligand engineering strategy to construct a Pt6 nanocluster catalyst stabilized on oxidized carbon nanotubes (Pt6/OCNT). The structural analysis revealed that Pt6 nanoclusters in Pt6/OCNT were fully exposed and exhibited a planar structure. Furthermore, the obtained Pt6/OCNT exhibited outstanding acidic HOR performances with a high mass activity of 18.37 A·mgpt‐1 along with excellent stability during a 24 h constant operation and good CO tolerance, surpassing those of the commercial Pt/C. Density functional theory (DFT) calculations demonstrated that the unique geometric and electronic structures of Pt6 nanoclusters on OCNT altered the hydrogen adsorption energies on catalytic sites and thus lowered the HOR theoretical overpotential. This work presents a new prospect for designing and synthesizing advanced APSNCs for efficient energy electrocatalysis.

Hexa-atom Pt Catalyst Fabricated by a Ligand Engineering Strategy for Efficient Hydrogen Oxidation Reaction.

Atomically precise supported nanocluster catalysts (APSNCs), which feature exact atomic composition, well-defined structures, and unique catalytic properties, offer an exceptional platform for understanding the structure-performance relationship at the atomic level. However, fabricating APSNCs with precisely controlled and uniform metal atom numbers, as well as maintaining a stable structure, remains a significant challenge due to uncontrollable dispersion and easy aggregation during synthetic and catalytic processes. Herein, we developed an effective ligand engineering strategy to construct a Pt6 nanocluster catalyst stabilized on oxidized carbon nanotubes (Pt6/OCNT). The structural analysis revealed that Pt6 nanoclusters in Pt6/OCNT were fully exposed and exhibited a planar structure. Furthermore, the obtained Pt6/OCNT exhibited outstanding acidic HOR performances with a high mass activity of 18.37 A·mgpt-1 along with excellent stability during a 24 h constant operation and good CO tolerance, surpassing those of the commercial Pt/C. Density functional theory (DFT) calculations demonstrated that the unique geometric and electronic structures of Pt6 nanoclusters on OCNT altered the hydrogen adsorption energies on catalytic sites and thus lowered the HOR theoretical overpotential. This work presents a new prospect for designing and synthesizing advanced APSNCs for efficient energy electrocatalysis.

Solid-State Surface-Anchoring Strategy to Prepare Anti-Sintering Supported Metal Cluster Catalysts.

Due to the unique geometric and electronic structures, supported metal clusters with sizes below 3 nm have appealed to great interest in heterogeneous catalysis. However, these supported ultrasmall metal clusters would endure severe particle coalescences under high reaction temperatures. Herein, based on the technology of ball-milling processing, we propose a solid-state "surface-anchoring" strategy to synthesize thermally stabilized Al2O3-supported Ni nanoclusters. Interestingly, when the theoretical Ni loading weight was 1 wt %, highly dispersed Ni species were found where no Ni nanoparticles would be seen after 500 °C calcination. Until the Ni loading weight increased to 5 wt % and the calcination temperature increased to 750 °C, the Ni nanoparticles became significant but still with a size of only about 6.8 nm. With the small Ni nanoparticles, the final 5-Ni-Al2O3-OAm-750 sample worked well as methane dry reforming catalysts with excellent anticoking performance during a 500 h stability test.

Hexa-atom Pt Catalyst Fabricated by a Ligand Engineering Strategy for Efficient Hydrogen Oxidation Reaction.

Atomically precise supported nanocluster catalysts (APSNCs), which feature exact atomic composition, well-defined structures, and unique catalytic properties, offer an exceptional platform for understanding the structure-performance relationship at the atomic level. However, fabricating APSNCs with precisely controlled and uniform metal atom numbers, as well as maintaining a stable structure, remains a significant challenge due to uncontrollable dispersion and easy aggregation during synthetic and catalytic processes. Herein, we developed an effective ligand engineering strategy to construct a Pt6 nanocluster catalyst stabilized on oxidized carbon nanotubes (Pt6/OCNT). The structural analysis revealed that Pt6 nanoclusters in Pt6/OCNT were fully exposed and exhibited a planar structure. Furthermore, the obtained Pt6/OCNT exhibited outstanding acidic HOR performances with a high mass activity of 18.37 A·mgpt-1 along with excellent stability during a 24 h constant operation and good CO tolerance, surpassing those of the commercial Pt/C. Density functional theory (DFT) calculations demonstrated that the unique geometric and electronic structures of Pt6 nanoclusters on OCNT altered the hydrogen adsorption energies on catalytic sites and thus lowered the HOR theoretical overpotential. This work presents a new prospect for designing and synthesizing advanced APSNCs for efficient energy electrocatalysis.

Numerical calculation of air permeability of warp-knitted jacquard spacer shoe-upper materials based on CFD

Abstract In order to realize the rapid design of warp-knitted jacquard spacer shoe-upper materials and to explore the influence of different fabric surface organization structures on their air permeability, it is proposed to numerically simulate the air permeability performance of jacquard shoe-upper materials using ANSYS fluid analysis. First, four kinds of shoe material specimens with different upper and lower layer mesh alignment relationships are designed and tested, then the corresponding geometric structure models of the fabric are established based on the samples and appropriately processed, and finally the air permeability of the shoe material is numerically simulated and calculated based on computational fluid dynamics. The results show that, under the same conditions, the air permeability of double-sided mesh structure is significantly better than that of single-sided mesh structure, and its air permeability can be optimized by adjusting the relationship between the upper and lower layers of the fabric mesh; the airflow mainly passes through the fabric pores and gaps between the yarns, and the closer the fabric structure is, the more the air flow is hindered, and the worse the air permeability is. The simulation results of the established numerical model are consistent with the measured results, and the error is less than 10%, so it can be used to simulate and predict the air permeability of the warp-knitted jacquard spacer footwear materials, and provide theoretical support for the design and performance research of footwear materials.

Numerical calculation of air permeability of warp-knitted jacquard spacer shoe-upper materials based on CFD

Abstract In order to realize the rapid design of warp-knitted jacquard spacer shoe-upper materials and to explore the influence of different fabric surface organization structures on their air permeability, it is proposed to numerically simulate the air permeability performance of jacquard shoe-upper materials using ANSYS fluid analysis. First, four kinds of shoe material specimens with different upper and lower layer mesh alignment relationships are designed and tested, then the corresponding geometric structure models of the fabric are established based on the samples and appropriately processed, and finally the air permeability of the shoe material is numerically simulated and calculated based on computational fluid dynamics. The results show that, under the same conditions, the air permeability of double-sided mesh structure is significantly better than that of single-sided mesh structure, and its air permeability can be optimized by adjusting the relationship between the upper and lower layers of the fabric mesh; the airflow mainly passes through the fabric pores and gaps between the yarns, and the closer the fabric structure is, the more the air flow is hindered, and the worse the air permeability is. The simulation results of the established numerical model are consistent with the measured results, and the error is less than 10%, so it can be used to simulate and predict the air permeability of the warp-knitted jacquard spacer footwear materials, and provide theoretical support for the design and performance research of footwear materials.

Multi-Objective Optimization Design of Dynamic Performance of Hydrofoil with Gurney Flap

The horizontal axis tidal turbine, as a crucial device for capturing tidal energy, has gained significant attention because it has better energy efficiency performance. Enhancing the performance of foils, a vital part of tidal turbine blades, can significantly improve tidal turbine performance. Among numerous methods to enhance the foil performance, the Gurney flap has gained significant attention due to its avoidance of complex structural design. Currently, there is limited research on optimizing the design of Gurney flaps while considering the dynamic performance of foils. In this study, the S809 foil with a blade cross-section was selected as the research subject, a multi-objective optimization design platform was created by integrating a multi-objective optimization algorithm with Computational Fluid Dy-namics (CFD) numerical simulation techniques. The objective of this platform is to enhance the dynamic performance of the hydrofoil by optimizing the geometric structure of the Gurney flap. The improvement of dynamic lift and the size of the dynamic stall hysteresis loop are used as objective variables in this study to evaluate the hydrofoil’s dynamic performance. The optimal Latin hypercube design method is used in the optimization process to choose sample locations, and the Kriging approximation model is used to determine the relationship between the design variables and the objective variables. Meanwhile, the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) is used to create a multi-objective optimization platform for solving the optimization problem, and the optimized results are validated using CFD. Comparative validation results show that quantifying the dynamic performance during hydrofoil pitching oscillation and using the optimal Latin hypercube design method and Kriging approximation model for optimizing the Gurney flap structure is rational and accurate. This study explores the mechanism of the Gurney flap through in-depth CFD numerical simulations and finds that the Gurney flap affects the flow characteristics at the hydrofoil’s trailing edge, thereby influencing the performance. It increases the pressure difference between the pressure and suction surfaces, thus enhancing the hydrofoil’s lift. Finally, this article provides three recommended parameters to improve the dynamic performance of the hydrofoil. This research can serve as a reference for the application of Gurney flaps in tidal turbine blade design.

Multi-Objective Optimization Design of Dynamic Performance of Hydrofoil with Gurney Flap

The horizontal axis tidal turbine, as a crucial device for capturing tidal energy, has gained significant attention because it has better energy efficiency performance. Enhancing the performance of foils, a vital part of tidal turbine blades, can significantly improve tidal turbine performance. Among numerous methods to enhance the foil performance, the Gurney flap has gained significant attention due to its avoidance of complex structural design. Currently, there is limited research on optimizing the design of Gurney flaps while considering the dynamic performance of foils. In this study, the S809 foil with a blade cross-section was selected as the research subject, a multi-objective optimization design platform was created by integrating a multi-objective optimization algorithm with Computational Fluid Dy-namics (CFD) numerical simulation techniques. The objective of this platform is to enhance the dynamic performance of the hydrofoil by optimizing the geometric structure of the Gurney flap. The improvement of dynamic lift and the size of the dynamic stall hysteresis loop are used as objective variables in this study to evaluate the hydrofoil’s dynamic performance. The optimal Latin hypercube design method is used in the optimization process to choose sample locations, and the Kriging approximation model is used to determine the relationship between the design variables and the objective variables. Meanwhile, the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) is used to create a multi-objective optimization platform for solving the optimization problem, and the optimized results are validated using CFD. Comparative validation results show that quantifying the dynamic performance during hydrofoil pitching oscillation and using the optimal Latin hypercube design method and Kriging approximation model for optimizing the Gurney flap structure is rational and accurate. This study explores the mechanism of the Gurney flap through in-depth CFD numerical simulations and finds that the Gurney flap affects the flow characteristics at the hydrofoil’s trailing edge, thereby influencing the performance. It increases the pressure difference between the pressure and suction surfaces, thus enhancing the hydrofoil’s lift. Finally, this article provides three recommended parameters to improve the dynamic performance of the hydrofoil. This research can serve as a reference for the application of Gurney flaps in tidal turbine blade design.

Low-frequency energy scavenging by a stacked tri-stable piezoelectric energy harvester

We present a portable, stacked and tri-stable energy harvester based on the unique geometrical structure of L-shaped beams and the nonlinearity of magnetic force to harness low-frequency and low-level excitations. In this device, multiple L-shaped piezoelectric beams are connected in a head-to-tail manner. Such a special stacked structure reduces the natural frequency of the system while realizing the series connection of multiple piezoelectric sheets in a limited space, thus improving the acquisition efficiency. Additionally, magnetic coupling endows the harvester with tri-stability and consequently enhances the bandwidth responses in low-frequency environments. The mathematical model is developed, and the static behaviors are explored by the bifurcation theory. Furthermore, the complex dynamic frequency method combined with the arc length method is conducted to investigate the dynamic responses with different parameters. Sinusoidal excitation experiments on the prototype verify the rationality of the mathematical model and the low-frequency broadband capability of the harvester. In the experiments, the designed harvester can reach a maximum output power of 1.49 mW at the low excitation amplitude of 0.4 g, and the magnet coupling significantly enhanced the electrical responses. The human motion tests demonstrate that the designed harvester has the potential to convert biomechanical energy into electrical energy for continuous powering of wireless devices.

Lagrangian descriptors with uncertainty

Lagrangian descriptors provide a global dynamical picture of the geometric structures for arbitrarily time-dependent flows with broad applications. This paper develops a mathematical framework for computing Lagrangian descriptors when uncertainty appears. The uncertainty originates from estimating the underlying flow field as a natural consequence of data assimilation or statistical forecast. It also appears in the resulting Lagrangian trajectories. The uncertainty in the flow field directly affects the path integration of the crucial nonlinear positive scalar function in computing the Lagrangian descriptor, making it fundamentally different from many other diagnostic methods. Despite being highly nonlinear and non-Gaussian, closed analytic formulae are developed to efficiently compute the expectation of such a scalar function due to the uncertain velocity field by exploiting suitable approximations. A rapid and accurate sampling algorithm is then built to assist the forecast of the probability density function (PDF) of the Lagrangian trajectories. Such a PDF provides the weight to combine the Lagrangian descriptors along different paths. Simple but illustrative examples are designed to show the distinguished behavior of using Lagrangian descriptors in revealing the flow field when uncertainty appears. Uncertainty can either completely erode the flow3 structure or barely affect the underlying geometry of the flow field. The method is also applied for eddy identification, indicating that uncertainty has distinct impacts on detecting eddies at different time scales. Finally, when uncertainty is incorporated into the Lagrangian descriptor for inferring the source target, the likelihood criterion provides a very different conclusion from the deterministic methods.