Geometric Integration Research Articles

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics.

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

Numerical methods in Poisson geometry and their application to mechanics

We recall the question of geometric integrators in the context of Poisson geometry and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and compared to traditional methods.

Variational Integrators for Stochastic Mechanical Hybrid Systems

This paper introduces stochastic variational impact integrators for the class of hybrid mechanical systems that incorporate random noise. The governing equations are obtained by the application of the variational principle to the stochastic action integral, where both the continuous-time dynamics as well as the discrete transitions are considered. Furthermore, structure-preserving geometric integrators are derived through the discretization of the stochastic variational principle. This ensures the consistency in comparison to the continuous versions of the Euler-Lagrange or Hamilton’s equations. The effectiveness of the proposed methods in capturing the long-term energy behavior of a stochastic mechanical hybrid system is illustrated by numerical examples.

Analysis and application of rectified complex t-spherical fuzzy Dombi-Choquet integral operators for diabetic retinopathy detection through fundus images

This paper proposes a rectified complex spherical fuzzy set (rCTSFS) model that enables the phase term of a complex number to function truthfully to its inherent meaning of representing directions, phases, or color hues. In addition, this paper proposes the score and accuracy functions for rectified complex spherical fuzzy numbers (rCTSFn), which allows the three types of membership degrees of an rCTSFn to fulfill human judgment/intuition. The proposed rCTSFS model proves to be a productive extension of the complex spherical fuzzy set (CSFS), complex fuzzy set (CFS), and spherical fuzzy set (SFS) models. On the other hand, Dombi t-norms prove more flexible and comprehensive than some of the other families of triangular norms, such as the algebraic t-norms and the Einstein t-norms, due to the presence of a parameter γ. The parameter γ determines the amount of aggressiveness at estimating the maximum and the minimum of a population based on a sample obtained. Therefore, this paper proposes two Dombi-Choquet integral operators, namely, the rectified complex t-spherical fuzzy arithmetic Dombi-Choquet integral (rCTSFAγ,wλ) operator and the rectified complex t-spherical fuzzy geometric Dombi-Choquet integral (rCTSFGγ,wλ) operator. A multi-criteria decision making (abbr. MCDM) algorithm utilizing the two Dombi-Choquet integral operators is innovated. The proposed Dombi-Choquet MCDM algorithm for the rCTSFS model is applied to an MCDM problem related to diabetic retinopathy detection on five real-life fundus images of different severity levels taken from the Messidor2 dataset. Our newly proposed algorithm proves to be the only algorithm that yields the correct diagnostic results that match the hard truth. On the other hand, none of the 50 algorithms observed among recent works in literature can produce the correct diagnostic results, even after lending the fuzzification procedure innovated in this work to them.

Second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and an extension of the Krupka-Betounes equivalent

We determine second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and we discuss some aspects concerning conservation laws associated with invariance properties of their extended ‘full’ equivalents, in particular of Krupka–Betounes type. Such equivalents are constructed by means of a recursive formula involving geometric integration by parts formulae.

Time Integration Methods for Nonlinear Dynamic Problems

The accurate simulation and analysis of nonlinear dynamic problems across various scientific disciplines are essential for understanding complex physical phenomena. Time integration methods play a crucial role in numerically solving these problems, providing numerical approximations to the time-evolution of systems governed by ordinary or partial differential equations. This paper presents a comprehensive exploration of time integration methods tailored for addressing nonlinear dynamic problems encountered in diverse fields such as physics, engineering, and computational sciences. The review encompasses a detailed analysis of fundamental time integration techniques, including explicit and implicit schemes, highlighting their respective advantages and limitations in handling nonlinearities. Furthermore, it examines advanced time integration approaches specifically designed to tackle the challenges posed by nonlinear systems. This includes discussions on adaptive time-stepping methods, geometric and symplectic integrators, and other specialized techniques aimed at enhancing accuracy and stability while mitigating computational costs. Through illustrative examples and comparative analyses, this paper evaluates the performance of various time integration methods in addressing nonlinear dynamic problems. It delineates the significance of these methods in accurately capturing system behavior and elucidates their implications for practical applications. Additionally, this review identifies current challenges and outlines prospective directions for further advancements in time integration methodologies tailored to the intricacies of nonlinear dynamic systems. This comprehensive review aims to provide researchers, practitioners, and computational scientists with valuable insights into selecting and implementing appropriate time integration methods for effectively addressing nonlinear dynamic problems across diverse scientific domains. Key Words: Nonlinear Dynamics, Time Integration Techniques, Numerical Simulation, Computational Methods.

Integration of nonsmooth 2-forms: From Young to Itô and Stratonovich

We show that geometric integrals of the type ∫Ωfdg1∧dg2 can be defined over a two-dimensional domain Ω when the functions f, g1, g2:R2→R are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map g=(g1,g2) when f(x)=F(x,g(x)) with F sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.

Research on Johnson-Cook Constitutive Model of γ-TiAl Alloy with Improved Parameters.

Due to its excellent physical properties, γ-TiAl alloy has been widely used in thin-walled components of aerospace engines. However, issues such as low thermal conductivity, poor machinability, and high cutting temperatures often result in difficulties in ensuring the geometric accuracy and surface integrity of the parts. This paper focuses on the study of the thermal deformation behavior of γ-TiAl alloy within a range of higher temperatures and strain rates. Firstly, by conducting quasi-static tests and Hopkinson bar tests on γ-TiAl alloy, the true stress-strain curves of γ-TiAl alloy are obtained within a temperature range of 20~500 °C and a strain rate range of 3000~11,000/s. Based on the Johnson-Cook model, the true stress-strain curves are fitted and analyzed with consideration of the coupling effect of strain rate, temperature, and strain. The strain rate hardening coefficient C and thermal softening exponent m are polynomialized, improving the Johnson-Cook constitutive model of γ-TiAl alloy. The improved model shows significant improvements in the correlation coefficient and absolute errors between the predicted values and experimental values, providing a better reflection of the thermal deformation behavior of γ-TiAl alloy within a range of higher temperatures and strain rates.

Contouring Analysis on Synthetic Contrast-Enhanced MR from GRMM-GAN and Implications on MR-Guide Radiation Therapy

MR-guided linear accelerators have been commercialized making MR-only planning and adaptation an appealing alternative circumventing MR-CT registration. However, obtaining daily contrast-enhanced MR images can be prohibitive due to the increased risk of side effects from repeated contrast injections. In this work, we evaluate the quality of contrast-enhanced multi-modal MR image synthesis network GRMM-GAN (gradient regularized multi-modal multi-discrimination sparse-attention fusion generative adversarial network) for MR-guided radiation therapy. With IRB approval, we trained the GRMM-GAN based on 165 abdominal MR studies from 65 patients. Each study included T2, T1 pre-contrast (T1pre), and T1 contrast enhanced (T1ce) images. The two pre-contrast MR modalities, T2 and T1pre images were adopted as inputs for GRMM-GAN, and the T1ce image at the portal venous phase was used as an output. Ten MR scans containing 21 liver tumors were selected for contouring analysis. A Turing test was first given to six radiation oncologists, in which 100 real T1ce and synthetic T1ce image slices are randomly given to the radiation oncologists to determine the authenticity of the synthesis. We then invited two radiation oncologists (RadOnc 1 and RadOnc2) to manually contour the 21 liver tumors independently on the real T1ce images. RadOnc2 then performed contouring on the respective synthetic T1ce MRs. DICE coefficient (defined as the intersection over the average of two volumes) and Hausdorff distance (HD, measuring how far two volumes are from each other) were used as analysis metrics. The DICE coefficients were calculated from the two radiation oncologists' contours on the real T1ce MR for each tumor. The DICE coefficients were also calculated from RadOnc 2's contours on real and synthetic MRs. Besides, tumor center shifts were extracted. The tumor center of mass coordinates was extracted from real and synthetic volumes. The difference in the coordinates indicated the shifts in the superior-inferior (SI), right-left (RL), and anterior-posterior (AP) directions between real and synthetic tumor volumes. An average of 52.3% test score was achieved from the six radiation oncologists, which is close to random guessing. RadOnc 1 and RadOnc 2, who had participated in the contouring analysis, achieved an average DICE of 0.91±0.02 from tumor volumes drawn on the real T1ce MRs. This result sets the inter-operator uncertainty baseline in the real clinical setting. RadOnc 2 achieved an average DICE (real vs. synth) of 0.90±0.04 and HD of 4.76±1.82 mm. Only sub-millimeter (SI: 0.67 mm, RL: 0.41 mm, AP: 0.39 mm) tumor center shifts were observed in all three directions. The GRMM-GAN method has the potential for MR-guided liver radiation when contrast agents cannot be administered daily and provide synthetic contrast-enhanced MR for better tumor targeting. The network can produce synthetic MR images with satisfactory contour agreement and geometric integrity.

A bending-active gridshell as falsework and integrated reinforcement for a ribbed concrete shell with textile shuttering: Design, engineering, and construction of KnitNervi

This paper presents a formwork system consisting of a bending-active gridshell that simultaneously serves as falsework and integrated reinforcement for realising a ribbed funicular concrete skeleton shell. Encased by a knitted textile shuttering, the formwork system was demonstrated through KnitNervi, an architectural-scale, funnel-shaped demonstrator measuring 9 m in diameter and 3.3 m in height. The gridshell is materialised from straight steel rebar actively bent into curvilinear, double-layered rebar cages. Regular stirrups and pairs of inclined stirrups forming triangulated shear connectors provide the necessary shape control and stiffness for the load-bearing falsework. The rebar cages define the shape of the concrete ribs by supporting a knitted closed sectional mould and stay in place to structurally reinforce the resulting ribs.The focus of this paper lies on the falsework and reinforcement system with its interrelated design drivers. The geometric design includes the funicular form finding of the target shell with Thrust Network Analysis and the incremental form finding of the bending-active gridshell with its informed assembly sequence towards the funicular target with Finite Element Analysis. The engineering of the falsework demonstrates its sufficient load-bearing capacity and deflection control to support the weight of the wet concrete at an architectural scale. Sensitivity studies reveal the effectiveness of activating the double layer through the shear-connecting stirrups, the relevance of the internal connection design, and the geometric integrity during a potential stepwise casting sequence. The construction of the demonstrator verified the shape control and fabrication design.In only 36 h, the bespoke falsework gridshell was efficiently assembled from its kit-of-parts of standard rebar elements with adequate precision, logistics, time, and material resources. It was relatively lightweight, compact for transport, and employed low-tech construction techniques common to the rebar industry. Its structural geometry and informed bending-active logic enabled its efficient construction without digital fabrication or wasteful, costly moulds, which typically present the bottleneck for custom concrete structures. The resulting funicular concrete skeleton shell saves structural mass, hence embodied carbon, compared to unarticulated bending-dominant typologies. The overarching motivation of the research is to outline a strategy that could mitigate the environmental impact of the construction sector, applicable to a broad range of technological contexts.

Modelling the loosening of rock in the experimental rock mining method

Analytical description of the process of rock discontinuity caused by the operation of an experimental cutting head, in which a combination of cutting and loosening methods was used, is presented. Loosening of the rock fragment takes place after reaching the tensile or shear strength of the rock, what is more favourable compared to the classical mining methods. An analytical model of the stress in the crack initiation plane was developed using the models of rock loosening using the disc tools and relationships formulated in fracture mechanism. The special integral of the differential equation of the loosening line was determined by geometric integration. Numerical simulations of the chip loosening were also carried out using the model of the solid rock built using the Finite Element Method. The developed models of losing the rock continuity were used in numerical simulations of loosening the rock fragment to analyze the impact of the selected parameters of the mining technology, of the rock properties and geometrical features of the experimental cutting head, on the shape of the loosening line and its extend. Using the stand test results, the qualitative compliance of the shape of the loosening line, determined using the analytical model and the numerical model, with the geometrical features of the rock chips obtained in the laboratory, was confirmed. It was found that the proper selection of the design features of the cutting head, such as: a small angle of the loosening disc and a small angle of inclination of the head axis, cause a qualitative increase in the loosening range.

Research on 3D Visualization of Real Scene in Subway Engineering Based on 3D Model

As an urban infrastructure project, subway engineering exhibits the characteristics of large scale, trans-regional presence, and complex environmental conditions. The three-dimensional (3D) real scene of subway engineering aims to visually present the project features by creating a virtual environment in the 3D form by integrating numerous geographical scenes and entities. However, existing methods of geometric data integration and model visualization are insufficient for handling the diverse and heterogeneous data types encountered in this process. To address these challenges, this study proposes a framework for real-scene 3D visualization management in subway engineering, establishing a technical process of multi-source data integration for organizing 3D models within the scene, and developing a visualization display strategy that incorporates octree and level of detail principles for multi-scale hierarchical dynamic loading of 3D Scene. To validate the proposed method, a prototype system is developed with Geographic Information System (GIS) and Building Information Modeling (BIM) technology and applied to an actual subway project. The results demonstrate that this method ensures effective integration and management of data, enabling the display of real scenes in an integrated 3D model in a web environment.

A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Itô-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.

High-order geometric integrators for the variational Gaussian approximation.

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it is symplectic, conserves energy exactly, and may partially account for tunneling. However, the variational method is also much more expensive. To improve its efficiency, we symmetrically compose the second-order symplectic integrator of Faou and Lubich and obtain geometric integrators that can achieve an arbitrary even order of convergence in the time step. We demonstrate that the high-order integrators can speed up convergence drastically compared to the second-order algorithm and, in contrast to the popular fourth-order Runge-Kutta method, are time-reversible and conserve the normand the symplectic structure exactly, regardless of the time step. To show that the method is not restricted to low-dimensional systems, we perform most of the analysis on a non-separable twenty-dimensional model of coupled Morse oscillators. We also show that the variational method may capture tunneling and, in general, improves accuracy over the non-variational thawed Gaussian approximation.

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).

The flow method for the Baker-Campbell-Hausdorff formula: exact results

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the Baker-Campbell-Hausdorff (BCH) formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.

Enhanced cooling performance of stacked chips by structural modification for fractal micro-protrusions

The dissipation of heat in stacked chips has garnered significant interest in numerous scientific investigations and electronic implementations in recent decades. As a result of increased power and reduced size, stacked chips with substantial heat generation necessitate improved cooling capabilities within limited volume, thereby minimizing the utilization of liquid and forced air cooling methods. Additionally, there is currently a heightened emphasis on passive heat dissipation performance to achieve carbon neutrality, representing a novel requirement for environmentally-friendly chip thermal management in the realm of green energy. To enhance the passive cooling of stacked chips with higher heat but smaller volume, and reduce the energy consumption of liquid cooling and forced air cooling, the proportion of green heat dissipation needs to be increased. However, this increases the difficulty of thermal management. While passive cooling can reduce energy consumption, it often requires a larger cooling structure to dissipate the same amount of heat, which fails to meet the miniaturization requirements of stacked chip cooling systems. To address this issue, heat dissipation structures in the form of fractal micro-protrusions are adopted to enhance the passive cooling of the chips. Utilizing principles from fractal geometry, this study incorporates two classical fractal graphics, namely the “Sierpinski” triangle and the “Box” rectangle, as heat dissipation structures. These fractal micro-protrusions are fabricated on the chip. Furthermore, the enhancement of heat dissipation is evaluated through structural modifications to the chip's micro-protrusions, taking into account alterations in the fractal categories of said micro-protrusions. The investigation primarily focuses on four parameters related to structural fractal categories, namely fractal iteration, fractal geometric forms, fractal integration, and fractal thickness, in order to determine and compare the maximum temperatures achieved under the same heating condition and heat dissipation environment. Applying fractal micro-protrusions to the chips results in improved cooling performance. In the absence of liquid cooling and forced air cooling, the maximum temperature of the chip without fractal micro-protrusions is 149.08 °C when subjected to a heat flux of 200 W/cm2. However, this temperature reduces to 135.32 °C when square fractal micro-protrusions with S1 iterations are adopted, 125.67 °C when square fractal micro-protrusions with S3 iterations are adopted, 108.4 °C when triangle fractal micro-protrusions with T3 iterations are adopted, and 80.018 °C when Tri-integration fractal micro-protrusions are adopted. These findings suggest that the application of fractal micro-protrusions has a significant impact on temperature reduction. It is promising to be applied to the actual stacked chips industry.

Quantum State Assignment Flows.

This paper introduces assignment flows for density matrices as state spaces for representation and analysis of data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian-Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions that can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorical probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment flows can also be characterized as Riemannian gradient flows with respect to a non-local, non-convex potential after proper reparameterization and under mild conditions on the underlying weight function. This weight function generates the parameters of the layers of a neural network corresponding to and generated by each step of the geometric integration scheme. Numerical results indicate and illustrate the potential of the novel approach for data representation and analysis, including the representation of correlations of data across the graph by entanglement and tensorization.

Investigation of the 3D Printing Process Utilizing a Heterophase System.

Direct ink writing (DIW) requires careful selection of ink composition with specific rheological properties, and it has limitations, such as the inability to create overhanging parts or branched geometries. This study presents an investigation into enhancing the 3D printing process through the use of a heterophase system, aiming to overcome these limitations. A modification was carried out in the 3D printer construction, involving adjustments to the structural elements responsible for the extrusion device's movement. Additionally, a method for obtaining a heterophase system based on gelatin microparticles was developed to enable the 3D printing process with the upgraded printer. The structure and rheological properties of the heterophase system, varying in gelatin concentration, were thoroughly examined. The material's viscosity ranged from 5.4 to 32.8 kPa·s, exhibiting thixotropic properties, pseudoplastic behavior, and long-term stability at 20 °C. The developed 3D printing technology was successfully implemented using a heterophase system based on different gelatin concentrations. The highest product quality was achieved with a heterophase system consisting of 4.5 wt.% gelatin, which exhibited a viscosity of 22.4 kPa·s, enabling the production of products without spreading or compromising geometrical integrity.

Camera-Aided Orientation of Mobile Lidar Point Clouds Acquired from an Uncrewed Water Vehicle.

This article presents a system for recording 3D point clouds of riverbanks with a mobile lidar mounted on an uncrewed water vehicle. The focus is on the orientation of the platform and the lidar sensor. Rivers are areas where the conditions for highly accurate GNSS can be sub-optimal due to multipath effects from the water and shadowing effects by bridges, steep valleys, trees, or other objects at the riverbanks. Furthermore, a small measurement platform may have an effect on the accuracy of orientations measured by an IMU; for instance, caused by electromagnetic fields emitted by the boat rotors, the lidar, and other hardware decreasing IMU accuracy. As an alternative, we use exterior orientation parameters obtained by photogrammetric methods from the images of a camera on the boat capturing the riverbanks in time-lapse mode. Using control points and tie points on the riverbanks enables georeferenced position and orientation determination from the image data, which can then be used to transform the lidar data into a global coordinate system. The main influences on the accuracy of the camera orientations are the distance to the riverbanks, the size of the banks, and the amount of vegetation on them. Moreover, the quality of the camera orientation-based lidar point cloud also depends on the time synchronization of camera and lidar. The paper describes the data processing steps for the geometric lidar-camera integration and delivers a validation of the accuracy potential. For quality assessment of a point cloud acquired with the described method, a comparison with terrestrial laser scanning has been carried out.