Differential Geometry Research Articles

Engineering flexible machine learning systems by traversing functionally invariant paths

Contemporary machine learning algorithms train artificial neural networks by setting network weights to a single optimized configuration through gradient descent on task-specific training data. The resulting networks can achieve human-level performance on natural language processing, image analysis and agent-based tasks, but lack the flexibility and robustness characteristic of human intelligence. Here we introduce a differential geometry framework—functionally invariant paths—that provides flexible and continuous adaptation of trained neural networks so that secondary tasks can be achieved beyond the main machine learning goal, including increased network sparsification and adversarial robustness. We formulate the weight space of a neural network as a curved Riemannian manifold equipped with a metric tensor whose spectrum defines low-rank subspaces in weight space that accommodate network adaptation without loss of prior knowledge. We formalize adaptation as movement along a geodesic path in weight space while searching for networks that accommodate secondary objectives. With modest computational resources, the functionally invariant path algorithm achieves performance comparable with or exceeding state-of-the-art methods including low-rank adaptation on continual learning, sparsification and adversarial robustness tasks for large language models (bidirectional encoder representations from transformers), vision transformers (ViT and DeIT) and convolutional neural networks.

Noncommutative differential geometry on crossed product algebras

We provide a differential structure on arbitrary cleft extensions B:=AcoH⊆A for an H-comodule algebra A. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra B#σH from the data of a bicovariant calculus on the structure Hopf algebra H and a calculus on the base algebra B, which is compatible with the 2-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. All results specialize to trivial extensions and smash product algebras B#H and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map j:H→A and the inclusion B↪A. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras H(r,n,q) is spelled out in detail.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

Simplified discrete model for axisymmetric dielectric elastomer membranes with robotic applications

Soft robots utilizing inflatable dielectric membranes can realize intricate functionalities through the application of non-mechanical fields. However, given the current limitations in simulations, including low computational efficiency and difficulty in dealing with complex external interactions, the design and control of such soft robots often require trial and error. Thus, a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical model is developed for analyzing the highly nonlinear mechanics in axisymmetric inflatable dielectric membranes. The model captures the intricate dynamics of these membranes under both inflationary pressure and electrical stimulation. Comprehensive validations using hyperelastic benchmarks demonstrate the model’s accuracy and reliability. Additionally, the focus on the electro-mechanical coupling elucidates critical insights into the membrane’s behavior under varying internal pressures and electrical loads. The research further translates these findings into innovative soft robotic applications, including a spherical soft actuator, a soft circular fluid pump, and a soft toroidal gripper, where the snap-through of electroelastic membrane plays a crucial role. Our analyses reveal that the functional ranges of soft robots are amplified by the snap-through of an electroelastic membrane upon electrical stimuli. This study underscores the potential of DDG-based simulations to advance the understanding of the nonlinear mechanics of electroelastic membranes and guide the design of electroelastic actuators in soft robotics applications.

Information geometry and parameter sensitivity of non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate the energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that the detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.

Universal mechanical modeling of fiber bundle under torsion: An experimental and numerical study

AbstractIn this paper, a universal mechanical model for the torsional behavior of fiber bundle was established, grounded in the principles of spatial geometry and continuum mechanics, incorporating differential geometry formulas. The model was validated through self‐designed experimental equipment and finite element simulations using Ansys Workbench, showing strong agreement between predicted torsional fracture behavior and experimental results. The study further examined the contributions of fiber bending, torsion, and tension to the overall torque, revealing that torsion plays the dominant role. Additionally, the effect of filament count on the torsional performance of fiber bundles was analyzed, demonstrating that while the number of fibers significantly influences stress distribution, its impact on overall torque is minimal.Highlights A universal mechanical model for fiber bundle torsional is proposed. High consistency between simulation and experimental results. Examined filament count effect on torsional strength. More filaments reduce stress concentration, improving performance.

Study on contact characteristics of double-disk arc-contact lapping for cylindrical rollers based on influence coefficient method

The double-disk arc-contact lapping (DDACL) method is a novel for precision machining of the rolling surface of cylindrical roller. The contact characteristics of DDACL process can affect the lapping quality of the bearing roller. However, there is a limited amount of study on the contact theory of DDACL. Therefore, the objective of this research is to create a semi-analytical modeling approach that can effectively and accurately forecast the contact characteristics of DDACL. In the initial step, the spatial coordinate transformation was employed to establish accurate mathematical models that represents the surfaces of the cylindrical roller and arc groove. Subsequently, the potential contact area was determined and subdivided into discrete elements based on the differential geometry principles. The accurate flexibility of the potential contact area is achieved by utilizing the interpolation theory and considering both the global and local deformations of the roller and groove. By employing the influence coefficient method and the force-deformation coordination relation, the deformation and contact force can be calculated. Finally, the accuracy and efficiency of this calculation is confirmed through a comparison between the finite element method (FEM) and the proposed method.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Deformable surface reconstruction via Riemannian metric preservation

Estimating the pose of an object from a monocular image is a fundamental inverse problem in computer vision. Due to its ill-posed nature, solving this problem requires incorporating deformation priors. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a reliable and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach for inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and achieves state-of-the-art performance without the need for offline training. Being a method that performs per-frame optimization, our method can refine its estimates, contrary to those based on performing a single inference step. Despite enforcing differential geometry constraints at each update, our approach is the fastest of all the tested optimization-based methods.

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Methods of correction to extrinsic parameters in a stereoscopic vision with the assistance of two known-distance points.

Stereoscopic vision plays a significant role in a three-dimensional measurement. With the calibrated intrinsic and extrinsic parameters, stereoscopic vision can complete an accurate measurement. However, the extrinsic parameters are inevitably disturbed by variations in the environment, such as vibration and assembly stress, resulting in a huge measurement error. To overcome the problem, with the assistance of two known-distance points, this Letter proposes correction methods based on triangulation and differential geometry, respectively. The methods formulate the distance and solve the corrected extrinsic parameters. Simulated and actual experiments are carried out, and the results show high accuracy and stability of the proposed methods.

Semi-analytical layerwise solutions for FG MEE complex shell structures

This paper presents a unique analytical layerwise solution for functionally graded magneto-electric-elastic shell with complex geometry. The middle surface of the shell is modeled by a parametric equation and the Lamé parameter and radii of curvature is modeled by Differential Geometry. The mechanical displacements, electrical and magnetic potentials are written in term of a simple cosine layerwise based on a unified formulation. The highly coupled differential equations are discretized by the Chebyshev-Gauss-Lobatto grid points and solved numerically via the Differential Quadrature Method (DQM). Lagrange interpolation polynomials are employed as the basis functions. The highly coupled differential equations are solved for shells subjected to different loads and boundary conditions. Since extremely few results on this topic is available in the literature, benchmarks complex shell problems and their solutions are introduced in this paper for the very first time.

Differentials of the basis in Clifford Geometric Algebra

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

On the Curvature Functions of Ruled Surface

We considered differential geometries of helical curves for ruled and developable surfaces in Euclidean 3-space E 3. We clarify the conditions for ruled surface to become stationary-axis ruled surface. Then, we concentrate on ruled surface generated by a line undergoing a one-parameter screw movement. Finally, we provide some modules to illustrate the main results.

Discrete differential geometry-based model for nonlinear analysis of axisymmetric shells

In this paper, we propose a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical method for geometrically nonlinear mechanics analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our numerical model leverages differential geometry principles to accurately capture the complex nonlinear deformation patterns exhibited by axisymmetric shells. By discretizing the axisymmetric shell into interconnected 1D elements along the meridional direction, the in-plane stretching and out-of-bending potentials are formulated based on the geometric principles of 1D nodes and edges under the Kirchhoff–Love hypothesis, and elastic force vector and associated Hessian matrix required by equations of motion are later derived based on symbolic calculation. Through extensive validation with available theoretical solutions and finite element method (FEM) simulations in literature, our model demonstrates high accuracy in predicting the nonlinear behavior of axisymmetric shells. Importantly, compared to the classical theoretical model and three-dimensional (3D) FEM simulation, our model is highly computationally efficient, making it suitable for large-scale real-time simulations of nonlinear problems of shell structures such as instability and snap-through phenomena. Moreover, our framework can easily incorporate complex loading conditions, e.g., boundary nonlinear contact and multi-physics actuation, which play an essential role in the use of engineering applications, such as soft robots and flexible devices. This study demonstrates that the simplicity and effectiveness of the 1D discrete differential geometry-based approach render it a powerful tool for engineers and researchers interested in nonlinear mechanics analysis of axisymmetric shells, with potential applications in various engineering fields.

Electromagnetic forces and their finite element computation

AbstractThis paper introduces the concepts of differential geometry that are necessary to establish a systematic definition for electromagnetic forces by means of a natural thermodynamic approach. It is shown that standard electromagnetic force formulae used in finite element computational electromagnetism are particular instances of that general approach. Finally, the paper offers a complete conceptual framework, as well as a mathematical toolbox, to derive electromagnetic force formulae for multi‐physics finite element models with complex materials.

Modeling and experiment for the roller hemming of variable curvature aluminum alloy panel with adhesive

The autobody closures have high requirements on the appearance accuracy of A-class surfaces, smooth contours, and continuous edges, so the forming quality of the autobody closures has long been the center of attention in the roller hemming process. To facilitate weight reduction in automotive manufacturing, lightweight materials such as aluminum alloy and hemming adhesive are commonly used. However, the heterogeneous coupling effect between the adhesive and aluminum alloy panel during roller hemming can significantly influence the forming quality of the panel. Additionally, the precision of the roller posture and trajectory plays a crucial role in determining the final quality of the panels, given the complex geometry of the autobody closures. Therefore, in order to continuously improve the forming quality of the autobody closures, this paper investigates the roller hemming process applied to variable curvature panel with adhesive on autobody doors. Firstly, the kinematic model for the roller posture and trajectory during roller hemming of the curved surface-curved edge panel was developed based on differential geometry theory. Secondly, a simulation model for roller hemming of the panel with adhesive was established using the finite element method and the smoothed particle hydrodynamics method (FEM-SPH). The validity of the simulation was confirmed by comparing the simulated results with experimental outcomes. In the experiment, the algorithm for solving the posture and trajectory of the roller was applied, and the forming quality of the panel was evaluated by using the two indexes of roll-in/out coefficient and surface wave coefficient. Thirdly, the impact of the hemming factors on surface wave and roll-in/out was analyzed using the response surface methodology, leading to the development of a mapping relationship between the hemming factors and forming quality. This study provides valuable support for predicting the forming quality of variable curvature panels with adhesive and continuously improving the forming quality of autobody closures in the roller hemming manufacturing process.

A study of conical and curved shock waves via the streamline transformation method

The axisymmetric conical and curved shock waves are studied theoretically via the streamline transformation method (STM), which integrates the differential geometries of streamlines into the conservation laws and solves the flow field geometrically. Under the condition of a straight shock wave, the governing equation is significantly simplified, from which the assumptions of the conical flow and geometric self-similarity are mathematically proved. The simplified equation is also demonstrated as equivalent to the classical Taylor–MacColl equation. The flow mechanism after a conical shock wave is discussed from a geometric perspective, where the key factor is the dimensionless lateral distance between symmetric planes. The corresponding geometric and flow parameters, e.g., the distance between streamlines, the streamline curvature, and the Mach number, are computed for the demonstration of this mechanism. In the flow field over a circular cone, the isentropic compression is modeled geometrically. Due to the isentropic compression, the conical shock wave remains approximately a Mach wave at small semi-angles and the detachment is suspended at large semi-angles. For the axisymmetric curved shock wave, the flow field is computed directly by the streamline transformation method, where both the concave and convex shock waves are considered. Along the wall streamline, an explicit proportional relation between the shock curvature and the orthogonal derivatives of flow parameters is also provided and verified numerically. These results are applicable in the theoretical analysis and inverse design of axisymmetric shock waves.

Frequency-domain axisymmetric and thermomechanical viscoelastic analysis of thermoplastic composite pipe

ABSTRACT Thermoplastic composite pipe (TCP), composed of polymers and composite laminates, shows significant potential for deep-water offshore oil field development due to its lightweight and high strength. However, current mechanical studies primarily focus on the elastic domain, ignoring the critical viscoelasticity of non-metallic materials. To solve this, we introduce a comprehensive theoretical model that incorporates viscoelastic properties under both axisymmetric and thermal loads, considering radial convective thermal gradients and axial heat conduction. This model accurately predicts TCP's mechanical behavior by using stiffness coefficients, differential geometry, and radial-axial temperature transfer curves. By applying the principle of elastic-viscoelastic correspondence, we obtain viscoelastic solutions in the frequency domain, showing strong agreement with Finite Element (FE) models particularly under cyclic loading. Viscoelasticity introduces larger equivalent axial stiffness and viscoelastic damping within complex structures, which become more obvious with temperature fluctuations.

Brane mechanics and gapped Lie n-algebroids

We draw a parallel between the BV/BRST formalism for higher-dimensional (≥ 2) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie n-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and “generalised R-flux” deformations thereof, such as models with an (n + 2)-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie n-algebroid. Studying covariant derivatives along n-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological n-branes to differential geometry on Lie n-algebroids.