Geometric Equations Research Articles

Geometric Neural Operators (GNPs) for data-driven deep learning in non-euclidean settings

Abstract We introduce Geometric Neural Operators (GNPs) for data-driven deep learning of geometric features for tasks in non-euclidean settings. We present a formulation for accounting for geometric contributions along with practical neural network architectures and factorizations for training. We then demonstrate how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve Bayesian inverse problems for identifying manifold shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry in the data-driven learning of operators.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Convergence of a stabilized parametric finite element method of the Barrett–Garcke–Nürnberg type for curve shortening flow

The parametric finite element methods of the Barrett–Garcke–Nürnberg (BGN) type have been successful in preventing mesh distortion/ degeneration in approximating the evolution of surfaces under various geometric flows, including mean curvature flow, Willmore flow, Helfrich flow, surface diffusion, and so on. However, the rigorous justification of convergence of the BGN-type methods and the characeterization of the particle trajectories produced by these methods still remain open since this class of methods was proposed in 2007. The main difficulty lies in the stability of the artificial tangential velocity implicitly determined by the BGN methods. In this paper, we give the first proof of convergence of a stabilized BGN method for curve shortening flow, with optimal-order convergence in L 2 L^2 norm for finite elements of degree k ≥ 2 k \geq 2 under the stepsize condition τ ≤ c h k + 1 \tau \leq c h^{k+1} (for any fixed constant c c ). Moreover, we give the first rigorous characterization of the particle trajectories produced by the BGN-type methods for one-dimensional curves, i.e., we prove that the particle trajectories produced by the stabilized BGN methods converge to the particle trajectories determined by a system of geometric partial differential equations which differs from the standard curve shortening flow by a tangential motion. The characterization of the particle trajectories also rigorously explains, for one-dimensional curves, why the BGN-type methods could maintain the quality of the underlying evolving mesh.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Interfaces, Free Boundaries and Geometric Partial Differential Equations

Partial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research. The workshop focused on new developments and emerging new themes. At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed. By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions.

An analytical approach for determining contact angle hysteresis on smooth, micropillared, and micropored homogeneous surfaces

HypothesisNumerous theoretical models have been developed from various research perspectives, including free energy analysis, force balance, and contact line dynamics, to elucidate the contact angle hysteresis on solid surfaces, especially for superhydrophobic surfaces. However, these models can produce inconsistent predictions, and few of them account for contact angle hysteresis on smooth and microstructured surfaces simultaneously. TheoryFormulas for advancing and receding free energy barriers of drops on different solid surfaces were derived, and then these formulas were simplified by incorporating the geometric constraint equation of drops, leading to an establishment of analytical models. FindingsThis study presented a unified approach for deriving analytical models of contact angle hysteresis for various wetting systems, including drops on smooth homogeneous surfaces, Cassie drops on micropillared and micropored homogeneous surfaces, and Wenzel drops on micropillared homogeneous surfaces. The established models revealed a significant impact of frictional tension, and of the change in free energy during drop motion, on the free energy barriers. These models were fully derived thermodynamically without subsequent theoretical modifications and found to agree well with experimental results. Some of the models in this study were validated using other existing models, despite the models having been developed using completely different approaches.

Periodic homogenization of geometric equations without perturbed correctors

Periodic homogenization of geometric equations without perturbed correctors

CIELAB Color Space as a Field for Tracking Color-Changing Chemical Reactions of Polymeric pH Indicators.

Two types of cross-linked polymeric dye films that exhibit reversible color changes in response to pH were prepared. Upon protonation, films with a nitrophenol dye moiety (NP) changed from yellow to colorless, whereas films with an azobenzene dye moiety (AB) changed from yellow to reddish-purple. The colored light transmitted through these films from a D65 white-light source was measured colorimetrically, and the transition of the chromaticity point in CIELAB upon protonation of the polymeric dyes was observed. At low dye concentrations ([NP] < 4.7 × 10-5 and [AB] < 2.7 × 10-5 mol L-1), linear loci of the chromaticity points were displayed for both polymeric dye films, indicating perceptual linearity. This linearity is caused by the avoidance of nonlinear perceptual phenomena such as the Abney effect and the Bezold-Brücke phenomenon. The perceptual linearity of space is a necessary condition for achieving stoichiometric linearity, which requires an additional linear relationship between the dye concentration and each component, a*, b*, and L*. A geometric equation that included the AB concentration and the protonation ratio of AB was used to successfully depict, on the basis of stoichiometric linearity, the protonation of the polymeric AB films three-dimensionally in the restricted area of CIELAB.

FLRW cosmology in metric-affine F(R,Q) gravity* *Supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan (AP14870191)

We investigated some Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological models in the context of metric-affine gravity, as proposed in [arXiv: 1205.5266v6]. Here, R and Q are the curvature and nonmetricity scalars using non-special connections, respectively. We obtained the modified field equations using a flat FLRW metric. We then found a connection between the Hubble constant , density parameter , and other model parameters in two different situations involving scalars u and w. Next, we used new observational datasets, such as the cosmic chronometer (CC) Hubble and Pantheon SNe Ia datasets, to determine the optimal model parameter values through a Markov chain Monte Carlo (MCMC) analysis. Using these best-fit values of the model parameters, we discussed the results and behavior of the derived models. Further, we discussed the Akaike information criterion (AIC) and Bayesian information criterion (BIC) for the derived models in the context of the Lambda cold dark matter (ΛCDM). We found that the geometrical sector dark equation of state parameter behaves just like a dark energy candidate. We also found that both models are transit phase models. Model-I approaches the ΛCDM model in the late-time universe, whereas Model-II approaches quintessence scenarios.

A Web-Based Interactive Application to Simulate and Correct Distortion in Multibeam Sonars

Multibeam sonars are advanced scientific tools for estimating fish school volume and density, using multiple beams to provide comprehensive size measurements of detected targets. However, challenges remain in accurately estimating target dimensions due to beam geometric expansion and overlap, particularly in athwart-beam measurements, which tend to be overestimated with increasing distance from the transducer. We present an interactive web application that simulates distortion caused by beam overlap and expansion in multibeam sonars using simple geometric equations. Users can define sonar characteristics, such as the number of beams, swath opening, or degree of overlap, as well as specify an elliptical target’s dimensions, orientation, and distance from the transducer. The application estimates and visualises the true and distorted shapes of the target, calculating the level of distortion. It can run simulations in both forward and inverse directions, either simulating the distortion of a true school or correcting the shape of a distorted school. This tool aims to enhance the interpretation of multibeam sonar signals and improve the accuracy of target dimension estimates, facilitating more effective use of these sonars in scientific research.

Three-Dimensional Human Posture Recognition by Extremity Angle Estimation with Minimal IMU Sensor.

Recently, posture recognition technology has advanced rapidly. Herein, we present a novel posture angle calculation system utilizing a single inertial measurement unit and a spatial geometric equation to accurately identify the three-dimensional (3D) motion angles and postures of both the upper and lower limbs of the human body. This wearable system facilitates continuous monitoring of body movements without the spatial limitations or occlusion issues associated with camera-based methods. This posture-recognition system has many benefits. Providing precise posture change information helps users assess the accuracy of their movements, prevent sports injuries, and enhance sports performance. This system employs a single inertial sensor, coupled with a filtering mechanism, to calculate the sensor's trajectory and coordinates in 3D space. Subsequently, the spatial geometry equation devised herein accurately computed the joint angles for changing body postures. To validate its effectiveness, the joint angles estimated from the proposed system were compared with those from dual inertial sensors and image recognition technology. The joint angle discrepancies for this system were within 10° and 5° when compared with dual inertial sensors and image recognition technology, respectively. Such reliability and accuracy of the proposed angle estimation system make it a valuable reference for assessing joint angles.

Motion Branch Transformation of 3(Rc)PU Parallel Mechanism with Reconfigurable Joint and Kinematic Performance Index

A novel 3(Rc)PU parallel mechanism based on the reconfigurable circular groove cross joint (Rc joint) is proposed. The motion branch transformation relationship and constraint performance evaluation method for the mechanism are studied. According to constrained screw analysis, the (Rc)PU limb provides a constraint couple to the moving platform in one motion mode and no constraint to the platform in another motion mode. According to the two configurations of the limb, the 3(Rc)PU mechanism has four different motion branches, namely 3T, 1R3T, 2R3T, and 3R3T. Based on the input selection principle, the input selection scheme of the 3(Rc)PU mechanism is determined, and it is concluded that, to achieve stable motion, at least two input drive pairs need to be applied on each limb. A unified kinematic model with four motion branches is established, and the geometric constraint equations of the 3(Rc)PU parallel mechanism are derived. After selecting the input actuated joints, the local minimization transmission index (LMTI) of the reconfigurable mechanism under different motion branches is established to evaluate the motion/force transmission performance of the reconfigurable mechanism under different motion branches. The conclusion shows that the LMTI values satisfy the corresponding constraint conditions, and the 3(Rc)PU parallel mechanism has good motion/force transmission performance.

Magneto-thermoelastic nonlinear dynamic modeling of a rotating functionally graded shell

Research is conducted on the dynamic modeling of a rotating functionally graded (FG) cylindrical shell subjected to magnetic and temperature fields. Based on the elasticity theory and generalized Hooke's law on the physical neutral surface, nonlinear geometric equations and thermoelastic constitutive relations are determined. According to the Kirchhoff-Love theory, variational formulas of strain energies for deformation, temperature, and centrifugal force are obtained. Considering the rotational effect, the kinetic energy and its variational formula are derived. The electromagnetic force model incorporating magnetization effect of the ferromagnetic FG shell is established by utilizing the electromagnetic theory. Subsequently, the magneto-thermoelastic dynamic model of the rotating FG shell is developed by adopting the Hamilton's principle. The model can reveal the coupling mechanisms of the interaction and superposition of multi-physical fields. Finally, taking the primary resonance as example, detailed numerical analyses are performed to investigate the effects of different parameters on vibration response and dynamical stability.

Analytical calculation method of circular-section rocking steel piers under horizontal cyclic load

The research object of this paper is to investigate the mechanical behavior and develop an analytical model of the prestressed circular-section rocking steel piers with external energy-dissipating devices under the horizontal cyclic load. Based on the method of monolithic beam analogy under monotonic loading, the concept of generalized rotation angle and intercept strain in the pre-decompression state was introduced. The geometrical equations for the pre- and post-decompression stages are deduced. The calculation formulas for internal forces of each component and the analytical calculation method for the entire process of the rocking steel pier under horizontal cyclic loading are established in detail. To reproduce the hysteresis curve of the test, some special test conditions, such as the retraction effect of prestressed tendons and increase of the compressed area due to the vertical plates, were considered in the analytical model. The results show that the analytical method based on generalized rotation angle and intercept strain can ensure an ideal transition of the hysteresis curve of the pier in pre- and post-decompression stages. Except for the pier with particularly severe local buckling and ovalization deformation, the calculated results obtained from the proposed analytical model agrees well with the test results for the rocking steel bridge piers. The effect of prestressed tendons and the P–Δ effect on the lateral load–displacement curve of rocking steel bridge biers was further analyzed through parametric study using the proposed model. The analytical calculation method presented in this paper provides a theoretical tool for seismic design of the rocking steel pier in engineering practice. This paper provides important guidance for the establishment of seismic design method of rocking steel piers.

Dynamic Analysis Strategy of Generalized Deployable Units via NCF-IGA with Force Constraint

Space deployable mechanisms typically employ modular designs, where each unit that can be independently deployed is defined as a generalized deployable unit (GDU), typically consisting of rigid components, flexible components and generalized kinematic pairs. Given the high-performance demands of spatial transmission mechanisms, generalized kinematic pairs frequently integrate flexible factors like torsion springs and flexible hinges, which control the motion according to the attitude and position of the component. In this investigation, a new modeling and solution strategy for rigid-flexible coupled dynamics is established by combining NCF and IGA. It can not only accurately describe the large deformation and large rotation of flexible components, but also consider the influence of flexible factors in the kinematic pairs. The rigid components are modeled by the Natural Coordinate Formula (NCF), and the flexible components are discretized in the inertial coordinate system using spline curves in Isogeometric Analysis (IGA). The intermediate reference coordinates were integrated into the NCF-IGA framework, overcoming the boundary constraints between B-spline and NCF elements, and the geometric constraint equation of the kinematic pair was established. Subsequently, this paper proposes the concept of force constraint in controllable deployable mechanisms. Based on the force constraint equation, the mechanical model of flexible joints is established. The dynamic equations of GDUs are derived considering both geometric and force constraints, and generalized-[Formula: see text] method is introduced to solve the equations. Finally, three numerical examples are provided to illustrate the applicability and superiority of the proposed method.

Geometric Design Methodology for Deployable Self-Locking Semicylindrical Structures

Due to their unique bistable characteristics, deployable self-locking structures are suitable for many engineering fields. Without changing the geometrical composition, such structures can be unfolded and locked solely by the elastic deformation of materials. However, their further applications are hampered by the lack of simple and systematic geometric design methodologies that consider arbitrary structural curvature profiles. This paper proposes such a methodology for double-layer semicylindrical grid structures to simplify their cumbersome geometric design. The proposed methodology takes joint sizes into account to ensure that the design results can be applied to actual projects without further adjustments. By introducing symmetry into the structural units (SUs) and selecting reasonable geometric parameters that describe the structural side elevation profile, a concise set of simultaneous nonlinear geometric constraint equations is established, the solution of which provides the geometric parameter values of the grid shape. On this basis, the remaining geometric parameter values, i.e., the geometric parameter values of the inner scissor-like elements (SLEs) of each SU, can be achieved from the equations derived from general SLEs. Design examples and the assembled physical grid structure indicate the feasibility and wide applicability of the proposed geometric design methodology.

Rough solutions of the relativistic Euler equations

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.

Research and Application of Multiwave Number Line measurement Model based on Geometric Algebraic Plane Number Equation

Research and Application of Multiwave Number Line measurement Model based on Geometric Algebraic Plane Number Equation

The problem of the structure and the soil plastic environment interaction in the conditions of dynamic evolutionary processes

Dynamic influences are one of the important factors affecting project decisions during the construction of buildings and structures. In the case of dynamic analysis, the range of possible variations in structural response is very difficult to determine. The complexity is determined not only by the parameters of the load itself (intensity, the law of change over time), but also by the characteristics of the research object (their values and change over time), which is an interconnected system of the structure and the soil base. Dynamic factors of influence can include periodic vibration or shock loads, the effects of explosions that cause a sharp change in pressure on the structure of the building, seismic fluctuations, etc. In many cases, the load is the result of stress waves propagating in the soil base. These factors can act on the natural environment long-term, relatively short-term, short-term and instantaneous. Therefore, the development of refined approaches for assessing the stress-strain state of structures during the interaction of the structure and the soil environment under the conditions of dynamic evolutionary processes remains an actual task today. In order to effectively find a solution to the dynamics problems, it is necessary to possess the entire set of analysis tools, to clearly present the rules and limits of their application. The ability to correctly assess the engineering and geological conditions of the construction site, the properties of the soils, the joint work of these soils with the foundations and structures of the building has always been and remains the basis of reliable and effective engineering design. The given statement of the problem is a theoretical basis to build the numerical approaches to the study of the vibrational processes of heterogeneous spatial bodies during their contact interaction with an elastic-plastic medium. The paper considers the modeling aspects of the structure and the soil elastic-plastic environment interaction under the conditions of dynamic evolutionary processes. The basic geometric and physical equations of the elasticity theory for heterogeneous circular and prismatic spatial bodies are given. In the range of solving the dynamics problem, the peculiarities of the mathematical model implementation, regarding the representation of the variation of kinetic energy as part of the equations of dynamic equilibrium of linear and nonlinear mechanical systems, are considered.

Active vibration control of piezoelectric multilayers FG‐CNTRC and FG‐GPLRC plates

AbstractThis study investigates the active vibration control of multilayer Functionally Graded Carbon NanoTube Reinforced polymer Composite (FG‐CNTRC) plates and multilayer functionally graded graphene platelets reinforced polymer composite (FG‐GPLRC) plates covered with a piezoelectric layer (PZTG1195N) serve as both actuators and sensors. The plate's heart is supposed to be reinforced nonlinearly using both types of reinforcement. Active vibration control of plates under uniform load was achieved by utilizing a velocity feedback control algorithm with a specific gain value. The finite element method is employed to analyze both the static and dynamic behavior of a square plate discretized into 9‐node quadratic finite elements with 5 ° of freedom per node. The material properties are assumed to change across the thickness direction following a power law distribution, exhibiting various patterns: Uniform Distribution (UD), as well as Functionally Graded types X, O, and A (FG‐X, FG‐O, and FG‐A). The geometrical nonlinear equations are solved by the Newmark integration method. This work begins by validating the natural frequencies of a Functionally Graded (FG) composite plate reinforced with four different distributions of Nano‐filler and then aims to show the effect of the nonlinear distribution of nanofillers on the plate's stiffness. The Results obtained after the execution of a developed code highlight the significance of employing a nonlinear distribution of nanofillers to attenuate vibrations.Highlights FG composite and sandwich plates reinforced with GPLs and CNTs. Effect of nonlinear nanofillers distribution of reinforcement on free and forced vibration. The ability of the nonlinear distribution of nanofillers to attenuate forced vibrations.