Mechanical Problems Research Articles

Simultaneous measurement of multiple incompatible observables and tradeoff in multiparameter quantum estimation

How well can multiple incompatible observables be implemented by a single measurement? This is a fundamental problem in quantum mechanics with wide implications for the performance optimization of numerous tasks in quantum information science. While existing studies have been mostly focusing on the approximation of two observables with a single measurement, in practice multiple observables are often encountered, for which the errors of the approximations are little understood. Here we provide a framework to study the implementation of an arbitrary finite number of observables with a single measurement. Our methodology yields novel analytical bounds on the errors of these implementations, significantly advancing our understanding of this fundamental problem. Additionally, we introduce a more stringent bound utilizing semi-definite programming that, in the context of two observables, generates an analytical bound tighter than previously known bounds. The derived bounds have direct applications in assessing the trade-off between the precision of estimating multiple parameters in quantum metrology, an area with crucial theoretical and practical implications. To validate the validity of our findings, we conducted experimental verification using a superconducting quantum processor. This experimental validation not only confirms the theoretical results but also effectively bridges the gap between the derived bounds and empirical data obtained from real-world experiments. Our work paves the way for optimizing various tasks in quantum information science that involve multiple noncommutative observables.

Asymptotic solutions for heat transfer and stresses in functionally graded porous sandwich pipes subjected to nonuniform pressures and thermal loads

In this work, thermomechanical behaviors of temperature-dependent (TD) functionally graded porous (FGP) sandwich pipes subjected to nonuniform pressures and thermal loads are studied. Because the material properties are variable across the wall of the pipe, seeking exact solutions for the pipe is almost impossible. We use a slice model where the pipe is partitioned into plenty of annular slices and each slice is assumed to have uniform material properties. First, the nonlinear heat transfer along the pipe wall thickness is obtained by introducing an iterative procedure. Then, by using the Fourier series, the mechanical problem is decomposed into axisymmetric and nonaxisymmetric parts. Both the parts can be treated by the state space method and transfer matrix method, and then the superposition principle is used to obtain the displacement and stress distributions. The model results are in good consistency with those obtained from the numerical simulation and those reported in the literature. Finally, an FGP sandwich pipe is considered to discuss the effects of the temperature dependence of material properties (TDMP), power-law index, porosity, and pressure distribution on its thermomechanical behaviors.

The p : q resonance for dissipative spin–orbit problem in celestial mechanics

The p : q resonance for dissipative spin–orbit problem in celestial mechanics

Effect of a gunite lining on the stability of salt caverns in bedded formations for hydrogen storage

In this paper, a novel methodology to improve the structural behaviour and durability of bedded salt caverns for hydrogen storage is proposed. The suggested technology consists in applying a gunite lining over the whole cavern surface by means of a pneumatic air blowing. This continuous flow projected at high speed onto the cavern surface produces a self-compacted gunite facing. The effect of adding a gunite lining on the structural performance of a bedded salt cavern is analysed. The nonlinear finite element method is then applied to solve the inherent nonlinear solid mechanics problem. It is shown that reinforcing salt caverns with a gunite lining improves the cavern behaviour, specially in the case of caverns located within a horizontal salt stratum confined between other strata. When dealing with these salt formations, the stratum geometry limits the cavern height. In general, wider caverns that exhibit larger displacements are required to achieve a huge storage capacity. Moreover, these caverns can be located at a considerable depth, where the temperature has an important role on the creep deformation process. Therefore, the proposition of stabilization techniques is required to ensure both their structural integrity and the ground subsidence. This technology allows the safe exploitation of salt caverns located at bedded-type stratified horizontal salt formations.

Self-similar solution for laminar bubbly flow evolving from a vertical plate

The development of a bubble plume from a vertical gas-evolving electrode is driven by buoyancy and hydrodynamic bubble dispersion. This canonical fluid mechanics problem is relevant for both thermal and electrochemical processes. We adopt a mixture model formulation for the two-phase flow, considering variable density (beyond Boussinesq), viscosity and hydrodynamic bubble dispersion. Introducing a new change of coordinates, inspired by the Lees–Dorodnitsyn transformation, we obtain a new self-similar solution for the laminar boundary layer equations. The results predict a wall gas fraction and gas plume thickness that increase with height to the power of 1/5 before asymptotically reaching unity and scaling with height to the power 2/5, respectively. The vertical velocity scales with height to the power of 3/5. Our analysis shows that self-similarity is only possible if gas conservation is entirely formulated in terms of the gas specific volume instead of the gas fraction.

An Efficient Numerical Algorithm for Abnormal Integrals with Arbitrary Order Double Bessel Functions of the First Type

Abstract The integrals of the first type of double Bessel functions have a wide range of applications in geological exploration, mechanical and electromagnetic responses, signal processing, scattering, and wetting. In this paper, we develop a linear transformation accelerated convergence algorithm (LTACA) that combines the large argument approximate expression of the Bessel function (LAAEBF) and the integral accumulation to provide an efficient numerical algorithm for abnormal integrals with arbitrary order double Bessel functions of the first type. The effectiveness and high efficiency of the algorithm are verified by numerical examples, and its high accuracy is demonstrated by comparison with the Gaver-Stehfest inverse Laplace transform method (GSILTM). This offers a reliable and efficient computational method for the study of signal processing and mechanical problems.

Chaos: From celestial mechanics to climate

Abstract Chaos is unpredictable behaviour arising in systems governed by deterministic laws. Since its ‘discovery’ in the 1960s by Edward Lorenz in his studies of meteorological systems, chaos has become an important way of understanding seemingly random behaviour in many other areas. These include the double pendulum, the climate, El Niño and electrical oscillators. Understanding chaos allows the limits of the predictability of a system to be determined. However, despite this recent activity this paper will show that the theory of chaos has a rich history and chaotic behaviour was originally identified in the 1890s by Henri Poincaré in his investigations of the three-body problem in celestial mechanics. But remarkably this paper will also show that chaotic behaviour can be found in such a simple problem as multiplication by 10 and it is closely linked to the question (studied by the ancient Greeks) of whether a number is rational or irrational.

Physics simulation capabilities of LLMs

Large Language Models (LLMs) can solve some undergraduate-level to graduate-level physics textbook problems and are proficient at coding. Combining these two capabilities could one day enable AI systems to simulate and predict the physical world. We present an evaluation of state-of-the-art (SOTA) LLMs on PhD-level to research-level computational physics problems. We condition LLM generation on the use of well-documented and widely-used packages to elicit coding capabilities in the physics and astrophysics domains. We contribute ∼50 original and challenging problems in celestial mechanics (with REBOUND), stellar physics (with MESA), 1D fluid dynamics (with Dedalus) and non-linear dynamics (with SciPy). Since our problems do not admit unique solutions, we evaluate LLM performance on several soft metrics: counts of lines that contain different types of errors (coding, physics, necessity and sufficiency) as well as a more educational’ Pass-Fail metric focused on capturing the salient physical ingredients of the problem at hand. As expected, today's SOTA LLM (GPT4) zero-shot fails most of our problems, although about 40% of the solutions could plausibly get a passing grade. About 70%–90% of the code lines produced are necessary, sufficient and correct (coding & physics). Physics and coding errors are the most common, with some unnecessary or insufficient lines. We observe significant variations across problem class and difficulty. We identify several failure modes of GPT4 in the computational physics domain, such as poor physical units handling, poor code versioning, tendency to hallucinate plausible sub-modules, lack of physical justification for global run parameters (e.g., simulation time, or upper-lower bounds for parametric exploration) and inability to define steady-state or stopping conditions reliably. Our reconnaissance work provides a snapshot of current computational capabilities in classical physics and points to obvious improvement targets if AI systems are ever to reach a basic level of autonomy in physics simulation capabilities.

Development of Multiple Representation Models in Physics Mechanics Learning Using a Self-Regulated Learning Approach

Abstract The challenge of learning mechanical physics is to help students with different characteristics learn to solve problems. Backgrounds education, initial abilities, understanding concepts, and self-regulated learning which can have an impact on representational abilities in solving physics problems. Model development needs to be carried out in an effort to facilitate students in learning to solve physics problems through a multiple representation model approach. Development uses the Dick & Carey model with descriptive analysis with pre and post-test design. The research was conducted at the Mathematics Education Departement, Mathematics and Science Faculty of Universitas Indraprasta PGRI, with a sample of 27 students taking physics mechanics course. Development has produced the VIFOCA learning model which consists of Visualization, Formulation, and Calculation. Formative evaluation involving instructional design experts, material and language experts obtained an average score of 4.75, while the formative evaluation of students showed an average score of 4.35, with a mean combined score of 4.55. The result of the evaluation formative show that the VIFOCA model is feasible for use in physics learning mechanics. The ability of physics problem solving is known based on the pre-test results of 61.33, while the post-test average is 68.74, with an increase of 12.08%. The results of this research indicate that the VIFOCA learning model can help facilitate learning and improve physics mechanics problem solving abilities.

A review of multiscale numerical modeling of rock mechanics and rock engineering

AbstractRock is geometrically and mechanically multiscale in nature, and the traditional phenomenological laws at the macroscale cannot render a quantitative relationship between microscopic damage of rocks and overall rock structural degradation. This may lead to problems in the evaluation of rock structure stability and safe life. Multiscale numerical modeling is regarded as an effective way to gain insight into factors affecting rock properties from a cross‐scale view. This study compiles the history of theoretical developments and numerical techniques related to rock multiscale issues according to different modeling architectures, that is, the homogenization theory, the hierarchical approach, and the concurrent approach. For these approaches, their benefits, drawbacks, and application scope are underlined. Despite the considerable attempts that have been made, some key issues still result in multiple challenges. Therefore, this study points out the perspectives of rock multiscale issues so as to provide a research direction for the future. The review results show that, in addition to numerical techniques, for example, high‐performance computing, more attention should be paid to the development of an advanced constitutive model with consideration of fine geometrical descriptions of rock to facilitate solutions to multiscale problems in rock mechanics and rock engineering.

PERTURBED MOTIONS OF A NEARLY DYNAMICALLY SPHERICAL RIGID BODY WITH A MOVABLE MASS SUBJECT TO CONSTANT BODY-FIXED TORQUE

The problem of motion of a rigid body about a fixed point is one of the classical problems of mechanics. The interest to the problems of the rigid body dynamics has increased in the second half of the XX century in connection with the development of rocket and space technologies. A spacecraft or satellite, while orbiting about its center of mass, experiences torques from forces of diverse physical nature. This includes torques generated by the motion of internal masses, which can arise from factors such as presence of rotating components (like rotors or gyroscopes), and the activities of crew members aboard the crew vehicle. The dynamics of rigid body incorporated moving masses is a significant focal point in classical mechanics. Extensive research is dedicated to investigating the rotation of a rigid body featuring motion of internal masses. It is assumed that the body contains a viscoelastic element that is modeled by a moving mass connected to the body by a strong damper. The moving mass model loosely attached elements in a space vehicles, which can significantly affect the vehicle’s motion about its center of mass during a long period of time. Some cases are considered of the motion of a rigid body containing internal masses connected to the body by means of elastic and dissipative elements. A number of works are devoted to the analysis of various problems of the dynamics of space vehicles containing internal movable masses. The paper develops an approximate solution by means of averaging method to the system of Euler’s equation terms for a nearly dynamically spherical rigid body containing a viscoelastic element under the action of constant body-fixed torque. We obtained the system of motion equations in the standard form which refined in square-approximation by small parameter. Asymptotic approach permits to obtain some qualitative results and to describe evolution of angular motion using simplified averaged equations and numerical solution. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a rigid body under the influence of small internal torque (cavity filled with a fluid of high viscosity) or external torque (resistive medium). The importance of the results is in progress of moving mass control motion of spinning projectiles.

Effective toughness based on Eshelby transformation theory for heterogeneous composites

Predicting fracture toughness of heterogeneous composites is an important and challenging problem in physics and mechanics. The dependence of effective toughness on elastic properties of phases remains unclear. Considering that energy plays an essential role in crack propagation, an energy approach is proposed to obtain effective toughness in this study. We built the relationship between effective toughness and the homogenized local surface energy. The energy is constructed by generalizing Eshelby’s equivalent inclusion formulation to heterogeneous case, which couples physical features with elastic properties. An analytical formula of effective toughness can be derived for heterogeneous composites. Based on this formula, effects of toughness and elastic properties of the phases are discussed in depth, which reveals that how elastic heterogeneity can influence the effective toughness fundamentally. It is demonstrated that the predictions of concretes and metal toughening glasses agree well with experimental evidences.

Dissipation Effects in the Tea Leaf Paradox

The Tea Leaf Paradox (TLP) describes unsteady fluid motions which help entrain and deposit suspended particles at the center of rotation. Various applications depend on the TLP for particle separations—spanning orders of magnitude in length scales—making it an important problem in fluid mechanics. Despite papers describing the phenomenon, the efficacy of particle separation using the TLP remains unclear as to the relative importance of, for example, hydrostatics, particle-fluid density ratio, wall friction, liquid bath aspect ratio and the rotation speed. The dynamics involved are notably complex and require a careful tuning of each variable. In this study, we have investigated the role of the limit of the aggregation dynamics in rotational flows within 3D-printed vessels of various sizes in tandem with particle imaging to probe the dissipation effects on the particle motions. We have found that the liquid bath aspect ratio limits how much aggregation may occur for a particle-fluid density ratio greater than unity (e.g., ρp/ρf>1), where ρp is the density of the particle and ρf is the ambient fluid density.

IDENTIFICATION OF AN ARBITRARY MOVING AXISYMMETRIC LOAD ACTING ON A CYLINDRICAL SHELL

Various types of external non-stationary loads can act on various structural elements and cylindrical shells of finite length in particular distributed and concentrated, stationary and moving load. When using various external load identification methods, the type of external load is usually known. In prac- tice this is not always the case. The purpose of the study is to develop a method for identifying an arbitrary axisymmetric load acting on an elastic cy- lindrical shell of finite length, which can be used to identify a moving load. To model the non-stationary load of a cylindrical shell, a system of differ- ential equations of the refined Timoshenko theory for medium thickness shells was used. The solution to this system of differential equations is ob- tained by expanding the unknown functions into Fourier series and applying the integral Laplace transform. The solution of the inverse problem was obtained using the theory of integral equations and the Tikhonov regularization method. As a result of the study, a solution of the inverse problem of solid mechanics to identify an arbitrary axisymmetric non-stationary load was obtained. A numerical experiment was carried out to use the developed method to identify a moving non-stationary load acting on a simply supported cylindrical shell of medium thickness. The simulation results indicate a fairly accurate identification of both the change in time and the distribution along the shell of a non-stationary axisymmetric moving load. A method has been developed for identifying an external non-stationary load randomly distributed along a cylindrical shell, which makes it possible to identify a moving load without preliminary information about the type of this load. The developed method allows us to reproduce moving non-stationary loads, which are often encountered in practice, and expand it to other types of structural elements.

An adaptive gravitational search algorithm for optimizing mechanical engineering design and machining problems

The gravitational search algorithm (GSA) is widely used for solving optimization problems because it performs in a superior manner as compared to various competing evolutionary and swarm-based metaheuristics. However, GSA frequently gets trapped in local optima due to a lack of solution diversity. Although chaotic gravitational search algorithm (CGSA) can resolve this issue to some extent but its degraded exploitation rate and convergence speed may not result in desired outcome. To this end, diversity-based chaotic GSA (DCGSA) has exhibited its capability to resolve the issues encountered with GSA and CGSA to a certain extent for unconstrained problems. DCGSA achieves this characteristic through the use of an adaptive gravitational constant, which varies according to the diversity values of the population. Since most real-world problems are subjected to some constraints, it is prudent to improve the performance of GSA in solving constrained optimization problems. The present study integrates the enhanced search capability of DCGSA with a generalized constraint handling mechanism to improve the performance of GSA in solving constrained problems. It is observed that DCGSA significantly outperforms GSA on both CEC (Congress on evolutionary computation) 2006, 2010 and 2017 functions and competes strongly with CGSA. Diversity analysis shows that the capability to balance exploration and exploitation rates is enhanced using CGSA and DCGSA. Furthermore, DCGSA algorithms outperform GSA and CGSA on real-world machining and CEC 2020 mechanical design problems. Comparison with state-of-the-art algorithms is made to analyze the performance of the algorithms from a larger perspective.

Viewing Explosion Models of Type Ia Supernovae through Insights from Terrestrial Cellular Detonation.

The cellular structure is considered to be a key as a criterion in initiation, propagation, and quenching of terrestrial detonation. While a few studies on type Ia supernovae, which are known to involve detonation, have addressed the importance of the cellular structure, further detailed treatment will benefit enhanced understanding of the explosion outcomes. In the present study, we bridge this gap in the astrophysics and engineering fields, focusing on the detonation in a helium-rich white dwarf envelope as the triggering process for the so-called double-detonation model. The cellular structures are quantified via high-resolution two-dimensional simulations. We demonstrate that widely accepted terrestrial-experimental criteria for quenching and initiation of detonation can indeed explain the results of previous hydrodynamic simulations very well. The present study highlights the potential of continuing to apply the insight from terrestrial detonation experiments to astrophysical problems, specifically the long unresolved problem of the explosion mechanism of type Ia supernovae.

Projected state ensemble of a generic model of many-body quantum chaos

Abstract The projected ensemble is based on the study of the quantum state of a subsystem A conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design, i.e. a system thermalizes when it becomes indistinguishable, up to the kth moment, from a Haar ensemble of uniformly distributed pure states. Here we consider a random unitary circuit with the brick-wall geometry and analyze its convergence to the Haar ensemble through the frame potential and its mapping to a statistical mechanical problem. This approach allows us to highlight a geometric interpretation of the frame potential based on the existence of a fluctuating membrane, similar to those appearing in the study of entanglement entropies. At large local Hilbert space dimension q, we find that all moments converge simultaneously with a time scaling linearly in the size of region A, a feature previously observed in dual unitary models. However, based on the geometric interpretation, we argue that the scaling at finite q on the basis of rare membrane fluctuations, finding the logarithmic scaling of design times t k = O ( log ⁡ k ) . Our results are supported with numerical simulations performed at q = 2.

Efficient BFGS quasi-Newton method for large deformation phase-field modeling of fracture in hyperelastic materials

The prediction of crack propagation in materials is a crucial problem in solid mechanics, with many practical applications ranging from structural integrity assessment to the design of advanced materials. The phase-field method has emerged as a powerful tool for modeling crack propagation in materials, due to its ability to accurately capture the propagation of cracks. However, current phase-field algorithms suffer from the elevated computational cost associated with the so-called staggered solution scheme, which requires extremely small time increments to advance the crack due to its inherent conditional stability. In this paper, we present, for the first time, a quantitative analysis detailing the numerical implementation and comparison of two common solution strategies for the coupled large-deformation solid-mechanics-phase-field problem, namely the quasi-Newton based Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) and the full Newton based alternating minimization (or staggered) (AM/staggered) algorithm. We demonstrate that the BFGS algorithm is a more efficient and advantageous alternative to the traditional AM/staggered approach for solving coupled large-deformation solid-mechanics-phase-field problems. Our results highlight the potential of the quasi-Newton BFGS algorithm to significantly reduce the computational cost of predicting crack propagation in hyperelastic materials while maintaining the accuracy and robustness of the phase-field method.

An extended lumped damage mechanics IGABEM formulation for quasi-brittle material failure

This study proposes a new formulation for the mechanical modeling of quasi-brittle materials. The material degradation due to cracking is addressed through the Extended Lumped Damage Mechanics (XLDM) approach. The model is inserted into an IGABEM formulation, where Non-Uniform Rational B-Splines (NURBS) are the basis functions. A novel nonlinear solution technique has been developed for the numerical implementation. Crack propagation is captured using the initial stress field approach, widely utilized by the BEM community to account for nonlinear material behavior. The XLDM is coupled into an IGABEM formulation by discretizing part of the domain into cells, placed only where damage is expected to grow. Classical benchmark problems are presented to demonstrate the method’s capability and effectiveness. The results show excellent agreement with both experimental and numerical findings from the literature. Damage evolution is assessed through the band thickness opening, leading to material degradation. This method offers a new way of addressing damage mechanics problems within the BEM framework. It could benefit the BEM community, as traditional damage analysis within this numerical method often leads to significant time-consuming and elevated computational costs, which are mitigated by the formulation proposed herein.

Role of grain-level chemo-mechanics in composite cathode degradation of solid-state lithium batteries

Solid-state Li-ion batteries, based on Ni-rich oxide cathodes and Li-metal anodes, can theoretically reach a high specific energy of 393 Wh kg−1 and hold promise for electrochemical storage. However, Li intercalation-induced dimensional changes can lead to crystal defect formation in these cathodes, and contact mechanics problems between cathode and solid electrolyte. Understanding the interplay between cathode microstructure, operating conditions, micromechanics of battery materials, and capacity decay remains a challenge. Here, we present a microstructure-sensitive chemo-mechanical model to study the impact of grain-level chemo-mechanics on the degradation of composite cathodes. We reveal that crystalline anisotropy, state-of-charge-dependent Li diffusion rates, and lattice dimension changes drive dislocation formation in cathodes and contact loss at the cathode/electrolyte interface. These dislocations induce large lattice strain and trigger oxygen loss and structural degradation preferentially near the surface area of cathode particles. Moreover, contact loss is caused by the micromechanics resulting from the crystalline anisotropy of cathodes and the mechanical properties of solid electrolytes, not just operating conditions. These findings highlight the significance of grain-level cathode microstructures in causing cracking, formation of crystal defects, and chemo-mechanical degradation of solid-state batteries.