Minimum Potential Energy Research Articles

Minimum energy principles in electrostatics

Abstract We employ the principle of minimum energy of the electrostatic field to derive Laplace's equation for electric potential in charge-free space. We also apply the principle of minimum electrical potential energy of a system of discrete charges to explain the possibility of the existence of their stable equilibrium configuration on the surface of a conductor.

Static and Transient Response Analysis of Arbitrary Laminated Composite and Sandwich Shells by the FSDT Meshless Method

This paper, for the first time, presents a novel meshless method based on the first-order shear deformation theory (FSDT) and moving-least squares (MLS) approximation to carry out static and transient response analysis on arbitrary laminated and sandwich shell structures. The novelty of the method lies in the implementation of MLS in the curvilinear shell formulation, and the derivation of meshless governing equations on the static and transient response of arbitrary laminated and sandwich shells according to the principle of minimum potential energy and Hamilton’s principle, respectively. The concept of a Convected coordinate system is adopted to deal with arbitrary curvilinear surfaces. Both the position and displacement vectors of shells are approximated by MLS, which is conceptually the same procedure as that in the isoparametric finite element method (FEM). The numerical integration of the stiffness matrices is conducted by the simple Gaussian integral in the Convected coordinate system. Because the moving-least squares (MLS) approximation does not satisfy the Kronecker delta condition, the full transformation method is used to modify the meshless governing equations. Numerical examples are calculated to investigate the static and transient response analysis of laminated and sandwich structures with different geometrical shell shapes, including square plates, spherical shells, doubly-curved shells, and cylindrical shells. The numerical tests show that the proposed method is reliable, and robust and produces accurate results compared to the analytical and numerical solutions taken from the previous literature.

Self-aggregation and microhydration mechanisms of monoethanolamine: Far-infrared identification of large-amplitude hydrogen bond libration.

The strong tendency for self-aggregation together with an intriguing mechanism for the microhydration of monoethanolamine (MEA) have been explored by low-temperature far-infrared cluster spectroscopy in doped neon "quantum" matrices at 4K complemented by high-level quantum chemical modeling. In addition to the assignment of new mid-infrared perturbed intramolecular transitions, a distinct far-infrared transition is unambiguously assigned to the concerted large-amplitude hydrogen bond librational motion of the MEA homodimer. This observation confirms a global "head-to-head" intermolecular potential energy minimum associated with the formation of a compact doubly intermolecular OH⋯N hydrogen-bonded cyclic structure, where both monomeric intramolecular OH⋯N hydrogen bonds are broken upon complexation. By means of relative mixing ratio dependencies, dedicated annealing procedures, and selective complexation between MEA and isotopic H216O and H218O samples, distinct far-infrared transitions associated with large-amplitude intra-molecular hindered OH torsional motion and inter-molecular H2O librational (hindered c-type overall rotational) motion of the MEA monohydrate are furthermore assigned unambiguously for the first time. These spectroscopic observations reveal an intriguing metastable conformation, where H2O acts as a OH⋯O hydrogen bond donor to the hydroxy group instead of the amino group of MEA upon microhydration in the cryogenic neon environment, where the microhydration strengthens the intramolecular OH⋯N hydrogen bond of MEA due to hydrogen bond cooperativity.

New Insight into Quantity-Dependent Self-Assembly Pattern of Light Levitating Droplet Clusters.

It has been reported that the self-assembly pattern of light levitating droplet clusters above the hot gas-liquid interface is dependent on the quantity of droplets. However, the already-reported theoretical explanation of the quantity-dependent self-assembly pattern cannot work well when the quantity of the light levitating droplet exceeds 15. Herein, we propose a new theoretical perspective to understand the self-assembly of a light levitating droplet cluster by referring to the classical densest packing problem of identical rigid circles in a larger circle with the introduction of the minimum total potential energy principle. Amazingly, the theoretical results obtained by this new approach agree well with experimental results, even though the quantity of the light levitating droplet is up to 142. This study deepens our understanding of the quantity-dependent self-assembly pattern of the light levitating droplet clusters and provides significant inspiration for other analogous self-assembly phenomena.

Design of piezoelectric quasi— zero—stiffness metastructures for improved low—frequency vibration isolation

Abstract In this paper, a novel type of piezoelectric quasi–zero stiffness (QZS) metastructure is proposed and the improved vibration isolation performance is investigated. A piezoelectric QZS metastructure is composed of curved beams covered with piezoelectric macro–fiber composite patches, to which digital circuits are connected. Based on the principle of minimum potential energy and the mode superposition method, the constrain on the geometry and material parameters of the piezo–curved beams to achieve QZS is given. According to this design criterion, 1D, 2D and 3D piezoelectric QZS metastructures are designed, and their improved low–frequency vibration isolation characteristics are comprehensively analyzed through theoretical, numerical and experimental studies. It is demonstrated that by optimizing the parameters of the resonant transfer functions implemented in the digital circuits, high–level damping localized near the first resonant peak can be introduced into the curved beams. As a result, the resonant peaks of the metastructures can be reduced without compromising the isolation performance beyond those peaks.

Noncircular Slip Surface Search on Slopes Based on Minimum Potential Energy Method and Improved SA Algorithm

ABSTRACTThe limit equilibrium method has been widely used in the study of searching the slip surface of slopes. However, the method ignores the deformation characteristics of the rock mass and assumes that the shape of the slip surface is circular, which is quite different from the actual situation of the slope. For this reason, this paper proposes a fast search method for noncircular slip surface considering the deformation characteristics of the rock mass. The method is able to calculate the compression and shear deformation energies stored in the slip surface, as well as the virtual displacement generated by the slide mass when the slope is in a critical equilibrium state. The direction of motion of the slide mass is further calculated from the magnitude of the virtual displacement. In addition, this paper improves the generation of new solutions in the simulated annealing (SA) algorithm for the structural characteristics of the slip surface of the slope, thus achieving a fast search of the slip surface. Finally, the method of this paper is compared with the test question of ACADS and the simulation results of the finite difference method (FDM) to verify the effectiveness of the method of this paper.

Influence of Transverse Normal Strain on Buckling and Vibration of Sandwich Plates

This study presents the buckling and free vibration behavior of simply supported square and rectangular sandwich plates. A third-order shear deformation theory considering both the transverse shear and normal deformations is used. The condition of zero transverse shear stresses on the top and bottom surfaces of the plate is satisfied. Hence, the present theory does not require the shear correction factor generally associated with the first-order shear deformation theory. Governing equations and boundary conditions are obtained using the principle of minimum potential energy. The closed-form solutions of simply supported sandwich plates are obtained. The effect of the side-to-thickness ratio and plate aspect ratio on buckling and free vibration responses of simply supported square and rectangular sandwich plates is examined. Parametric effects of face sheet thickness ratios and load factor upon the critical buckling load for uniaxial and biaxial buckling analysis of square and rectangular sandwich plates for different staking sequences are studied. All numerical solutions are obtained using MATLAB® programs. The results obtained from the present theory are compared with those from classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, and the exact three-dimensional elasticity solutions as applicable.

A modified solution for evaluating the sliding direction and stability of 3D slopes

In this paper, we present an improved and efficient analytical solution for evaluating the stability of 3D slopes based on the minimum potential energy method. The equilibrium equation of the landslide is employed to determine the shear stress on the failure surface, and a new formulation for shear potential energy is introduced. The vector-combined direction of the force acting on the landslide is considered equivalent to the sliding direction (SD). Two classic examples are reanalyzed to validate the accuracy and performance of the proposed method. The analysis demonstrates that the obtained results align well with reference solutions and tend to be conservative in engineering applications. Parameter investigations are conducted to explore the influences of SD, shear strength parameters, slope angle, and soil unit weight on the safety factor (SF) of 3D slopes. The findings indicate that a larger SD corresponds to a less stable slope. Finally, the effects of optimization parameters, landslide volume, and surcharge on the location of the failure surface, SF, and SD are also examined. The developed analytical approach offers a novel and straightforward solution with sufficient accuracy and eliminates the need for iteration in determining the SF, making it a valuable complementary method for assessing the stability of 3D slopes.

Analytical and numerical predictions of elastoplastic buckling behaviors of the subsea lined pipelines with ovality defects under hydrostatic pressure

The subsea pipelines may deteriorate and/or be cracked after a long period of service. A cost-effective trenchless rehabilitation technology is to install a thin-walled steel liner inside the pipelines. However, elliptical defects may occur in the liner due to issues such as insufficient inflation, deformation of the host pipelines, and joint misalignment in practical engineering. Thus, this study aims to develop a systematic analytical model and numerical model for the elastic and inelastic buckling behaviors of the thin-walled oval steel liner, respectively. An admissible radial displacement function is introduced to describe the single-lobe deformation of the liner. The governing equations are established to trace the equilibrium paths by applying the theory of thin-walled shells and the principle of minimum potential energy. A two-dimensional finite element model is developed for the elastic and inelastic buckling performance of the liner. The analytical solutions of elastic buckling are in good agreement with other closed-form solutions, numerical results, and respective test results available in the literature. Finally, parametric evaluations are carried out in terms of ovality, non-uniform liner-pipeline gap, internal pressure, and liner yield strength.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

Level-set based shape optimization for plane elastic structures using radial basis functions and Hilbertian descent direction

Structural optimization problems are often associated with the so-called shape functionals depending on a shape through its geometry and the state being a solution of given partial differential equation. In such a framework it is convenient to work with the gradient-like method based on a concept of a shape derivative and level set method. The key idea of level set method is to represent the structural boundary with zero level set of given function (level set function—LSF). Now, changing the shape of a structure under optimization is equivalent to transport the LSF in such a direction that ensures decreasing the value of the objective functional. To this end, we make use of coercive bilinear form taken from the weak formulation of elasticity problem to obtain descent direction at each iteration. This descent direction is a solution of an additional variational problem, involving the bilinear form mentioned above and the volumetric expression of the shape derivative plays the role of a linear form. In this paper, we combine level set method with radial basis functions (RBFs) used to approximate LSF. We focus on the so-called multiquadric RBFs, but other classes of RBFs are also briefly considered. This eventually leads to transformation of partial differential equation (linear transport equation governing the evolution of shapes) to a system of linear ordinary differential equations which admits analytical formula for the solution. We apply our method to compliance minimization of a cantilever problem as well as to total potential energy minimization of a structure with kinematic loading. To run all the numerical experiments, we wrote our own code in Wolfram Mathematica environment.

Minimum Energy Atomic Deposition: A novel, efficient atomistic simulation method for thin film growth

Thin-film growth is an area of research concerned with complex phenomena happening at atomic scales. Therefore, molecular simulation has been an important tool to confront experimental results to theoretical assumptions. However, the traditional thin film growth simulation methods, i.e., Molecular Dynamics (MD) and kinetic Monte-Carlo (kMC) and combinations thereof, suffer from limitations inherent to their design, i.e., limitations in system size and simulation time for MD and predetermined reaction rates and reaction sites for kMC. Consequently, it is practically impossible to simulate the evolution of polycrystalline growth resulting in ∼100nm thick films with realistic stress fields and defect structures, such as grain boundaries, stacking faults, etc. In this work, we propose a versatile and efficient atomistic simulation method (Minimum Energy Atomic Deposition) which works by direct insertion of atoms at points of minimal potential energy through efficient scanning of candidate positions and rapid relaxation of the system. This method allows simulating ≥100nm film thickness while mimicking experimental growth rates and high crystallinity and low-defect concentration and enables in-depth studies of atomic growth mechanisms, the evolution of crystal defects, and residual stress build-up. We demonstrate the efficiency and versatility of the method through the deposition of Al on Si, Al on Al, and Si on Si. The simulation results are systematically compared with experimental observations of thin-film deposition, yielding consistent observations. The method has been implemented in open-source LAMMPS software, making it easily accessible to the research community.

Piezoelectric Interaction Analysis and Load‐Deflection Prediction for Nonlinear Bending and Snap‐Through of Post‐Buckled FG Piezoelectric Beams

Revealing the interaction effect in the nonlinear bending of post‐buckled functionally graded (FG) piezoelectric beams promotes the intelligent development of structural design. The nonlinear bending behavior of post‐buckled piezoelectric beams is studied. Based on the displacement field constructed by the refined beam model, the post‐buckling configuration is considered. The nonlinear geometric relationship and the constitutive relationship in the multi‐physical field are modeled. In the framework of the principle of minimum potential energy, the nonlinear static models of post‐buckled FG piezoelectric beams are established. The post‐buckling configuration is obtained by the extended two‐step perturbation method. To obtain the approximate solution for the static problems of the post‐buckled FG piezoelectric beam, the perturbation scheme of electric potential and generalized displacement is proposed. The detailed parametric analysis indicates the interaction mechanism in the nonlinear bending of the post‐buckled FG piezoelectric beam. The snap‐through process and its sensitive interval are further captured. The back propagation neural network (BPNN) technique developed for the nonlinear bending problem of post‐buckled FG piezoelectric beams achieves accurate prediction.

Low-velocity impact response and damage mechanism of cosine function cell-based lattice core sandwich panels

This study aims to investigate the impact response and damage mechanism of cosine-function cell-based (CFCB) lattice core sandwich panels. Several low-velocity impact tests were conducted to explore their advantages in impact resistance by comparing them with traditional body-centered cubic (BCC) lattice-core sandwich panels. The impact response and deformation patterns of CFCB lattice core sandwich panels with two different faceplate materials (aluminum alloy and carbon fiber reinforced plastic composite) were experimentally investigated. The CFCB lattice core sandwich panels exhibited higher impact resistance and energy absorption capacity than their BCC lattice core counterparts. Furthermore, sandwich panels with aluminum alloy faceplates provided better impact resistance capacity than those with CFRP faceplates. Subsequently, several numerical models were developed to explore the deformation mechanisms and energy absorption characteristics of CFCB lattice core sandwich panels. In addition, the effects of the core structural parameters on mechanical performance were numerically investigated. Results indicated that increasing the cell rod diameter and/or reducing the cosine period length could decrease the indentation depth and enhance crush force efficiency. Finally, based on the principle of minimum potential energy, a theoretical model was developed to predict the initial peak load of CFCB lattice core sandwich panels with isotropic faceplates. This study aimed to explore the impact response and provide design guidance for CFCB lattice core sandwich panels.

Effect of Leaving Centrosymmetric Character on Spectral Properties in Mono-, Bi-, and Triphotonic Absorption Spectroscopies.

Numerical simulations of the absorption bands of photoswitch E-o-tetrafluoroazobenzene in DMSO solution under one-, two-, and three-photon absorption conditions combined with the analysis of the behavior of transition probability under distortion of planarity reveal many similarities between the mono- and triphoton spectroscopic behaviors with a two-photon spectrum being set apart. The position of the absorption peak for the studied nπ* and ππ* transitions appears shifted to lower energies (longer wavelengths) than the conventional estimate based on vertical excitation from the ground-state potential energy minimum.

Analytical Solution to Estimate the Effect of Underlying Tunnel Excavations on the Existing Tunnel

ABSTRACTThe tunneling underlying inevitably leads to the displacement of adjacent soil, greatly influencing the deformation of the tunnel above. Most theoretical studies primarily concentrate on analyzing the mechanical equilibrium of individual tunnel sections, ignoring the energy generated by the system during tunnel deformation. Based on this, in view of the energy relationship, the overlying tunnel's deformation is simulated by using the Rayleigh–Ritz method. Further, its potential energy equation can be established based on the Vlasov foundation. The corresponding energy variational solution is solved according to the minimum potential energy principle, leading to an analytical solution for the shield tunneling‐induced overlying tunnel's response. The efficacy of the suggested method is verified by contrasting it with centrifuge experiments and field case studies derived from prior studies. Relative to the Winkler foundation model, which deviated from the suggested approach, the results derived from the suggested method show a closer correlation with the collected measurement data. Further parameter studies show that the vertical clearance and skew angle between two tunnels, the volume loss rate, and elastic modulus are significant factors affecting the tunnel behaviors due to tunneling underneath. The suggested theoretical model can be applied to forecast potential risks that an existing tunnel may encounter during the excavation of a new tunnel underlying in similar engineering projects.

Determination of Chip Compression Ratio for the Orthogonal Cutting Process

The chip compression ratio is the most important characteristic of various machining processes with chip generation. This characteristic enables the determination of kinetic and other energy loads on the tool and the machined material. This provides an overall evaluation of the machining process and the possibility of its subsequent optimization. This paper presents the results of determining this cutting characteristic by experimental method, analytical calculation, and numerical modeling. For the analytical calculation of the chip compression ratio, an analytical cutting model developed based on the variational principle of the minimum potential energy was used. A finite element model of orthogonal cutting was used for the numerical simulation of the above process characteristic. Experimentally, the chip compression ratio was determined by the ratio of the chip thickness to the cutting depth (undeformed cutting thickness). The chip thickness was determined by direct measurement using chip slices obtained during the cutting process. The Johnson–Cook constitutive equation was used as the machined material model and the Coulomb model was used as the friction model. The generalized parameters’ determination of the constitutive equation was performed through a DOE (Design of Experiment) sensitivity analysis. The variation range of these parameters was chosen based on the analysis of the effect of individual parameters of the constitutive equation on the chip compression ratio value. The largest deviations between the experimental and analytically calculated values of the chip compression ratio did not exceed 21%. At the same time, the largest deviations of simulated values of the indicated cutting characteristic and its experimental values did not exceed 20%. When comparing the experimental values of the chip compression ratio with the corresponding calculated and simulated values, the deviations were within 22%.

Modeling magnetic soft continuum robot in nonuniform magnetic fields via energy minimization

The emerging magnetic soft continuum robots (MSCRs) – a type of slender and soft rods with embedded hard-magnetic particles – hold great promise in endovascular intervention via remote magnetic actuation. Although numerous advantages of using permanent magnets have been demonstrated for manipulating MSCRs (e.g., simple systems with high actuation force, large operating workspace), the magneto-mechanical behavior of MSCRs in nonuniform magnetic fields, particularly those generated by permanent magnets, remains largely unexplored. In this work, a systematic study of MSCRs in the nonuniform field is presented, which includes theoretical modeling using hard-magnetic elastica theory, numerical analyses by energy minimization method, finite element simulations using ABAQUS user element (UEL), and experimental validation. Without solving governing differential equations, the large deflection of MSCRs is efficiently obtained via the minimization of the total potential energy using sequential quadratic programming (SQP). This efficient modeling method offers insights into the control of MSCRs using nonuniform magnetic fields. Two practical strategies are provided for precisely controlling MSCRs by manipulating a cubic magnet through the adjustment of the actuation distance, rotation angle, and spin angle, laying the foundation for applications of magnetically-assisted endovascular intervention.

Optimization of tuned mass dampers – minimization of potential energy of elastic deformation

A tuned mass damper (TMD) optimization can be performed under various assumptions and objectives. All the variables of the optimization, such as structural model, performance index and load type affect the optimal parameters of the TMD. This paper presents a new optimization method that implements straightforward performance index and allows taking load spectral characteristics into account. Thanks to the usage of modal coordinates, the method allows fast numerical optimization of TMD attached to large or complicated structures with numerous degrees of freedom. One of the complicated tasks while optimizing TMD is the choice of a performance index. In this paper, the mean value of potential energy stored in the elastic deformation of a structure under periodic load serves as a performance index, which leads to a low numerical complexity task if the optimization is performed in the frequency domain. The new method also allows a simple inclusion of load spectral characteristics and permits TMD optimization for any loading spectral range. When applied to a structure with a single degree of freedom, this method leads to H2 optimization in the case of white noise excitation. However, it is applicable to multiple degrees of freedom structures with single or multiple TMDs and any given load. The paper also presents several examples of numerical optimization of the TMD attached to both single and multiple degrees of freedom structures under various loads, including white noise excitation, pedestrian load, and earthquake strong motion.

Forced-Vibration Characteristics of Bowtie-Shaped Honeycomb Composite Sandwich Panel with Viscoelastic Damping Layer.

The incorporation of viscoelastic layers in laminates can markedly enhance the damped dynamic characteristics. This study focuses on integrating viscoelastic layers into the composite facesheet of the bowtie-shaped honeycomb core composite sandwich panel (BHC-CSP). The homogenization of the damped BHC-CSP is performed by employing the variational asymptotic method. Based on the generalized total energy equation, the energy functional of the representative unit cell of the damped BHC-CSP is asymptotically analyzed. The warping function, derived following the principle of minimum potential energy, provides a basis for obtaining the corresponding Euler-Lagrange equation to ascertain the equivalent elastic properties of the damped BHC-CSP. Utilizing the developed two-dimensional equivalent model, the free-vibration characteristics of the damped BHC-CSP are examined across diverse boundary conditions while delving into the impact of an external viscous damping layer on the natural frequency of the damped BHC-CSP. The results reveal that intensified boundary constraints effectively diminish the effective vibration region of the damped BHC-CSP, thereby enhancing its overall stability. The introduction of a PMI foam layer proves effective in adjusting the stiffness and mass distribution of the damped BHC-CSP. Resonance characteristics are explored through frequency and time-domain analyses, highlighting the pivotal roles of the excitation position and receiver point in influencing the displacement and velocity responses. Although the stiffness is improved by incorporating a PMI foam layer, its effect on the damping performance of the damped BHC-CSP is minimal when compared to the T-SW308 foam layer.