Total Potential Function Research Articles

Data-driven enhanced phase field models for highly accurate prediction of Mode I and Mode II fracture

By coupling the elastic and crack surface energy in the total potential function, phase field methods favor a natural track of crack initiation and propagation within the continuum mechanics framework. However, in turn, it raises a critical issue in determining the coupling factor—energy degradation function. This work aims to tackle the problems induced by assigning a degradation function empirically, including the inaccurate reproduction of critical loads and unphysical stiffness reduction prior to fracture. Inspired by the concepts of data-driven computational mechanics, we propose a data-driven phase field scheme to enhance the numerical reproduction of physically consistent fracture responses. This method collaborates a datasets generator, a classifier, a physical-constrained learning algorithm, and the phase field solver in an extensible modular framework, allowing searching for the optimized damage constitutive relation in a carefully designed degradation function space with flexible form. In the case studies, mode I and mode II fracture characteristics of linear elastic and hyperelastic materials are well predicted by the data-driven phase field algorithm, which learns and derives the results from simple scalar datasets, controlling the average error lower than 10%. When introducing the input dataset noise, the model still shows robustness. The results demonstrate the model’s interpretability, thermodynamical consistency, and accuracy.

Gradient Play in Stochastic Games: Stationary Points and Local Geometry

We study the stationary points and local geometry of gradient play for stochastic games (SGs), where each agent tries to maximize its own total discounted reward by making decisions independently based on current state information which is shared between agents. Policies are directly parameterized by the probability of choosing a certain action at a given state. We show that Nash equilibria (NEs) and first-order stationary policies are equivalent in this setting by establishing a gradient domination condition for SGs. We characterize the structure of strict NEs and show that gradient play locally converges to strict NEs within finite steps. Further, for a subclass of SGs called Markov potential games, we prove that strict NEs are local maxima of the total potential function, thus locally stable under gradient play, and fully-mixed NEs are saddle points, thus unstable under gradient play.

Artificial Potential Function Safety and Obstacle Avoidance Guidance for Autonomous Rendezvous and Docking with Noncooperative Target

The problem of artificial potential function (APF) safety and obstacle avoidance guidance for autonomous rendezvous and docking of chaser spacecraft with noncooperative spacecraft is studied. The relative motion equation of the chaser and the target is established based on the line-of-sight coordinate system, the reference state is designed, and the corresponding state error is deduced. The attitude motion equation of the noncooperative target spacecraft in space is established. The safety and obstacle avoidance guidance problem of autonomous rendezvous and docking with noncooperative target is transformed into a path planning problem in a dynamic environment. The attractive potential function is designed according to the state error. In order to ensure that the chaser can safely approach the noncooperative target spacecraft, a safe corridor with ellipse cissoid is designed in the final approaching stage of autonomous rendezvous and docking. The obstacle is assumed to be a sphere with a certain radius to avoid its influence in the approach, and the obstacle potential function is designed based on the Gaussian function method. The total potential function of the system is designed according to the attractive potential function, the safe potential function, and the obstacle potential function. The total potential function of the system is modified to ensure that the reference state is the minimum of the total potential function of the system. The stability of the system is proven according to the Lyapunov stability principle, and the conditions for satisfying the monotonic decrease in the total potential function of the system are deduced. Finally, the effectiveness of the proposed method is verified by three sets of numerical simulations.

REAL-TIME OBSTACLE AVOIDANCE FOR AN UNDERACTUATED FLAT-FISH TYPE AUTONOMOUS UNDERWATER VEHICLE IN 3D SPACE

This paper discusses a real-time obstacle avoidance algorithm and its implementation for an underactuated flat-fish type autonomous underwater vehicle (AUV) in 3D space. The algorithm has been developed using multi-point potential field (MPPF) method and its real-time testing is carried out using hardware-in-loop (HIL) simulations. In MPPF method, a region of predefined radius on a hemisphere in the positive x-axis around the bow of an AUV is discretized into equiangular points with centre as the current position. By determining the point at which the minimum total potential exists, the vehicle can be moved towards that point. Here the analytical gradient of the total potential function is not calculated as it is not essentially required for moving the vehicle to the next position. The MPPF method is interfaced with dynamic model of an underactuated flat-fish type AUV and it is tested and verified using HIL simulation tool. The details of the dynamics of AUV, MPPF method and its implementation, development of HIL test bench and the simulation results are presented in this paper. The results show that the proposed MPPF method is very effective for obstacle avoidance in 3D space and can be used in the real-time control of the AUV.

Multipoint potential field method for path planning of autonomous underwater vehicles in 3D space

A multipoint potential field method (MPPF) for path planning of autonomous underwater vehicles (AUV) in 3D space is presented in this paper. The algorithm is developed based on potential field method by incorporating a directed search method for sampling the potential field. In this approach, the analytical gradient of the total potential function is not computed, as it is not essentially required for moving the vehicle to the next position. Rather, a hemispherical region in the direction of motion around the AUV's bow is discretized into equiangular points with center as the current position. By determining the point at which the minimum potential exists, the vehicle can be moved towards that point in 3D space. This method is very simple and applicable for real-time implementation. The problem of local minima is also analyzed and found that the local minima in 2D space can be easily overcome with the MPPF. A simple strategy to avoid the local minima in 3D space is also proposed. The proposed method reduces the burden of fine-tuning the positive scaling factors of potential functions to avoid local minimum. The algorithm development and the simulation results are presented.

Sloshing of Liquid in Rigid Cylindrical Container with a Rigid Annular Baffle. Part II: Lateral Excitation

Sloshing response of liquid in a rigid cylindrical container with a rigid annual baffle subjected to lateral excitation has been studied. The complicated liquid domain is separated into several simple sub-domains by introducing the artificial interfaces. The analytical solutions of potential function corresponding to every sub-domain are obtained by using the method of separation of variables and the superposition principle. The total potential function under lateral excitation is taken as the sum of the container potential function and the liquid perturbed function. The expression of the liquid perturbed function is obtained by introducing the generalized coordinates. On the base of the natural frequencies and modes having been obtained by the sub-domain method, the orthogonality among the sloshing modes has been demonstrated. Substituting the potential functions into the free surface wave equation establishes the dynamic response equation of liquid. Then, the generalized coordinates are solved. The sloshing surface displacement, the hydrodynamic pressure distribution, the resultant hydrodynamic force and moment are discussed for the containers subjected to harmonic and seismic lateral excitation, respectively.

Sloshing of Liquid in Rigid Cylindrical Container with a Rigid Annular Baffle. Part II: Lateral Excitation

Sloshing response of liquid in a rigid cylindrical container with a rigid annual baffle subjected to lateral excitation has been studied. The complicated liquid domain is separated into several simple sub-domains by introducing the artificial interfaces. The analytical solutions of potential function corresponding to every sub-domain are obtained by using the method of separation of variables and the superposition principle. The total potential function under lateral excitation is taken as the sum of the container potential function and the liquid perturbed function. The expression of the liquid perturbed function is obtained by introducing the generalized coordinates. On the base of the natural frequencies and modes having been obtained by the sub-domain method, the orthogonality among the sloshing modes has been demonstrated. Substituting the potential functions into the free surface wave equation establishes the dynamic response equation of liquid. Then, the generalized coordinates are solved. The sloshing surface displacement, the hydrodynamic pressure distribution, the resultant hydrodynamic force and moment are discussed for the containers subjected to harmonic and seismic lateral excitation, respectively.

Catastrophe Theory Analysis on Asymmetric Mining Pillar Dynamic Failure

Based on the simplified analytic model of asymmetric mining rock beam-pillar system, the catastrophe model of dynamic failure of the rock beam-pillar system in asymmetric mining has been established by way of the differential form of total potential function deduced by law of conservation of energy. Analysis and illustration show it is consistent in discriminating the system stability by energy importing rate J—presenting the descending or increasing tendency. Outside or suroundings imports energy into rock beam-pillar system when J> 0; and the system exports energy to outside or surroundings when J<0.J=0 is the critical condition of system destabilization,which implicates Cook stiffness criterion and is a dynamic generalization of Cook criterion in the form of energy.

Response of Liquid in Cylindrical Tank with Rigid Annual Baffle Considering Damping Effect

Sloshing response of liquid in a rigid cylindrical tank with a rigid annual baffle under horizontal sinusoidal loads was studied. The effect of the damping was considered in the analysis. Natural frequencies and modes of the system have been calculated by using the Sub-domain method. The total potential function under horizontal loads is assumed to be the sum of the tank potential function and the liquid perturbed function. The expression of the liquid perturbed function is obtained by introducing the generalized coordinates. Substituting potential functions into the free surface wave conditions, the dynamic response equations including the damping effect are established. The damping ratio is calculated by Maleki method. The liquid potential are obtained by solving the dynamic response equations of the system.

Catastrophic mechanism of coal and gas outbursts and their prevention and control

Based on the engineering observations of coal and gas outbursts during mining processes and the experimental results, we built a thin plate mechanical model for layered and spalled coal bodies. We studied the mechanical mechanism of outbursts, due to instability, of thin plates of coal rocks under the action of in-plane load and normal load, by using the catastrophe theory. The total potential function is derived for the layered rock system, the cusp catastrophe model for the system is established, the bifurcation set that makes the system unstable is given, the process in which gradual change of action forces leads to catastrophic change of state is analyzed, and the effect of movement path of point (P, q) in the control space on the stability of rock plate is analyzed. The study results show that during the process of coal mining, the stability of the layered coal bodies depends not only on its physical properties and dimensions but also on the magnitudes and changing paths of the in-plane load and the normal load. When the gas in the coal bodies ahead of the mining face is pre-drained, the gas pressure can be reduced and the normal load q can be lowered. Consequently, disasters such as coal and gas outbursts can be effectively prevented.

Fold catastrophe model of dynamic pillar failure in asymmetric mining

A rock burst disaster not only destroys the pit facilities and results in economic loss but it also threatens the life of the miners. Pillar rock burst has a higher frequency of occurrence in the pit compared to other kinds of rock burst. Understanding the cause, magnitude and prevention of pillar rock burst is a significant undertaking. Equations describing the bending moment and displacement of the rock beam in asymmetric mining have been deduced for simplified asymmetric beam-pillar systems. Using the symbolic operation software MAPLE 9.5 a catastrophe model of the dynamic failure of an asymmetric rock-beam pillar system has been established. The differential form of the total potential function deduced from the law of conservation of energy was used for this deduction. The critical conditions and the initial and final positions of the pillar during failure have been given in analytical form. The amount of elastic energy released by the rock beam at the instant of failure is determined as well as. A diagrammatic form showing the pillar failure was plotted using MATLAB software. This plot contains a wealth of information and is important for understanding the behavior during each deformation phase of the rock-beam pillar system. The graphic also aids in distinguishing the equivalent stiffness of the rock beam in different directions.

Element behavior in post-critical plane frame analysis

Depending on the type of uni-modal (cuspoid) catastrophe which governs the local behavior of a structure at a simple critical point, correct prediction of the post-critical path requires an accurate estimation of a number of terms in the Taylor series expansion of the total potential function at the critical point. This imposes certain conditions on the accuracy of the adopted non-linear elements. These conditions are mainly related to the quality of the kinematic assumptions which underlie the strain definition used in the model. If these assumptions are not accurate enough, the fourth and higher order terms in the Taylor expansion of the total potential function will not be correctly represented. As a consequence, the model will fail to predict correctly even the initial post-buckling behavior whenever the criticality at hand is more complex than the fold (limit point, asymmetric bifurcation). This proves to be the case for the so-called ‘technical’ beam models. This inability is inherited by any constructed finite element model, regardless of the interpolation function used in its definition and of the number of elements used in the discretization. An improved non-linear model based on the treatment by Antman (Bifurcation problems in non-linearly elastic structures, in: P.H. Rabinowitz, ed., Application of Bifurcation Theory, Academic Press, NY, 1977), is adopted, and several finite elements are developed on the basis of this model. These elements are tested for a number of problems for which the critical behavior is governed by fold, cusp and butterfly singularities. The numerical results outline the importance of the ‘small’ kinematic terms, especially in conjunction with the occurrence of higher-order uni-modal singularities. Some risks for numerical locking are pointed out and remedied. The recommended element leads to a computational cost, which is comparable to an implemented ‘shallow arch’ element.

A deterministic global optimization approach for molecular structure determination

A deterministic global optimization algorithm is introduced for locating global minimum potential energy molecular conformations. The proposed branch and bound type algorithm attains finite ε-convergence to the global minimum through the successive refinement of converging lower and upper bounds on the solution. These bounds are obtained through a novel convex lowering bounding of the total potential function and the subsequent solution of a series of nonlinear convex optimization problems. The minimization of the total potential energy function is performed on an independent set of internal coordinates involving only dihedral angles. A number of example problems illustrate the proposed approach.

Interaction potential for SiO2: A molecular-dynamics study of structural correlations.

An interaction potential consisting of two-body and three-body covalent interactions is proposed for ${\mathrm{SiO}}_{2}$. The interaction potential is used in molecular-dynamics studies of structural and dynamical correlations of crystalline, molten, and vitreous states under various conditions of densities and temperatures. The two-body contribution to the interaction potential consists of steric repulsion due to atomic sizes, Coulomb interactions resulting from charge transfer, and charge-dipole interaction to include the effects of large electronic polarizability of anions. The three-body covalent contributions include O-Si-O and Si-O-Si interactions which are angle dependent and functions of Si-O distance. In lattice-structure calculations with the total potential function, \ensuremath{\alpha}-cristobalite and \ensuremath{\alpha}-quartz are found to have the lowest and almost degenerate energies, in agreement with experiments. The energies for \ensuremath{\beta}-cristobalite, \ensuremath{\beta}-quartz, and keatite are found to be higher than those for \ensuremath{\alpha}-cristobalite and \ensuremath{\alpha}-quartz. Molecular-dynamics calculations with this potential function correctly describe the short- and intermediate-range order in molten and vitreous states.In the latter, partial pair-distribution functions give Si-O, O-O, and Si-Si bond lengths of 1.62, 2.65, and 3.05 \AA{}, respectively. The vitreous state consists of nearly ideal Si(${\mathrm{O}}_{1/2}$${)}_{4}$ tetrahedra in corner-sharing configurations. The Si-O-Si bond-angle distribution has a peak at 142\ifmmode^\circ\else\textdegree\fi{} and a full width at half maximum (FWHM) of 25\ifmmode^\circ\else\textdegree\fi{} in good agreement with nuclear magnetic resonance experiments. The calculated static structure factor is also in agreement with neutron-diffraction experiments. Partial static structure factors reveal that intermediate-range Si-Si, O-O, and Si-O correlations between 4 and 8 \AA{} give rise to the first sharp diffraction peak (FSDP). The FSDP is absent in charge-charge structure factor, which indicates that charge neutrality prevails over length scales between 4 and 8 \AA{}. Dynamical correlations in vitreous and molten states, phonon densities of states of crystalline and vitreous ${\mathrm{SiO}}_{2}$, infrared spectra of crystalline, vitreous and molten states, isotope effect, distribution of rings and their structure in molten and vitreous states, and structural transformations at high pressures will be discussed in subsequent papers.

Spectral line broadening and shift by collisions in an air-acetylene flame

Collisional broadenings and shifts of profiles have been investigated for the following emission lines: 5889.95 A Na, 4047.20 A K, 4226.73 A Ca, 3273.96 A Cu, and 4030.76 A Mn in an air-acetylene flame. The Lorentz widths δλ L at half-maximum intensity, the shifts δλ s , and the ratio η ≡δλ L /δλ s were determined experimentally. The values of C 6 in the Lennard-Jones (LJ) potential were calculated from a Coulomb approximation. For the LJ-potential, δλ c ,δλ s and the repulsive constants C 12 were calculated from η exp, C 6 and compared with experiments. A method is proposed for η<2.78 to obtain suitable values of the broadening parameters. We recommend calculation of the total potential function because the LJ-potential may not be suitable for all systems, particularly when η<2.78. The natural (δλ N ) and quenching (δλ q ) widths were calculated and used as correction factors.

AB-initio studies of N-O-F compounds. Internal rotation in hydroxylamine and its fluorinated derivatives

Ab-initio molecular orbital theory at both the minimal and extended basis set levels have been applied to the study of internal rotation in hydroxylamine and its fluorinated derivatives. The computed energies are analyzed in terms of a Fourier-type expansion of the potential function. The total potential function V(φ)can be dissected into onefold (V 1), twofold (V 2) and threefold (V 3) components and plots of these components together with V(φ) are given for each of the molecules studied herein. Additionally geometry optimizations have been carried out as a function of the internal rotation angle φ (φ = : NOX dihedral angle) for H 2NOH and F 2NOF. For H 2NOH geometry optimizations are found to be less important than for F 2NOF. In general the fluorinated hydroxylamines prefer a trans -conformation (φ = 180°) while hydroxylamine itself adopts the cis -conformation (φ = 0°) largely as a result of a lower dipole interaction (V 1 term) in the cis -conformation.

Crystal Dynamics of Potassium. I. Pseudopotential Analysis of Phonon Dispersion Curves at 9°K

The frequencies of normal modes of vibration of potassium at 9\ifmmode^\circ\else\textdegree\fi{}K have been measured by inelastic-neutron-scattering techniques. Certain selected frequencies are (units ${10}^{12}$ cps): ${H}_{15}$ 2.21\ifmmode\pm\else\textpm\fi{}0.02, ${P}_{4}$ ${1.78}_{5}$\ifmmode\pm\else\textpm\fi{}0.02, ${{N}_{1}}^{\ensuremath{'}}$ 2.40\ifmmode\pm\else\textpm\fi{}0.04, ${{N}_{3}}^{\ensuremath{'}}$ 1.50\ifmmode\pm\else\textpm\fi{}${0.02}_{5}$, and ${{N}_{4}}^{\ensuremath{'}}$ 0.53\ifmmode\pm\else\textpm\fi{}0.02. The results are very similar, apart from a scale factor 1.65, to those for sodium. Analysis of the results has been carried out in terms both of conventional Born-von K\'arm\'an models and of potential functions defined in reciprocal space. A fifth-neighbor, axially symmetric force model has been used to compute the frequency distribution function for the normal modes and the associated heat capacity. The reciprocal-space analysis was performed in two ways: (a) in terms of a total potential function, whose Fourier transform is the effective interatomic potential between "neutral pseudo-atoms" of potassium, and (b) in terms of the screened pseudopotential for the conduction-electron-ion interaction. Analysis (a) shows that a wide variety of interatomic potentials, both with and without long-range oscillatory character, can be found which give a satisfactory fit to the results. These potentials are compared with those obtained from an analysis of x-ray scattering data for liquid potassium. The pseudopotentials obtained from analysis (b) are in good agreement with that derived by Bortolani from the Heine-Abarenkov model. A reanalysis of the phonon dispersion curves for sodium leads to very similar conclusions, confirming earlier work by Cochran.