Direct Integral Approach Research Articles

Hydrostatic Equation of State of bcc Bi by Directly Solving the Partition Function

Body-centered cubic bismuth (Bi) is considered to be an enticing pressure marker, and, therefore, it is highly desirable to command its accurate equation of state (EOS). However, significant discrepancies are noted among the previous experimental EOSs. In the present work, an EOS of up to 300 GPa is theoretically obtained by solving the partition function via a direct integral approach (DIA). The calculated results nearly reproduce the hydrostatic experimental measurements below 75 GPa, and the deviations from the measurements gradually become larger with increasing pressure. Based on the ensemble theory of equilibrium state, the DIA works with high precision particularly in high-pressure conditions, so the hydrostatic EOS presented in this work is expected to be a reliable pressure standard.

Direct integration bias-compensated maximum correntropy criterion algorithm independent of measurement noise samples

In system identification, if the filter input and the desired signal are both interfered by noise, traditional adaptive filtering algorithms will not work effectively. The recently presented bias-compensated normalized maximum correntropy criterion (BC-NMCC) algorithm, which was developed based on the information theory learning (ITL) and Taylor expansion, can deal with this case. However, it needs to use measurement noise samples to update the adaptive filter weight vector, whose values are usually unavailable. To solve this problem, we propose a direct integration bias-compensated maximum correntropy criterion (DI-BC-MCC) algorithm under an unbiasedness criterion (UC). Specifically, we utilize a direct integral approach to calculate the expected values instead of using Taylor expansion. With this idea, the update equation of DI-BC-MCC does not involve measurement noise samples, unlike BC-NMCC. The derived compensation term can reduce the impact of input noise significantly, and DI-BC-MCC can perform well in impulsive noise environments. In addition, the steady-state performance of DI-BC-MCC is analyzed under some frequently used assumptions. Finally, simulations are provided to demonstrate the good performance of DI-BC-MCC and to verify the accuracy of the theoretical analysis.

Equation of state for tungsten obtained by direct solving the partition function

Utilization of metal tungsten (W) as the structural material or pressure scale requires accurate knowledge of the equation of state (EOS), which is far beyond the available experimental measurements. In the present work, a direct integral approach (DIA) with ultrahigh efficiency was applied to calculate the EOS of W up to 500 GPa and 3000 K with ab initio calculations. Compared with previous static compression experiments up to 150 GPa under room temperature and 35 GPa at high temperatures up to 1673 K, all the deviations of the calculated pressure are within or comparable to the uncertainty of experiments. Predictions for higher-temperature and simultaneously higher-pressure EOS up to 300 GPa and 3000 K differ slightly from the comprehensive analysis by Litasov et al. [J. Appl. Phys. 113, 133505 (2013)] via fitting available experimental data with the empirical equation. These results indicate that the EOS of crystal W obtained from DIA should be convincible, and DIA without any empirical or artificial parameters may find its wide applications for predicting thermodynamic properties of condensed matter in the future.

An ab initio study of structural phase transitions of crystalline aluminium under ultrahigh pressures based on ensemble theory

We recently put forward a direct integral approach to solve the partition function with ultrahigh efficiency and precision so that the rigorous ensemble theory can be applied to study the phase behaviors of realistic condensed-matter materials. In this work, the approach is used to investigate the structural phase transitions of crystalline aluminium up to 600GPa at room temperature and the obtained equation of state is in an excellent agreement with recent experiments within 4% deviations. For the disputed transition pressures of the FCC→HCP→BCC, our calculated results are 194 and 361GPa, which solely coincide with those measured by one of the experiments (Dewaele et al., 2018) within 2% and 0.3% differences respectively but deviate 13% and 5% from the others. In addition, our computations show that the axial ratio of the stable HCP structure is 1.62 and there exists discontinuities in the equation of state at the phase boundaries accompanied with subtle volume changes of −0.67% and −0.90%, all of which exactly support the observations from the experiment by Dewaele et al.

Pressure-induced structural phase transitions of zirconium: an ab initio study based on statistical ensemble theory

Recently, we put forward a direct integral approach to solve the partition function with ultrahigh efficiency and precision, which enables the rigorous ensemble theory to investigate phase behaviors of realistic condensed matters and has been successfully applied to the phase transition of vanadium metal (Ning et al 2022 J. Phys.: Condens. Matter 34 425404). In this work, the approach is applied to the structural phase transitions of zirconium metal under compressions up to 160 GPa and ultrahigh calculation precision is achieved. For the obtained equation of state with pressure over 40 GPa, the deviations from latest experiments are within and the computed transition pressure of is 6.93 GPa, which is about five times larger than previous theoretical predictions and in excellent agreement with the measured range of 5–15 GPa. Our results support the argument that there is no existence of the isostructural phase transition of Zr metal that was asserted by recent experimental observations.

Multiple internal resonances in nonlinear vibrations of rotating thin-walled cylindrical shells

Multiple internal resonances in nonlinear vibrations of rotating thin-walled cylindrical shells are investigated in the paper. Considering the Coriolis forces as well as the initial hoop tension, the dynamic model of rotating thin-walled cylindrical shells is derived by using Lagrange equations. Based on this model, linear vibration characteristics are given and two kinds of multiple internal resonances are then proposed and illustrated. One is the internal resonances caused by opportune geometric parameters, and the other is the internal resonances caused by the rotation of shells. Finally, further numerical investigations are carried out using harmonic balance method combined with arc length continuation technique, as well as direct integral approach. Nonlinear vibration characteristics of rotating thin-walled cylindrical shells with 1:1:1:1 internal resonances are shown in detail.

Efficient approaches to solutions of partition function for condensed matters

The key problem of statistical physics standing over one hundred years is how to exactly calculate the partition function (or free energy), which severely hinders the theory to be applied to predict the thermodynamic properties of condensed matters. Very recently, we developed a direct integral approach (DIA) to the solutions and achieved ultrahigh computational efficiency and precision. In the present work, the background and the limitations of DIA were examined in details, and another method with the same efficiency was established to overcome the shortage of DIA for condensed system with lower density. The two methods were demonstrated with empirical potentials for solid and liquid cooper, solid argon and C60 molecules by comparing the derived internal energy or pressure with the results of vast molecular dynamics simulations, showing that the precision is about ten times higher than previous methods in a temperature range up to melting point. The ultrahigh efficiency enables the two methods to be performed with ab initio calculations and the experimental equation of state of solid copper up to ∼600 GPa was well reproduced, for the first time, from the partition function via density functional theory implemented.

Free Vibration Analysis of Axially Functionally Graded (AFG) Cantilever Columns of a Regular Polygon Cross-Section with Constant Volume

This paper deals with the transverse free vibration of axially functionally graded (AFG) cantilever columns under the influence of axial compressive load. The columns possessing a regular polygon in their cross-section are tapered and their material properties vary along the axis of the column. An emphasis is placed on the columns with constant volume for admissible geometries and materials. The governing differential equation of the problem is derived and solved using the direct integral approach in conjunction with the determinant search technique. The obtained results are in good agreement with those in the available literature and computed by finite element analysis. Numerical examples for the natural frequency and mode shape of the columns are presented to investigate the effects of parameters related to geometrical nonuniformity and material inhomogeneity.

Comparison of Two Efficient Methods for Calculating Partition Functions

In the long-time pursuit of the solution to calculating the partition function (or free energy) of condensed matter, Monte-Carlo-based nested sampling should be the state-of-the-art method, and very recently, we established a direct integral approach that works at least four orders faster. In present work, the above two methods were applied to solid argon at temperatures up to 300 K. The derived internal energy and pressure were compared with the molecular dynamics simulation as well as experimental measurements, showing that the calculation precision of our approach is about 10 times higher than that of the nested sampling method.

The Dynamic Analysis of a Slab on a Poroviscoelastic Halfspace under Vertical Load via BEM

Mechanics of advanced materials, such as poro-, visco-or poroviscoelastic materials, is relevant to such disciplines as geophysics, geo-and biomechanics, seismology, constricting. Because of the complexity of the inertial viscosity and mechanical phases coupling in porous media most transient response problems can only be solved via numerical methods. The present work is dedicated to numerical modelling of a problem of a Heaviside-type impact load acting on a brittle slab situated above a fluid saturated foundation. Slab is treated as elastic or poroelastic rock. Fluid saturated foundation is a soil and modeled as a poroviscoelastic media. Poroviscoelastic formulation is based on Biot’s theory of poroelasticity in combination with elastic-viscoelastic correspondence principle. Classical models of viscoelasticity are employed, such as Kelvin-Voight model, standard linear solid model and model with weakly singular kernel. The problem is treated in Laplace domain. Direct boundary integral method approach is used to obtain solution. Modified Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. A problem of Heaviside-type vertical load acting on a slab bonded on a poroviscoelastic halfspace is considered. The comparison of dynamic responses when poroviscoelastic halfspace is described by different viscoelactic models is presented.

On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain

We consider the Cauchy problem for the Laplace equation, i.e. the reconstruction of a harmonic function from knowledge of the value of the function and its normal derivative given on a part of the boundary of the solution domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. Since the Cauchy problem is ill-posed, as a regularizing method we generalize the novel direct integral equation approach in [1], originally proposed for a circular outer boundary curve, to a more general simply connected domain. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves and in which both densities are unknown. To identify these densities, the representation is matched with the given Cauchy data to generate a system to solve for the densities. It is shown that the operator corresponding to this system is injective and has dense range, thus Tikhonov regularization is applied to solve it in a stable way. For the discretisation, the Nyström method is employed generating a linear system to solve, and via Tikhonov regularization a stable discrete approximation to these densities are obtained. Using these one can then find an approximation to the solution of the Cauchy problem. A numerical example is included and we compare with other regularizing methods as well (implemented via integral methods). These results show that the proposed direct method gives accurate reconstructions with little computational effort (the computational time is of order of seconds). Moreover, the obtained approximation can be used as an initial guess in more involved regularizing methods to further improve the accuracy.

Scattering of a plane harmonic SH wave by a rough multilayered inclusion of arbitrary shape

Steady-state scattering of a plane harmonic SH wave by a multilayered inclusion of arbitrary shape consisting of a general number of homogeneous isotropic layers, embedded within a half space, is investigated by using a direct boundary integral equation approach. The layer interfaces are assumed to be either smooth or periodically corrugated without any sharp corners being present. The peak corrugation amplitude is arbitrarily prescribed. The corresponding integral equations are derived in terms of the unknown boundary displacements and tractions. The integral equations are discretized using linear elements leading to a system of linear algebraic equations which are solved by using the standard Gauss elimination procedure.Testing of the numerical results is performed by comparing the surface response of a cavity with those available in the literature. The tests show excellent agreement between the two results for different frequencies and angles of incidence.The numerical results include surface response of multilayered inclusions and pipes. The inclusion layers are assumed to be either softer or stiffer from the half space. The pipe layers include both soft–stiff and stiff–soft combinations of the materials.The layered inclusion results demonstrate that for the soft materials the presence of layers may significantly change the surface response when compared to the corresponding homogeneous inclusion motion. For stiff layers this effect is greatly reduced. The same phenomenon has been observed for the inclusions with periodically corrugated interfaces as well.The surface response of a layered pipe is found to strongly depend upon the nature of the incident wave and the materials of the layers.

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R 3, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R 3 with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.

On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions, and properties of these equations are shown in an $L^2$-setting. An effective way of discretizing these boundary integral equations based on the Nyström method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

Scattering of plane harmonic SH waves by multiple inclusions

SUMMARY Scattering of a plane harmonic SH wave by an arbitrary number of inclusions of general shape and placements within an elastic half-space is investigated by using a direct boundary integral equation approach. The integral equations have been reduced to a system of linear algebraic equations by using the linear elements. The structure of the resulting algebraic system is investigated and compared to the single-inclusion model. For convenience the multiple-inclusion models are assumed to consist of the reference and the auxiliary inclusions. The reference inclusion has the same shape, placement and the material properties as those of the corresponding single-inclusion model. The auxiliary inclusions then surround the reference inclusion assuming different patterns. Extensive testing of the proposed solution is performed. These tests include the transparency and the single-inclusion tests. In the former the material properties of the inclusions are assumed to be the same as those of the half-space and the surface response is found to be equal to the free field one. In the single-inclusion test the material properties of the auxiliary inclusions are assumed to be the same as those of the half-space while those of the reference inclusion are different. Then the resulting surface response is found to reduce to that of the single-inclusion model. For different number of multiple inclusions the surface motion atop the reference inclusion is compared with that of the corresponding single-inclusion model. In particular, a specific measure is introduced to assess accurately the importance of the multiple scattering upon the surface response. The effect of the auxiliary inclusion locations with respect to the reference inclusion upon the surface motion is investigated for two-, three- and nine-inclusion models. The results clearly show the importance of the interactions between the inclusions for a wide range of parameters present in the problem (e.g. nature of the incident wave, geometry of the model and the number of inclusions). For the problems considered here it appears that by ignoring these interactions, that is, using the single-inclusion model, will result in underestimation of the peak surface motion amplification.

Influence of Cylindrical Harmonic SH Waves on Motion along a Random Interface between Two Elastic Half-Spaces

Abstract This study investigates how a system of incident harmonic waves affects the interface motion between two materials. To that end diffraction of a system of cylindrical harmonic SH waves by a random interface between two elastic half-spaces is investigated using a direct boundary integral equation approach. In particular, the incident system consists of a primary source of unit amplitude and an arbitrary number of secondary sources placed along an ellipse centered at the primary source. The interface peak amplitude, the number of the secondary sources, and the frequency are assumed to be arbitrary. Response along the random interface is computed for different frequencies and the source configurations. The interface motion is found to strongly depend upon the incident system. Two incident system configurations are identified that produce the interface motion that is considerably amplified or reduced when compared to the corresponding displacements due to a single unit primary source. The two systems share the same geometry and the source amplitudes but have different source amplitude phases.

A FAST POTENTIAL AND SELF-GRAVITY SOLVER FOR NONAXISYMMETRIC DISKS

Disk self-gravity could play an important role in the dynamic evolution of interaction between disks and embedded protoplanets. We have developed a fast and accurate solver to calculate the disk potential and disk self-gravity forces for disk systems on a uniform polar grid. Our method closely follows the method given by Chan et al., in which a fast Fourier transform in the azimuthal direction is performed and a direct integral approach in the frequency domain in the radial direction is implemented on a uniform polar grid. This method can be very effective for disks with vertical structures that depend only on the disk radius, achieving the same computational efficiency as for zero-thickness disks. We describe how to parallelize the solver efficiently on distributed parallel computers. We propose a mode-cutoff procedure to reduce the parallel communication cost and achieve nearly linear scalability for a large number of processors. For comparison, we have also developed a particle-based fast tree code to calculate the self-gravity of the disk system with a vertical structure. The numerical results show that our direct integral method is at least two orders of magnitude faster than our optimized tree-code approach.

Calculation of Rydberg potential energy curve from Murrell–Sorbie parameters

Potential functions that that possess larger number of parameters generally provide better descriptions of interatomic energy owing to the greater flexibility in curve fitting. However, most molecular mechanics and computational solid state softwares adopt simple potential functions in order to reduce the computation time. This paper discusses three methods for converting the extended-Rydberg parameters to the Rydberg parameter, i.e. the direct approach, derivative approach and integral approach. The derivative and integral methods possess comparative advantages for small and large bond stretching, respectively. Although the direct method is generally the least accurate, a criterion for its validity has been established via reference to the integral method. These conversions allow the better description of bond-stretching energy to be reflected without drastic change in the molecular mechanics force fields.

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderon projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new method.

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.