Radial Integrals Research Articles

Free vibration analysis of two-dimensional functionally graded coated and undercoated substrate structures

In this paper, the free vibration behaviors of the functionally graded (FG) coated and undercoated substrate structures are studied by a meshfree boundary–domain integral equation method. Based on the two-dimensional elasticity theory, the boundary–domain integral equations for each single layer of these coating–substrate structures are derived initially by using elastostatic fundamental solutions. Employ the radial integration method to transform the domain integrals into boundary integrals and achieve a meshfree scheme. By applying the multi-region boundary element method, obtain the generalized eigenvalue system of the whole structure, which involves system matrices with boundary integrals only and the complete solutions for natural frequency and vibration modes are rigidly resolved. A comparative study of FG versus homogeneous coating is conducted. The influences of material composition, material gradient, coating thickness ratio, substrate structure aspect ratio and the boundary conditions on the natural frequencies of the FG coated and undercoated substrate structure are evaluated and discussed.

Semi-empirical studies of atomic transition probabilities, oscillator strengths and radiative lifetimes in Hf II

Over the past few years, laser induced fluorescence and Fourier Transform techniques have been applied to measure radiative lifetimes and branching fractions in Hf II in order to derive oscillator strengths and transition probabilities. In the present work, we propose to compare for the first time these experimental data to computed values obtained by two different semi-empirical approaches, respectively based on a parametrization of the oscillator strengths and on a pseudo-relativistic Hartree–Fock model including core-polarization effects. The overall agreement between all sets of data is found to be good. We furthermore give radial integrals of the main atomic transitions in this study: 〈5d6s6p|r1|5d26s〉=0.1504 (0.0064), 〈6s26p|r1|5d6s2〉=1.299 (0.012), 〈5d26p|r1|5d26s〉=−0.298 (0.013), 〈5d26p|r1|5d3〉=2.025 (0.027). Finally a new set of oscillator strengths and transition probabilities is reported for many transitions in Hf II.

Radial integration BEM for solving non-Fourier heat conduction problems

The radial integral BEM (RIBEM) with a step-by-step integration method is presented for solving non-Fourier heat conduction problems in this paper. First, the system of second-order ordinary differential equations is obtained by using the RIBEM to discretize the space domain. Then, the Newmark method and the central difference method are adopted to solve the system of ordinary differential equations with respect to time. Finally, several numerical examples with laser heat sources are performed to demonstrate the performance of the present method. The results show that the present approach can obtain accurate and stable numerical results.

New analytical expressions in radial integration BEM for stress computation with several kinds of variable coefficients

Abstract This paper presents a set of new analytical expressions for evaluating radial integrals appearing in the stress computation of several kinds of variable coefficient elastic problems using the radial integration boundary element method (RIBEM). The strong singularity involved in the stress integral equation is explicitly removed from the derivation of the analytical expressions. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. In addition, because it can solve many kinds of variable coefficient elastic problems, this approach has a very wide applicability. The fourth-order spline (Radial Basis Function) RBF is employed to approximate the unknowns appearing in domain integrals caused by the variation of the shear modulus. The radial integration method is utilized to convert domain integrals to the boundary, which results in a pure boundary discretization algorithm. Numerical examples are given to demonstrate the efficiency of the presented formulations.

A new analytical approach of functionally graded material structures for thermal stress BEM analysis

This paper presents a new approach using analytical expressions in the radial integration boundary element method (RIBEM) for 2D and 3D thermal stress analysis of structures consisting of functionally graded materials. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. Using this equation together with the three-step multi-domain BEM technique, thermal stress analysis can be carried out for complicated structures comprised of arbitrarily many numbers of media with varying material properties. The Kelvin fundamental solutions independent of elastic modulus are used in this paper. As a result, the resulting integral equations include domain integrals due to the non-homogeneity and temperature variations of the material. All these domain integrals are transformed into equivalent boundary integrals using the radial integration method (RIM), resulting in a pure boundary element analysis algorithm. Numerical examples are given to demonstrate the efficiency of the presented approach.

Unified treatment of the bound states of the Schiöberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schiöberg and Eckart potentials using an approximation of the centrifugal term. In order to determine the ℓ-states solutions, we use the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space–time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions. We show that the Eckart potential can be derived from the Schiöberg potential. The obtained results are compared to those produced by other methods and are found to be consistent.

On the products of bipolar harmonics

The products of two and three bipolar harmonics \({\mathcal {Y}}^{\ell _1 \ell _2}_{LM}(\mathbf{r}_{31}, \mathbf{r}_{32})\) are represented as the finite sums of powers of the three relative coordinates \(r_{32}, r_{31}\) and \(r_{21}\). The complete (angular + radial) integrals of the products of the two and three bipolar harmonics in the basis of exponential radial functions are expressed as finite sums of the auxiliary three-particle integrals \(\Gamma _{n,k,l}(\alpha , \beta , \gamma )\). The formulas derived in this study can be used to accelerate highly accurate computations of rotationally excited (bound) states in arbitrary three-body systems. In particular, we have constructed compact (400-term) variational wave functions for the triplet and singlet \(2P(L = 1)\)-states in light two-electron atoms and ions. Highly accurate calculations (20–21 stable decimal digits in the total energy) of the triplet and singlet \(2P(L = 1)\)-states in the two-electron Li\(^{+}\), Be\(^{2+}\), B\(^{3+}\) and C\(^{4+}\) ions are performed for the first time.

Synchronous phase demodulation algorithm for conic carrier Hartmann topographer

In this paper, a theoretical analysis of a new algorithm for measuring phase objects (PO) is presented. The algorithm analyses images captured with a modified Hartmann plate. The proposed method demodulates a signal with a conic-wavefront carrier by using synchronous interferometric techniques. Typically a Hartmann plate is a mask with an array of holes, the proposal is to replace those holes with a series of circular concentric rings created by a conic carrier, named Conic Hartmann plate mask (CHM). The proposed algorithm computes a dense correspondence between the pattern source and its interferogram on the PO. This dense correspondence is computed as the phase change of the refracted waveform with respect to the reference one. This approach is more robust than standard techniques that computes the center of each spot in order to estimate its position; which involves pattern recognition and segmentation tasks, something normal in a typical Hartmann test. We show in this work that the new method overcomes different problems such as irregular background illumination, spots overlapping with pupil border and spots matching due to rapid changes on the surface slopes. This work shows a general procedure for radial derivatives integration based on base functions. As far as the authors know, the here presented demonstration has not been reported before. In particular, we show that Zernike polynomials are suitable for PO. Numerical experiments with real and simulated data demonstrate the algorithm performance.

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method. To solve this problem, a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term. The singular domain integral is transformed into a boundary integral using the radial integration method. Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates. The regular domain integral, after expressing the displacements in terms of the radial basis functions, is also transformed to the boundary using the radial integration method. Finally, a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.

Failure Analysis of Anisotropic Plates by the Boundary Element Method

AbstractThis paper presents a dynamic formulation of the boundary element method for stress and failure criterion analyses of anisotropic thin plates. The elastostatic fundamental solutions are used in the formulations and inertia terms are treated as body forces. The radial integration method (RIM) is used to obtain a boundary element formulationithout any domain integral for general anisotropic plate problems. In the RIM, the augmented thin plate spline is used as the approximation function. A formulation for transient analysis is implemented. The time integration is carried out using the Houbolt method. Integral equations for the second derivatives of deflection are developed and all derivatives of fundamental solutions are computed analytically. Only the boundary is discretized in the formulation. Numerical results show good agreement with results available in literature as well as finite element results.

New analytical expressions in radial integration BEM for solving heat conduction problems with variable coefficients

In this paper, a new approach using analytical expressions in the radial integration boundary element method (RIBEM) is presented for solving three kinds of representative variable coefficient heat conduction problems. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. Also, because it can solve any kinds of variable coefficient heat conduction problems, this approach has a very wide applicability. The fourth-order spline RBF is employed to approximate the unknowns appearing in domain integrals arising from the varying heat conductivity. The radial integration method is utilized to convert domain integrals to the boundary, which results in a pure boundary discretization algorithm. Numerical examples are given to demonstrate the efficiency of the presented approach.

Application of the Radial Integration Method into Dynamic Formulation of Anisotropic Shallow Shells Using Boundary Element Method

The formulation developed in this work is based on the coupling of plane elasticity formulation and thin plate formulation for plates (Kirchhoff plates). Both formulations use elastostatic fundamental solutions. Curvature effects are considered as body forces, which generates domain integrals. Domain integrals are transformed into boundary integrals using the radial integration method. Thus, only the boundary is discretized. A radial basis function is used as approximation function in domain integrals. The developed formulation is applied to the dynamic analysis of anisotropic and composite laminate shallow shells under time dependent loads. A computational implementation was performed for the formulation developed and results were compared with results from literature.

A Precise Time-Domain Expanding Boundary-Element Method for Solving Three-Dimensional Transient Heat Conduction Problems with Variable Thermal Conductivity

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving three-dimensional transient heat conduction problems with variable thermal conductivity. First, by expanding physical quantities at discrete time intervals, the recursive formulation of the governing equation is derived. Then, the recursive equation is solved by the RIBEM, and a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time step. Finally, three numerical examples show that the present approach can obtain very stable and accurate results for different time-step size.

A precise integration boundary element method for solving transient heat conduction problems

Abstract In this paper, a combined approach of boundary element method and precise integration method is presented for solving transient heat conduction problems with heat sources. The boundary integral equation is derived by means of the Green’s function for the Laplace equation, and as a result, two domain integrals are involved in the derived integral equations. Firstly, the radial integration method is used to convert the domain integrals into equivalent boundary integrals, so the system of ordinary differential equations on the boundary integral equation can be obtained by the boundary element method. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, several numerical examples are given to demonstrate the performance of the present method. The results show that the present approach gives satisfactory performance even for very large time step size, and the results given by the present approach are independent of the time step size if some integrals involves boundary conditions and heat sources can be integrated analytically.

Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method

Free vibration of the functionally graded (FG) sandwich beams are studied by a meshfree boundary-domain integral equation method. Two sandwich beams, namely, FG core with homogeneous face sheets and homogeneous core with FG face sheets are considered in the study. Based on the two-dimensional elasticity theory, the boundary-domain integral equations are derived by applying the elastostatic fundamental solutions. Radial integration method is employed to transform the domain integrals related to material nonhomogeneity and inertia effect into boundary integrals, hence a meshfree scheme involving boundary integrals only is achieved. Each layer in the sandwich beam is modeled separately and the equilibrium and compatibility conditions are enforced at the interfaces to derive the equations for the whole sandwich beam. Extensive parametric study is presented to examine the influence of material composition, material gradient, layer thickness proportion, thickness to length ratio and boundary conditions on the free vibration of FG sandwich beams.

Spherical radial approximation for nested mixed effects models

We consider a likelihood approximation in generalized linear mixed-effects models (GLMM) with multilevel nested random effects. Likelihood evaluation in such models is difficult, hindered by the need for high dimensional integration, where the dimension is proportional to the number of units per level and the number of random effects per unit. Various integration approaches have been proposed, including the penalized quasi-likelihood method, Laplace approximation, quadrature approximation, simulation, and MCMC algorithms. We propose a new quadrature approximation method, which is based on the spherical radial integration approach of Monahan and Genz (J Am Stat Assoc 92:664---674 1997), and at the same time takes advantage of the hierarchical structure of the integration. Our new hierarchical spherical radial method has a time complexity that is linear in the number of units per level and the number of random effects per unit, in contrast to the exponential complexity of the adaptive Gaussian quadrature method of Pinheiro and Chao (J Comput Graph Stat 15:58---81 2006) for the same problem. Using a spline approximation, the generalized additive mixed models (GAMM) are GLMMs with two levels of nested random effects. We apply our method to estimation of GAMMs. We compare it with competing methods through simulations and apply our method to analyze virologic and immunologic responses in an AIDS clinical trial. An R package is written and available at http://users.wpi.edu/~jgagnon/software.html.

Analytical expressions for evaluation of radial integrals in stress computation of functionally graded material problems using RIBEM

This paper presents a set of new analytical expressions for evaluating radial integrals appearing in the stress computation of variable coefficient elastic problems using the radial integration boundary element method (RIBEM). The strong singularity involved in the stress integral equation is explicitly removed in the derivation of the analytical expressions. The fourth-order spline RBF is employed to approximate unknowns appearing in domain integrals from variation of the shear modulus. Using these analytical expressions, considerable computational efficiency can be improved by overcoming the time-consuming deficiency of using the radial integration method (RIM) to convert domain integrals to the boundary which results in a pure boundary discretization algorithm in solving variable coefficient problems. Numerical examples are given to demonstrate the efficiency of the presented formulations.

Voxel-based partial volume correction for accurate quantitative voxel values

Purpose: The accuracy of voxelized information in emission imaging is limited by spatial resolution (FWHM = 2.35σ) producing biases for objects smaller than 3 FWHM. If the signal distribution is non‐uniform within 3σ of the voxel of interest then equilibrium does not exist and partial volume effect (PVE) compromises voxel accuracy. We propose a mathematical model to improve the accuracy of quantitative images of arbitrary distribution by bounding true voxel signal and estimating PVE for each voxel. Methods : A monotonically increasing parametric dataset is created for each voxel of an emission image by radial integration from the voxel center to radius = 6σ. Each cumulative integration plot from r = 3σ to 6σ is fit to a function A*4π /3*r 3 + B*ΔV derived assuming a local uniform signal distribution (A) where ΔV is the voxel volume. The constant BΔV represents the converged within 3σ integral of PVE. B > 0 implies spill‐out, B 99% in simulated 1D & 2D datasets. For the tumor, the original maximum value was 10kBq/ml. We obtained A = 3.5kBq/ml and B = 14kBq/ml for a total of 17.7kBq/ml. This yields (A+B)/original = 1.8 indicating substantial spill‐out of ~80% and a large error for the original voxel value. For a voxel in the center of the bladder, the original value was 46kBq/ml with A = 44kBq/ml, B = 7kBq/ml. (A+B)/original = 1.11 indicating near‐equilibrium at center of bladder and low spill-out of ~11% as expected. Local signal images (A) resemble low‐pass filtered original image and (B) shows the magnitude and direction of PVE. Conclusion : A new mathematical model to estimate the accuracy of voxels in quantitative images of arbitrary distribution has been developed. Analysis of additional patients is underway. ------------------------------------- Cite this article as: Mikell J, Kappadath SC. Voxel-based partial volume correction for accurate quantitative voxel values. Int J Cancer Ther Oncol 2014; 2(2):020229. DOI: 10.14319/ijcto.0202.29

Improvement in acute contractility and hemodynamics with multipoint pacing via a left ventricular quadripolar pacing lead

A quadripolar left ventricular (LV) pacing can deliver multipoint pacing (MPP). It is unknown if this confers improved cardiac function compared to conventional cardiac resynchronization therapy (CRT). We aimed to characterize changes in acute cardiac contractility and hemodynamics with multisite left ventricular "multipoint" pacing (MPP) in a prospective multicenter study in patients implanted with a CRT-defibrillator incorporating a quadripolar LV lead. The device was programmed to deliver MPP acutely pacing with eight configurations of varying timing delays. Global peak LV radial strain and LV outflow velocity time integral (LVOT VTI) were measured for conventional CRT and each MPP configuration. Out of the eight tested MPP configurations, the one that yielded the best echocardiographic measurement for each patient was defined as "optimal MPP". Forty CRT recipients had complete radial strain datasets suitable for analysis. Compared to conventional CRT, the mean peak radial strain was significantly higher for the optimal MPP configuration (18.3 ± 7.4 vs. 9.3 ± 5.3%, p < 0.001), and at least one MPP configuration was significantly superior (>20%) in 63% of patients. LVOT VTI data were collected in a subset of 13 patients. In these patients, mean VTI was significantly higher for optimal MPP compared to conventional CRT (13.5 ± 2.7 vs. 10.9 ± 3.3 cm, p < 0.01). MPP delivered via a quadripolar LV lead resulted in a significant improvement in acute cardiac contractility and hemodynamics compared to conventional CRT in the majority of patients studied. Clinicaltrials.gov identifier NCT01044784.

Feynman integral treatement of the Hyperbolical potential with a centrifugal term approximation

We obtain an analytical expression for the energy eigenvalues of the Hyperbolical potential using an approximation of the centrifugal term. In order to obtain the l-states solutions, we use the the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space-time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the eigenfunctions are given in terms of hypergeometric functions. The present results are consistent with those obtained by others methods.