Circumferential Radius Research Articles

A More Precise Measurement of the Radius of PSR J0740+6620 Using Updated NICER Data

PSR J0740+6620 is the neutron star with the highest precisely determined mass, inferred from radio observations to be 2.08 ± 0.07 M ⊙. Measurements of its radius therefore hold promise to constrain the properties of the cold, catalyzed, high-density matter in neutron star cores. Previously, Miller et al. and Riley et al. reported measurements of the radius of PSR J0740+6620 based on Neutron Star Interior Composition Explorer (NICER) observations accumulated through 2020 April 17, and an exploratory analysis utilizing NICER background estimates and a data set accumulated through 2021 December 28 was presented in Salmi et al. Here we report an updated radius measurement, derived by fitting models of X-ray emission from the neutron star surface to NICER data accumulated through 2022 April 21, totaling ∼1.1 Ms additional exposure compared to the data set analyzed in Miller et al. and Riley et al., and to data from XMM-Newton observations. We find that the equatorial circumferential radius of PSR J0740+6620 is 12.92−1.13+2.09 km (68% credibility), a fractional uncertainty ∼83% the width of that reported in Miller et al., in line with statistical expectations given the additional data. If we were to require the radius to be less than 16 km, as was done in Salmi et al., then our 68% credible region would become R=12.76−1.02+1.49 km, which is close to the headline result of Salmi et al. Our updated measurements, along with other laboratory and astrophysical constraints, imply a slightly softer equation of state than that inferred from our previous measurements.

A semi-analytical spectral element model for guided wave propagation in composite laminated conical shells

The conical shell is a typical non-uniform waveguide with its circumferential radius varies linearly. This work aims to provide physical insights for the composite laminated conical shell in terms of waves. A semi-analytical spectral element model is presented, which adopts the Fourier series in the circumferential direction and the higher-order shape functions in the axial direction. The first-order shear deformation theory and Hamilton principle are employed to derive the energy functions of the conical shell, and they are discretized using the spectral elements to achieve a weak-form variational problem. The dispersion equation and the group velocity expression are derived via the piecewise approximation concept and Bloch's periodic theorem. The accuracy of the developed spectral element is verified by comparison with published results. The deformation evolution mechanisms of the composite conical shell are revealed, and the wave propagation characteristics of uniform and ply drop-off composite conical shells are investigated. The results demonstrated that wavenumber distribution at 0–5 kHz is spatially dependent, and wavenumber becomes cut off and bifurcate near the cone vertex. When the frequency exceeds 2 kHz, the deformation of conical shell changes from rigid to flexible, and the structure tends to be a uniform waveguide. Compared to the geometric curvature effect, the asymmetric lay-up leads to more significant mode coupling.

Quasi-universality of the magnetic deformation of neutron stars in general relativity and beyond

Neutron stars are known to host extremely powerful magnetic fields. Among its effects, one of the consequences of harbouring such fields is the deformation of the neutron star structure, leading, together with rotation, to the emission of continuous gravitational waves. On the one hand, the details of their internal magnetic fields are mostly unknown. Likewise, their internal structure, encoded by the equation of state, is highly uncertain. Here, we present a study of axisymmetric models of isolated magnetised neutron stars for various realistic equations of state considered viable by observations and nuclear physics constraints. We show that it is possible to find simple relations between the magnetic deformation of a neutron star, its Komar mass, and its circumferential radius in the case of purely poloidal and purely toroidal magnetic configurations that satisfy the criterion for equilibrium in the Bernoulli formalism. Such relations are quasi-universal, meaning that they are mostly independent from the equation of state of the neutron star. Thanks to their formulation in terms of potentially observable quantities, as we discuss, our results could help to constrain the magnetic properties of the neutron star interior and to better assess the detectability of continuous gravitational waves by isolated neutron stars, without knowing their equation of state. Our results are derived both in general relativity and in scalar-tensor theories (one of the most promising extensions of general relativity), in this case by also considering the scalar charge. We show that even in this case, general relations that account for deviations from general relativity still hold, which could potentially be used to set constraints on the gravitational theory.

The Radius of PSR J0740+6620 from NICER and XMM-Newton Data

Abstract PSR J0740+6620 has a gravitational mass of 2.08 ± 0.07 M ⊙, which is the highest reliably determined mass of any neutron star. As a result, a measurement of its radius will provide unique insight into the properties of neutron star core matter at high densities. Here we report a radius measurement based on fits of rotating hot spot patterns to Neutron Star Interior Composition Explorer (NICER) and X-ray Multi-Mirror (XMM-Newton) X-ray observations. We find that the equatorial circumferential radius of PSR J0740+6620 is 13.7 − 1.5 + 2.6 km (68%). We apply our measurement, combined with the previous NICER mass and radius measurement of PSR J0030+0451, the masses of two other ∼2 M ⊙ pulsars, and the tidal deformability constraints from two gravitational wave events, to three different frameworks for equation-of-state modeling, and find consistent results at ∼1.5–5 times nuclear saturation density. For a given framework, when all measurements are included, the radius of a 1.4 M ⊙ neutron star is known to ±4% (68% credibility) and the radius of a 2.08 M ⊙ neutron star is known to ±5%. The full radius range that spans the ±1σ credible intervals of all the radius estimates in the three frameworks is 12.45 ± 0.65 km for a 1.4 M ⊙ neutron star and 12.35 ± 0.75 km for a 2.08 M ⊙ neutron star.

Ellis wormholes in anti-de Sitter space

We construct traversable wormholes with anti-de Sitter asymptotics supported by a phantom field. These wormholes are massless and symmetric with respect to reflection of the radial coordinate eta rightarrow - eta . Their circumferential radius decreases monotonically from radial infinity to their single throat. Analogous to their asymptotically flat counterparts, these anti-de Sitter wormholes possess an unstable radial mode.

Computer modeling for frequency performance of viscoelastic magneto-electro-elastic annular micro/nanosystem via adaptive tuned deep learning neural network optimization

The presented paper is the first attempt to apply deep learning for predicting the frequency characteristics of a magneto-electro-elastic (MEE) annular nano/microdisk (MEEAD). The optimum amount of the factors participating in the mechanism of the fully connected neural network are achieved through the optimizer based on the momentum. The positive side of the mentioned approach employed in this investigation would be due to its high accuracy along with lower epochs required for training the multi-layered network. This scrutinization would be semi-computational research that estimates the vibrational behavior of a MEEAD employing a non-classical continuum model known as the modified couple stress (MCS) model. First-order shear deformation theory (FSDT) and shell model would be provided for presenting their displacement fields. Then, Kelvin-Voight theory has been applied to model the viscoelastic foundation. Its non-classical governing equations, as well as related boundary conditions (BCs) of small-scaled MEEAD, would be achieved by considering the symmetric spinning gradient along with higher-order stress tensors for the strain energy. The provided non-classical theory would be able to capture the small scale in the MEEAD employing just one length scale of a material factor, then, the mathematical modeling of MEEAD according to the classical theory would be able to be recovered from the provided model by eliminating the material length scale factor. Ultimately, the non-classical governing equations would be solved by applying the generalized differential quadrature (GDQ) approach for multifarious BCs. Moreover, parametric research has been conducted to analyze the influences of the viscoelastic foundation, length scale factor, geometry of MEE, radial and circumferential mode number, radius ratio, and BCs on the frequency behavior of the MEEAD by applying MCST. The outcomes reveal that there would be a crucial radius ratio in which the relationship between these elements and crucial inserted voltage alters from direct to indirect relation.

Magnetic deformation of neutron stars in scalar-tensor theories

Scalar-tensor theories are among the most promising alternatives to general relativity that have been developed to account for some long-standing issues in our understanding of gravity. Some of these theories predict the existence of a non-linear phenomenon that is spontaneous scalarisation, which can lead to the appearance of sizable modifications to general relativity in the presence of compact matter distributions, namely neutron stars. On the one hand, one of the effects of the scalar field is to modify the emission of gravitational waves that are due to both variations in the quadrupolar deformation of the star and the presence of additional modes of emission. On the other hand, neutron stars are known to harbour extremely powerful magnetic fields which can affect their structure and shape, leading, in turn, to the emission of gravitational waves – in this case due to a magnetic quadrupolar deformation. In this work, we investigate how the presence of spontaneous scalarisation can affect the magnetic deformation of neutron stars and their emission of quadrupolar gravitational waves, both of tensor and scalar nature. We show that it is possible to provide simple parametrisations of the magnetic deformation and gravitational wave power of neutron stars in terms of their baryonic mass, circumferential radius, and scalar charge, while also demonstrating that a universal scaling exists independently of the magnetic field geometry and of the parameters of the scalar-tensor theory. Finally, we comment on the observability of the deviations in the strain of gravitational waves from general relativity by current and future observatories.

Axial and lateral stiffness of spherical self-balancing fiber reinforced rubber pipes under internal pressure

Abstract Fiber reinforced rubber pipes are widely used to transport fluid at locations requiring flexible connections in pipeline systems. The spherical self-balancing fiber reinforced rubber pipes with low stiffness are drawing attention because of their vibration suppression performance under high internal pressure. In this paper, a theoretical model is proposed to calculate the axial stiffness and lateral stiffness of spherical self-balancing fiber reinforced rubber pipes. The inhomogeneous anisotropy of the reinforced layer and the nonlinear stress-strain relationship of the reinforced fiber are considered in the model. The accuracy of the model is verified by experimental results. Theoretical calculation finds that both the axial and lateral stiffness are influenced significantly by the key structural parameters of the pipe (the axial length, the circumferential radius at the end, the meridional radius, and the initial winding angle). The stiffness can be reduced remarkably with optimal meridional radius and initial winding angle, without any side effect on the self-balance of the pipe. The investigation methods and results presented in this paper will provide guidance for design of fiber reinforced rubber pipes in the future.

Burst pressure prediction of fiber-reinforced flexible pipes with arbitrary generatrix

Fiber-reinforced flexible pipes are widely used to transport the fluid at locations requiring flexible connection in pipeline systems. It is important to predict the burst pressure to guarantee the reliability of the flexible pipes. Based on the composite shell theory and the transfer-matrix method, the burst pressure of flexible pipes with arbitrary generatrix under internal pressure is investigated. Firstly, a novel method is proposed to simplify the theoretical derivation of the transfer matrix by solving symbolic linear equations. The method is accurate and much faster than the manual derivation of the transfer matrix. The anisotropy dependency on the circumferential radius of the pipe is considered in the theoretical approach, along with the nonlinear stretch of the unidirectional fabric in the reinforced layer. Secondly, the burst pressure is predicted with the Tsai-Hill failure criterion and verified by burst tests of six different prototypes of the flexible pipe. It is found that the burst pressure is increased significantly with an optimal winding angle of the unidirectional fabric. The optimal result is determined by the geometric parameters of the pipe. The investigation method and results presented in this paper will guide the design and optimization of novel fiber-reinforced flexible pipes.

Differentiation of hypertensive heart disease and hypertrophic cardiomyopathy with myocardial stiffness measurements: a shear wave imaging study using ultra-high frame rate echocardiography

Abstract Background Recently, cardiac shear wave (SW) elastography, based on high frame rate (HFR) echocardiography, has been proposed as new non-invasive technique for assessing myocardial stiffness. As myocardial stiffness increases with increasing wall stress, differences in measured operating myocardial stiffness do not necessarily reflect differences in intrinsic myocardial properties, but can also be caused by mere changes in loading or chamber geometry. This complicates myocardial stiffness interpretation for different types of pathologic hypertrophy. Purpose To explore the relationship between myocardial stiffness and underlying pathological substrates for cardiac hypertrophy. Methods We included 20 patients with hypertension (HT) and myocardial remodelling (59±14 years, 75% male), 20 patients with hypertrophic cardiomyopathy (HCM) (59±16 years, 60% male) and 20 healthy controls (56±14 years, 75% male). Left ventricular (LV) parasternal long axis views were acquired with an experimental HFR scanner at 1293±362 frames per seconds. Propagation velocity of SW occurring after mitral valve closure in the interventricular septum (IVS) served as measure of operating myocardial stiffness (Figure A). To compare myocardial stiffness among hearts with differing loading conditions and chamber geometry, SW velocities were normalized to end-diastolic wall stress, estimated at IVS from regional wall thickness, longitudinal and circumferential regional radii of curvature, and non-invasively estimated LV end-diastolic pressure (EDP). Results SW velocities differed significantly between groups (p<0.001). The controls had the lowest SW velocities (4.02±0.97 m/s), whereas values between HT and HCM group were comparable (6.46±0.99 m/s vs. 7.00±2.10 m/s; p=0.738). Considering end-diastolic wall stress, HCM patients had the same SW velocity at lower wall stress compared to HT (Figure B), indicating higher myocardial stiffness in the HCM group. SW velocities normalized for wall stress indicated significantly different myocardial stiffness among all groups (p<0.001) (Figure C). In a multiple linear regression model, the underlying pathological substrate independently influenced SW velocity (beta 1.37, 95% CI (0.78–1.96); p<0.001), while wall stress did not significantly affect its value (p=0.479). Conclusions Our study demonstrated that SW elastography can detect differences in myocardial stiffness in hypertensive heart and hypertrophic cardiomyopathy. Additionally, our results suggest that SW velocity is dominated by underlying myocardial tissue properties. We hypothesize that differential changes in cardiomyocytes and/or the extracellular matrix contribute to the differential myocardial stiffening in different pathologic entities of LV hypertrophy. Thus, SW elastography could provide useful novel diagnostic information in the evaluation of LV hypertrophy. Figure A, B, C Funding Acknowledgement Type of funding source: None

Self-gravitating perfect-fluid tori around black holes: Bifurcations, ergoregions, and geometrical properties

We investigate models of stationary, selfgravitating, perfect-fluid tori (disks) rotating around black holes, focusing on geometric properties of spacetime. The models are constructed within the general-relativistic hydrodynamics, assuming differential (Keplerian) rotation of the fluid. We discuss a parametric bifurcation occurring in the solution space, different possible configurations of ergoregions (including toroidal ergoregions associated with the tori), nonmonotonicity of the circumferential radius, as well as the impact of the torus gravity on the location of the innermost stable circular orbit.

417 Can myocardial stiffness measurements distinguish the underlying pathology in hearts with thick walls? A shear wave imaging study using ultra-high frame rate echocardiography

Abstract Background Different pathophysiologic pathways in the development of left ventricular (LV) hypertrophy may alter passive myocardial stiffness differently. Recently, cardiac shear wave (SW) elastography has been proposed as new non-invasive technique for assessing myocardial stiffness. Purpose To explore the relationship between myocardial stiffness and the underlying pathological substrates for cardiac hypertrophy. Methods We included 17 patients with cardiac amyloidosis (AML) (69 ± 10 years, 41% male), 17 patients with hypertrophic cardiomyopathy (HCM) (59 ± 16 years, 65% male) matched for interventricular septum (IVS) thickness and 17 hypertensive patients (HT) with prominent myocardial remodelling (56 ± 15 years, 71% male). LV parasternal long axis views were acquired with an experimental ultrasound scanner at 1255 ± 354 frames per seconds. Myocardial acceleration maps were created from the HFR-datasets and an anatomical M-mode line was drawn along the midline of the IVS (Figure A). The propagation velocity of natural SWs occurring at mitral valve closure (MVC) was measured on these M-modes in order to assess operating myocardial stiffness. To compare myocardial stiffness among hearts with differing loading conditions and chamber geometry, SW velocities were normalized to operating end-diastolic wall stress. The end-diastolic wall stress was estimated at the IVS from regional wall thickness, longitudinal and circumferential regional radii of curvature, and noninvasively estimated left ventricular end-diastolic pressure (EDP). Results IVS thickness was significant different among groups (AML: 1.63 ± 0.33 cm, HCM: 1.69 ± 0.21 cm, HT: 1.48 ± 0.14 cm; p = 0.037). HT patients had significant higher septal radius of curvature compared to other two groups (p < 0.05), while the AML patients had the highest estimated EDP (p < 0.05). All groups had comparable, elevated SW velocities at MVC (AML: 6.49 ± 1.00 m/s, HCM: 6.46 ± 1.45 m/s, HT: 6.22 ± 0.96 m/s; p = 0.752). Considering end-diastolic wall stress, HT patients had the same SW velocity at higher wall stress compared to AML and HCM (Figure B), indicating lower myocardial stiffness in the HT group. SW velocities normalized for wall stress indicated significantly different myocardial stiffness among groups (p = 0.003) (Figure C). The HT group had the lowest normalized myocardial stiffness, whereas values of the AML group overlapped with the HCM group (p = 1.00). Conclusions Our study demonstrated that shear wave elastography can detect differences in myocardial stiffness in hearts with thick walls. Considering the effect of wall stress, our results suggest that factors other than chamber geometry and loading condition mediate myocardial stiffness in hearts with thick walls. We hypothesize that differential changes in cardiomyocytes and/or the extracellular matrix contribute to the differential myocardial stiffening in different pathologic entities of LV hypertrophy. Abstract 417 Figure.

PSR J0030+0451 Mass and Radius from NICER Data and Implications for the Properties of Neutron Star Matter

Abstract Neutron stars are not only of astrophysical interest, but are also of great interest to nuclear physicists because their attributes can be used to determine the properties of the dense matter in their cores. One of the most informative approaches for determining the equation of state (EoS) of this dense matter is to measure both a star’s equatorial circumferential radius R e and its gravitational mass M. Here we report estimates of the mass and radius of the isolated 205.53 Hz millisecond pulsar PSR J0030+0451 obtained using a Bayesian inference approach to analyze its energy-dependent thermal X-ray waveform, which was observed using the Neutron Star Interior Composition Explorer (NICER). This approach is thought to be less subject to systematic errors than other approaches for estimating neutron star radii. We explored a variety of emission patterns on the stellar surface. Our best-fit model has three oval, uniform-temperature emitting spots and provides an excellent description of the pulse waveform observed using NICER. The radius and mass estimates given by this model are km and (68%). The independent analysis reported in the companion paper by Riley et al. explores different emitting spot models, but finds spot shapes and locations and estimates of R e and M that are consistent with those found in this work. We show that our measurements of R e and M for PSR J0030+0451 improve the astrophysical constraints on the EoS of cold, catalyzed matter above nuclear saturation density.

Two mass conjectures on axially symmetric black hole-disk systems

We analyze stationary self-gravitating disks around spinning black holes that satisfy the recently found general-relativistic Keplerian rotation law. There is a numerical evidence that the angular velocity, circumferential radius, and angular momenta yield a bound onto the asymptotic mass of the system. This bound is proven analytically in the special case of massless disks of dust in the Kerr spacetime.

Neutron star mass and radius measurements from atmospheric model fits to X-ray burst cooling tail spectra

Observations of thermonuclear X-ray bursts from accreting neutron stars (NSs) in low-mass X-ray binary systems can be used to constrain NS masses and radii. Most previous work of this type has set these constraints using Planck function fits as a proxy: both the models and the data are fit with diluted blackbody functions to yield normalizations and temperatures which are then compared against each other. Here, for the first time, we fit atmosphere models of X-ray bursting NSs directly to the observed spectra. We present a hierarchical Bayesian fitting framework that uses state-of-the-art X-ray bursting NS atmosphere models with realistic opacities and relativistic exact Compton scattering kernels as a model for the surface emission. We test our approach against synthetic data, and find that for data that are well-described by our model we can obtain robust radius, mass, distance, and composition measurements. We then apply our technique to Rossi X-ray Timing Explorer observations of five hard-state X-ray bursts from 4U 1702-429. Our joint fit to all five bursts shows that the theoretical atmosphere models describe the data well but there are still some unmodeled features in the spectrum corresponding to a relative error of 1-5% of the energy flux. After marginalizing over this intrinsic scatter, we find that at 68% credibility the circumferential radius of the NS in 4U 1702-429 is R = 12.4+-0.4 km, the gravitational mass is M=1.9+-0.3 Msun, the distance is 5.1 < D/kpc < 6.2, and the hydrogen mass fraction is X < 0.09.

A shear deformation theory for bending and buckling of undersea sandwich pipes

In this paper, a first order shear deformation theory is proposed for cylindrical sandwich pipes subjected to undersea water pressure. The new model examines the change of the circumferential radius due to the radial deflection of the cylindrical sandwich shell and its effect on the bending moments. Differential equations of equilibrium for the non-shallow cylindrical sandwich shell are derived and are expressed in terms of the components of displacement and the rotation angles of the middle planes of the face sheets with respect to the neutral surface. The proposed theory is applied to investigate the buckling of a thermal insulating sandwich pipe laid on the seabed with fully constraint ends. Numerical analyses are conducted by developing MATLAB program to obtain the buckling pressures as well as the corresponding half wave numbers in circumferential direction for cylindrical sandwich pipes for different slenderness ratio, radius-to-thickness ratio and core-to-facesheet thickness ratio. The efficiency and accuracy of the presented theory is confirmed by comparing the analytical solutions with finite element results from ABAQUS, both showing a good agreement with each other.

Neutron stars with hyperon cores: stellar radii and equation of state near nuclear density

The existence of 2 Msun pulsars puts very strong constraints on the equation of state (EOS) of neutron stars (NSs) with hyperon cores, which can be satisfied only by special models of hadronic matter. The radius-mass relation for these models is sufficiently specific that it could be subjected to an observational test with future X-ray observatories. We want to study the impact of the presence of hyperon cores on the radius-mass relation for NS. We aim to find out how, and for which particular stellar mass range, a specific relation R(M), where M is the gravitational mass, and R is the circumferential radius, is associated with the presence of a hyperon core. We consider a set of 14 theoretical EOS of dense matter, based on the relativistic mean-field (RMF) approximation, allowing for the presence of hyperons in NSs. We seek correlations between R(M) and the stiffness of the EOS below the hyperon threshold needed to pass the 2 Msun test. For NS masses 1.0<M/Msun<1.6, we get R>13km, because of a very stiff pre-hyperon segment of the EOS. At nuclear density, the pressure is significantly higher than a robust upper bound obtained recently using chiral effective field theory. If massive NSs do have a sizable hyperon core, then according to current models the radii for M=1.0-1.6 Msun are necessarily >13km. If, on the contrary, a NS with a radius R<12 km is observed in this mass domain, then sizable hyperon cores in NSs, as we model them now, are ruled out. Future X-ray missions with <5% precision for a simultaneous M and R measurement will have the potential to solve the problem with observations of NSs. Irrespective of this observational test, present EOS allowing for hyperons that fulfill condition M_max>2 Msun yield a pressure at nuclear density that is too high relative to up-to-date microscopic calculations of this quantity.

Two Dimensional Analysis of Functionally Graded Partial Annular Disk under Radial Thermal Shock Using Hybrid Fourier-Laplace Transform

In this paper, a dynamic two dimensional analysis of functionally graded partial annular disk under radial thermal shock is carried out by employing a hybrid Fourier-Laplace transform in conjunction with finite element approach. The material properties of functionally graded partial annular disk are assumed to vary continuously through the radius of the disk in power law form. The governing equations, including the equation of the motion and energy equation are obtained based on Lord-Shulman theory. These two equations are solved simultaneously to obtain the displacement components and temperature distributions. Using Fourier series expansion along circumstantial direction and then Laplace transfer technique to transfer the governing equations into the space domain, where the Galerkin finite element method is employed to obtain the solution in space domain. A simply support boundary condition through circumferential and outer radius edges is assumed. The inverse of Laplace transfer is performed numerically to achieve the final solution in the real time domain. Finally, the results are validated with the known data reported in the literature.

Evaluation of left anterior descending coronary artery stenosis severity from myocardial end-diastolic wall stress estimated by tissue-Doppler imaging

To identify patients with significant coronary artery disease by the noninvasive quantification of myocardial wall stress in diastole. We studied 60 male subjects in sinus rhythm with significant (n = 30) or moderate (n = 30) proximal left anterior descending coronary artery stenosis, and 30 healthy subjects (control group). The average end-diastolic wall stress was estimated at left ventricle anterior and interventricular septum wall segments from regional wall thickness, meridional and circumferential regional radii of curvature, and noninvasively estimated left ventricular end-diastolic pressure. There were significant differences in left ventricular end-diastolic pressure between patients and controls (p < 0.05). End-diastolic myocardial wall stress was significantly different between patients with significant and moderate coronary stenosis and healthy subjects in all anterior and septal wall segments (p < 0.05) except for the anterior wall at mid level. The receiver-operating characteristic curves showed that septum apex wall stress has the highest discriminatory power for predicting significant stenosis versus healthy coronary artery with 83% area under the curve. Estimated end-diastolic myocardial wall stress may help in evaluating regional myocardial dysfunction due to coronary artery disease.

An Approach for Analysis of Bulged Profile and Thickness Distribution of Tube in Free Hydro-Bulging Process

Tubular components are widely used in the areas of automotive and aerospace industries due to their excellent properties. A mathematical model considering the bulged region as a parabola curve is proposed to examine the plastic deformation behavior of a thin-walled tube during the free hydro-bulged process. The finite element simulations of the free hydro-bulging process are carried out to verify the approach indirectly. The results indicate that the model is accurate and acceptable to figure out the circumferential radius, wall thickness and axial radius of the bulged profile.