Geometric Assumptions Research Articles

Wave-optics limit of the stochastic gravitational wave background

The stochastic gravitational wave background (SGWB) is a rich resource of cosmological information, encoded both in its source statistics and anisotropies induced by propagation effects. We provide a theoretical description of it, without employing tools which rely on the geometric optics assumption. Our formalism is based on the so-called classical matter approximation and it is able to capture wave-optics effects, such as interference and diffraction. We show that the interaction between the gravitational waves and the cosmic structures along the line-of-sight produce observable scalar and vector polarization modes, on top of modulating the tensorial ones. We build the two point correlation function describing the statistics of the SGWB, and introduce the intensity and gravitational Stokes parameters for all its components. In the case of an unpolarized, Gaussian and statistically homogeneous SGWB, we show that the interaction with matter modulates its intensity and does not generate a net difference between left- and right-helicity tensor and vector modes, as expected. We demonstrate that, in order to produce QT- and UT-tensor polarization modes the background must possess a hexadecapole anisotropy, while a quadrupole anisotropy can source the vector QV- and UV-Stokes parameters.

Local data inverse problem for the polyharmonic operator with anisotropic perturbations

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

Evolving Role of Three-Dimensional Echocardiography for Right Ventricular Volume Analysis in Pediatric Heart Disease: Literature Review and Clinical Applications

Accurate knowledge of right ventricular (RV) volumes and ejection fraction is fundamental to providing optimal care for pediatric patients with congenital and acquired heart disease, as well as pulmonary hypertension. Traditionally, these volumes have been measured using cardiac magnetic resonance because of its accuracy, reproducibility, and freedom from geometric assumptions. More recently, an increasing number of studies have described the measurement of RV volumes using three-dimensional (3D) echocardiography. In addition, volumes by 3D echocardiography have also been used for outcome research studies in congenital heart surgery. Importantly, 3D echocardiographic acquisitions can be obtained over a small number of cardiac cycles, do not require general anesthesia, and are less costly than CMR. The ease and safety of the 3D echocardiographic acquisitions allow serial studies in the same patient. Moreover, the studies can be performed in various locations, including the intensive care unit, catheterization laboratory, and general clinic. Because of these advantages, 3D echocardiography is ideal for serial evaluation of the same patient. Despite these potential advantages, 3D echocardiography has not become a standard practice in children with congenital and acquired heart conditions. In this report, the authors review the literature on the feasibility, reproducibility, and accuracy of 3D echocardiography in pediatric patients. In addition, the authors investigate the advantages and limitations of 3D echocardiography in RV quantification and offer a pathway for its potential to become a standard practice in the assessment, planning, and follow-up of congenital and acquired heart disease.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

Ancient Solutions of Ricci Flow with Type I Curvature Growth

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a simply connected, noncompact, κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution with nonnegative sectional curvature necessarily splits a Euclidean factor. It follows that a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.

On semidefinite descriptions for convex hulls of quadratic programs

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of Γ, the cone of convex Lagrange multipliers, and its relatives Γ1 and Γ∘.

Effects of edge-crack orientation and depth on non-linear dynamics of laminated nanocomposite single-variable-edge plates

This paper presents an investigation into the free vibration and nonlinear dynamic analyses of new complex edge contour plate structures, known as single-variable-edge (SVE) plates. Cracks emerge at the center of the plate’s surface, and the effects of edge crack orientation and depth on the dynamical characteristics of the laminated nanocomposite SVE plates have been considered in this study. The functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) material was utilized for the edge-cracked plate that rests on a Winker-Pasternak elastic foundation. The Halpin-Tsai micromechanical model is used to predict the material characteristics, which are presumptively temperature-dependent. The rule of mixtures is employed to determine Poisson's ratio, density mass, and thermal expansion. The governing equations are established based on the theory of elasticity, and Von Karman-Donnell geometric nonlinearity assumption. To confirm the reliability and accuracy of this approach, the acquired findings are validated with those produced using Finite Element Analysis (FEA). Furthermore, the geometrical and material parameters, fracture properties, and environmental (temperature and elastic foundation) factors are also investigated in the study. The research findings hold significant implications for applications in aerospace and aeronautical, civil, and mechanical engineering.

The efficient data-driven solution to estimate the nonlinear bending of sandwich doubly curved panel subjected to transient loading

This paper investigates nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading which is extensively used in aerospace structures. Due to the lack of a dataset for the nonlinear transient bending of the sandwich doubly curved panel for the data-driven solution, mathematical modeling is used. The sandwich doubly curved panel is made of two graphene nanoplatelets (GPLs) reinforced face sheets, and a polymer core. In the mathematical modeling section, the effective Young's modulus is obtained via modified Halpin-Tsai methodology, and the role of the mixture is used to estimate other material properties of the composite face sheets. The presented structure is analyzed using Reddy's third-order shear deformation theory (RTSDT), which incorporates von Kármán nonlinear geometric assumptions and Hamilton's principle. The governing equations and associated boundary conditions are derived accurately by imposing shear stress-free conditions on the upper and lower surfaces of the panel. The meshfree radial point interpolation method (RPIM) with interpolation functions is utilized to determine the unknown degrees of freedom to numerically simulate the mid-plane of the sandwich doubly-curved panel. The Newton-Raphson approach is used to solve the nonlinear ordinary differential using the constant-average acceleration strategy from Newmark's time integration method. After obtaining the dataset, the data is collected, and using an efficient data-driven approach, the results are tested, validated, and trained. In future studies, the presented efficient data-driven approach can be used with low computational cost to simulate nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading for related aerospace industries.

A biomechanical model of a vehicle passenger in the sagittal plane

Musculoskeletal biomechanical models have wide applications in ergonomics, rehabilitation, and injury estimation. Their use can be extended to enable quantitatively explaining and estimating ride comfort for a vehicle's passenger. A biomechanical model of the upper body in the sagittal plane is constructed, which allows for curved motion to simulate the propagation of disturbance energy within a seated passenger aboard a moving vehicle. The dynamic predictions of the model are validated against experimental results within the literature. Frequency responses show that within the vehicular frequency range, the L4L5 and the L5S1 discs in the lower lumbar region are susceptible to the highest vibration transmission. It was also found that vibration transmission is maximized at around 4.5 Hz. The model provides analytical and geometric intuition into the motion of the various segments of the upper body using a few simple geometric assumptions and can be employed to develop a quantitative ride-comfort metric, such that the most comfortable ride would be that which would induce the least internal motion within the passenger model.

Echocardiographic Assessment of Left Ventricular Systolic Function in Lymphoma Patient After Initial Dose of Chemotherapy in Baghdad Teaching Hospital

Despite cancer therapeutics-related cardiac dysfunction (CTRCD) can be initially asymptomatic, if not detected and properly managed, it may progress to severe and irreversible heart failure. Therefore, identification of high-risk patients and early detection of subclinical myocardial dysfunction are fundamental tasks for the management of cancer patients undergoing chemotherapy, involving both cardiologists and oncologists. LVEF has low sensitivity for the detection of small changes in LV function. LVEF calculated by conventional 2DE often fails to detect small changes in LV contractility because of several factors. These factors include LV geometric assumptions, inadequate visualization of the true LV apex, lack of consideration of subtle regional wall motion abnormalities, and inherent variability of the measurement. Accordingly, strategies using newer echocardiographic technology, such as STE-derived strain imaging for the early detection of subclinical LV systolic dysfunction. GLS being the most robust and reproducible among the myocardial strain parameters, this parameter has raised the interest of the investigators to detect early and subclinical myocardial dysfunction in patients at risk of CTCRD

Nonlinear torsional vibration and dynamic post-buckling responses of spiral stiffened functionally graded porous cylindrical shells

This study employs a semi-analytical approach to investigate the nonlinear torsional vibration and dynamic torsional post-buckling (DTPB) responses of spiral stiffened functionally graded (FG) porous (SSFGP) cylindrical shells. These shells are resting on a generalized nonlinear viscoelastic foundation (GNVEF). This foundation consists of a dual-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model. The model includes nonlinear cubic stiffness and takes damping effects into consideration. Within the scope of this research, two variations of SSFGP cylindrical shells are examined: those characterized by non-uniform and uniform porosity distributions. Employing the Donnell shell theory, von-Kármán nonlinear geometric assumptions, and Galerkin’s method, a discretized nonlinear governing equation is derived to analyze the behaviors of the shells. Consequently, explicit formulations for dynamic torsional load are meticulously obtained. The findings of the present study are validated by comparing them with the outcomes documented in existing literature, as well as through alignment with the P-T method. This research delves into the system’s nonlinear behaviors, with scrutinization of the effects of diverse factors such as material and geometrical parameters. The researchers and engineers in this field may use the findings of this research as benchmarks for their design and research of SSFGP cylindrical shells.

STUDY OF CARDIAC FLUID DYNAMICS IN THE RIGHT SIDE OF THE HEART WITH AI PIV

Particle image velocimetry (PIV), when used to investigate cardiac flows, has been restricted to the exploration of left-heart hemodynamics. Easy phantom modeling, simple geometrical assumptions, and the use of modified ultrahigh-frequency 3-D echocardiographic probes dedicated to the left heart are the main reasons for this restriction. However, hemodynamics of the right heart, due to its complex geometry, is still poorly understood. AI PIV, based on deep learning and convolutional neural networks, offers a super-resolution view of the velocity fields. In this paper, we apply this new technique to agitated saline bubble echocardiographic recordings of the right heart. The obtained higher-resolution results show promising patterns and vortices throughout the cardiac cycle, circumventing the abovementioned obstacles. For instance, the annular tricuspid excursion, which has been used for decades as a marker of systolic function of the right ventricle, seems to be crucial for the formation of two diastolic vortices in the right atrium. A lateral counterclockwise vortex and a medial clockwise vortex that direct flow from the low-pressure right atrium to the middle of the tricuspid valve were noted repeatedly throughout different cardiac cycles within the same patients and between different patients.

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.

Growth of Subsolutions of Δpu=V|u|p-2u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _p u = V|u|^{p-2}u$$\\end{document} and of a General Class of Quasilinear Equations

In this paper we prove some integral estimates on the minimal growth of the positive part u+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_+$$\\end{document} of subsolutions of quasilinear equations divA(x,u,∇u)=V|u|p-2u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{div}A(x,u,\ abla u) = V|u|^{p-2}u \\end{aligned}$$\\end{document}on complete Riemannian manifolds M, in the non-trivial case u+≢0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_+\ ot \\equiv 0$$\\end{document}. Here A satisfies the structural assumption |A(x,u,∇u)|p/(p-1)≤k⟨A(x,u,∇u),∇u⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|A(x,u,\ abla u)|^{p/(p-1)} \\le k \\langle A(x,u,\ abla u),\ abla u\\rangle $$\\end{document} for some constant k>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k>0$$\\end{document} and for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} the same exponent appearing on the RHS of the equation, and V is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on M beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.

A new study for dynamical characteristics of double-variable-edge and variable thickness plates made of open-cell porous metal

This article explores the dynamic characteristics of a specific plate known as 'double-variable-edge' (DVE) plate with varying thickness, constructed from functionally graded porous material. The edge profiles of the plate and its thickness variation are defined by optional mathematical functions (polynomial and arc functions are chosen in this paper). The nonlinear dynamics of the double-variable-edge and variable thickness (DVEVT) plate, encompassing periodic and chaotic behaviors, have been investigated utilizing classical plate theory, incorporating the Von Karman-Donnell geometrical nonlinearity assumption, and employing Galerkin's method. Furthermore, this paper examines the linear characteristics by assessing fundamental frequencies. To establish the reliability of this approach, the linear frequencies and nonlinear curves are validated against prior literature and finite element analysis (FEA). Additionally, the study reveals the emergence of chaotic phenomena, as evidenced by the phase plane and Poincaré map, which become increasingly pronounced as the external load approaches the critical load. This critical load point signifies the juncture at which the structural behaviors transitions into a state of chaos. This research is anticipated to unlock significant potential within the aerospace, mechanical, and construction engineering, particularly for structures characterized by intricate profiles and varying thicknesses.

Shallow current of viscous fluid flowing between diverging or converging walls

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power law ( $x^n$ ), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$ , where $t$ is time and $\alpha$ is a constant. The invading fluid has a higher viscosity than the ambient fluid, thus avoiding Saffman–Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al. (J. Fluid Mech., vol. 747, 2014, pp. 218–246) studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow 1$ and $n\rightarrow 3$ , when fluid flows into and out of the channel, respectively. No divergence is found in the shallow case presented here up to the breakdown of the geometric assumption. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming an inverse dependence on the ratio disagreement between the current front using the method of lines. An inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase-plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta }$ , which does not show any divergent behaviour for large $n$ . An asymptotic behaviour is found for $\delta$ that approaches $1/2$ for $n\gg 1$ .

Spline retracker: a geometrical retracking algorithm for coastal and open ocean altimetry

Satellite altimetry has enhanced the understanding of ocean dynamics through high-rate sampling and global coverage. However, land contamination and bad reflection effects limit its accuracy. We present a geometrical method for retracking altimetry waveforms in coastal areas. Our method follows a geometrical assumption related to the symmetrical reciprocal motion of the radar pulse. Based on this assumption, the altimetry waveform is modelled as a continuous and differentiable third-order spline function, and the symmetry point of this function is considered as the retracking gate. The spline retracking algorithm is validated against the tide gauges at Onsala, Halmstad, and Muscat stations in Sweden and Oman, and its performance is compared with existing retracking algorithms. Our results showed a remarkable reduction of 50-91% in the unbiased-Root-Mean-Squared-Error (ubRMSE) and an increase of at least 13% in correlation coefficients when compared with other algorithms in Swedish coast. This algorithm presented equivalent results with the threshold and improved threshold retrackings in Muscat station, based on Jason-2 measurements. However, along the Jason-3 pass, our spline method showed a considerable reduction of 80% in ubRMSE and the minimum increase of 42% in correlation coefficients than the empirical algorithms. This method also outperformed the ALES algorithm in most cases.

Robust optimization for geometrical design of 2D sequential interlocking assemblies

In the realm of sustainable construction within a circular economy, the study of demountable buildings garners significant interest. Achieving the full potential of reusing materials from disassembled structures demands innovative assembly methods surpassing conventional fasteners like nails. Although traditional joinery has addressed this challenge to some extent, it faces design limitations. Modern digital fabrication technologies, such as CNC milling and additive manufacturing, have expanded the horizons for manufacturable assemblies. This paper builds upon prior research to introduce a flexible method for crafting 2D assemblies adaptable to various geometric assumptions. It offers two contributions. Firstly, it provides a versatile numerical model for analyzing the mechanical properties of diverse designs, uncovering novel assemblies with superior mechanical performance compared to traditional configurations. Secondly, it presents an optimization approach enabling precise control over assembly and disassembly of imperfect parts by optimizing joint geometry. By integrating advanced fabrication techniques, adaptable design methods, and mechanical analysis, this research paves the way for the development of sustainable and mechanically efficient demountable buildings.

On the selected problems of time-dependent dynamics of composite truncated conical shells-like aerospace structures

In this paper, the nonlinear dynamic responses of ring-stiffened truncated conical sandwich shells-like aerospace structures fabricated by perfectly bonded graphene platelets (GPLs) reinforced composite porous core and face sheets (GPLRC-PC-FS) under asymmetric internal ring-shaped moving load are investigated. The sandwich shells have edge point supports, which are simulated using the translational and rotational artificial springs. In addition to the open cell porosities, closed cells are also considered and analyzed. The spatially discretized nonlinear motion equations are derived based on the first-order shear deformation theory (FSDT) under the von Kármán geometric nonlinearity assumptions using the Ritz method with Chebyshev polynomials as its admissible basis functions. Then, Newmark's time integration scheme along with the Newton–Raphson method is employed to obtain the shell dynamic responses. After demonstrating the robustness and accuracy of the approach, the influence of rings number, their size and arrangement, core porosity distribution and structure, face sheets, GPLs parameters, moving load velocity and boundary conditions on the shell responses are studied. The results show that the shell dynamic behaviors are significantly affected by porosity, but almost are independent of the type of porosity distribution pattern and its structure (open and closed cell porosities). It is shown that the shell deflection can be minimized by suitably choosing the number of rings, their dimensions and arrangement. Moreover, the nonlinear behaviors of the shells have a significant correlation with the number of point supports and by adjusting the support stiffnesses, the nonlinear dynamic responses of the shells can be reduced.

Some Conditions of Non-Blow-Up of Generalized Inviscid Surface Quasigeostrophic Equation

In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) θ t + u · ∇ θ = 0 , u = ∇ ⊥ ψ , − Λ β ψ = θ , θ x , 0 = θ 0 x . . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field u related to the scalar θ by u = − ∇ ⊥ Λ − β θ , where 1 ≤ β ≤ 2 . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.