Intrinsic Equations Research Articles

On dual r-elastic line

In this paper, we derive intrinsic equations for dual r-elastic line deformed on non-null surface by external field in dual Lorentzian space.

Chebyshev collocation method for static intrinsic equations of geometrically exact beams

A Chebyshev collocation method is presented for the static analysis of the geometrically exact beams undergoing arbitrary large deflections using intrinsic formulation. In the intrinsic formulation of the governing equations of the spatial beams, neither displacement nor rotation variables appear. The proposed collocation technique is based on the Chebyshev points as the collocation points and the orthogonal Chebyshev polynomials as the trial functions. This method is successfully applied to the static solution of the strong form of the intrinsic governing equations of the nonlinear spatial beam and in order to show the applicability, accuracy and computational efficiency of the proposed method the numerical results are compared to the experimental, analytical and other numerical results for a number of test cases.

Branch switching at Hopf bifurcation analysis via asymptotic numerical method: Application to nonlinear free vibrations of rotating beams

This paper deals with the computation of backbone curves bifurcated from a Hopf bifurcation point in the framework of nonlinear free vibrations of a rotating flexible beams. The intrinsic and geometrical equations of motion for anisotropic beams subjected to large displacements are used and transformed with Galerkin and harmonic balance methods to one quadratic algebraic equation involving one parameter, the pulsation. The latter is treated with the asymptotic numerical method using Padé approximants. An algorithm, equivalent to the Lyapunov–Schmidt reduction is proposed, to compute the bifurcated branches accurately from a Hopf bifurcation point, with singularity of co-rank 2, related to a conservative and gyroscopic dynamical system steady state, toward a nonlinear periodic state. Numerical tests dealing with clamped, isotropic and composite, rotating beams show the reliability of the proposed method reinforced by accurate results.

Explicit Examples of Motions of Inextensible Curves in Spherical Space S3

In this paper we study the motion of curves in 3-dimensional spherical space S 3 . We derive the evolution equations of the orthonormal frame and evolution equations for curvatures. Moreover, we give some explicit examples of motions of inextensible curves in S 3 and we determine the curves from their intrinsic equations (curvature and torsion). Then we determine the surfaces that are generated by the motion of these curves. To visualize these surfaces in S 3 , we use the stereographic projection in R 4 .

Practical Camera Calibration System on Mobile Platforms

We propose a practical camera calibration system on mobile platforms to calibrate the camera’s intrinsic parameters, based on the geometrical property of the vanishing points, which are determined by two perpendicular groups of parallel lines. In this system, first, we detect and cluster the lines of the input images to compute the vanishing points. Then, we establish the intrinsic parameter constraint equations based on the perpendicular property of the vectors that connect the optical center and the vanishing points, and solve them using the least squares method. This system only requires the camera to observe a rectangular card shown in a few (at least four) different orientations. The experimental results of real images show that the proposed calibration system is robust, simple, and practicable.

Determination of permeability index using Stoneley slowness analysis, NMR models, and formation evaluations: a case study from a gas reservoir, south of Iran

In hydrocarbon reservoirs, permeability is one of the most critical parameters with a significant role in the production of hydrocarbon resources. Direct determination of permeability using Stoneley waves has always had some difficulties. In addition, some un-calibrated empirical models such as Nuclear Magnetic Resonance (NMR) models and petrophysical evaluation model (intrinsic permeability) do not provide reliable estimates of permeability in carbonate formations. Therefore, utilizing an appropriate numerical method for direct determination of permeability using Stoneley waves as well as an appropriate calibration method for the empirical models is necessary to have reliable results. This paper shows the application of a numerical method, called bisection method, in the direct determination of permeability from Stoneley wave slowness. In addition, a linear regression (least squares) method was used to calibrate the NMR models including Schlumberger Doll Research (SDR) and Timur–Coates models as well as the intrinsic permeability equation (permeability from petrophysical evaluations). The Express Pressure Tester (XPT) permeability was considered as an option for the reference permeability. Therefore, all permeability models were validated for the Stoneley permeability and calibrated for the empirical models with the XPT permeability. In order to have a quantitative assessment on the results and compare the results before and after the calibration, the Root Mean Squares Error (RMSE) was calculated for each of the used models. The results for the Stoneley permeability showed that, in many points there was not much difference between the Stoneley permeability calculated by the bisection method and the XPT permeability. Comparing the results showed that the calibration of the empirical models reduced their RMSE values. As a result of the calibration, the RMSE was decreased by about 39% for the SDR model, 18% for the Timur–Coates model, and 91% for the petrophysical evaluations model. Presented bisection method calculates permeability directly without of any inversion or external calibration.

POSITION VECTOR OF SPACELIKE SLANT HELICES IN MINKOWSKI 3-SPACE

In this paper, position vector of a spacelike slant he- lix with respect to standard frame are deduced in Minkowski space E 3. Some new characterizations of a spacelike slant helices are presented. Also, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike curve. In terms of solution, we determine the parametric repre- sentation of the spacelike slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric represen- tation of some special spacelike slant helices such as: Salkowski and anti-Salkowski curves.

Relaxed Hyperelastic Curves on a Non-Degenerate Surface

We study the variational problem belonging to a relaxed hyperelastic curve for non-null curve on a non-degenerate surface in Minkowski three-space $${E_{1}^{3}}$$ . Firstly, we derive the intrinsic equations for a relaxed hyperelastic curve and we give the necessary condition for being relaxed hyperelastic curve of any non-null geodesic on the surface in $${E_{1}^{3}}$$ . Then, we examine this formulation on non-null geodesics of pseudo-plane, pseudo-sphere $${S_{1}^{2}(r) }$$ , hyperbolic space $${H_{0}^{2}(r)}$$ and pseudo-cylinder $${C_{1}^{2}(r)}$$ .

Passive morphing of flying wing aircraft: Z-shaped configuration

High Altitude, Long Endurance (HALE) aircraft can achieve sustained, uninterrupted flight time if they use solar power. Wing morphing of solar powered HALE aircraft can significantly increase solar energy absorbency. An example of the kind of morphing considered in this paper requires the wings to fold so as to orient a solar panel to be hit more directly by the sun's rays at specific times of the day. An example of the kind of morphing considered in this paper requires the wings to fold so as to orient a solar panel that increases the absorption of solar energy by decreasing the angle of incidence of the solar radiation at specific times of the day. In this paper solar powered HALE flying wing aircraft are modeled with three beams with lockable hinge connections. Such aircraft are shown to be capable of morphing passively, following the sun by means of aerodynamic forces and engine thrusts. The analysis underlying NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Aircraft), a computer program that is based on geometrically exact, fully intrinsic beam equations and a finite-state induced flow model, was extended to include the ability to simulate morphing of the aircraft into a “Z” configuration. Because of the “long endurance” feature of HALE aircraft, such morphing needs to be done without relying on actuators and at as near zero energy cost as possible. The emphasis of this study is to substantially demonstrate the processes required to passively morph a flying wing into a Z-shaped configuration and back again.

A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy

In this paper we develop a new efficient and stable Lagrangian numerical method for computing the evolution of 3D curves driven in the normal plane by a driving force and curvature. This new method contains asymptotically uniform tangential redistribution of grid points designed originally for 3D curve evolution in this paper which makes our computations stable and is crucial for the presented application. Together with the design of new tangentially stabilized algorithm for 3D evolving curves, we develop a new method for the fully automatic finding of the optimal trajectory of the camera in the virtual colonoscopy. The proposed method consists of three steps: 3D segmentation of the colon from CT images, finding an initial trajectory guess inside the segmented 3D subvolumes, and driving the initial 3D curve to its optimal position. To that goal, a suitable intrinsic advection-diffusion partial differential equation with a driving force is designed and solved numerically in fast and robust way in order to find a smooth uniformly discretized 3D curve representing the ideal path of the camera in the virtual colonoscopy.

Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part I: Formulation and Verification of Single-Load-Path Configurations

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. In Part I, the theory and the formulation are presented, along with verification of single-load-path configurations. Part II is devoted to the verification of dual-load-path configurations.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part II: Dual-Load-Path Configurations

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. Part II is devoted to verification of certain dual-load-path configurations suggested by Bell Helicopter Textron.

Three-dimensional equilibria of nonlinear pre-curved beams using an intrinsic formulation and shooting

This article describes a shooting method for computing three-dimensional equilibria of pre-curved nonlinear beams with axial and shear flexibility using the intrinsic beam formulation. For distributed and concentrated follower loads acting on a cantilevered beam, the method amounts to a direct solution approach requiring only a single shot (zero iterations) to compute the equilibria. This is possible since the system equations are defined in a local coordinate system that rotates and translates with the beam, akin to the follower loads themselves. A general procedure employing nonconservative follower loads, which invokes the Picard–Lindelöf theorem on uniqueness and existence of solutions, is also introduced for finding all solutions for three-dimensional pre-curved beam problems with conservative loading. This is particularly useful in beam buckling problems where multiple stable and unstable solutions exist. Three-dimensional equilibrium solutions are generated for many loading cases and boundary conditions, including three-dimensional helical beams, and are compared to similar solutions where available in the literature. Excellent agreement is documented in all comparison cases. For buckling examples, the stability of the computed solutions is assessed using a dynamic finite element code based on the same intrinsic beam equations. Due to the ability to avoid iteration, the presented approach may find application in model-based control for practical three-dimensional problems such as the control of manipulators utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables.

Intrinsic equations for a generalized relaxed elastic line on an oriented surface in the galilean space

In this article, we derive the intrinsic equations for a generalized relaxed elastic line on an oriented surface in the Galilean 3-dimensional space G3. These equations will give direct and more geometric approach to questions concerning about generalized relaxed elastic lines on an oriented surface in G3.

A NEW MODEL FOR SHALLOW VISCOELASTIC FLUIDS

We propose a new reduced model for gravity-driven free-surface flows of shallow viscoelastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for viscoelastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. In addition to the classical layer depth and velocity in shallow models, our system describes also the evolution of two components of the stress. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.

Study of the Electrical Properties of Monolayer MoS<sub>2</sub> Semiconductor

We present the study of the electrical properties of monolayer MoS2 in terms of semiconductor theory. The free electron and hole concentrations formulas in two-dimensional (2D) semiconductors have been developed based on three-dimensional (3D) semiconductors theory, and derived the intrinsic carrier concentration equation of 2D system. Using these equations, we simulated the intrinsic carrier concentration in monolayer MoS2 with temperature. The intrinsic carrier density in monolayer MoS2 increases exponentially with temperature, but it lows a few orders of magnitude than that of 3D semiconductor. It means that monolayer MoS2 based devices can operated at very high temperatures. Accordingly, the conductivity and resistivity were simulated for 2D MoS2, the former increases exponentially while the latter decreases with temperature or carrier concentration.

Index-Aware Model Order Reduction for Linear Index-2 DAEs with Constant Coefficients

A model order reduction method for index-2 differential-algebraic equations (DAEs) is introduced, which is based on the intrinsic differential equations and on the remaining algebraic constraints. This extends the method introduced in a previous paper for index-1 DAEs. This procedure is implemented numerically and the results show numerical evidence of its robustness over the traditional methods.

Intrinsic Equations of Some Special Curves in Euclidean 3-Space

In this paper, we obtain the intrinsic equations of some special curves in Euclidean 3-space characterized by det = 0, where r i < r i+1,r 1 ∈ {1, 2, 3}, r 2 ∈ {2, 3, 4} and r 3 ∈ {3, 4, 5}.

True intrinsic mechanical behaviour of semi-crystalline and amorphous polymers: Influences of volume deformation and cavities shape

A model enabling the determination of the intrinsic mechanical constitutive equations of uniaxially stretched polymers is presented. This model takes into account the cavitation-induced volume strain which can occur during the deformation of such materials. In particular, the true intrinsic axial stress and strain depends on the overall volume strain and a form factor depicting the evolution of the voids shape. Based on our model, the true intrinsic behaviour of high-density polyethylene (HDPE), polypropylene/ethylene-propylene rubber (PP/EPR), and polyethylene terephtalate (PET) was assessed in tension. Compared to the overall true behaviour, the intrinsic true behaviour of the materials did not exhibit anomalies at large strain levels with changing experimental parameters (strain rate and temperature), and can be accurately predicted by means of phenomenological constitutive equations as the one proposed by G’sell and Jonas (1979).