Three-Dimensional Equilibria and Stability of Nonlinear Curved Beams Using Intrinsic Equations and Shooting

2012 ◽  
Author(s):  
Kyle N. Karlson ◽  
Michael J. Leamy
Keyword(s):  
Boundary Condition ◽  
Numerical Solutions ◽  
General Procedure ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Curved Beams ◽  
Equilibrium Solutions ◽  
Equilibrium Equations ◽  
All Solutions ◽  
Similar Solutions

This article describes a shooting method that provides numerical solutions to static equilibrium equations for intrinsically curved beams in three-dimensions. Notably, the method avoids iteration for cantilever beams subjected to distributed or point follower loads. This is due to the governing equations being given in first-order form such that the specification of a single boundary condition on the forced end results in automatic satisfaction of the fixed boundary condition. Also documented is a general procedure for finding all solutions to static beam problems with conservative loading. This is particularly useful in beam buckling problems where multiple stable and unstable solutions exist. The procedure for finding all solutions is built around the Picard-Lindelöf theorem on the uniqueness and existence of solutions to initial value problems. Using the presented approach, three-dimensional equilibrium solutions are generated for many loading cases and boundary conditions, including a three-dimensional helical beam, and are compared to similar solutions available in the literature. The stability of the generated solutions is assessed using a dynamic finite element code based on the same intrinsic beam equations. Due to the absent need for iteration, the presented approach may find application in model-based control for practical problems such as the control of equipment utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables.

COMMENTS ON SUPERSPACE SUPERGRAVITY IN THREE DIMENSIONS

1989 ◽  
Vol 04 (14) ◽  
pp. 1307-1314 ◽  
Author(s):  
ROBERT N. OERTER
Keyword(s):  
General Procedure ◽  
Three Dimensional ◽  
Three Dimensions ◽  
World Volume ◽  
Linearized Theory

The spinning membrane is formulated with manifest world-volume supersymmetry by introducing an extra scalar multiplet. The three-dimensional superconformal transformations are constructed. Prepotentials are found for the three-dimensional supergravity, and the linearized theory is quantized via the Fadeev-Popov procedure. The superspace formulation of N=2 supergravity in three dimensions is derived. An appendix gives a general procedure for deriving density projectors which project superspace actions down to components in any number of dimensions.

SOME APPLICATIONS OF LINEAR PROGRAMMING TO THE INVERSE GRAVITY PROBLEM

Geophysics ◽  
1977 ◽  
Vol 42 (6) ◽  
pp. 1215-1229 ◽  
Author(s):  
Claude Safon ◽  
Guy Vasseur ◽  
Michel Cuer
Keyword(s):  
Linear Programming ◽  
Numerical Solutions ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Physical Parameters ◽  
Cylindrical Structures ◽  
All Solutions ◽  
Finite Set ◽  
Ideal Body ◽  
Programming Algorithms

An approach is presented for solving the inverse gravity problem in the presence of various constraints such as bounds on density. This approach takes into account the nonuniqueness of the solution: for a finite set of measurements, the region studied is divided into a great number of rectangular prisms of unknown density. The set of all solutions of this undetermined problem may be described through various convex diagrams of moments; plots of these moments give bounds on some physical parameters such as the partial and total mass or the position of the center of mass. Numerical solutions are obtained using linear programming algorithms. Also, particular solutions such as the so‐called ideal body may readily be obtained using this technique. Only two‐dimensional cylindrical structures are considered, but application of this technique to three‐dimensional bodies is straight‐forward.

On the nature of unsteady three-dimensional laminar boundary-layer separation

1978 ◽  
Vol 88 (2) ◽  
pp. 241-258 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
The Body ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Unsteady Separation ◽  
Similar Solutions

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

scholarly journals Determination of load distribution in a given medium according to the values of the loads at certain points

2021 ◽  
pp. 167-175
Author(s):  
Oleksandr Mostovenko ◽  
Serhii Kovalov ◽  
Svitlana Botvinovska
Keyword(s):  
Boundary Conditions ◽  
Load Distribution ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Geometric Model ◽  
Three Dimensional ◽  
The Body ◽  
Three Dimensions ◽  
Two Dimensional ◽  
One Dimensional

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2]. In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7]. From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes. Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions. If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid. If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes. The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells. As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

Three-Dimensional Diffusion in Arbitrary Domain Using Generalized Coordinates for the Boundary Condition of the First Kind: Application in Drying

2012 ◽  
Vol 326-328 ◽  
pp. 120-125 ◽  
Author(s):  
V.S.O. Farias ◽  
Wilton Pereira Silva ◽  
C.M.D.P. Silva e Silva ◽  
Antônio Gilson Barbosa de Lima
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Transient State ◽  
Arbitrary Domain ◽  
Time Step ◽  
Generalized Coordinates ◽  
Computational Code

This work presents a three-dimensional numerical solution for the diffusion equation in transient state, in an arbitrary domain. For this end, the diffusion equation was discretized using the finite volume method with a fully implicit formulation and generalized coordinates, for the equilibrium boundary condition. For each time step, the system of equations obtained for a given structured mesh was solved by the Gauss-Seidel method. The computational code was developed in FORTRAN, using the CFV 6.6.0 Studio, in a Windows platform. The proposed solution was validated using analytical and numerical solutions of the diffusion equation for different geometries (orthogonal and non-orthogonal meshes). The analysis and comparison of the results showed that the proposed solution provides correct results for the cases investigated. The developed computational code was applied in the simulation of the drying of ceramic roof tiles for the following temperature: 55.6 °C. The analysis of the results makes it possible to affirm that the developed numerical solution satisfactorily describes the drying processes in this temperature.

Unsteady separation and heat transfer upstream of obstacles

1995 ◽  
Vol 305 ◽  
pp. 1-27 ◽  
Author(s):  
R. I. Puhak ◽  
A. T. Degani ◽  
J. D. A. Walker
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Symmetry Plane ◽  
Plane Wall ◽  
Three Dimensions ◽  
Spiral Vortex ◽  
Layer Flow

The development of a laminar boundary layer upstream of both two- and threedimensional obstacles mounted on a plane wall is considered. The motion is impulsively started from rest, and it is shown that the boundary layer upstream of the obstacle initially develops independently from that on the obstacle itself. Numerical solutions for the unsteady boundary-layer flow on the plane wall are obtained in both Eulerian and Lagrangian coordinates. It is demonstrated that in both situations the flow focuses into a narrow-band eruption characteristic of separation phenomena at high Reynolds number. For the three-dimensional problem, results are obtained on a symmetry plane upstream of the obstacle which indicate the evolution, and subsequent sharp compression, of a spiral vortex in the near-wall flow in a manner consistent with recent experimental studies. The eruptive response of the two-dimensional boundary layer is found to be considerably stronger than the corresponding event in three dimensions. Calculated results for the temperature distribution are obtained for the situation where the wall temperature is constant but different from that of the mainstream. It is shown that a concentrated response develops in the surface heat transfer rate as the boundary layer starts to separate from the surface.

Numerical Solution of the Three-Dimensional Diffusion Equation in Solids with Arbitrary Geometry for the Convective Boundary Condition: Application in Drying

2013 ◽  
Vol 334-335 ◽  
pp. 149-154 ◽  
Author(s):  
V.S.O. Farias ◽  
Wilton Pereira Silva ◽  
C.M.D.P. Silva e Silva ◽  
L.D. Silva ◽  
F.J.A. Gama ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Convective Boundary Condition ◽  
Finite Cylinder ◽  
Transport Parameters ◽  
Time Step ◽  
Convective Boundary

In this work, a numerical solution for the diffusion equation applied to solids with arbitrary shape considering convective boundary condition is presented. To this end, the diffusion equation, written in generalized coordinates, was discretized by the finite-volume method with a fully implicit formulation. The transport parameters and the dimensions of the solids are considered constant during all process. For each time step, the system of equations obtained for a given non-orthogonal structured mesh was solved by the Gauss-Seidel method. One computational code was developed in FORTRAN, using the CFV 6.6.0 Studio, in a Windows platform. The proposed solution was validated using analytical and numerical solutions of the diffusion equation for different geometries (parallelepiped and finite cylinder). The analysis and comparison of the results showed that the proposed solution provides correct results for the cases investigated. In order to verify the potential of the proposed numerical solution, we used experimental data of the drying of ceramic roof tiles for the following temperature: T = 55.6 °C. The analysis of the results and the statistical indicators enables to affirm that the developed numerical solution satisfactorily describes the drying processes in this temperature for the convective boundary condition.

Multi-Level Boundary Element Methods for Steady Heat Diffusion in Three-Dimensions

2005 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Element Methods ◽  
Three Dimensions ◽  
Heat Diffusion ◽  
Numerical Heat Transfer ◽  
Multi Level ◽  
Solution Accuracy ◽  
Steady Heat

We have recently developed a novel multi-level boundary element method (MLBEM) for steady heat diffusion in irregular two-dimensional domains (Numerical Heat Transfer Part B: Fundamentals, 46: 329–356, 2004). This presentation extends the MLBEM methodology to three-dimensional problems. First, we outline a 3-D MLBEM formulation for steady heat diffusion and discuss the differences between multi-level algorithms for two and three dimensions. Then, we consider an example problem that involves heat conduction in a semi-infinite three-dimensional domain. We investigate the performance of the MLBEM formulation using a single-patch approach. The MLBEM algorithms are shown facilitate fast and accurate numerical solutions with no loss of the solution accuracy. More dramatic speed-ups can be achieved provided that patch-edge corrections are also evaluated using multi-level technique.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

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