Large Deformation Analysis of Orthodontic Appliances

1990 ◽  
Vol 112 (1) ◽  
pp. 29-37 ◽  
Author(s):  
A. W. Lipsett ◽  
M. G. Faulkner ◽  
K. El-Rayes
Keyword(s):  
Boundary Conditions ◽  
Local Coordinate System ◽  
Iterative Solution ◽  
Computational Effort ◽  
Deformation Analysis ◽  
Orthodontic Appliances ◽  
Equilibrium Equations ◽  
Nonlinear Approach ◽  
Space Closure ◽  
Time Required

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.

Hyperelastic Thin Shells: Equilibrium Equations and Boundary Conditions

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
Chi Zhang ◽  
Jian Wu ◽  
Keh-Chih Hwang
Keyword(s):  
Boundary Conditions ◽  
Finite Deformation ◽  
Deformation Analysis ◽  
Thin Shells ◽  
Membrane Stress ◽  
Equilibrium Equations

The equilibrium equations and boundary conditions in terms of the second Piola–Kirchhoff membrane stress and moment are given in this note, which are necessary for the finite deformation analysis of shells.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

A Variational Approach to Loaded Ply Structures

2003 ◽  
Vol 9 (1-2) ◽  
pp. 175-185 ◽  
Author(s):  
G. H.M. Van Der Heijden ◽  
J. M.T. Thompson ◽  
S. Neukirch
Keyword(s):  
Boundary Conditions ◽  
Energy Method ◽  
Variational Approach ◽  
Energy Analysis ◽  
Numerical Results ◽  
Equilibrium Equations

We show how an energy analysis can be used to derive the equilibrium equations and boundary conditions for an end-loaded variable ply much more efficiently than in previous works. Numerical results are then presented for a clamped balanced ply approaching lock-up. We also use the energy method to derive the equations for a more general ply made of imperfect anisotropic rods and we briefly consider their helical solutions.

Comparison of Various Pressure Based Boundary Conditions for Three-Dimensional Subsonic DSMC Simulation

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Niraj Shah ◽  
Abhimanyu Gavasane ◽  
Amit Agrawal ◽  
Upendra Bhandarkar
Keyword(s):  
Boundary Conditions ◽  
Cell Size ◽  
Pressure Ratio ◽  
Three Dimensional ◽  
Direct Simulation Monte Carlo ◽  
Statistical Error ◽  
Nonequilibrium Effect ◽  
Time Required ◽  
Characteristic Scheme ◽  
Simulation Time

Three-dimensional (3D) direct simulation Monte Carlo (DSMC) has been used to simulate flow in a straight microchannel using an in-house parallelized code. In the present work, a comparative study of seven boundary conditions is carried out with respect to time required for achieving steady-state, accuracy in predicting the specified pressure at the boundaries, and the total simulation time required for attaining a statistical error within one percent. The effect of changing the Knudsen number, pressure ratio (PR), and cross aspect ratio (CAR) on these parameters is also studied. The presence of a boundary is seen to affect the simulated pressure in a cell when compared to the specified pressure, the difference being highest for corner cells and least for cells away from walls. All boundary conditions tested work well at the inlet boundary; however, similar results are not obtained at the outlet boundary. For the same cell size, the schemes that employ first- and second-order corrections lead to a smaller pressure difference compared to schemes applying no corrections. The best predictions can be obtained by using first-order corrections with finer cell size close to the boundary. For most of the simulated cases, the boundary condition employing the characteristic scheme with nonequilibrium effect leads to the minimum simulation time. Considering the nonequilibrium effect, prediction of inlet and outlet pressures and the speed of simulation, the characteristic scheme with nonequilibrium effect performs better than all the other schemes, at least over the range of parameters investigated herein.

Equilibrium equations and boundary conditions of strain gradient theory in arbitrary curvilinear coordinates

2014 ◽  
Vol 23 (5-6) ◽  
pp. 169-176
Author(s):  
Mikhail Guzev ◽  
Chengzhi Qi ◽  
Jiping Bai ◽  
Kairui Li
Keyword(s):  
Boundary Conditions ◽  
Plane Strain ◽  
Strain Gradient ◽  
Special Form ◽  
Curvilinear Coordinates ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Equilibrium Equations ◽  
Plane Strain Problem

AbstractEquilibrium equations and boundary conditions of the strain gradient theory in arbitrary curvilinear coordinates have been obtained. Their special form for an axisymmetric plane strain problem is also given.

Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

2021 ◽  
Vol 103 (3) ◽  
pp. 53-62
Author(s):  
Victor Revenko ◽  
Andrian Revenko
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Partial Solution ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Normal Stresses ◽  
Equilibrium Equations ◽  
Dimensional Boundary ◽  
The Right

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

Free vibration analysis of porous laminated rotating circular cylindrical shells

2019 ◽  
Vol 25 (18) ◽  
pp. 2494-2508 ◽  
Author(s):  
Ahmad Reza Ghasemi ◽  
Mohammad Meskini
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Cylindrical Shells ◽  
Free Vibration Analysis ◽  
Equilibrium Equations ◽  
Rotating Speed ◽  
Simply Supported ◽  
Numerical Result ◽  
Love’S Shell Theory ◽  
Circular Cylindrical Shells

In this research, investigations are presented of the free vibration of porous laminated rotating circular cylindrical shells based on Love’s shell theory with simply supported boundary conditions. The equilibrium equations for circular cylindrical shells are obtained using Hamilton’s principle. Also, Navier’s solution is used to solve the equations of the cylindrical shell due to the simply supported boundary conditions. The results are compared with previous results of other researchers. The numerical result of this study indicates that with increase of the porosity coefficient the nondimensional backward and forward frequency decreased. Then the results of the free vibration of rotating cylindrical shells are presented in terms of the effects of porous coefficients, porous type, length to radius ratio, rotating speed, and axial and circumferential wave numbers.

The Validity of Approximate Boundary Conditions for Natural Convection With Thermal Radiation in Open Cavities

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
Om Singh ◽  
Shireesh B. Kedare ◽  
Suneet Singh
Keyword(s):  
Natural Convection ◽  
Boundary Conditions ◽  
Significant Loss ◽  
Computational Effort ◽  
Computational Domain ◽  
Maximum Difference ◽  
Open Cavities ◽  
Shallow Cavities ◽  
Approximate Boundary Conditions ◽  
The Difference

Abstract The use of approximate boundary conditions at the opening of the cavities leads to restriction of the computational domain and, hence, the reduction in computational effort. However, the accuracy of the restricted domain approach (RDA) had been evaluated only for the natural convection inside open cavities and that too only for one aspect ratio (AR). The validity of the approach had not been evaluated for inclined, as well as, shallow cavities. This study focuses on the analysis of the accuracy of RDA against extended domain approach (EDA) in open cavities of different ARs, at different inclinations and different Rayleigh numbers (Ra). The results show that the difference between the approaches is only significant in very shallow cavities (AR is defined as the height of the hot wall divided by the depth of the cavity) at low Ra. For Ra higher than  106 and an AR greater than 0.2, the maximum difference between the two approaches is around 5% and hence RDA can be recommended in these ranges, resulting in increased computational efficiency without significant loss in the accuracy. Moreover, the maximum difference in the results for the two methods is for intermediate inclinations. Even there, an increase in the difference is more pronounced at lower Ra. Furthermore, distribution of the exit velocity and temperature at the opening as well as the distribution of the Nusselt number at the hot wall is compared for RDA and EDA to explain the behavior of error at different ARs and inclinations.

Variable Fidelity Modeling in Closed Loop Dynamical Systems

2014 ◽  
Author(s):  
Matthew A. Williams ◽  
Andrew G. Alleyne
Keyword(s):  
Dynamical Systems ◽  
Closed Loop ◽  
System Development ◽  
Computational Cost ◽  
Computational Effort ◽  
High Fidelity ◽  
Vapor Compression System ◽  
Variable Fidelity ◽  
Time Required ◽  
Fidelity Model

In the early stages of control system development, designers often require multiple iterations for purposes of validating control designs in simulation. This has the potential to make high fidelity models undesirable due to increased computational complexity and time required for simulation. As a solution, lower fidelity or simplified models are used for initial designs before controllers are tested on higher fidelity models. In the event that unmodeled dynamics cause the controller to fail when applied on a higher fidelity model, an iterative approach involving designing and validating a controller’s performance may be required. In this paper, a switched-fidelity modeling formulation for closed loop dynamical systems is proposed to reduce computational effort while maintaining elevated accuracy levels of system outputs and control inputs. The effects on computational effort and accuracy are investigated by applying the formulation to a traditional vapor compression system with high and low fidelity models of the evaporator and condenser. This sample case showed the ability of the switched fidelity framework to closely match the outputs and inputs of the high fidelity model while decreasing computational cost by 32% from the high fidelity model. For contrast, the low fidelity model decreases computational cost by 48% relative to the high fidelity model.

scholarly journals Study on Buckling Behavior of Tapered Friction Piles in Soft Soils with Linear Shaft Friction

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Junxiu Liu ◽  
Xianfeng Shao ◽  
Baoquan Cheng ◽  
Guangyong Cao ◽  
Kai Li
Keyword(s):  
Boundary Conditions ◽  
Soft Soil ◽  
Buckling Load ◽  
Lateral Stiffness ◽  
Equilibrium Equations ◽  
Tree Roots ◽  
Soft Soils ◽  
Buckling Behavior ◽  
Friction Pile ◽  
Shaft Friction

The buckling instability of long slender piles in soft soils is a key consideration in geoengineering design. By considering both the linear shaft friction and linear lateral stiffness of the soft soil, the buckling behaviors of a tapered friction pile embedded in heterogeneous soil are extensively studied. This study establishes and validates an analytical model to formulate the equilibrium equations and boundary conditions and then numerically solves the boundary value problem to obtain the critical buckling load and buckling shape by using software Matlab. The effects of boundary conditions, tapered ratio, stiffness ratio, friction ratio, lateral stiffness, and shaft friction on the buckling behavior of the friction pile are extensively explored. This study demonstrates that the buckling load decreases with the increase of friction ratio of the linear shaft friction. There exists an optimal tapered ratio corresponding to the maximum dimensionless buckling load in the tapered friction pile with linear shaft friction. The result means that the linear shaft friction should be considered in designing the tapered friction piles in heterogeneous soils. The results also have potential applications in the fields of growing of tree roots in soils, moving of slender rods in viscous fluids, penetrating of fine rods in soft elastomers, etc.

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