scholarly journals On the determination of shifting operators along geodesics on a surface

2013 ◽  
Vol 40 (1) ◽  
pp. 17-26
Author(s):  
Zoran Draskovic
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Algebraic Equations ◽  
Geodesic Line ◽  
Euclidean Spaces ◽  
Tensor Fields ◽  
Finite Element Approximations ◽  
Linear Algebraic Equations ◽  
The Future

A procedure to obtain a closed form of the shifting operators along a known geodesic line on a surface as a solution of a system of linear algebraic equations is proposed. Its correctness is numerically demonstrated in the case of a helicoid surface and a spherical one. The future use of these operators in finite element approximations of tensor fields in non-Euclidean spaces is announced.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Schwarz Type Solvers for -FEM Discretizations of Mixed Problems

2012 ◽  
Vol 12 (4) ◽  
pp. 369-390
Author(s):  
Sven Beuchler ◽  
Martin Purrucker
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Schur Complement ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Contraction Ratio ◽  
Linear Algebraic Equations ◽  
Stokes Problems ◽  
Elasticity Problems ◽  
Element Pair

AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

Determination of critical angle of circumferential throughwall crack for structurally distorted 90° pipe bends under bending moment with and without internal pressure

2017 ◽  
Vol 233 (5) ◽  
pp. 817-830 ◽  
Author(s):  
S Sumesh ◽  
AR Veerappan ◽  
S Shanmugam
Keyword(s):  
Finite Element ◽  
Internal Pressure ◽  
Closed Form ◽  
Bending Moment ◽  
External Loading ◽  
Element Analysis ◽  
Critical Crack ◽  
Structural Distortions ◽  
Internal Pressures

Throughwall circumferential cracks (TWC) in elbows can considerably minimize its collapse load when subjected to in-plane bending moment. The existing closed-form collapse moment equations do not adequately quantify critical crack angles for structurally distorted cracked pipe bends subjected to external loading. Therefore, the present study has been conducted to examine utilizing elastic-plastic finite element analysis, the influence of structural distortions on the variation of critical TWC of 90° pipe bends under in-plane closing bending moment without and with internal pressure. With a mean radius ( r) of 50 mm, cracked pipe bends were modeled for three different wall thickness, t (for pipe ratios of r/ t = 5,10,20), each with two different bend radius, R (for bend ratios of R/r = 2,3) and with varying degrees of ovality and thinning (0 to 20% with increments of 5%). Finite element analyses were performed for two loading cases namely pure in-plane closing moment and in-plane closing bending with internal pressure. Normalized internal pressures of 0.2, 0.4, and 0.6 were applied. Results indicate the modification in the critical crack angle due to the pronounced effect of ovality compared to thinning on the plastic loads of pipe bends. From the finite element results, improved closed-form equations are proposed to evaluate plastic collapse moment of throughwall circumferential cracked pipe bends under the two loading conditions.

Variational and Finite Element Analysis of Vibroequilibria

2004 ◽  
Vol 4 (3) ◽  
pp. 290-323 ◽  
Author(s):  
K. Beyer ◽  
M. Günther ◽  
K. Timokha
Keyword(s):  
Finite Element ◽  
Shape Design ◽  
Stationary States ◽  
Optimal Shape Design ◽  
Free Boundaries ◽  
Element Analysis ◽  
Finite Element Approximations ◽  
Potential Flows ◽  
2D And 3D

AbstractWe adapt, via asymptotic expansion, Kapitsa's formula for the effective potential of a pendulum with vibrating suspension to rapidly forced potential flows with free boundaries. Determination of time-averaged stationary states leads to an optimal shape design problem. Under periodic boundary conditions existence and uniqueness of smooth minimizers to the averaged energy is proved using local coerciveness. In the numerical part of the article, 2D and 3D finite element approximations including related error estimates are discussed. Some illustrating examples are sketched.

On the Analytical Determination of the Nominal Stresses in Dovetail Attachments

2020 ◽  
Author(s):  
Glenn Sinclair
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Contact Region ◽  
Essential Elements ◽  
Contact Stresses ◽  
Physical Models ◽  
Analytical Determination ◽  
Bending Stress ◽  
Attachment Model

Abstract Simple physical models are developed for the nominal contact stresses in dovetail attachments. These nominal stresses include the pressure, the shear traction, and the bending stress in the contact region, both during loading up and unloading. The models furnish closed-form expressions for these stresses. For a specific dovetail attachment, model values are compared with verified finite element values. As a result of the simplifications introduced to make the models tractable, model values only approximately equal finite element values. Nonetheless, the models capture the essential elements of the response of nominal stresses in dovetail attachments.

scholarly journals A High-Frequency Model of a Circular Beam with a T-Shaped Cross Section

Acoustics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 295-336
Author(s):  
Andrew Hull ◽  
Daniel Perez
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Cross Section ◽  
High Frequency ◽  
Algebraic Equations ◽  
Frequency Range ◽  
Linear Algebraic Equations ◽  
Circular Beam ◽  
The Web ◽  
Shaped Cross Section

This paper derives an analytical model of a circular beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 50 kHz. The T-shaped cross section is composed of an outer web and an inner flange. The web in-plane motion is modeled with two-dimensional elasticity equations of motion, and the left portion and right portion of the flange are modeled separately with Timoshenko shell equations. The differential equations are solved with unknown wave propagation coefficients multiplied by Bessel and exponential spatial domain functions. These are inserted into constraint and equilibrium equations at the intersection of the web and flange and into boundary conditions at the edges of the system. Two separate cases are formulated: structural axisymmetric motion and structural non-axisymmetric motion and these results are added together for the total solution. The axisymmetric case produces 14 linear algebraic equations and the non-axisymmetric case produces 24 linear algebraic equations. These are solved to yield the wave propagation coefficients, and this gives a corresponding solution to the displacement field in the radial and tangential directions. The dynamics of the longitudinal direction are discussed but are not solved in this paper. An example problem is formulated and compared to solutions from fully elastic finite element modeling. It is shown that the accurate frequency range of this new model compares very favorably to finite element analysis up to 47 kHz. This new analytical model is about four magnitudes faster in computation time than the corresponding finite element models.

An Analysis of the Large Deflections of Beams using the Rayleigh-Ritz Finite Element Method

1968 ◽  
Vol 19 (4) ◽  
pp. 357-367 ◽  
Author(s):  
A. C. Walker ◽  
D. G. Hall
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Large Deflection ◽  
Energy Functional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Comparison Of The Results ◽  
Element Method

SummaryThe Rayleigh-Ritz finite element method is used to obtain an approximate solution of the exact non-linear energy functional describing the large deflection bending behaviour of a simply-supported inextensible uniform beam subjected to point loads. The solution of the non-linear algebraic equations resulting from the use of this method is effected, using three different techniques, and comparisons are made regarding the accuracy and computing effort involved in each. A description is given of an experimental investigation of the problem and comparison of the results with those of the numerical method, and of the available exact continuum analyses, indicates that the numerical method provides satisfactory predictions for the non-linear beam behaviour for deflections up to one quarter of the beam’s length.

scholarly journals Development of a finite-element-based design sensitivity analysis for buckling and postbuckling of composite plates

1995 ◽  
Vol 1 (3) ◽  
pp. 255-274 ◽  
Author(s):  
Ruijiang Guo ◽  
Aditi Chattopadhyay
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Linear Equations ◽  
Composite Plates ◽  
Material Model ◽  
Design Sensitivity ◽  
Algebraic Equations ◽  
Sensitivity Derivatives ◽  
Analysis Technique ◽  
Linear Algebraic Equations

A finite element based sensitivity analysis procedure is developed for buckling and postbuckling of composite plates. This procedure is based on the direct differentiation approach combined with the reference volume concept. Linear elastic material model and nonlinear geometric relations are used. The sensitivity analysis technique results in a set of linear algebraic equations which are easy to solve. The procedure developed provides the sensitivity derivatives directly from the current load and responses by solving the set of linear equations. Numerical results are presented and are compared with those obtained using finite difference technique. The results show good agreement except at points near critical buckling load where discontinuities occur. The procedure is very efficient computationally.

scholarly journals Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

2021 ◽  
Vol 102 (2) ◽  
pp. 87-95
Author(s):  
Hryhorii Habrusiev ◽  
Iryna Habrusieva
Keyword(s):  
Contact Interaction ◽  
Rigid Base ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
Linear Algebraic Equations ◽  
Vertical Displacements ◽  
Smooth Contact ◽  
Incompressible Solids ◽  
Initial Strains

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

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