Final Synopsis Project Ppt

Introductory Seminar On

Paralinear Element Technique For The Finite

Element Analysis By

Miss. Shubhangi A. Chafale

On

Dt. 21th August 2012

guide

Shri. K.R.C. Reddy

Contents

1.Introduction2.Literature Review3.Outline of the project4.References

1.Introduction

1. INTRODUCTION

Selection of Element i. e. Size and Shape of Element.

Importance of Aspect Ratio

Number of Elements increases the results.

Order of Displacement field also increases the

results.

But in case of Rectangular Element, large Aspect Ratio deviate the results.

Use of Para Linear Element

In this study, it is interesting to see the effect of more Aspect Ratio in Para Linear, Cubic Linear, Cubic Para Elements.

1. INTRODUCTION

2. Literature Review

i. FEM – General formulationsii.Triangular elements in FEMiii.Discretization of structures in FEMiv.Effect of aspect ratio in modeling

i. FEM – General formulations Dunning et al (2009)

This paper introduces a new fixed mesh structural analysis technique.

This paper first introduces the Area Ratio Based Fixed Grid (AFG) formulation.

Later on they introduces new Isoparametric Based Fixed Grid (IFG) formulation.

2. Literature Review

A fixed grid mesh is generated by discretization of a rectangular domain into equal sized rectangular elements. This creates three types of elements are shown in figure.

2. Literature Review

Figure: Fixed Grid Elément Types

The two dimensional isoparametric method divides boundary elements into three types depending on their inside area shape.

Quadrilateral elements

Triangular elements

pentagonal elements

2. Literature Review

Figure: Isoparametric Fixed Grid Element Types

2. Literature Review

They concluded that, the IFG formulation shows potential improvement for displacement calculation compared to the AFG method along the boundary.

Clough and Wilson (1999)

2. Literature Review

In this paper they have attempted to summarize the early research at the University of California at Berkeley from 1957 to 1970.

Significant finite element research was conducted at the University of California at Berkeley during the period 1957 to 1970.

2. Literature Review

The initial research was a direct extension of classical methods of structural analysis which previously had been restricted to one-dimensional elements.

The majority of the research conducted was motivated by the need to solve practical problems in Aerospace Mechanical Civil Engineering.

2. Literature Review

Analysis models for both continuous structures and frame struct-ures were modeled as a system of elements interconnectedat joints or nodes as indicated in Figure

Figure : The Finite Element Idealization

As per this paper, during this short period the finite element method was extended to the solution of linear and nonlinear problems associated with

•creep,•incremental construction or excavation, •crack closing, •heat transfer, •flow of water in porous media, •soil consolidation, •dynamic response analysis and •Computer assisted learning of structural analysis.

2. Literature Review

Lee et al (1993)

They discuss the effects of element distortions on the isoparametric quadrilateral finite elements.

The objective of this paper is to present a systematic study with numerical and analytical results.

In the paper, they focus their attention on the types of element distortions.

2. Literature Review

2. Literature Review

They consider a square element with evenly spaced nodes to be the undistorted element.

Comparing with this configuration, they may identify the following basic types of distortions:

1. aspect-ratio and parallelogram distortions

2. unevenly-spaced-nodes distortion

3. angular distortion

4. curved-edge distortion

In this, they use the

Lagrangian isoparametric elements

Serendipity isoperametric elements.

2. Literature Review

Ergatoudis et al (1968) An increase of available parameters associated

with an element usually leads to improved accuracy of solution.

It is possible thus to use fewer elements for the solution.

This paper concerned with a method of generating a series of elements.

2. Literature Review

2. Literature Review

Figure: General quadrilateral elements

2. Literature Review

Beam flexure problems can be represented very closely using one element in the thickness of a beam figure.

2. Literature Review

Figure:Four elements used to represent a cantilever. Improvement of accuracy with element order is exident.

2. Literature Review

Li et al (2004)

In this paper, the problems involved the process of degeneration of quadrilateral element into triangular elements are thoroughly analyzed.

The contents include the formulation of geometry mapping induced by collapsing one side of the quadrilateral element and construction of the shape function.

ii Triangular elements in FEM

2. Literature Review

The study focuses:

4-node bilinear quadrilateral (Q4) element to 3-node constant strain triangular (CST) element.

8-node serendipity (Q8) element to 6-node triangular element (T6).

Auricchio and Taylor (1995)

They present a new formulation for a triangular finite element.

The element takes advantage of internal rotational degrees of freedom and a linked interpolation between the transverse displacement and the rotations.

2. Literature Review

2. Literature Review

The element has excellent interpolating capacity.

In the present paper they concluded that, with an appropriate choice of the shape functions they are able to obtain a constant shear strain along each side of an element.

Batoz and Ho (1980)

In this paper they present the results of a detailed theoretical and numerical study of triangular plate bending elements with 9 corner dof only.

The objective in this study was to identify or develop an optimum element for the general linear analysis of plate bending problems.

2. Literature Review

In this paper they first review alternative formulation of 3-noded triangular plate bending elements.

This review suggests giving specific attention to the theoretical formulation and numerical evaluation of three elements:

a DKT (discrete Kirchhoff theory) element, a HSM (hybrid stress model) element and a SRI (selective reduced integration) element.

2. Literature Review

Figure: Nine dof triangular plate bending element

2. Literature Review

James et al (1970)

For a plane polygonal domain and a corresponding (general) triangulation they define classes of functions.

These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method.

The case of second-order problems is discussed.

2. Literature Review

Maliki et al (2008)

Higher order finite element methods providing higher accuracy.

They have developed an efficient iterative method.

The Hierarchical Iterative method is similar to geometric multi grid methods.

2. Literature Review

iii. Discretization of structures in FEM

The performance of Hierarchical Iterative method is compared with the algebraic multi grid method and others iterative methods.

Their method requires less computing time and less memory storage.

The solutions produced by those FEM methods are more reliable.

In this paper, they have proposed an efficient iterative method to solve those problems.

2. Literature Review

Arnold (1981)

The discretization by finite elements of a model variational problem for a clamped loaded beam is studied.

It is shown that the approximation achieved by a standard finite element method degenerates for thin beams.

2. Literature Review

The most useful of these methods may be realized by replacing the integrals in the stiffness matrix by Gauss quadratures.

They examine the finite element discretization of a model variational problem in which the dependent variables represent the

vertical displacement and the rotation.

2. Literature Review

Burman et al (2004)

2. Literature Review

iv. Effect of aspect ratio in modeling

A multiphase-field model for the description of

coalescence is solved numerically using adaptive finite elements with high aspect ratio.

The unknown of the multiphase-field model are the three phase fields (solid phase1, solid phase 2, and liquid phase).

2. Literature Review

An adaptive finite element algorithm is proposed.

In order to reduce the number of mesh vertices,

the generated meshes contain elements with high aspect ratio.

They introduce an error indicator.

Numerical results on two test cases (1D and 2D) show the efficiency of the method.

2. Literature Review

Miller et al (1995)

In this paper, they give an algorithm for finding a well-conditioned hierarchical gradient of a two dimensional unstructured mesh.

Their algorithm can be used to generate both node-nested and non-nested gradients.

They use the maximal independent set (MIS) technique.

In the below figure, it is seen that, the choices of MIS of the original mesh degrade the aspect ratio of the coarser mesh.

Figure: Repeated application of MIS can degrade the aspect ratio

2. Literature Review

2. Literature Review

Rice (1985)

An experiment study was made of the effect of elements with large aspect ratios on the solution of second order elliptic partial differential equations.

They use Hermite bicubics and Galerkin with nine different piecewise polynomial basis functions on rectangular grids.

The aspect ratio of an element is defined by

a=R2/R1

where, R2 is the radius of the largest circle

R1 is the radius of the smallest circle

They choose one PDE and explore the effect of having one grid line move progressively closer to another with all the others remaining fixed.

2. Literature Review

2. Literature Review

This was done for ten finite element method and ordinary finite differences.

They conclude that large aspect ratios do not imply large losses of the accuracy.

They also conclude that the condition number of the matrix associated with the method does not give a reliable guide to the effects of round-off.

CONCLUSIONS ON LITERATURE By studying the literature as mentioned above it is

found that• the refinement of mesh and in some cases higher

order displacement functions are used.• The application of para linear, cubic linear elements

is not seen much in the literature. • In the present study it is aimed to use the para

linear element technique in finite element analysis.

2. Literature Review

3. OUTLINE OF PROJECT

3. OUTLINE OF PROJECT

The complete project work is divided in the following three stages:

In the first stage, problem will be taken up with aspect ratio of rectangular element as one and the result will be obtained.

In the second stage, by varying the aspect ratio into various ratios, by keeping the number of elements almost same, the results will be obtained and errors will be estimated to compare with the values obtained in the first stage.

In the third stage, to overcome the problem of aspect ratio, a para linear element technique will be applied to reduce the errors.

It is also decided to see the effect of cubic linear and cubic para elements, if time permits, to see the minimization of errors.

3. OUTLINE OF PROJECT

Plan of the project work

The first stage work of analyzing the problem by using the rectangular elements with aspect ratio as one will done in the present semester.

The remaining work of two stages, i. e. Analysis of problem with varied aspect ratio and the application of para linear elements in the analysis will be taken up in the next semester.

4. REFERECES

4.REFERECE

S Arnold, D. N., (1981), ‘Discretization by finite Elements of a Model

Paramater Dependent Problem’, Int. J. Numer. Math., 37, 405-421. Auricchio, F. and Taylor, R. L. (1995), ‘Triangular Thick Plate Finite Element

with An Exact Thin Limit’, Proceeding in Elsevier Finite Element in Analysis and Design 19, 57-68.

Batoz, K. J., Ho, L. W., (1980), ‘A Study of Three-node Triangular Plate Bending Elements’, Int. J. for Numer. Meth. In Engg., Vol. 15, 1771-1812.

Burman, E., Jacot, A., Picasso, M., (2004), ‘Adaptive Finite Elements with High Aspect Ratio for the Computation of Coalescence using a Phase-Field Model’, Journal of Computational Physics 195, 153-174.

Clough, R. W. and Wilson, E. L., (1999), ‘Early Finite Element Research At Berkeley’, Present at the fifth U.S. National Conference on Computational Mechanics, 1-35.

Dunning, P. D., Kim, H. A. and Mullineux, G., (2009), ‘Two Dimensional Fixed Grid Based Finite Element Structural Analysis’, Department of Mechanical Engineering, university of Bath, Bath, BA2 7AY, UK, 1-9.

Ergatoudis, I., Irons, B. M., and Zienkiewicz, O. C., (1968), ‘Curved, Isoparametric, Quadrilateral, Elements for Finite Element Analysis’, Int. J. Solids Structures, Vol. 4, 31-42.

James, Bramble, H., and Zlama, M., (1970), ‘Triangular Elements in the Finite Element Method’, Dept. of Mathematics of Computation, Vol 24, No. 112, 809-820.

Lee, R. and Cangellaris, A. C., (1992), ‘A Study of Discretization Error in the Finite Element Approxination of wave solutions’, IEEE Transactions on Antennas and Propagation, Vol. 40, No. 5, 542-549.

Lee, N. S. and Bathe, K. J., (1993), ‘Effects of Element Distortions on the Performance of Isoparametric Elements’, Int. J. for Numer. Meth. In Engg., Vol. 36, 3553-3576.

Maliki, A. E., Guenette, R., and Fortin, M., (2008), ’A Hierarchial Iterative Method for Quqdratic Discretizations of Finite Element Problems’, Numerical Linear Algebra Appl., 1-26.

Miller, G. L., Talmor, D. and Teng, S. H., (1995), ‘Optimal Good-Aspect–Ratio Coarsening for Unstructured Meshes’, in proc. 27th Annu. ACM Sympos. Theory Comput., 683-692.

4.REFERECES

Picasso, M., (2006), ‘Adaptive Finite Elements with Large Aspect Ratio based on an Anisotropic Error Estimator Involving First Order Derivatives’, Proc. In Elsevier Comput. Methods Appl. Mech. Engg. 196, 14-23.

Rice, J. R. (1985), ‘Is the Aspect Ratio Significant For Finite Element Problems?’, National Science Foundation grant MS-8301589, 1-14.

Thank you