ce 257 discrete methods of structural analysis · 2020. 6. 30. · ce 257 5. introduction •thus,...

CE 257 Discrete Methods of Structural Analysis

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This lecture material is not for sharing/distribution outside of CE257 and CE297 class.

Last meeting

• Time-dependent Problems

• 2D Triangular Elements

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For this meeting

• Single-variable problems in two dimensions

Reference for this lecture:

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 8)

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CE28 Analytical and Computational Methods in CEIII

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Introduction

Introduction

• FE analysis of two dimensional (2D) problemsinvolve the same basic steps as those describedfor one-dimensional problems.

• The analysis is somewhat complicated by the factthat 2D problems are described by PDEs overgeometrically complex regions. The boundary Γ ofa two-dimensional domain Ω is, in general, acurve. Therefore, finite elements are simple twodimensional geometric shapes that allowapproximations of a given two-dimensionaldomain as well as the solution over it

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Introduction

• Thus, in 2D problems we not only seek anapproximate solution to a given problem on adomain, but we also approximate the domain by asuitable finite element mesh.

• Consequently, we will have approximation errorsdue to the approximation solution as well asdiscretization errors due to the approximation ofthe domain in the FE analysis of 2D problems.

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Introduction

• FE mesh: triangles, rectangles/quadrilaterals

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Boundary Value Problems


The model equation:

• Consider finding the solution 𝑢(𝑥, 𝑦) of the2𝑛𝑑 order PDE

• for given data 𝑎𝑖𝑗 , 𝑎00, and 𝑓, and specified BCs.

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• As a special case, we can obtain the Poissonequation from the model DE by setting 𝑎11 =𝑎22 = 𝑘(𝑥, 𝑦) and 𝑎12 = 𝑎21 = 0

• where ∇ is the gradient operator.

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In Cartesian coordinate system:

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Major steps in developing the FE model of model DE:

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F. E. Discretization

FE Discretization

• The representation of a given region by a set ofelements (i.e. discretization or mesh generation) is animportant step in FE analysis.

• The choice of element type, number of elements, anddensity of elements, depends on the geometry of thedomain, the problem to be analyzed, and the degreeof accuracy desired.

• In general, the analyst is guided by his or her technicalbackground, insight into the physics of the problembeing modeled (e.g. qualitative understanding of thesolution) and experience with the FE modeling.

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FE Discretization

• The general rules of mesh generation for FEformulation include:

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Weak Form

Weak Form

• First step is to multiply the DE with a weightfunction 𝑤, which is assumed to be differentiableonce with respect to 𝑥 and 𝑦 and then integratethe equation over the element domain Ω𝑒 .

• where

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Weak Form

• Second step: IBP to distribute the differentiationamong 𝑢 and 𝑤 equally.

• We will use the following identities:

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Weak Form

• Next we use the component form of the gradient(or divergence) theorem

• where 𝑛𝑥 and 𝑛𝑦 are the components (i.e. the

direction cosines) of the unit normal vector

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Weak Form

• We obtain

Let 𝑞𝑛 be boundary expression:

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Weak Form

• By definition, 𝑞𝑛 is taken positive outward fromthe surface as we move counterclockwise alongthe boundary Γ𝑒.

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Weak Form

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Weak Form

• Rewriting the weak form:

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Finite Element Model


• The weak form requires that the approximationchosen for 𝑢 should be atleast linear in both 𝑥 and𝑦 so that there are no terms that are identicallyzero.

• Since the primary variable is simply the functionitself, the Lagrange family of interpolationfunctions is admissible.

• Approximation of 𝑢 over the typical finite elementΩ𝑒:

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• where

Lagrange interpolation property

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• Substituting the FE approximation for 𝑢 into theweak form, we obtain

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or

where


• In matrix notation

• where

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Interpolation Functions

Derivation of Interpolation functions

• The FE approximation 𝑢ℎ𝑒(𝑥, 𝑦) over an element

Ω𝑒 must satisfy the following conditions in orderfor the approximate solution to converge to thetrue solution:

1. 𝑢ℎ𝑒 must be continuous as required in the weak

form of the problem (i.e. all terms in the weak formare represented as nonzero values)

2. The polynomial used to represent 𝑢ℎ𝑒 must be

complete (i.e. all terms, beginning with a constantterm up to the highest order used in the polynomial)

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Derivation of Interpolation functions

• The FE approximation 𝑢ℎ𝑒(𝑥, 𝑦) over an element

Ω𝑒 must satisfy the following conditions in orderfor the approximate solution to converge to thetrue solution:

3. All terms in the polynomial should be linearlyindependent

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Triangular element

• Complete linear polynomial in 𝑥 and 𝑦 in Ω𝑒

• We can denote:

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Triangular element

• Approximation:

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Triangular element

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Triangular element

• Shapes to avoid:

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Triangular element

• inverting coefficient matrix

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Triangular element

• Substituting 𝑐𝑖 back to the approximation for 𝑢 ∶

• where:

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Triangular element

• Interpolation functions:

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Triangular element

• Properties of interpolation functions:

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Triangular element

• Representation of continuous function

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Linear Rectangular Element

• Consider the complete polynomial

• which contains four linearly independent termsand is linear in 𝑥 and 𝑦, with a bilinear term in 𝑥and 𝑦. This polynomial requires an element withfour nodes.

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• Incompatible four node triangular elements

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• Using a rectangular element with sides 𝑎 and 𝑏,we choose a local coordinate system, ( ҧ𝑥, ത𝑦) toderive the interpolation functions.

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Solving for 𝑐𝑖:


• Simplifying..

• where

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• Interpolation functions:

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• Properties of interpolation functions:

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• Quadratic Elements (personal reading)

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Evaluation of Matrices and Vectors

Evaluation of Element Matrices and Vectors

• We rewrite [𝐾𝑒] as the sum of the five basicmatrices

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• Element Matrices of a Linear Triangular Element(for a Poisson problem)

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• Element Matrices of a Linear Rectangular Element

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• Element Matrices of a Linear Rectangular Element

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Evaluation of Element Matrices andVectors

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Evaluation of Boundary Integrals

• Boundary Integral

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Assembly of Element Equations


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• In matrix form:

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Postcomputations

Postcomputations

• The finite element solution at any point (𝑥, 𝑦) inan element delta is given by

• and it derivatives are computed as

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Postcomputations

• The derivatives will not be continuous atinterelement boundaries because continuity ofderivatives is not imposed during the assemblyprocedure.

• The weak form of the equations suggests that theprimary variable is 𝑢, which is to be carried as thenodal variable.

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Finite Element Method

References:

1. J. Fish, T. Belytschko, A First Course in Finite Elements

2. J. N. Reddy, An Introduction to the Finite Element Method

3. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, Its Basis & Fundamentals

4. K-J. Bathe, Finite Element Procedures

5. K. Leet, C-M., Uang, A.M. Gilbert, Fundamentals of Structural Analysis

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This lecture

• Single-variable problems in two dimensions

Reference for this lecture:

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 8)

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This lecture material is not for sharing/distribution outside of CE257 and CE297 class.


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ce 257 discrete methods of structural analysis · 2020. 6. 30. · ce 257 5. introduction •thus,...

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