CE 257 Discrete Methods of Structural Analysis

(1)

CE 257 Discrete Methods of Structural Analysis

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

(2)

Last meeting

• Time-dependent Problems

• 2D Triangular Elements

CE 257 2

(3)

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)

(4)

CE28 Analytical and Computational

Methods in CEIII 4

Introduction

(5)

Introduction

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

• The analysis is somewhat complicated by the fact that 2D problems are described by PDEs over geometrically complex regions. The boundary Γ of a two-dimensional domain Ω is, in general, a curve. Therefore, finite elements are simple two dimensional geometric shapes that allow

(6)

Introduction

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

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

CE 257 6

(7)

Introduction

• FE mesh: triangles, rectangles/quadrilaterals

(8)

Methods in CEIII 8

Boundary Value Problems

(9)

Boundary Value Problems

The model equation:

• Consider finding the solution 𝑢(𝑥, 𝑦) of the 2^𝑛𝑑 order PDE

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

(10)

Boundary Value Problems

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

• where ∇ is the gradient operator.

CE 257 10

(11)

Boundary Value Problems

In Cartesian coordinate system:

(12)

Boundary Value Problems

Major steps in developing the FE model of model DE:

CE 257 12

(13)

F. E. Discretization

(14)

FE Discretization

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

• The choice of element type, number of elements, and density of elements, depends on the geometry of the domain, the problem to be analyzed, and the degree of accuracy desired.

• In general, the analyst is guided by his or her technical background, insight into the physics of the problem being modeled (e.g. qualitative understanding of the solution) and experience with the FE modeling.

CE 257 14

(15)

FE Discretization

• The general rules of mesh generation for FE formulation include:

(16)

Methods in CEIII 16

Weak Form

(17)

Weak Form

• First step is to multiply the DE with a weight function 𝑤, which is assumed to be differentiable once with respect to 𝑥 and 𝑦 and then integrate the equation over the element domain Ω_𝑒 .

• where

(18)

Weak Form

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

• We will use the following identities:

CE 257 18

(19)

Weak Form

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

• where 𝑛_𝑥 and 𝑛_𝑦 are the components (i.e. the direction cosines) of the unit normal vector

(20)

Weak Form

• We obtain

Let 𝑞_𝑛 be boundary expression:

CE 257 20

(21)

Weak Form

• By definition, 𝑞_𝑛 is taken positive outward from the surface as we move counterclockwise along the boundary Γ_𝑒.

(22)

Weak Form

CE 257 22

(23)

Weak Form

• Rewriting the weak form:

(24)

Methods in CEIII 24

Finite Element Model

(25)

Finite Element Model

• The weak form requires that the approximation chosen for 𝑢 should be atleast linear in both 𝑥 and 𝑦 so that there are no terms that are identically zero.

• Since the primary variable is simply the function itself, the Lagrange family of interpolation functions is admissible.

• Approximation of 𝑢 over the typical finite element Ω_𝑒:

(26)

Finite Element Model

• where

Lagrange interpolation property

CE 257 26

(27)

Finite Element Model

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

(28)

Finite Element Model

CE 257 28

or where

(29)

Finite Element Model

• In matrix notation

• where

(30)

Methods in CEIII 30

Interpolation Functions

(31)

Derivation of Interpolation functions

• The FE approximation 𝑢_ℎ^𝑒(𝑥, 𝑦) over an element Ω_𝑒 must satisfy the following conditions in order for the approximate solution to converge to the true solution:

1. 𝑢_ℎ^𝑒 must be continuous as required in the weak form of the problem (i.e. all terms in the weak form are represented as nonzero values)

2. The polynomial used to represent 𝑢_ℎ^𝑒 must be complete (i.e. all terms, beginning with a constant

(32)

Derivation of Interpolation functions

• The FE approximation 𝑢_ℎ^𝑒(𝑥, 𝑦) over an element Ω_𝑒 must satisfy the following conditions in order for the approximate solution to converge to the true solution:

3. All terms in the polynomial should be linearly independent

CE 257 32

(33)

Triangular element

• Complete linear polynomial in 𝑥 and 𝑦 in Ω_𝑒

• We can denote:

(34)

Triangular element

• Approximation:

CE 257 34

(35)

Triangular element

(36)

Triangular element

• Shapes to avoid:

CE 257 36

(37)

Triangular element

• inverting coefficient matrix

(38)

Triangular element

• Substituting 𝑐_𝑖 back to the approximation for 𝑢 ∶

• where:

CE 257 38

(39)

Triangular element

• Interpolation functions:

(40)

Triangular element

• Properties of interpolation functions:

CE 257 40

(41)

Triangular element

• Representation of continuous function

(42)

Linear Rectangular Element

• Consider the complete polynomial

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

CE 257 42

(43)

Linear Rectangular Element

• Incompatible four node triangular elements

(44)

Linear Rectangular Element

• Using a rectangular element with sides 𝑎 and 𝑏, we choose a local coordinate system, (𝑥,ҧ 𝑦) toത derive the interpolation functions.

CE 257 44

(45)

Linear Rectangular Element

(46)

Linear Rectangular Element

CE 257 46

Solving for 𝑐

_𝑖

:

(47)

Linear Rectangular Element

• Simplifying..

• where

(48)

Linear Rectangular Element

• Interpolation functions:

CE 257 48

(49)

Linear Rectangular Element

• Properties of interpolation functions:

(50)

Linear Rectangular Element

• Quadratic Elements (personal reading)

CE 257 50

(51)

Evaluation of Matrices and Vectors

(52)

Evaluation of Element Matrices and Vectors

• We rewrite [𝐾^𝑒] as the sum of the five basic matrices

CE 257 52

(53)

Evaluation of Element Matrices and Vectors

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

(54)

Evaluation of Element Matrices and Vectors

• Element Matrices of a Linear Rectangular Element

CE 257 54

(55)

Evaluation of Element Matrices and Vectors

• Element Matrices of a Linear Rectangular Element

(56)

Evaluation of Element Matrices and Vectors

CE 257 56

(57)

Evaluation of Boundary Integrals

• Boundary Integral

(58)

Methods in CEIII 58

Assembly of Element Equations

(59)

Assembly of Element Equations

(60)

Assembly of Element Equations

• In matrix form:

CE 257 60

(61)

Postcomputations

(62)

Postcomputations

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

• and it derivatives are computed as

CE 257 62

(63)

Postcomputations

• The derivatives will not be continuous at interelement boundaries because continuity of derivatives is not imposed during the assembly procedure.

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

(64)

Methods in CEIII 64

(65)

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

(66)

This lecture

• Single-variable problems in two dimensions

Reference for this lecture:

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

CE 257 66

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

(67)