MCE 561 Computational Methods in Solid Mechanics

Introduction

Need for Computational Methods

• Stress Analysis Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry

• Most Real World Problems Involve Cases with Complex Shape, Boundary Conditions and Material Behavior

• Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed-form Methods

• This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element, Boundary Element, Discrete Element Methods

Finite Difference MethodFinite Difference Method (FDM)

The finite difference method replaces the derivatives in governing field equations bydifference quotients which involve values of the solution at discrete mesh points in the domain understudy. For example, the first derivative with respect to x may be represented by

Repeated applications of this representation set up algebraic systems of equations in termsof the unknown nodal values ui . The major difficulty with this method lies in the inaccuracies indealing with regions of complex shape, although this problem can be eliminated through the use ofcoordinate transformation techniques.

Finite Element Method (FEM)

(i-1) x(i) (i+1)

xx

du

dx

u u

xbackw ard d ifference

u u

xforw ard d ifference

u u

xcen tra l d ifference

i i

i i

i i

1

1

1 1

2

. . .

. . .

. . .

Finite Element MethodFinite Element Method (FEM)

The fundamental concept of the finite element method lies in dividing the body under studyinto a finite number of pieces (subdomains) called elements . Particular assumptions are then madeon the variation of the unknown dependent variable(s) across each element using so-calledinterpolation or shape functions. This approximated variation is quantified in terms of solutionvalues at special locations within the element called the nodes. Through this discretization process,the method sets up an algebraic system of equations for unknown nodal values which approximatethe continuous solution. Because element size and shape are variable, the finite element method canhandle problem domains of quite complicated shape, and thus it has become a very useful andpractical tool.

Boundary Element Method

Boundary Element Node

Bounday Element Method (BEM)The boundary element method is based upon an integral statement of the governing equations ofthe problem under study. The integral statement may be cast into a form which containsunknowns only over the boundary of the body domain. This boundary integral equation maythen be solved using concepts from the finite element method; i.e. the boundary may bediscretized into a number of elements and the interpolation approximation concept may then beapplied. This method again produces an algebraic system of equations to solve for the unknownboundary nodal values, but the system is generally much smaller than that generated by internaldiscretization techniques such as finite element method. By avoiding interior discretization, theboundary element method can have significant advantages over finite element schemes forinfinite or very large domains, and for cases where only boundary information is sought.

Discrete Element MethodDiscrete Element Method (DEM)When the continuum under study is discrete in nature such as that found in particulate and granularmaterials, it is convenient to allow the element sub-domains to move freely subject to the mechanicsof the problem. For such cases a technique called the discrete element method allows continuity ofthe element assembly to be relaxed and contact conditions between elements determine thedeformations and motions of the material. Individual elements can therefore have finite motions(displacements and rotations), and simplifying constitutive assumptions (rigid or linear elastic) arenormally made for each element. Other similar techniques include block element methods ordiscontinuous deformation analysis.

DiscreteElements

Introduction to Finite Element AnalysisThe finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.

Advantages of Finite Element Analysis

- Models Bodies of Complex Shape

- Can Handle General Loading/Boundary Conditions

- Models Bodies Composed of Composite and Multiphase Materials

- Model is Easily Refined for Improved Accuracy by Varying

Element Size and Type (Approximation Scheme)

- Time Dependent and Dynamic Effects Can Be Included

- Can Handle a Variety Nonlinear Effects Including Material

Behavior, Large Deformations, Boundary Conditions, Etc.

Finite Element CodesBased on the success and usefulness of the finite element method, numerous computer codes have been developed that implement the numerical scheme. Some codes are special purpose in-house while others are general purpose commercial codes. Some of the more popular commercial codes include:

- ABAQUS- ADINA- ALGOR- ANSYS- COSMOS- NASTRAN

Basic Concept of Finite Element Method

Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains.

Exact Analytical Solution

x

T

Approximate Piecewise Linear Solution

x

T

One-Dimensional Temperature Distribution

Two-Dimensional Discretization

-1-0.5

00.5

11.5

22.5

3

1

1.5

2

2.5

3

3.5

4-3

-2

-1

0

1

2

xy

u(x,y)

Approximate Piecewise Linear Representation

Discretization Concepts

x

T

Exact Temperature Distribution, T(x)

Finite Element Discretization

Linear Interpolation Model (Four Elements)

Quadratic Interpolation Model (Two Elements)

T1

T2 T2 T3 T3 T4 T4T5

T1

T2

T3T4 T5

Piecewise Linear Approximation

T

x

T1

T2T3 T3

T4 T5

T

T1

T2

T3T4 T5

Piecewise Quadratic Approximation

x

Temperature Continuous but with Discontinuous Temperature Gradients

Temperature and Temperature GradientsContinuous

Common Types of Elements

One-Dimensional ElementsLine

Rods, Beams, Trusses, Frames

Two-Dimensional ElementsTriangular, Quadrilateral

Plates, Shells, 2-D Continua

Three-Dimensional ElementsTetrahedral, Rectangular Prism (Brick)

3-D Continua

Discretization Examples

One-Dimensional Frame Elements

Two-Dimensional Triangular Elements

Three-Dimensional Brick Elements

Multi-Phase Material Problems

Material 1 Material 2

Material 3

Interface Behavior

Basic Steps in Finite Element Method

- Domain Discretization- Select Element Type (Shape and Approximation)- Derive Element Equations (Variational and Energy Methods)- Assemble Element Equations to Form Global System 

[K]{U} = {F}  [K] = Stiffness or Property Matrix {U} = Nodal Displacement Vector {F} = Nodal Force Vector  - Incorporate Boundary and Initial Conditions - Solve Assembled System of Equations for Unknown Nodal Displacements and Secondary Unknowns of Stress and Strain Values

Common Sources of Error in FEA

• Domain Approximation

• Element Interpolation/Approximation

• Numerical Integration Errors

• Computer Errors (Round-Off, Etc., )

Measures of Accuracy in FEA

Accuracy

Error = |(Exact Solution)-(FEM Solution)|

Convergence

Limit of Error as:

Number of Elements (h-convergence) or

Approximation Order (p-convergence)

Increases

Ideally, Error 0 as Number of Elements or Approximation Order

Two-Dimensional Discretization Refinement Plate With a Circular Hole Example

(Discretization with 228 Elements)

(Discretization with 912 Elements)

(Triangular Element)

(Node)

One Dimensional Examples

1 2

u1 u2

Bar Element

Uniaxial Deformation of BarsUsing Strength of Materials Theory

Beam Element

Deflection of Elastic BeamsUsing Euler-Bernouli Theory

1 2

w1w2

2

1

dx

duau

qcuaudx

d

,

:ionSpecificat ConditionsBoundary

0)(

:Equation alDifferenti

)(,,,

:ionSpecificat ConditionsBoundary

)()(

:Equation alDifferenti

2

2

2

2

2

2

2

2

dx

wdb

dx

d

dx

wdb

dx

dww

xfdx

wdb

dx

d

Two DimensionalScalar-Valued Problems

1

2

3

Triangular Element

yx ny

nxdn

d

yxfyx

,

:ionSpecificat CondtionsBoundary

),(

:Equation alDifferenti Example

2

2

2

2

One Degree of Freedom per Node

Two-DimensionalVector/Tensor-Valued Problems

u1

u2

1

2

3 u3

v1

v2

v3

Triangular Element

yxy

yxx

y

x

ny

vC

x

uCn

x

v

y

uCT

nx

v

y

uCn

y

vC

x

uCT

vu

Fy

v

x

u

y

Ev

Fy

v

x

u

x

Eu

221266

661211

2

2

,

ConditonsBoundary

0)1(2

0)1(2

ntsDisplaceme of Termsin Equations Field Elasticity

Two Degrees of Freedom per Node

Development of Finite Element Equation• The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem

• For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from Either Strength of Materials or Theory of Elasticity

• FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or Weighted Residual Methods

Variational-Virtual Work Method

Based on the concept of virtual displacements, leads to relations between internal and external virtual work and to minimization of system potential energy for equilibrium

Weighted Residual Method

Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement. For stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational methods.

Direct Method

Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process

Simple Element Equation ExampleDirect Stiffness Derivation

1 2k

u1 u2

F1 F2

}{}]{[

rm Matrix Foinor

2 Nodeat mEquilibriu

1 Nodeat mEquilibriu

2

1

2

1

212

211

FuK

F

F

u

u

kk

kk

kukuF

kukuF

Stiffness Matrix Nodal Force Vector

Common Approximation SchemesOne-Dimensional Examples

Linear Quadratic Cubic

Polynomial Approximation

Most often polynomials are used to construct approximation functions for each element. Depending on the order of approximation, different numbers of element parameters are needed to construct the appropriate function.

Special Approximation

For some cases (e.g. infinite elements, crack or other singular elements) the approximation function is chosen to have special properties as determined from theoretical considerations