UNIT-1: Fundamentals Concepts of FEA 1.0 Introduction€¦ · UNIT-1: Fundamentals Concepts of FEA...

402050 B Finite Element Analysis

UNIT-1: Fundamentals Concepts of FEA 1.0 Introduction: The finite element method is a numerical method for solving problems of engineering and

mathematical physics. Typical problem areas of interest in engineering and mathematical

physics that are solvable by use of the finite element method include structural analysis, heat

transfer, fluid flow, mass transport, and electromagnetic potential. For problems involving complicated geometries, loadings, and material properties, it is

generally not possible to obtain analytical mathematical solutions. Analytical solutions are

those given by a mathematical expression that yields the values of the desired unknown

quantities at any location in a body (here total structure or physical system of interest) and are

thus valid for an infinite number of locations in the body. These analytical solutions generally require the solution of ordinary or partial differential

equations, which, because of the complicated geometries, loadings, and material properties,

are not usually obtainable. Hence we need to rely on numerical methods, such as the finite

element method, for acceptable solutions. The finite element formulation of the problem

results in a system of simultaneous algebraic equations for solution, rather than requiring the

solution of differential equations. These numerical methods yield approximate values of the

unknowns at discrete numbers of points in the continuum. Hence this process of modeling a

body by dividing it into an equivalent system of smaller bodies or units (finite elements)

interconnected at points common to two or more elements (nodal points or nodes) and/or

boundary lines and/or surfaces is called discretization. In the finite element method, instead

of solving the problem for the entire body in one operation, we formulate the equations for

each finite element and combine them to obtain the solution of the whole body. Briefly, the solution for structural problems typically refers to determining the displacements

at each node and the stresses within each element making up the structure that is subjected to

applied loads. In non structural problems, the nodal unknowns may, for instance, be

temperatures or fluid pressures due to thermal or fluid fluxes.

2.0 Brief History of FEM Engineers, physicists and mathematicians have developed finite element method

independently. In 1943 Courant made an effort to use piecewise continuous functions defined

over triangular domain.


After that it took nearly a decade to use this distribution idea. In fifties renewed interest in

this field was shown by Polya, Hersh and Weinberger. Argyris and Kelsey introduced the

concept of applying energy principles to the formation of structural analysis problems in

1960. In the same year Clough introduced the word ‘Finite Element Method’. In sixties convergence aspect of the finite element method was pursued more rigorously. One

such study by Melesh led to the formulation of the finite element method based on the

principles of minimum potential energy. Soon after that de Veubeke introduced equilibrium

elements based on the principles of minimum potential energy, Pion introduced the concept

of hybrid element using the duel principle of minimum potential energy and minimum

complementary energy. In Late 1960’s and 1970’s, considerable progress was made in the field of finite element

analysis. The improvements in the speed and memory capacity of computers largely

contributed to the progress and success of this method. In the field of solid mechanics from

the initial attention focused on the elastic analysis of plane stress and plane strain problems,

the method has been successfully extended to the cases of the analysis of three dimensional

problems, stability and vibration problems, non-linear analysis.

2.1 Methods to Solve Any Engineering Problem


2.2 Introduction to Different Numerical Methods

1) Finite Element Method (FEM):

FEM is the most popular numerical method. The Finite Element Method (FEM) is a

numerical technique used to determine the approximated solution for partial differential

equations (PDE) on a defined domain (W). To solve the PDE, the primary challenge is to

create a function base that can approximate the solution. There are many ways of building the

approximation base and how this is done is determined by the formulation selected. The

Finite Element Method has a very good performance to solve partial differential equations

over complex domains that can vary with time. Applications - Linear, nonlinear, buckling,

thermal, dynamic and fatigue analysis. 2) Boundary Element Method (BEM):

This is a very powerful and efficient technique to solve acoustics or NVH problems. Just like

the finite element method, it also requires nodes and elements, but as the name suggests it

only considers the outer boundary of the domain. So when the problem is of a volume, only

the outer surfaces are considered. If the domain is of an area, then only the outer periphery is

considered. This way it reduces the dimensionality of the problem by a degree of one and

thus solving the problem faster.

The Boundary Element Method (BEM) is a numerical method of solving linear PDE which

have been formulated as integral equations. The integral equation may be regarded as an

exact solution of the governing partial differential equation. The BEM attempts to use the

given boundary conditions to fit boundary values into the integral equation, rather than values

throughout the space defined by a partial differential equation. Once this is done, in the post-

processing stage, the integral equation can then be used again to calculate numerically the

solution directly at any desired point in the interior of the solution domain. The boundary

element method is often more efficient than other methods, including finite elements, in

terms of computational resources for problems where there is a small surface/volume ratio.

Conceptually, it works by constructing a “mesh” over the modeled surface.

However, for many problems boundary element methods are significantly less efficient than

volume-discretization methods like FDM, FVM or FEM. 3) Finite Volume Method (FVM):

The Finite Volume Method (FVM) is a method for representing and evaluating partial

differential equations as algebraic equations. It is very similar to FDM, where the values are

calculated at discrete volumes on a generic geometry. In the FVM, volume integrals in a


partial differential equation that contain a divergence term are converted to surface integrals,

using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of

each finite volume. Because the flux entering a given volume is identical to that leaving the

adjacent volume, these methods are conservative. Another advantage of the finite volume

method is that it is easily formulated to allow for unstructured meshes. The method is used in

many computational fluid dynamics packages. 4) Finite Difference Method (FDM):

Finite Element and Finite Difference Methods share many common things. In general the

Finite Difference Method is described as a way to solve differential equation. It uses Taylor’s

series to convert a differential equation to an algebraic equation. In the conversion process,

higher order terms are neglected. It is used in combination with BEM or FVM to solve

thermal and CFD coupled problems.

3.0 Finite Element Terminology

1) The minimum number of parameters (motion, coordinates, temperature, etc.) required

defining the position and state of any entity completely in space is known as degrees of

freedom (dof). 2) The total DOFs for a given mesh model is equal to the number of nodes multiplied by the

number of dof per node. 3) All of the elements do not always have 6 dofs per node. The number of dofs depends on

the type of element (1D, 2D, 3D), the family of element (thin shell, plane stress, plane

strain, membrane, etc.), and the type of analysis. For example, for a structural analysis, a

thin shell element has 6 dof/node (displacement unknown, 3 translations and 3 rotations)

while the same element when used for thermal analysis has single dof /node (temperature

unknown). 4) Discretization of Problem:

All real life objects are continuous. This means there is no physical gap between any two

consecutive particles. As per material science, any object is made up of small particles,

particles of molecules, molecules of atoms, and so on and they are bonded together by the

force of attraction. Solving a real life problem with the continuous material approach is

difficult. The basis of all numerical methods is to simplify the problem by discretizing

(discontinuation) it. In other words, nodes work like atoms and the gap in between the


nodes is filled by an entity called an element. Calculations are made at the nodes and

results are interpolated for the elements. 5) Each part of the discretized body is called as an element. Every element has one or more

nodes. Elements are connected to each other at these nodes. 6) Finite – Any continuous object has infinite degrees of freedom and it is not possible to

solve the problem in this format. The Finite Element Method reduces the degrees of

freedom from infinite to finite with the help of discretization or meshing (nodes and

elements).

7) Element – All of the calculations are made at a limited number of points known as nodes.

The entity joining nodes and forming a specific shape such as quadrilateral or triangular

is known as an Element. To get the value of a variable (say displacement) anywhere in

between the calculation points, an interpolation function (as per the shape of the element)

is used. 8) Method - There are 3 methods to solve any engineering problem. Finite element analysis

belongs to the numerical method category.

4.0 General FEM Procedure

In engineering problems there are some basic unknowns. If they are found, the behaviour of

the entire structure can be predicted. The basic unknowns or the Field variables which are

encountered in the engineering problems are displacements in solid mechanics, velocities in

fluid mechanics, electric and magnetic potentials in electrical engineering and temperatures

in heat flow problems. In a continuum, these unknowns are infinite. The finite element procedure reduces such

unknowns to a finite number by dividing the solution region into small parts called elements

and by expressing the unknown field variables in terms of assumed approximating functions

(Interpolating functions/Shape functions) within each element. The approximating functions

are defined in terms of field variables of specified points called nodes or nodal points. Thus

in the finite element analysis the unknowns are the field variables of the nodal points. Once

these are found the field variables at any point can be found by using interpolation functions. After selecting elements and nodal unknowns next step in finite element analysis is to

assemble element properties for each element. For example, in solid mechanics, we have to

find the force-displacement i.e. stiffness characteristics of each individual element.

Mathematically this relationship is of the form


[k]e {δ}e = {F}e

where [k]e is element stiffness matrix, {δ}e is nodal displacement vector of the element and

{F}e is nodal force vector. The element of stiffness matrix kij represent the force in

coordinates direction ‘i’ due to a unit displacement in coordinate direction ‘j’. Four methods

are available for formulating these element properties viz. direct approach, variational

approach, weighted residual approach and energy balance approach. Any one of these

methods can be used for assembling element properties. In solid mechanics variational

approach is commonly employed to assemble stiffness matrix and nodal force vector

(consistant loads). Element properties are used to assemble global properties/structure properties to get system

equations [k] {δ} = {F}. Then the boundary conditions are imposed. The solution of these simultaneous equations

gives the nodal unknowns. Using these nodal values additional calculations are made to get

the required values e.g. stresses, strains, moments, etc. in solid mechanics problems. Thus the

various steps involved in the finite element analysis are: (i) Select suitable field variables and the elements. (ii) Discritise the continua. (iii) Select interpolation functions. (iv) Find the element properties. (v) Assemble element properties to get global properties. (vi) Impose the boundary conditions. (vii) Solve the system equations to get the nodal unknowns. (viii) Make the additional calculations to get the required values.

5.0 Applications of FEM in various fields

The finite element method can be used to analyze both structural and non-structural

problems. Typical structural areas include

1. Stress analysis, including truss and frame analysis, and stress concentration

problems typically associated with holes, fillets, or other changes in geometry in a

body

2. Buckling

3. Vibration analysis


Nonstructural problems include

1. Heat transfer

2. Fluid flow, including seepage through porous media

3. Distribution of electric or magnetic potential

Some biomechanical engineering problems (which may include stress analysis) typically

include analyses of human spine, skull, hip joints, jaw/gum tooth implants, heart, and eye.

6.0 Advantages and disadvantages of FEM

Advantages of FEM

i. Finite element method can deal with complex material models such as homogeneous,

heterogeneous, isotropic, anisotropic, nonlinear models.

ii. Any complex geometry can be analyzed using finite element method. iii. Any type of boundary conditions can be accommodated in FEM.

iv. Non-linearity, time-dependent problems and any arbitrary loading conditions can be

handled easily.

v. Finite element method can be used for any type of physical problems. vi. FE formulation and development of codes are possible. The formulation and

developed codes for one class can be used for other class of problem with little

modifications. vii. Software’s are available and are affordable.

i. Parametric study is difficult as finite element method cannot provide closed form

solution.

ii. Skilled user, reliable program and computer are essential iii. Many input data are required and voluminous output needs to be sorted and analyzed.

iv. To define a good model experience an engineering judgement is required.

v. A general purpose program has extensive documentation which cannot be ignored.

7.0 P & h Formulation

‘h’ is for the linear dimension that characterizes the size of an element. An h-refinement

consists of adding elements of the same type as shown in figure.


P is the degree of the highest complete polynomial in the element field quantity.

It increases the degree of polynomial within elements without changing the number of

elements.

This can be done in following ways :

Adding dot to the existing nodes.

Adding nodes on the existing inter-element

boundaries Adding internal dot.

8.0 Consistent Units System

Careless use of units in engineering calculations leads to numerous errors. Such errors can be

minimized by using consistent set of units, while solving any problem. The fundamental units

used in engineering problems are mass, length, time and temperature. All other units are

based on these fundamental units. Nowadays the International System of Units (SI) is used

generally to mark the dimension unit as shown in below table:

Table Physical Quantities and Their Units

symbol SI measurement units symbol unit dimensions

distance d meter m m

mass m kilogram kg kg

time t second s s

temperature T Kelvin K K

acceleration a meter per second squared m/s2

m/s2

area A square meter m2

m2

electric current I ampere A C/s

pressure p pascal Pa kg/m.s2

velocity v meter per second m/s m/s

volume V cubic meter m3

m3

work W joule J kg.m

2/s

2


9.0 Review of Solid Mechanics

Stresses:

Strain:

The linear strain -displacement relation is given by:

For a three dimensional problem, the strain stress relation for isotropic material is,


Plane Stress Problems:

The thin plates subject to forces in their plane only, fall under this category of the problems.

Plane Strain Problems:

A long body subject to significant lateral forces but very little longitudinal forces falls under

this category of problems. Examples of such problems are pipes, long strip footings, retaining

walls, gravity dams, tunnels, etc. In these problems, except for a small distance at the ends,

state of stress is represented by any small longitudinal strip. The displacement in longitudinal

direction (z-direction) is zero in typical strip.

Axi-symmetric Problems:

Axi-symmetric structures are those which can be generated by rotating a line or curve about

an axis. Cylinder are the common examples of axi-symmetric structures. If such structures

are subjected to axi-symmetric loadings like uniform internal or external pressures, uniform

self weight or live load uniform over the surface, there exists symmetry about any axis. The


advantage of symmetry may be made use to simplify the analysis. In these problems

cylindrical coordinates can be used advantageously. Because of symmetry, the stress

components are independent of the angular (θ) coordinate.

Strain Energy:

The energy stored in the deformation process. Strain energy / unit volume

=

1

(force) (deflection)

2

= 1 (stress Area)

deflection length

2

length 1 1

=

(A) (L) =

AL

2 2 The internal strain energy of the beam member accounts only for bending moment deformation.

Total Potential Energy:

Potential energy is the capacity to do the work by the forces acting on deformable bodies.

The forces acting on a body may be classified as external forces and internal forces. External

forces are the applied loads while internal forces are the stresses developed in the body.

Hence the total potential energy is the sum of internal and external potential energies.

1 T T T – = 2 {} { } dv – {u} {b} dv – {u} { t } dA – ui Pi V V A


Where, {} = Strain matrix

{} = Stress matrix

{u} = Displacement matrix

{b} = body force matrix –

= surface force matrix

{ t } V = total volume of the body

A = portion of surface area over which surface force is acting

Pi = point load

ui = corresponding displacement due to point load.

10.0 Essential and natural boundary conditions

Essential Boundary Conditions

These boundary conditions place constraints on the magnitude of the displacements and slope at any given position on the boundary.

At point (1), x = 0 Displacement, y = 0

dy

Slope, dx = 0

Natural Boundary Conditions

The constraints placed on the boundaries with respect to the magnitude of forces or moments.

At point ‘2’ where x = L

d3y

Shear force, EI dx3 = 0 d

2y

Bending moment, EI dx2 = 0

11.0 Review of Matrix Algebra

Simultaneous equations: are of the form

a11 x1 + a12 x2 + a13 x3 + … + a1n xn = b1

a21 x1 + a22 x2 + a23 x3 + … + a2n xn = b2

am1x1 + am2 x2 + am3 x3 + … + amn xn = bm

Which can be written as a matrix equation


a 11

a 12

a 13

a 1n x b

1 1 a a a a x b 21 22 23 2n 2

2

a

a

a

a

x

m1 m2 m3 mn n b

n size:

mxn nx1 mx1

A x b

Matrix Addition:

a A B

a

11 a

12

a

21 22

b

b 11 12 b b (2 x 2) 21 22

a b a b 11 11 12 12 a b a b 21 22 (2 x 2) 21 22

(2 x 2)

Matrix Transpose:

a

11

a

a

A

a

a

A

T

21 11 21 31 (1x 3)

a 31

( 3x1)

a

11 a

12

a

a

a

11 21 31

A a21 a22

A T

a

a

a

a

a

12 22 32 31 32 ( 3x 2)

( 2 x 3)

Determinant of a matrix (as it relates to matrix singularity)

2 D:

a 11

a

a 21 a

3D:

12 a11a22 a21a12 22

a

a

a

11

21

31

a

a

a

12

22

32

a

a

a

13

a

a 23 11

a 33

22

32

a

a

a 33 a12 a23

21

31

a

a

a 33 a13 a23

21

31

a

a

22 32


Diagonal Matrix

d 1

0 0

d2

D 0 0

0 0

d

3

D d1d2d3

Identity Matrix

1 0

0

I 0 1

0

0 0

1

I 1 ( for any size)

Triangular Matrix

d 1

0 0

LowerTriangular d

2

t1 0

t3

t2

d3

d 1

t 1

t 2

UpperTriangular 0 d2

t3

0 0

d3

UT LT d1d2 d3

Matrix Inverses:

The inverse of a n n matrix A , if it exists, is denoted by A 1 , and is defined to be the n n

matrix where

AA

1 A

1 A I ,

where I is the n n identity matrix. It can be shown that A1 is unique. Note the inverse only

exists for square matrices where the row and column number are the same.

Banded Skyline Solutions:

Symmetric Banded Matrix

A symmetric matrix is a square matrix whose elements satisfy aij = aji or A = AT


In a bonded matrix, all of the non-zero elements are contained within a band and

outside of the band all elements are zero. Consider an (n x n) symmetric banded matrix.

Symmetric

0

nbw is called the half-band width. Since only the nonzero elements need to be stored,

0

1.8.1.2 Skyline Storage

If there are zeroes at the top of a column, only the elements starting from the first

nonzero value need to be stored. The line separating the top zeroes from the first

nonzero element is called the skyline. Consider the following example.


Column 1 2 2 4 4 4 3 5

height

a

11

a12 0

a14 0 0 0 0

a

22

a23

a24

a25 0 0 0

a

33

a34 0

a36 0 0 Skyline

a

44 0 a

46 0 a

48

a

55

a56

a57 0

a

66

a67

a68

a

77

a78

a

88

For efficiency only the active column need to be stored as follows

a

11 1 (Diagonal pointex ID)

a

12 1

a

22 3 3

A = a

23 ID = 5

a

33 5 9

: 13

: 17

: 20

a

88 25 25

12.0 Introduction to Solvers

There are three types of linear equation solvers:

Direct (Sparse) solver

It is a default solver in Ansys software for virtually all analyses.

It is design to run in different modes of operation, depending on the amount of memory

available.


It is a robust solver and solver’s mode of operation can have significant impact on run

time. Memory usage for a direct sparse solver is determined by several steps.

In software, the matrix that is input to the sparse solver is assembled entirely in

memory before being written to the full file. Then the solver reads the full file, process the matrix, factors the matrix and computes

the solution. Direct method solvers factor the input matrix into the produce of a lower triangular

matrix in order to solve the system of equations. For symmetric input matrices, only the lower triangular factor is required since it is

equivalent to the transpose of the upper triangular factor. The process of factorization produces matrix factors which are 10 to 20 times larger

than the input matrix. The calculation of this factor is computationally intensive.

Iterative (PCG) solver

Iterative solvers offer a powerful alternative to sparse solver.

They do not require a costly matrix factorization of the assembled matrix. Iterative solvers are not default solver due to following reason.

It proceeds from an initial random guess to the solution by an iterative process and are

dependent on matrix properties that can cause the iterative solver to fail to converge in

some case. The important factor in iterative solver is the preconditioning step.

The preconditioned conjugate gradient (PC4) iterative solver uses two different

proprietary pre-conditioners which have been specifically developed for a wide range

of element types. It is effective for problems with poor element aspect ratios.

Block lanczos solver

Finding the natural frequencies and mode shapes of a structure is one of the most

computationally demanding task. Specific equation solvers called eigen solvers are used to solve for the natural

frequencies and mode shapes.


There are three eigen solvers for modal analyses of undamped systems :

(a) Spares solver - based Block Lanczos

(b) PCG Lanczos solver

(c) Super node solver Block Lanczos solver uses the sparese direct solver.

Block lanczos algorithm computes blocks of vectors apart from two matrix

factorizations. These vectors are stored on files during Block Lanczos iterations. The size of these files grows as more modes are computed.

Each Block Lanczos iteration requires multiple solves using the large matrix factor file

and one in-memory block of vectors. The PCG Lanczos solver represents a breakthrough in modal analysis capability

because it allows users to extend the maximum size of models used in modal analyses

well beyond the capacity of direct solver – based eigen solvers.

13.0 Introduction to different approaches used in FEA

Methods for formulating the stiffness matrix

1) Direct Method: It is an extension of matrix displacement approach 2) Variational Method 3) Weighted Residual Method

The direct method is easy to understand but difficult to formulate using computer

programming. While the Variational and Weighted Residual Methods are difficult to

understand, but easy from a programming point of view. That’s the reason why all software

codes either use the Variational or Weighted Residual Method formulation.

Direct Approach

The standard form of matrix displacement equation is [K] {x} = {F}

Where [K] = Stiffness matrix {x} =

displacement vector {F} =

Force vector

Variational approach: Raleigh Ritz approach / energy approach

A structure consists of different members such as link, truss or frame which can be

represented by finite number of degrees of freedom.


Whereas an elastic solid has infinite number of degrees of freedom where these degrees

of freedom show the displacement of every point in the solid body. Such an elastic body is mathematically represented in terms of partial differential

equations. There is a very few chance that a stress field or displacement field can solve a

differential equation and an satisfy boundary conditions. This difficulty of solving the differential equation can be eliminated by using Rayleigh

– Ritz technique to a functional such as potential energy that describes the problem. Rayleigh – Ritz technique converts a problem with finite number of degrees of freedom

and is represented by algebraic equation rather than differential equations. For an elastic solid body, the total potential energy ‘’ will be used to find out a trial

solution.

Rayleigh-Ritz technique is composed of assuming an displacement field by,

u = bi i (x, y, z) i = 1 to p

v = bj j (x, y, z) j = p + 1 to q

w = bk k (x, y, z) k = q + 1 to r

p > q > r

where i are assumed as polynomials.

u, v, w are displacements that satisfy the single – valued nature of displacements and

specified boundary conditions. Potential energy for an elastic body can be given as

= (b1, b2, b3…bn)

Where n = number of independent unknowns. Now for ‘’ to be minimum, a set of ‘n’ equations are obtained as

= 0 i = 1, 2, 3…n

bi Weighted Residual Method

Weighted residual method is based upon minimization of the residual left after an trail

or approximate solution is substituted into the differential equation. Let us consider an differential equation as follows :

2 u*

–

u*

= f(x) …1(a) x2

t

Where, u* = unknown,


x = co-ordinate

t = time

f(x) = forcing function

Rewriting Equation (1.12.1(a)) as

Lu* = f …1(b)

Where, L =

2

+

is the differential operator.

x2

t

We are seeking an approximate solution to Equation (1) and denote an approximate

function ‘u’ for u* as

u = o + 1 1 + 2 2 + ….. (2)

Where, o , 1 , 2 … are known functions chosen in such a way as to satisfy boundary

conditions.

Now, if the approximate solution ‘u’ is substituted into Equation (1), we are left with a

residual.

R(x) = Lu – f (3)

If u = u*, then R(x) = 0

In the method of weighted residual the aim is to find an approximate solution ‘u’ for

‘u*’ such that the residual R(x) in Equation (3) is made as small as possible or minimized.

A number of schemes are possible to achieve the aim of minimization of residue R(x).

Mathematically the idea of minimization can be expressed as

R(x) W (x) dx = 0 …(4)

D i

Where, ‘D’ denotes the domain of the system under consideration,

Wi denotes weighting functions,

i = 1, 2, … n.

Various residual schemes use different weighting functions. There are four different

schemes available to choose weighting functions. (i) Collocation method

This method minimizes the residual or vanishes at a specific point.

In this method the number of points selected is equal to the number of

coefficients in the approximate solution.

In this method weighting functions are in the form of impulse functions

Wi (x) = (x–xi).


(ii) Subdomain method

In this method the aim is to vanish the integral of the residue over an selected

region.

And the number of integration interval is equal to the number of coefficients in

the trail solution.

In this method the weighting function over a specified region is selected as

Wi (x) = 1.

(iii) Least square method

In this method the residue is used as weighting function.

An error function is given by

A

e = [R(x)]2dx

0

With respect to coefficients in the trail solution this error function is minimized.(iv) Galerkin’s method

In this method the weighting functions are same as that used in the trial functions.

This method finds application in problems involving first derivative terms.

UNIT-1: Fundamentals Concepts of FEA 1.0 Introduction€¦ · UNIT-1: Fundamentals Concepts of FEA...

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