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engineering your designs www.pro-sim.com Basics of FEM Do’s and Don’ts, common mistakes (Workshop on Advanced NonLinear Finite Element Analysis April 9-10th 2010: COEP Pune) B Sreehari Kumar Email: [email protected] [email protected]
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Page 1: Basics of FEM

engineering your designswww.pro-sim.com

Basics of FEM

Do’s and Don’ts, common mistakes(Workshop on Advanced NonLinear Finite Element Analysis

April 9-10th 2010: COEP Pune)

B Sreehari KumarEmail: [email protected]

[email protected]

Page 2: Basics of FEM

engineering your designswww.pro-sim.com

Overview

• Background

• Basics of Solid Mechanics

• Finite Element Method - A numerical tool for

designers

• Sources of Error in the FEM

• Technologies that Compete With the FEM

• Future Trends in the FEM and Simulation

• Selected FEM Resources on the Internet

Page 3: Basics of FEM

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BACKGROUND

• The finite element method is a numerical and computer-intensive technique of

solving a variety of practical engineering problems that arise in different fields

including, heat transfer, fluid mechanics, and solid and structural mechanics.

• Computer implementation of the method is modular, and allows general

purpose as well as special purpose computer code development with different

types of elements and degrees of approximation.

• There exist a large number of general purpose finite element computer

programs (e.g., ABAQUS, ADINA, ANSYS, MSCNASTRAN, NISA, etc) with varying

degree of sophistication and analysis capabilities to analyze physical problems

with complex domains, physical features (e.g., geometric and material

nonlinearities), and subjected to thermal, mechanical and/or hydrodynamic

loads.

• The main task of an engineer or scientist in using a readily available finite--

element computer program or developing a finite-element computer program

for his or her specific problem lies in the understanding of the basic theory

governing the problem, underlying assumptions and limitations of the theory,

and the details of the finite element model.

Page 4: Basics of FEM

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Engineering mechanics and Industry

Engineering mechanics

Stress Analysis

Seismic Analysis

Vibration Analysis

Laminated Composite Analysis

Fatigue and Fracture Analysis

Optimization

Motion and Linkage Analysis

Rotor Dynamics

Thermal Analysis

Printed Circuit Board (PCB) Analysis

Computational Fluid Dynamics (CFD) Analysis

Electromagnetic Analysis

MEMS & NANO Devices

Crash Simulations

Engineering Industry

Automobile

Rail, Road

Space

Nuclear

Defense

Heavy Engineering

Ship Technology

Civil Engg. Structures

Bio Medical Engineering …

Page 5: Basics of FEM

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Stress

• Definition of Stress:

Stress is the internal resistance of a material to the distorting effects of an

external force or load.

Stress = σ = P / A

where: σ = stress ; P = applied force; A = cross-sectional area

• Three types of stress:

Tensile stress is the type of stress in which the two sections of material on

either side of a stress plane tend to pull apart or elongate.

Compressive stress is the reverse of tensile stress. Adjacent parts of the

material tend to press against each other.

Shear stress exists when two parts of a material tend to slide across each

other upon application of force parallel to that plane

• Types of Stresses:

Structural Stress, Residual Stresses, Pressure Stresses, Flow Stresses,

Thermal Stresses, Fatigue Stresses

Page 6: Basics of FEM

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Strain

• Definition of Strain.

Strain is the proportional dimensional change, or the intensity or degree of

distortion, in a material under stress.

Strain = Є = ∆/L

where: Є = strain; ∆ = total elongation; L = original length

• Two types of strain:

Elastic strain is a transitory dimensional change that exists only while the

initiating stress is applied and disappears immediately upon removal of

the stress.

Plastic strain (plastic deformation) is a dimensional change that does not

disappear when the initiating stress is removed.

• The phenomenon of elastic strain and plastic deformation in a material are

called elasticity and plasticity, respectively.

Page 7: Basics of FEM

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Hook’s Law & Young’s Modulus

• Hook’s Law:

Hooke's Law states that in the elastic range of a material strain is

proportional to stress. It is measured by using the following equation:

Δ = Pl / AE

P = force producing extension of bar

l = length of bar

A = cross-sectional area of bar

Δ = total elongation of bar

E = elastic constant of the material, called the Modulus of Elasticity, or

Young's Modulus

• Young's Modulus:

Young's Modulus (Elastic Modulus) is the ratio of stress to strain, or the

gradient of the stress-strain graph. It is measured using the following

equation: E = σ / Є

Where E = Young's Modulus; σ = stress; Є = strain

Page 8: Basics of FEM

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Stress vs. Strain

Where:

E = Modulus of Elasticity

σ = stress

ε = strain

Strain: Stress:l

δε =

A

F=σ

Hooke’s law:

εσ E=

Page 9: Basics of FEM

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By superposition of the components of strain in the x, y,

and z directions, the strain along each axis can be

written as:

( )[ ]zyxx vE

σσσε +−=1

( )[ ]xzyy vE

σσσε +−=1

( )[ ]yxzz vE

σσσε +−=1

Generalized Hooke’s Law Contd..

Page 10: Basics of FEM

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The shearing stresses acting on the unit cube produce shearing

strains.

The proportionality constant G is the modulus of elasticity in

shear, or the modulus of rigidity. Values of G are usually

determined from a torsion test. For isotropic material

xyxy Gγτ =

yzyz Gγτ =

xzxz Gγτ =

Generalized Hooke’s Law Contd..

( )v

EG

+=

12

Page 11: Basics of FEM

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Force

Static Force

Dynamic Force

Transient,

Frequency response,

Random,

Shock Response

Force due to Contact

Force due to Creep

……

Page 12: Basics of FEM

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Theory of elasticity

equilibrium equations in X, Y and Z directions are:

Theory of plasticity

Theory of beams

Theory of plates

Theory of shells

Contact Mechanics

Theories associated with Solid mechanics

Page 13: Basics of FEM

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Classification of Solid-Mechanics Problems

Analysis of solids

Static Dynamics

Behavior of Solids

Linear Nonlinear

Material

Fracture

GeometricLarge Displacement

Instability

Plasticity

ViscoplasticityGeometric

Classification of solids

Skeletal Systems1D Elements

Plates and Shells2D Elements

Solid Blocks3D Elements

TrussesCablesPipes

Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elements

Brick ElementsTetrahedral ElementsGeneral Elements

Elementary Advanced

Stress Stiffening

Page 14: Basics of FEM

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Linear Analysis:

(1) The displacements are very small and that the material is linearly elastic.

(2) The geometry, loading, and constitutive behavior remain unchanged during

the entire deformation process.

F = c u

Non-Linear Analysis:

(1) Moderately large displacements and rotations but small strains (rotations of

line elements are moderately large, but their extensions and changes of angles

between two line elements are small).

(2) Large displacements, rotations, and strains (the extension of a line element

and angle changes between two line elements are large, and displacements and

rotations of a line element are also large).

[M] - Assembled Mass Matrix ;[C] - Overall Damping Matrix; {X} -

Displacement

[K] - Assembled Stiffness Matrix ;{F(t)} - Forcing Function

Assumptions

[ ] { } [ ] { } { })(),( tFXXGXCXM =++ &&&&

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Basic equation for a linear static analysis

[K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld}

[K] = total stiffness matrix

{u} = nodal displacement

{Fapp} = applied nodal force load vector

{Fth} = applied element thermal load vector

{Fpr} = applied element pressure load vector

{Fma} = applied element body force vector

{Fpl} = element plastic strain load vector

{Fcr} = element creep strain load vector

{Fsw} = element swelling strain load vector

{Fld} = element large deflection load vector

Page 16: Basics of FEM

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GEOMETRIC NONLINEARITY

• Large displacements, large rotations, finite strains

• Total and updated Lagrangian formulation

• Large strain deformation

• Stress stiffening

• Post buckling analysis

LOADING

• Conservative loading (fixed direction force, moment, and

pressure)

• Non-conservative loading (deformation dependent follower

concentrated force and follower pressure)

• Body forces (weight and inertia)

• Thermal loading (specified temperature vs. time curve)

Geometric Nonlinearity

Page 17: Basics of FEM

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MATERIAL NONLINEARITY

· Material models include von Mises, Tresca, Mohr-Coulomg, and

Drucker-Prager yield criterion

· Elastic perfectly plastic, elastoplastic with isotropic, kinematic or mixed

work hardening

· Uniaxial stress-strain curve description includes elastic perfectly

plastic, elastic linear hardening, elastic piece-wise linear hardening, and

Ramberg-Osgood curve

· Hyperelasticity and rubber-like material behavior, material models

include generalized Mooney-Rivlin, Blatz-Ko, Alexander, etc.

· Creep laws such as Norton, McVetty, Soderberg, Dorn, ORNL, etc. are

supported. These laws can be expressed as general functions of time, stress,

and temperature

· Anisotropic elastoplastic material model with linear of piece-wise

linear hardening for composite shell elements

· Temperature dependent inelastic properties

· User-defined material model

Material Nonlinearity

Page 18: Basics of FEM

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Material Nonlinearity

*PLASTIC

Card Set1: Plasticity properties for elastoplastic material models

Page 19: Basics of FEM

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Introduction to the Finite Element Method

• What is the finite element method?

The Finite Element Method (FEM) is a numerical technique to obtain

approximate solutions to a wide variety of engineering problems where the

variables are related by means of algebraic, differential and integral

equations.

• Mathematical Model:

A set of algebraic, differential, and/or integral equations that govern the

physical phenomenon of a particular system.

The model is based on a set of assumptions and restrictions placed on the

phenomenon and the laws of physics that govern it.

• Numerical Method:

An inexact procedure by which the governing equations can be solved for

the dependent (unknowns) variables.

Page 20: Basics of FEM

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Introduction to the Finite Element Method …

• Mathematical Models

A mathematical model of the problem being studied must represent the physics

consistent with the goals of the study

• Desirable Features of a Computational Approach:

Must preserve all features of the mathematical model in the formulation and

associated computational model

Avoid ad-hoc approaches to `fix’ numerical deficiencies of the computational

model (robustness)

Where is finite element analysis used?

Finite elements has become the defacto industry standard for solving multi-

disciplinary engineering problems that can be described by equations of calculus.

Applications cut across several industries by virtue of the applications – solid

mechanics (civil, aerospace, automotive, mechanical, biomedical, electronic), fluid

mechanics (geotechnical, aerospace, electronic, environmental, hydraulics,

biomedical, chemical), heat transfer (automotive, aerospace, electronic, chemical),

acoustics (automotive, mechanical, aerospace), electromagnetics (electronic,

aerospace) and many, many more.

Page 21: Basics of FEM

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Finite Element Method Defined

• Many problems in engineering and applied science are governed by differential or integral equations.

• The solutions to these equations would provide an exact, closed-formsolution to the particular problem being studied.

Analytical, Classical Methods

•Separation of variables

•General solution in terms of series functions

•Conformal Transformation

•Integral Transform Technique

• However, complexities in the geometry, properties and in the

boundary conditions that are seen in most real-world problems usually

means that an exact solution cannot be obtained or obtained in a

reasonable amount of time.

Page 22: Basics of FEM

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Finite Element Method Defined (cont.)

• Current product design cycle times imply that engineers must obtain design solutions in a ‘short’ amount of time.

• They are content to obtain approximate solutions that can be readily obtained in a reasonable time frame, and with reasonable effort. The FEM is one such approximate solution technique.

• The FEM is a numerical procedure for obtaining approximate solutions to many of the problems encountered in engineering analysis.

•Domain methods

•Boundary methods

•Mixed methods

Page 23: Basics of FEM

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Finite Element Method Defined (cont.)

• In the FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements.

• The properties and the governing relationships are assumed over these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes.

• An assembly process is used to link the individual elements to the given system. When the effects of loads and boundary conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained.

• Solution of these equations gives the approximate behavior of the continuum or system.

Page 24: Basics of FEM

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Finite Element Method Defined (cont.)

• The continuum has an infinite number of degrees-of-freedom

(DOF), while the discretized model has a finite number of DOF.

This is the origin of the name, finite element method.

• The number of equations is usually rather large for most real-

world applications of the FEM, and requires the computational

power of the digital computer. The FEM has little practical

value if the digital computer were not available.

• Advances in and ready availability of computers and software

has brought the FEM within reach of engineers working in

small industries, and even students.

Page 25: Basics of FEM

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Finite Element Method Defined (cont.)

Two features of the finite element method are worth noting.

• The piecewise approximation of the physical field

(continuum) on finite elements provides good precision even

with simple approximating functions. Simply increasing the

number of elements can achieve increasing precision.

• The locality of the approximation leads to sparse equation

systems for a discretized problem. This helps to ease the

solution of problems having very large numbers of nodal

unknowns. It is not uncommon today to solve systems

containing a million primary unknowns.

Page 26: Basics of FEM

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Origins of the Finite Element Method

• It is difficult to document the exact origin of the FEM, because the

basic concepts have evolved over a period of 150 or more years.

• The term finite element was first coined by Clough in 1960. In the

early 1960s, engineers used the method for approximate solution of

problems in stress analysis, fluid flow, heat transfer, and other areas.

• The first book on the FEM by Zienkiewicz and Chung was

published in 1967.

• In the late 1960s and early 1970s, the FEM was applied to a wide

variety of engineering problems.

Page 27: Basics of FEM

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Origins of the Finite Element Method (cont.)

• The 1970s marked advances in mathematical treatments, including

the development of new elements, and convergence studies.

• Most commercial FEM software packages originated in the 1970s

(ABAQUS, ADINA, ANSYS, MARK, NISA/DISPLAY, PAFEC)

and 1980s (FENRIS, LARSTRAN ‘80, SESAM ‘80.)

• The FEM is one of the most important developments in

computational methods to occur in the 20th century. In just a few

decades, the method has evolved from one with applications in

structural engineering to a widely utilized and richly varied

computational approach for many scientific and technological areas.

Page 28: Basics of FEM

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Steps in Finite Element Method

• Step 1 - Discretization: The problem domain is discretized into a collection of

simple shapes, or elements.

• Step 2 - Develop Element Equations: Developed using the physics of the problem,

and typically Galerkin’s Method or variational principles.

• Step 3 - Assembly: The element equations for each element in the FEM mesh are

assembled into a set of global equations that model the properties of the entire

system.

• Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless

boundary conditions are applied. They reflect the known values for certain primary

unknowns. Imposing the boundary conditions modifies the global equations.

• Step 5 - Solve for Primary Unknowns: The modified global equations are solved for

the primary unknowns at the nodes.

• Step 6 - Calculate Derived Variables: Calculated using the nodal values of the

primary variables.

Page 29: Basics of FEM

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Process Flow in a Typical FEM Analysis

StartProblemDefinition

Pre-processor

• Reads or generates nodes and elements (ex: ANSYS)

• Reads or generates material property data.

• Reads or generates boundary conditions (loads and constraints.)

Processor

• Generates element shape functions

• Calculates master element equations

• Calculates transformation matrices

• Maps element equations into global system

• Assembles element equations

• Introduces boundary conditions

• Performs solution procedures

Post-processor

• Prints or plots contours of stress components.

• Prints or plots contours of displacements.

• Evaluates and prints error bounds.

Analysis anddesign decisions

Stop

Step 1, Step 4

Step 6

Steps 2, 3, 5

Page 30: Basics of FEM

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Theoretical Basis: Formulating Element Equations

• Several approaches can be used to transform the physical

formulation of a problem to its finite element discrete analogue.

• If the physical formulation of the problem is described as a

differential equation, then the most popular solution method is

the Method of Weighted Residuals.

• If the physical problem can be formulated as the minimization

of a functional, then the Variational Formulation is usually

used.

Page 31: Basics of FEM

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Theoretical Basis: MWR

• One family of methods used to numerically solve differential equations are called the methods of weighted residuals (MWR).

• In the MWR, an approximate solution is substituted into the differential equation. Since the approximate solution does not identically satisfy the equation, a residual, or error term, results.

Consider a differential equationDy’’(x) + Q = 0 (1)

Suppose that y = h(x) is an approximate solution to (1). Substitution then gives Dh’’(x) + Q = R, where R is a nonzero residual. The MWR then requires that

∫ Wi(x)R(x) = 0 (2)

where W i(x) are the weighting functions. The number of weighting functions equals the number of unknown coefficients in the approximate solution.

Page 32: Basics of FEM

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Theoretical Basis: Galerkin’s Method

• There are several choices for the weighting functions, Wi.

• In the Galerkin’s method, the weighting functions are the same functions that were used in the approximating equation.

• The Galerkin’s method yields the same results as the variational method when applied to differential equations that are self-adjoint.

• The MWR is therefore an integral solution method.

• Many readers may find it unusual to see a numerical solution that is based on an integral formulation.

Page 33: Basics of FEM

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Advantages of the Finite Element Method

• Can readily handle complex geometry:

• The heart and power of the FEM.

• Can handle complex analysis types:

• Vibration

• Transients

• Nonlinear

• Heat transfer

• Fluids

• Can handle complex loading:

• Node-based loading (point loads).

• Element-based loading (pressure, thermal, inertial forces).

• Time or frequency dependent loading.

• Can handle complex restraints:

• Indeterminate structures can be analyzed.

Page 34: Basics of FEM

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Advantages of the Finite Element Method (cont.)

• Can handle bodies comprised of nonhomogeneous materials:

• Every element in the model could be assigned a different set of

material properties.

• Can handle bodies comprised of nonisotropic materials:

• Orthotropic

• Anisotropic

• Special material effects are handled:

• Temperature dependent properties.

• Plasticity

• Creep

• Swelling

• Special geometric effects can be modeled:

• Large displacements.

• Large rotations.

• Contact (gap) condition.

Page 35: Basics of FEM

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Disadvantages of the Finite Element Method

• A specific numerical result is obtained for a specific problem.

A general closed-form solution, which would permit one to

examine system response to changes in various parameters, is

not produced.

• The FEM is applied to an approximation of the mathematical

model of a system (the source of so-called inherited errors.)

• Experience and judgment are needed in order to construct a

good finite element model.

• A powerful computer and reliable FEM software are essential.

• Input and output data may be large and tedious to prepare and

interpret.

Page 36: Basics of FEM

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Disadvantages of the Finite Element Method (cont.)

• Numerical problems:

• Computers only carry a finite number of

significant digits.

• Round off and error accumulation.

• Can help the situation by not attaching stiff

(small) elements to flexible (large) elements.

Page 37: Basics of FEM

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FINITE ELEMENT LIBRARY

Page 38: Basics of FEM

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Information Available from Various Types of FEM Analysis

• Static analysis» Deflection » Stresses » Strains » Forces » Energies

• Dynamic analysis» Frequencies » Deflection (mode shape) » Stresses » Strains » Forces » Energies

• Heat transfer analysis

»Temperature

» Heat fluxes

» Thermal gradients

» Heat flow from convection faces

• Fluid analysis

» Pressures

» Gas temperatures

» Convection coefficients

» Velocities

Page 39: Basics of FEM

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Warning: The Computed Answer May Be Wrong

Page 40: Basics of FEM

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Sources of Error in the FEM

• The three main sources of error in a typical FEM solution are discretization errors,

formulation errors and numerical errors.

• Discretization error results from transforming the physical system (continuum) into a

finite element model, and can be related to modeling the boundary shape, the

boundary conditions, etc.

Discretization error due to poor geometry

representation.Discretization error effectively eliminated.

Page 41: Basics of FEM

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Susceptible to user-introduced modeling errors:

Poor choice of element types.

Distorted elements.

Geometry not adequately modeled.

Sources of Error in the FEM (cont.)

Page 42: Basics of FEM

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Sources of Error in the FEM (cont.)

• Formulation error results from the use of elements that don't precisely

describe the behavior of the physical problem.

• Elements which are used to model physical problems for which they are not

suited are sometimes referred to as ill-conditioned or mathematically

unsuitable elements.

For example a particular finite element might be formulated on the assumption that

displacements vary in a linear manner over the domain. Such an element will

produce no formulation error when it is used to model a linearly varying physical

problem (linear varying displacement field in this example), but would create a

significant formulation error if it used to represent a quadratic or cubic varying

displacement field.

Page 43: Basics of FEM

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Sources of Error in the FEM (cont.)

• Numerical error occurs as a result of numerical

calculation procedures, and includes truncation errors

and round off errors.

• Numerical error is therefore a problem mainly

concerning the FEM vendors and developers.

• The user can also contribute to the numerical

accuracy, for example, by specifying a physical

quantity, say Young’s modulus, E, to an inadequate

number of decimal places.

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Meshing Guidelines

Meshing of rounding (bents)

r ≤ 3mm Meshed with 1 element or 90deg.

r > 3mm Meshed with 2, 3 or 4 elements

Holes Modeling

The mesh around hole has 2 rosettes.

If there is not enough place, mesh with 1 rosette.

The physical property of rosettes is same as the

sheet around it.

Hole dia. d<6mm leave out holes

Hole dia 6mm ≤ d < 10mm use 6-edges

10mm ≤ d use 8 edges

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Bolt Modeling

Inner dia. : bore hole diameter

Outer dia. of first rosette : Bolt flange dia.

Outer dia. of second rosette : triple of bolt

dia.

Thickness of first rosette : sheet thickness +

bolt head height

Bolt Shank is modeled by CBAR with bolt

nominal dia.

The rosettes must be parallel to the part

direction.

Meshing Guidelines

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Meshing Crossover

If there exist a crossover between

smaller and larger parts, a mild

crossover is needed. In one element-

line the edge-length can be doubled

at most. Then follow a line with only

quadratic elements. And then the

crossover is allowed again.

Weld lines:

the quality of elements next to a weld

line is important. (triangles should

not be used)

shell thickness Min [d1:d2]

Contact: The contact parts meshed

with the Collinear nodes

The Nodes are connected with the

CBAR Elements with only axial

stiffness

Meshing Guidelines

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Technologies that compete with the FEM

• Other numerical solution methods:

– Finite differences

» Approximates the derivatives in the differential equation using

difference equations.

» Useful for solving heat transfer and fluid mechanics problems.

» Works well for two-dimensional regions with boundaries parallel

to the coordinate axes.

» Cumbersome when regions have curved boundaries.

– Weighted residual methods (not confined to a small subdomain):

» Collocation

» Subdomain

» Least squares*

» Galerkin’s method*

– Variational Methods* (not confined to a small subdomain)

* Denotes a method that has been used to formulate finite element

solutions.

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Technologies that Compete With the FEM (cont.)

• Prototype Testing

» Reliable. Well-understood.

» Trusted by regulatory agencies

» Results are essential for calibration of simulation software.

» Results are essential to verify modeled results from simulation.

» Non destructive testing (NDT) is lowering costs of testing in general.

» Expensive, compared to simulation.

» Time consuming.

» Development programs that rely too much on testing are increasingly less

competitive in today’s market.

» Faster product development schedules are pressuring the quality of

development test efforts.

» Data integrity is more difficult to maintain, compared to simulation.

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Future Trends in the FEM and Simulation

• The FEM in particular, and simulation in general, are becoming

integrated with the entire product development process (rather than just

another task in the product development process):

– FEM cannot become the bottleneck.

• A broader range of people are using the FEM:

– Not just hard-core analysts.

• Increased data sharing between analysis data sources (CAD, testing,

FEM software, ERM software.)

• FEM software is becoming easier to use:

– Improved GUIs, automeshers.

– Increased use of sophisticated shellscripts and “wizards.”, API for

customized development.

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Future Trends in the FEM and Simulation (cont.)

• Enhanced multiphysics capabilities are coming:

– Coupling between numerous physical phenomena.

» Ex: Fluid-structural interaction is the most common example.

» Ex: Semiconductor circuits, EMI and thermal buildup vary with current

densities.

»Ex: Smart Structures; Nano technology (MEMS-Micro-electro-

mechanical-systems), MEOMS(micro-opto-electro-mechanical systems).

Utilizes micro machanics and micro machanics

• Improved life predictors, improved service estimations.

• Increasing use of non-deterministic analysis and design methods:

– Statistical modeling of material properties, tolerances, and anticipated loads.

– Sensitivity analyses.

• Faster and more powerful computer hardware. Massively parallel processing, 64 bit

computing, Multi core systems.

•Decreasing reliance on testing.

• FEM and simulation software available via Internet subscription.

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Selected FEM Resources on the Internet

• http://www.engineeringzones.com - A website created to educate people in the latest engineering technologies, manufacturing techniques and software tools. Exellent FEM links, including links to all commercial providers of FEM software.

• http://www.comco.com/feaworld/feaworld.html - Extensive FEM links, categorized by analysis type (mechanical, fluids, electromagnetic, etc.)

• http://femur.wpi.edu - Extensive collection of elementary and advanced material relating to the FEM.

• http://www.engr.usask.ca/%7Emacphed/finite/fe_resources/fe_resources.html - Lists many public domain and shareware programs.

• http://sog1.me.qub.ac.uk/dermot/ferg/ferg.html#Finite - Home page of the the Finite Element Research Group at The Queen's University of Belfast. Excellent set of FEM links.

• http://www.tenlinks.com/cae/ - Hundreds of links to useful and interesting CAE cited, including FEM, CAE, free software, and career information.

•http://www.geocities.com/SiliconValley/5978/fea.html - Extensive FEM links.

• http://www.nafems.org/ - National Agency for Finite Element Methods and Standards (NAFEMS).

Page 52: Basics of FEM

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