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]
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]
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
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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.
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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 …
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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
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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.
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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
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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=
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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..
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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
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Force
Static Force
Dynamic Force
Transient,
Frequency response,
Random,
Shock Response
Force due to Contact
Force due to Creep
……
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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
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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
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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
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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
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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
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Material Nonlinearity
*PLASTIC
Card Set1: Plasticity properties for elastoplastic material models
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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FINITE ELEMENT LIBRARY
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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
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Warning: The Computed Answer May Be Wrong
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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.
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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.)
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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.
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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).
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Thank you