Finite Element Primer for Engineers

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• Finite Element Primer for Engineers
• Mike Barton S. D. Rajan

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• Finite Element Primer for Engineers

Contents

• Introduction to the Finite Element Method (FEM)
• Steps in Using the FEM An Example from Solid
  Mechanics
• Examples
• Commercial FEM Software
• Competing Technologies
• Future Trends
• Internet Resources
• References

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Foreword This document was submitted as a term
paper for the graduate engineering course CEE598
Finite Elements for Engineers, offered at Arizona
State University. The objective of this article
is to provide engineers with a brief introduction
to the finite element method (FEM). The article
includes an overview of the FEM, including a
brief history of its origins. The theoretical
basis for the FEM is discussed, with emphasis on
the basic methodologies, assumptions, and
advantages (and limitations) of the method.
Next, the basic steps that must be performed in
any FEM analysis are illustrated (using an
example from solid mechanics), and FEM examples
are provided for problems from other engineering
disciplines. To aid the reader in selecting a
FEM software package, a brief survey of currently
available FEM software is presented, together
with a discussion of alternative analysis
techniques that might be considered in lieu of
the FEM. Finally, we examine future trends in
the FEM. References are provided for those
desiring further information on the FEM
(including selected Internet references.)
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Contents

• Introduction to the Finite Element Method (FEM)
• Steps in Using the FEM (an Example from Solid
  Mechanics)
• Examples
• Commercial FEM Software
• Competing Technologies
• Future Trends
• Internet Resources
• References

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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-form solution to the particular
  problem being studied.
• 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.

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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, 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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How can the FEM Help the Design Engineer?

• The FEM offers many important advantages to the
  design engineer
• Easily applied to complex, irregular-shaped
  objects composed of several different materials
  and having complex boundary conditions.
• Applicable to steady-state, time dependent and
  eigenvalue problems.
• Applicable to linear and nonlinear problems.
• One method can solve a wide variety of problems,
  including problems in solid mechanics, fluid
  mechanics, chemical reactions, electromagnetics,
  biomechanics, heat transfer and acoustics, to
  name a few.

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How can the FEM Help the Design Engineer? (cont.)

• General-purpose FEM software packages are
  available at reasonable cost, and can be readily
  executed on microcomputers, including
  workstations and PCs.
• The FEM can be coupled to CAD programs to
  facilitate solid modeling and mesh generation.
• Many FEM software packages feature GUI
  interfaces, auto-meshers, and sophisticated
  postprocessors and graphics to speed the analysis
  and make pre and post-processing more
  user-friendly.

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How can the FEM Help the Design Organization?

• Simulation using the FEM also offers important
  business advantages to the design organization
• Reduced testing and redesign costs thereby
  shortening the product development time.
• Identify issues in designs before tooling is
  committed.
• Refine components before dependencies to other
  components prohibit changes.
• Optimize performance before prototyping.
• Discover design problems before litigation.
• Allow more time for designers to use engineering
  judgement, and less time turning the crank.

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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 equation
• Dy(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
• S Wi(x)R(x) 0 (2)
• where Wi(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 Galerkins Method

• There are several choices for the weighting
  functions, Wi.
• In the Galerkins method, the weighting
  functions are the same functions that were used
  in the approximating equation.
• The Galerkins 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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Theoretical Basis Variational Method

• The variational method involves the integral of
  a function that produces a number. Each new
  function produces a new number.
• The function that produces the lowest number has
  the additional property of satisfying a specific
  differential equation.
• Consider the integral
• p D/2 y(x) - Qydx 0. (1)
• The numerical value of p can be calculated given
  a specific equation y f(x). Variational
  calculus shows that the particular equation y
  g(x) which yields the lowest numerical value for
  p is the solution to the differential equation
• Dy(x) Q 0. (2)

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Theoretical Basis Variational Method (cont.)

• In solid mechanics, the so-called Rayeigh-Ritz
  technique uses the Theorem of Minimum Potential
  Energy (with the potential energy being the
  functional, p) to develop the element equations.
• The trial solution that gives the minimum value
  of p is the approximate solution.
• In other specialty areas, a variational
  principle can usually be found.

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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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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 Youngs modulus, E, to an
  inadequate number of decimal places.

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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.
• Susceptible to user-introduced modeling errors
• Poor choice of element types.
• Distorted elements.
• Geometry not adequately modeled.
• Certain effects not automatically included
• Buckling
• Large deflections and rotations.
• Material nonlinearities .
• Other nonlinearities.