Finite element method Among up-to-date methods of mechanics and specifically stress analyses, finite element method (abbreviated as FEM below, or often as FEA.

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Finite element
method
Among up-to-date methods of mechanics and specifically stress analyses, finite
element method (abbreviated as FEM below, or often as FEA for analyses as well)
dominates clearly nowadays; it is used also in other fields of engineering analyses
(heat transfer, streaming of liquids, electric and magnetic fields, etc.).
In mechanics, the FEM enables us to solve the following types of forward problems
(inverse problems cannot be solved at all!):
• stress-state analysis under static, cyclic or dynamic loading, incl. various non-linear
problems (large displacements and strains, elastic-plastic or other non-linear
material behaviour, etc.) ;
• natural as well as forced vibrations, with or without damping;
• contact problem (contact pressure distribution);
• stability problems (buckling of structures);
• stationary or non-stationary heat transfer and evaluation of temperature stresses
(incl. residual stresses);
• fracture mechanics (linear or non-linear, prediction of crack propagation incl. crack
shape).
Functional
Fundamentals of FEM are quite different from the analytical methods of
stress-strain analysis. While the analytical methods of stress-strain analysis
are based on the differential and integral calculus, FEM is based on the
calculus of variations which is generally not so well known; it seeks for a
minimum of some functional using variational methods.
Explanation of the concept – analogy with functions:
• Function - is a mapping between sets of numbers. It is a mathematical term
for a rule which enables us to assign unambiguously some numerical value
(from the image of mapping) to an initial numerical value (from the domain
of mapping).
• Functional - is a mapping from a set of functions to a set of numbers. It is
„function of a function“, i.e. a rule which enables us to assign
unambiguously some numerical value to a function (on the domain of the
function or on its part). It takes a function for its input argument and returns
a scalar. Definite integrals (e.g. strain energy or arc length) are examples of
a functional.
Principle of minimum of quadratic functional
Among all the allowable displacements (i.e. those which meet boundary conditions of
the problem, its geometric and physical equations) only those displacements can come
into existence between two close loading states (displacement change by a variation δu)
which minimize the quadratic functional ΠL. This functional (called Lagrange
potential) represents the total potential energy of the body, and the corresponding
fields of displacements, stresses and strains minimizing its value represent the searched
elasticity functions.
This principle is called Lagrange variation principle.
Lagrange potential ΠL can be described as follows:
ΠL = W – P
where
W - total strain energy of the body
P – total potential energy of external loads
The minimum can be searched in two ways:
• Ritz and Galerkin methods use continuous description and give specific solutions
dependent on the shape of the investigated body.
• FEM uses discrete description (replaces the continuum by a set of discrete points
and finite elements); a simple solution can be repeated many times and is independent
of the shape of the whole body.
Basic concepts of FEM
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Finite element – a subregion of the solved body with a simple geometry.
Node – a point used to describe geometry of the finite element and to define its base functions .
Base function – a function describing the distribution of deformation parameters
(displacements) inside the element (between the nodes).
Shape function – a function describing the distribution of strains inside the element, it
represents a derivative of the base function.
Discretization – transformation of a continuous problem to a solution of a finite number of
non-continuous (discrete) numerical values.
Mesh density – inverse of element magnitude, decissive for time consumption and accuracy of
the solution.
Matrixes (they are created by summarization of contributions of the individual elements)
– of displacements,
– of stiffness,
– of base functions.
Convergence – a property of the method, the solution tends to the real (continuous) solution
when the mesh (discretization) density increases.
Percentual energy error gives an assessment of the total inaccuracy of the solution, it
represents the difference between the calculated stress values and their values smoothed by
postprocessing tools used for their graphical representation, transformed into difference in
strain energies.
Isoparametric element – element with the same order of the polynomials used in description
of both geometry and base functions.
Overview of basic types of finite elements
They can be distinguished from the viewpoint of theory the element is based
on (general theory of elasticity, axisymmetry, Kirchhoff theory of plates,
theory of membrane shells, bars, etc.), or what family of problems the
element is proposed for .
• 2D elements (plane stress, plain strain, axisymmetry)
• 3D elements (volume elements - bricks)
• Bar-like (1D) elements (either for tension-compression only, or for flexion
or torsion as well)
• Shell elements
• Special elements (contact elements, crack elements, cohesive elements, etc.)
Type and size of elements is decissive for time consumption of the analysis
which increases with 2nd or 3rd power of the number of DOFs. Important
especially with non-linear analyses.
Types of elements – one-dimensional
Type of elements
LINK (bar element loaded in
tension-compression only)
in plane
in 3D space
BEAM (element loaded by in-plane flexion and shear)
BEAM (bar element loaded in
tension-compression, flexion
and torsion as well in 3D space)
in plane
in 3D space
sketch
parameters of
deformation DOFs)
Types of elements – two-dimensional
Type of elements
Membrane or 2D elements
Triangle linear
Triangle quadratic
Tetragon (bi)linear
Tetragon isoparametric quadratic
Plate element
Shell element
(general shell under both
membrane and bending load)
sketch
parameters of
deformation (DOFs)
Types of elements – three-dimensional
Volume elements
(with general 3D stress state)
Tetrahedron linear
Pentahedron linear
Hexahedron (brick) linear
Hexahedron (brick) quadratic,
isoparametric
sketch
parameters of
deformation (DOFs)
Basic types of constitutive relations in FEA
• linear elastic isotropic and anisotropic (elastic parameters are direction
dependent - monocrystals, wood, fibre composites or multilayer materials),
• elastic-plastic (steel above the yield stress) with different types of the
behaviour above the yield stress (perfect elastic-plastic materials, various
types of hardening),
• non-linear elastic (small strains are reversible , but non-linearly related to
the stresses),
• hyperelastic (showing large reversible strains on the order of 101 – 102 % ,
then stress-strain dependences are always non-linear as well),
• viscoelastic (the material shows creep, stress relaxation and hysteresis,
stress is related not only to train magnitude but to strain rate as well, the
response under steady load is time dependent, energy dissipation),
• viscoplastic (their plastic deformation is time-dependent),
etc.
Typical structure of FE software
consists of several parts with specific aims:
• Preprocessor
– Creation of model of geometry and its discretization (creating FE mesh – free
or mapped).
– Choice of material model and setting its parameters.
– Formulation of boundary conditions (loads and supports).
• Solver
– Choice of the type of analysis, setting the parameters and limitations of nonlinear solutions (number of substeps, maximum number iterations, accuracy
limits, etc.).
– Formulation of matrix equations of the problem and their mathematical
solution.
• Postprocessor
– Choice of the results to be presented.
– Processing of the results, their numerical and graphical interpretation.
Influence of base functions and mesh density on
the calculated stress concentration in a notch
Stress distribution
in the dangerous cross section of the notch
Example of a non-linear problem:
a FE mesh in a plastic guard of a pressure bottle valve
Example of a solution to a non-linear problem

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