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).

FunctionalFundamentals 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 functionalAmong 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• 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

LINK (bar element loaded in tension-compression only)

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 plane

in 3D space

in 3D space

Type of elements sketchparameters of deformation DOFs)

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)

Types of elements – two-dimensional

sketchType of elementsparameters of deformation (DOFs)

Volume elements

(with general 3D stress state)

Tetrahedron linear

Pentahedron linear

Hexahedron (brick) linear

Hexahedron (brick) quadratic, isoparametric

parameters of deformation (DOFs)sketch

Types of elements – three-dimensional

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 non-

linear 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