Finite Elements Analysis of Structures

Acknowledgement

This contribution has been prepared in the frame of the project MSM0021630519 Progressive reliable and durable civil engineering structures.

GLOSSARY ... 7

1 PRINCIPLE OF FINITE ELEMENT METHOD ... 13

1.1 MATHEMATICAL DEFINITION OF FEM... 13

1.2 DEFORMATION VARIANT OF FEM USED IN PRACTICAL STATICS... 14

1.3 MORE GENERAL FORM OF FEM... 15

1.4 MATHEMATICAL FORMULATION OF BOUNDARY ELEMENT METHOD –BEM... 18

1.5 PARTIAL DISCRETISATION OF THE PROBLEM -FINITE LAYER METHOD... 18

1.6 IMPACTS OF CURRENT DIVISION OF LABOUR ON FEM IN PRAKTICE... 19

2 BASIC TERMS AND ALGORITHMS OF FINITE ELEMENT METHOD... 22

2.1 EXPLANATION OF USED TERMINOLOGY... 22

2.2 EXPLANATION OF THE PROCEDURE ON AN EXAMPLE DEFORMATION VARIANT OF FEM ... 23

2.3 INDIVIDUAL STEPS OF DEFORMATION VARIANT OF FEM ... 27

2.4 SPECIFICATION OF SELECTED OPERATIONS THAT ARE USEFUL TO UNDERSTAND FEM TERMS... 29

2.5 PRINCIPLE OF VIRTUAL WORK APPLIED IN FEM PROGRAMS... 36

2.6 MAIN OUTCOME FOR THE USERS OF FEM PROGRAMS... 41

2.6.1 Selecting the elements of the FEM analysis model ... 41

2.6.2 Interpretation of FEM output data ... 45

3 PHYSICAL AXIOMS AND VARIATIONAL PRINCIPLES OF MORE COMPLEX FEM PROBLEMS... 48

3.1 PRINCIPLES OF APPROXIMATION OF THE SOUGHT DISTRIBUTION OF A QUANTITY IN FEM ... 48

3.2 ELEMENTARY PRINCIPLES OF PHYSICAL NATURE OF FEM... 55

3.3 VARIATIONAL PRINCIPLES OF MECHANICAL PROBLEMS OF FEM... 59

3.3.1 Position of variational principles in mechanics ... 59

3.3.2 Scalar, vector and tensor field in FEM inputs and outputs ... 62

3.3.3 General variational principle ... 69

3.3.4 Consequences for some variants of FEM procedures... 73

3.3.5 Special configurations used in FEM... 76

3.3.6 Example of evaluation of elements by means of the variational principle... 80

3.3.7 Bufler’s variational principles... 81

3.3.8 Inverse variational principles ... 84

4 FINITE ELEMENTS... 85

4.1 GEOMETRIC PROPERTIES OF ELEMENTS... 85

4.1.1 Differential elements and finite elements... 86

4.1.1.1 How many differential elements are there?...87

4.1.2 Advantages and disadvantages of finite elements... 89

4.1.3 How not to get lost in the collection of finite elements ... 91

4.1.4 1D elements ... 93

4.1.4.1 Hermite 1D polynomials in FEM...93

4.1.4.2 The most known 1D elements ...100

4.1.4.3 Thin-walled beams of open cross section...102

4.1.5 2D elements ... 105

4.1.5.1 Triangular elements...105

4.1.5.2 Triangular elements with polynomials in L1, L2, L3...113

4.1.5.3 Quadrilateral elements with polynomials in x,y...117

4.1.5.4 Iso-, hypo- and hyper-parametric elements ...121

4.1.5.5 Surface elements recommended by the authors...127

Glossary

4.1.6.1 Tetrahedron ...147

4.1.6.2 Bricks ...150

4.1.6.3 Toroid...157

4.1.6.4 Special 3D elements...157

4.1.6.5 Solid elements recommended by the authors ...158

4.2 PHYSICAL PROPERTIES OF ELEMENTSE... 174

4.2.1 Physical models of materials of elements ... 174

4.2.2 What is the effect of physical properties in FEM algorithms... 180

4.2.2.1 3D constitutive laws...180

4.2.2.2 Reduction of the dimension of a problem ...182

4.2.2.3 Description of deformation in a reduced problem...186

4.2.2.4 Components of deformation in a reduced problem ...193

4.2.2.5 Physical constants of 2D FEM elements ...197

4.2.2.6 Physical constants of 1D FEM elements ...202

4.2.2.7 Physical constants of toroids ...208

4.2.2.8 Gas elements ...209

5 MODELLING OF STRUCTURES FOR FEM ANALYSIS... 211

5.1 INTRODUCTION TO THE THEORY AND PRACTICE OF CREATION OF FEMMODELS... 211

5.1.1 Present-day Approach to Modelling of Structures and Soil Environment ... 211

5.1.1.1 Objects and Terms ...211

5.1.1.2 The Selection of an Effective FEM Model in Practice...218

5.1.2 Dimensions of the Model for FEM analysis... 221

5.1.2.1 The 1D Models ...226

5.1.2.2 2D Models...234

5.1.2.3 Systems Consisting of 1D and 2D Elements ...241

5.1.3 Numerical Stability of the Calculation of FEM Models... 250

5.1.3.1 Defective FEM Results Due to Arithmetics ...250

5.1.3.2 Present-day Possibilities of Improving Arithmetics in FEM Calculations ...258

5.1.4 Modelling of Non-linear Behaviour of Structures by means of FEM Algorithms... 260

5.1.4.1 User Approach to Non-linear FEM Problems ...260

5.1.4.2 Assembly of Equation Systems in Non-linear FEM problems ...263

5.1.4.3 User’s interventions into the execution of non-linear FEM programs...266

5.1.4.4 More Complicated Constitutive Relations and Projects Depending on the Path ...270

5.1.4.5 The Selection of the Number of Increments and the Course of the Equilibrium Iteration ...280

5.1.4.6 Newton-Raphson Method and its Modifications...281

5.1.5 Transformation of Physical Quantities ... 291

5.1.5.1 Transformation of Tensors of Stress, Deformation and Physical Constants ...292

5.1.5.2 Design Stress and Internal Forces ...309

5.2 NOTES CONCERNING THE PROBLEMS OF MODELLING OF CERTAIN STRUCTURES IN THE ENGINEERING PRACTICE... 311

5.2.1 Introductory note ... 311

5.2.2 Modelling of Stiffeners in Planar Structures ... 312

5.2.3 Modelling of Column-Supports of Floor Slabs ... 314

5.2.4 Boundary Effects in Slab Models... 317

5.2.5 Singularities in the Analyses of Structures ... 318

5.2.6 Modelling of the Interactions between Foundation Grids and Subsoil... 319

5.2.7 The Density of the Mesh... 320

5.2.8 Modelling of a “Double Beam” Bridge with wide Beams ... 322

5.3 PHYSICALANDSHAPEORTHOTROPYOFPLATES ... 323

5.3.1 Purpose of the guide ... 323

5.3.2 Method ... 323

5.3.3 Core principal of solution... 323

5.3.4 Stress components in physically orthotropic plates ... 325

5.3.5 Internal forces in physically orthotropic plates ... 328

5.3.5.1 Technical theory of plates with the effect of transverse shear not taken into account...328

5.3.5.2 Plates with the effect of transverse shear taken into account...331

5.3.6 Shape orthotropy of plates... 332

5.3.6.1 Main principles of the transformation into physical orthotropy ...332

5.3.6.2 Simple types of orthotropic plates...333

5.3.6.4 Box-sections...352

5.3.6.5 Multi-cell slabs with linear hinges in longitudinal direction ...356

5.3.6.6 Other plate types ...358

6 MODELLING OF STRUCTURE-SOIL INTERACTION ... 360

6.1 INTRODUCTION... 360

6.1.1 Origin and Development of the Efficient Subsoil Model ... 360

6.1.2 The Main Ideas of the Efficient Subsoil Model ... 362

6.1.3 The Efficient Structure-Soil Interaction Model Assuming an Arbitrary Shape of Structure-Soil Interface ... 364

6.1.4 Some Remarks about Soil-Foundation-Structure Interaction... 366

6.2 ENERGYDEFINITIONANDGENERALTHEORYOFTHEEFFICIENTSUBSOILMODEL.. 371

6.2.1 Reduction of the Three-dimensional Model to the Two-dimensional Model ... 371

6.2.1.1 Three-dimensional Models in Geomechanics ...371

6.2.1.2 Two-dimensional Efficient Subsoil Model...373

6.2.2 One-dimensional Efficient Subsoil or Soil Medium Model... 391

6.2.2.1 Introduction...391

6.2.2.2 An Example of the Relation Between the Constants of One- and Two-dimensional Models ...392

6.2.2.3 Basic One-dimensional Relations ...394

6.2.3 Three-dimensional Efficient Subsoil Model as an Improvement on the Two-dimensional Model396 6.2.3.1 Main Idea of the Improvement of the Two-dimensional Efficient Subsoil Model ...396

6.2.3.2 Basic Geometrical and Physical Relations ...397

6.2.3.3 Some Special Cases ...402

6.3 THEORYOFPLATESONTHEEFFICIENTSUBSOILMODEL... 405

6.3.1 Introductory Comment... 405

6.3.2 Variational Problem of the Plates on Efficient Subsoil Model ... 406

6.3.2.1 Total Virtual Work of the Structure-Soil System ...406

6.3.2.2 Potential Energy of the Plate-Soil System...409

6.3.2.3 Potential Energy of the Improved Subsoil Model ...415

6.3.2.4 Variational Principles of Structure-Soil Interaction ...416

6.3.2.5 Advantages of Lagrange's Variational Principle and Principle of Total Virtual Work...418

6.3.3 Implementation of the Soil-Structure Interaction Model Using the Finite Element Technique ... 420

6.3.3.1 Dimension and Compatibility of Finite Elements ...420

6.3.3.2 Kirchhoff's Plate on the 2D-Efficient Model of the Subsoil...423

6.3.3.3 General Remarks on the Finite Element Technique ...427

6.3.3.4 Conclusions for the Solution of the Plate-Subsoil Interaction...431

6.3.3.5 Mindlin's Plate on the 2D-Efficient Model of the Subsoil...432

6.3.3.6 Mindlin's Plate on the 3D-Efficient Model of the Subsoil...435

6.3.4 Nonlinear Analysis of Structure-Soil Interaction using the 2D Efficient Subsoil Model ... 448

6.3.4.1 Introduction...448

6.3.4.2 Stress in subsoil...448

6.3.4.3 Physical model of soil based on the formula stated in CSN 73 1001 ...449

6.3.4.4 Physical model of soil according to DIN 4019...452

6.3.4.5 Physical model of soil according to Eurocode 7 ...455

6.3.4.6 Variability of subsoil input data ...455

6.3.4.7 Reduction of the dimension of the interactive problem...459

6.3.4.8 Surface model of subsoil...459

6.3.4.9 The effect of subsoil outside of the structure ...464

6.3.4.10 Implementation into SCIA•ESA PT system...465

6.3.4.11 Statistical analysis of the structure-soil interaction ...469

6.3.4.12 Conclusion ...470

7 NONLINEAR MECHANICS OF CONTINUA AND STRUCTURES ... 471

7.1 INTRODUCTION... 471

7.1.1 Selected Mathematical Concepts and Notations... 471

7.1.1.1 Index, tensor and matrix notations ...471

7.1.1.2 Voigt notation ...474

7.1.1.3 Voigt rule for higher order tensors ...476

7.1.1.4 Tensors...477

7.1.1.5 Transformation of finite elements matrices...484

7.1.2 Classification of Nonlinearity ... 489

Glossary

7.2 GEOMETRICALNONLINEARITY... 494

7.2.1 Foundational Concepts... 494

7.2.1.1 Systems of coordinates in nonlinear mechanics ...494

7.2.1.2 Deformation gradient ...494

7.2.1.3 Rate of deformation ...497

7.2.2 Strain Measures ... 498

7.2.2.1 Green – Lagrange strain tensor

E

7.2.2.2 Euler - Almansi strain tensor (e) ...501

7.2.2.3 Logarithmic strain measure (

ε

7.2.2.4 Infinitesimal strain tensors (ε), (ˆe) ...502

7.2.2.5 Other strain measures ...503

7.2.2.6 Comparison of strain tensors...504

7.2.3 Stress Measures ... 506

7.2.3.1 Cauchy stress (σ) ...508

7.2.3.2 Nominal stress (N), First Piola – Kirchhoff stress (P) ...508

7.2.3.3 Second Piola – Kirchhoff stress (S) ...509

7.2.3.4 Corotation stress (σˆ) ...509

7.2.3.5 Kirchhoff stress (τ) ...509

7.2.3.6 Biot stress (T)...510

7.2.3.7 Transformations between different types of stress ...510

7.2.3.8 Objective stress rate ...510

7.2.4 Energetically Conjugate Stress And Strain Measures ... 511

7.2.5 Two Formulations of Geometrical Nonlinearity in FEM ... 514

7.2.5.1 Formulation based on current configuration (updated Lagrangean)...515

7.2.5.2 Formulation based on reference configuration (total Lagrangean)...522

7.3 MATERIALNONLINEARITY... 527

7.3.1 Uniaxial Stress... 527

7.3.1.1 Uniaxial nonlinear elasticity...531

7.3.2 General Stress... 533

7.3.2.1 Saint-Venant – Kirchhoff material ...534

7.3.2.2 Hyper-elastic materials...535

7.3.2.3 Hypo-elastic Materials ...536

7.4 SOLUTIONMETHODSFORNONLINEARALGEBRAICEQUATIONS... 537

7.4.1.1 Picard Iteration Method...538

7.4.1.2 Newton – Raphson Iteration Method ...539

7.4.1.3 Riks Method...542

7.5 LINEARANDNONLINEARSTABILITY,POST-CRITICALANALYSIS ... 547

7.5.1 Introduction ... 547

7.5.2 Linear stability... 548

7.5.3 Nonlinear Stability... 550

7.5.4 Post-critical Analysis... 551

8 LINEAR AND NONLINEAR DYNAMICS OF STRUCTURES ... 553

8.1 VARIATIONAL FORMULATION OF THE INERTIAL PROBLEM... 553

8.2 DYNAMICS OF FOUNDATION PLATES... 555

8.2.1 Consistent Mass Matrix of the Plate on the 2D Subsoil Model ... 555

8.2.1.1 Consistent Mass Matrix of the Plate...555

8.2.1.2 Consistent Mass Matrix of the Subsoil...556

8.2.1.3 Resulting Consistent Mass Matrix of the Plate on the 2D Subsoil Model...557

8.2.2 Consistent Mass Matrix of the Plate on the 3D Subsoil Model ... 558

8.2.2.1 Damping Properties of the Plate-Soil System ...559

8.3 LINEAR SOLUTION OF STRUCTURES SUBJECTED TO VIBRATION... 562

8.3.1 The decomposition into eigenmodes method ... 562

8.3.1.1 Calculation of seismic effects from response spectrum...563

8.3.2 Numerical methods of direct integration ... 565

8.3.3 Explicit methods... 565

8.3.4 Method of central differences ... 565

8.3.5.1 Newmark method...566

8.3.5.2 Wilson method ...568

8.4 NUMERICAL METHODS FOR NONLINEAR SOLUTION OF STRUCTURE MODELS SUBJECTED TO DYNAMIC LOAD 569 8.4.1 Modification of relations in motion equations... 569

8.4.2 Modification of relations between displacement, velocity and acceleration vectors ... 570

8.4.3 Variants of connection of methods... 572

8.4.3.1 Algorithm of linear solution...573

8.4.3.2 Variant I ...574

8.4.3.3 Variant II...579

8.4.3.4 Variant III ...580

9 BENCHMARKS AND ILLUSTRATIVE EXAMPLES ... 585

9.1 BENDING WITH THE SHEAR DEFORMATION... 585

9.1.1 General remarks ... 585

9.2 GEOMETRIC NONLINEARITY... 586

9.2.1 General remarks ... 586

9.2.2 Axially and transversally loaded cantilever beam ... 589

9.3 SUBSOIL... 594 9.3.1 General remarks ... 594 9.4 CABLES... 605 9.5 MEMBRANES... 610 9.6 MECHANISMS... 612 9.7 STABILITY (BUCKLING)... 617 9.8 DYNAMICS... 618 LITERATURE... 621

1.1 Mathematical definition of FEM

Glossary

SYMBOLS

&f For a field, the superposed dot denotes the material time derivative, i.e. &f t( , )X = ∂f( , )X t ∂t ; for a function of time only, it is the ordinary time derivative, i.e. ( )&f t df t dt = ( ) ' x

f Derivative with respect to the variable x ; when the comma is followed by an index, such as , ,i j k , to s , it is the derivative

with respect to the corresponding spatial coordinate, i.e. 'i = ∂ ∂ i

f f x

⋅ As in a b⋅ indicates contraction of inner indices; for vectors, ⋅

a b is the scalar product a b ; if one or more of the variables _{i i}

are tensors of second order or higher, the contraction is on the inner indices, i.e. A B represents ⋅ A B , _{ij jk} A a⋅ represents A a _{ij j}

: As in A B: indicates double contraction of inner indices: A B:

is given by A B , ij ij C D: is C D ; note the order of the ijkl kl indices! Note also that if A or B is symmetric A B: =A B_{ij ji} × As in a b× indicates cross-product, or vector product; in indicial

notation, a b× →e a b_{ijk j k}

⊗ As in a⊗b indicates matrix product of vectors, or Kronecker product of matrices; in indicial notation, a⊗ →b a bi j; in matrix notation, a⊗ →b

{ }{ }

VARIABLES

, _x, _y, _z

A A A A Cross section area of a beam, A full area, _x A A shear area in _y, _z the y and z directions

, I

B B Matrix of spatial derivatives of shape functions in Voigt notation

arranged so that δ

{ }

{ }

rectangular matrix

[

]

B bandwiceth of the overall (global) stiffness matrix ,

I

B B Matrix of material derivatives of shape functions in Voigt

notation arranged so that

{ }

{ }

rectangular matrix B B1, 2,K,Bn

C Cauchy Green tensor, C=FT⋅F ; it is distinguished from the

material response matrices which follow by absence of a superscript, or damping matrix

, ,_ _ SE SE SE

ijkl

C C C Material tangent moduli relating &S to &E

, ,

τ τ _{ }τ

 

ijkl

C C C Material tangent moduli relating convected rate of Kirchhoff stress τ∇c to D , , σ σ _ σ _   J J J ijkl

C C C Material tangent moduli relating Jaumann rate of Cauchy stress

∇ σ J to D , , στ στ _ στ_   ijkl

C C C Material tangent moduli relating Truesdell rate of Cauchy stress σ to D ∇τ 1 2 2 , , , S S S S x y C C C

C Stiffness of subsoil, C subsoil stiffness matrix, S 1, 2 , 2 S S S x y

C C C

subsoil stiffness parameters

{ }

,D_ij,

D D Rate of deformation, velocity strain, D=sym

( )

,E_ij

E Green strain tensor, 1

(

)

2

= T⋅ −

E F F I

,F_ij

1.1 Mathematical definition of FEM

J Determinant of Jacobian between spatial and material coordinates, J =det_∂ ∂x_i X_j_

0

K Linear stiffnes matrix

T

K Tangent stiffnes matrices

, _σ M

K K Material and geometric tangent stiffness, respectively ,L_ij

L Spatial gradient of velocity field

M Mass matrix

I

N Shape functions

,Pij

P Nominal stress (transpose of first Piola-Kirchhoff stress) ,R_ij

R Rotation matrix (rotation tensor)

,Sij

S Second Piola-Kirchhoff (PK2) stress

T Transformation matrix

ε

T Transformation matrix for rotation of strain vector σ

T Transformation matrix for rotation of stress vector d

T Transformation matrix for deformation parameters ,U_ij

U Right stretch tensor

W Work

virt

W Virtual work

, int ext

W W _{Internal and external work}

, i

X X Material (Lagrangian) coordinates ,

I XiI

X X_I =

[

]

d Nodal displacements stored in Voigt form ,e_ij

e Euler – Almansi strain tensor 1

(

)

2

− −

= − T⋅

e I F F

i

e _e e e_x, _y, _z_, base vectors of coordinates

, _I, f_iI

f f Nodal forces

, ,

int int int I fiI

f f Internal nodal forces

, ,

ext ext ext I fiI

f f External nodal forces

l length, span , ,i j k indices of the ,x y and 2 axes respectively

0 0 ,n_i, ,n_i

n n Unit normal in current (deformed) and initial (reference, undeformed) configurations

,q_i

q Heat flux, also collection of internal variables in constitutive models t time ,t_i t Surface tractions ,u_i u Displacement field , ,

u v w Displacements in the ,x y and 2 directions respectively

, I uiI

u Matrix of components of displacement at node I ,

I viI

v Matrix of components of velocity at node I ,v_i

v Velocity field

,

w w Hyperelastic potential on reference and intermediate configurations respectively, e.g. S= ∂ ∂w E

≡ −

IJ I J

x x x Difference in nodal coordinates ,x_i

x Spatial (Eulerian) coordinates

, I xiI

x x_I =

[

]

,

1.1 Mathematical definition of FEM

undeformed) configurations ,ξ

ξ i

[

]

,ε

ε ij Infinitesimal strain tensor

εn logarithmic strain tensor

Π Total potential energy

Πint

e potential energy of one element

Πint

potential energy of internal forces

Πext

potential energy of external forces 0

,

ρ ρ Current and original density

Σ Stress matrix from the relation σ Σ d= ⋅ , in Section 6 stress tensor, where each component is multiplied by the unit diagonal matrix I

{ }

σ ij σ Cauchy (physical) stress tensor

{ }

σ ij σ Corotational stress tensor

{ }

τ ij τ Kirchhoff stress tensor

( , )

Φ X t Mapping from the initial configuration Ω₀ to the current or spatial configuration Ω

ˆ ( , )

Φ X t Mapping from the referential configuration ˆΩ to the spatial configuration Ω

0 ,

Ω Ω Domain of current (deformed), initial (undeformed)

Ωe Domain of one element

∂ Matrix of differential operators, its application to vector u yield tensorε ε:

{ }

the number of components of the tensor

{ }

1.1 Mathematical definition of FEM

1 Principle of Finite Element Method

1.1 Mathematical definition of FEM

When speaking about a general concept of finite element methods – including FEM, BEM, finite layer method and strip method – the mathematical nature inheres in what is termed discretisation of the problem. The term discrete is the opposite of continuous. To be clear: searching for unknown functions in domain Ω with boundary Γ is replaced by searching for a finite number of values of these functions or parameters d from which an approximate solution can be formulated, as explained later. Older methods used the same approach: classical variational methods (W. Ritz, 1908) searched for coefficients of pre-selected functions with generally non-zero values across the whole domain Ω. The well-known sieving method, well-known also as a differential method, replaced derivatives by differences or, more generally, by combinations of several function-values in the nodes of a mesh. The collocation method limited itself to the requirement of satisfying the given conditions (roughly) in several selected points of Ω and Γ. Formally, the analytical solution of differential equations was always transformed to the solution of systems of algebraic

linear equations. The same applies to FEM. The improvement is in the way this

transformation is carried out, or mathematically speaking, in the selection of base functions into which the sought functions are decomposed. The decomposition is closely related to the division of domain Ω (or Γ in BEM) into subdomains Ω_e, briefly called finite elements, contrary to “infinitely small differentials” dΩ, dΓ of the exact analysis.

In the beginning, mathematicians were not interested into this approach. The first approximation of the function of two variables by a combination of linear functions over triangular elements (R. Courant 1943, termed plated surface) remained completely unnoticed and even significant accomplishments of engineers in 1956-1965 who started to use polynomials of second, third and even higher degree over the elements, attracted no attention of analysts. Only after the first international conference where the FEM was presented (1st Conference on Matrix Methods in Engineering, Ohio, 1965) even mathematicians noticed what engineers had thought for long – that a qualitatively new mechanism was created and that it should be thoroughly researched. And in around 1968, a quite exact mathematical

definition of FEM was already given.

FEM is a generalised Ritz-Galerkin variational method that uses base functions defined in a small compact domains closely linked to the selected division of the whole analysed domain to finite elements.

In the same period it was shown that FEM generates systems of linear equations that are numerically significantly better conditioned than the still commonly used sieving method; formally flawless definitions of many useful terms were presented, hierarchies of various 1D, 2D and 3D finite elements were established according to conditions of continuity, etc. For detailed information refer to [3] to [5]. Present-day engineers benefit from this extensive research in the following way: they do not have to doubt about the mathematical

unobjectionability of FEM and can rely on the convergence towards the exact solution with

compatible elements (brief overview in [5], Appendix). In order to understand the core principle of FEM, it is convenient to use the engineering explanation of its most often used variant: deformation method, which employs what is called Lagrangean finite elements. This explanation will be made in the following paragraph. It forms the basis of almost all (in 1995 estimated 90 to 95%) commercially successful FEM systems.

1.2 Deformation variant of FEM used in

practical statics

This method can be easily programmed and produces well conditioned equation systems. The simplicity lies in the energetic concept of the problem, generally in the variational formulation of the problem where we search for an extreme of an operator Π (a “functional” in mechanics) that is of additive nature. It means that its value is for the whole system (domain) equal to the sum of values in the parts or elements of the system (subdomains – finite elements). This nature is characteristic especially for all the quantities defined by means of any bounded integral in the domain. Thus, for example total potential energy Π = Π + Πint ext of internal and external forces in the body is – according to the Lagrange variational principle – minimal just for the real state of the body ( , , )u ε σ . The FEM equation can be in this particular situation obtained through the differentiation of Π with respect to individual deformation parameters d d₁, ₂,K,d_m,K,d_N. For example, the m –th equation is:

(

)

and can be obtained through the versatile addition theorem, the principle of which lies in the additivity of energy as a scalar, i.e. in the additivity of energy derivations:

( ) ( ) 1 1, ,1 1 P int e P int e int m e m _{N N} e m d d d = = ∂ Π ∂Π ∂ = = ∂Π ∂ = ∂

∑

∑

∑

∑

Notice that the geometric dimension of the elements e=1, 2,K is not important. ,P

Only their stiffness matrices K are involved together with vectors of load parameters _e f_e that may be of different dimension for different elements of the same system ( ,n n , ( ,1)_{e e}) n_e . The elements may be one-, two, or three-dimensional with no negative impact on the principle of the solution. One to-be-solved system can generally include e.g. beam, plate, wall and/or shell elements, as well as 3D elements. Before they can be summed, all the matrices K and _e f _e

1.3 More general form of FEM

number of deformation parameters. This applies to matrices K , _er f that are established _er

when elements of matrices K , _e f are written to those positions in matrices ( ,_e N N , ( ,1)) N

where they belong to according to their global code numbers that define the topology of the system. The remaining elements of matrices ( ,N N , ( ,1)) N are zero. The whole system of linear equations for the calculation of unknown parameters d=[ ,d d_{1 2},K,d_m,K,d_N]T is formed in a simple way:

(N NK, )(Nd,1)=(Nf ,1) (1.2.4) (N,N) _{( , )} ( ,1) _{( ,1)} 1 1 P P er er N N N N e= e= =

∑

∑

As the global code numbers (list of element nodes) hold complete information about the position of the matrix elements in the ( ,N N grid, it is not necessary to factually establish ) the extended matrices K , _er f , which would consume a considerable portion of the memory, _er

but the algorithm employs only the original matrices K , _e f . And it is just this simplicity and _e

universality together with the fact that the system of equations is well-conditioned, that represents the practical advantage of FEM in comparison with classical approaches and that is the main reason why it is so universally and widely used in the present advanced era of computer development. The generalisation through the orthogonalisation principle or through the weighed residua method is explained in the following text.

1.3 More general form of FEM

After a more detailed investigation, we can easily find that the additivity of the functional

e e

Π =

∑

or potentially of any bounded integral in domain e e Ω =

∑

∫

∑

∫

is the only condition for the application of a powerful apparatus for establishing the system of equations (i.e. the calculation of coefficients of unknown and absolute elements) developed in FEM. Therefore, even in the early period of FEM development, it became used for problems where no Π-nature quantity could be defined, but we know differential equation L u₁( )=0 in

domain Ω and boundary conditions L u₂( )=0 on its boundary Γ that must be met by the sought function u . This may be a set of functions u=[ , ,u vK , shortly a vector of functions, ]T

or a “vector function”. In that situation we also have more equations L and conditions ₁ L , ₂

which is indicated by the used matrix notation. Then, the problem may be parameterised by the substitution uˆ =Ua or directly by uˆ =Nd (more exact matrix notation will be discussed later) with unknown coefficients a or parameters d. After substituting a set of values a or d

into the above-mentioned equations, they will not be fulfilled for obvious reasons and the right-hand side will not give identical zeroes, but certain functions in Ω and Γ that can be called “residuum ε and ₁ ε ”; which, in general, represent vector functions: ₂

1ˆ = ≠1 inΩ

L u ε 0 (1.3.2)

2 = ≠2 onΓ

L û ε 0 (1.3.3)

Now, we can use the “orthogonalisation principle”, termed recently also the “principle of weighed residua”, that was in fact used already by Bubnov (1910) and Galerkin (1915) as the main improvement of the Ritz method (1908 - 1909), but without the modern parameterisation of the problem by a numerically suitable base functions with a small compact support (uˆ =Nd), in the classical Ritz form uˆ =Ua. From this point of view, FEM can be seen as a generalised Ritz - Galerkin method resulting in significantly better conditioned systems of linear equations. The fundamental idea is simple. First, it will be explained for one unknown function. For the exact solution of u , the residua (1.3.2), (1.3.3) are identical zeros, and consequently, also products ε ₁ g in ₁ Ω and ε ₂ g on ₂ Γ, for arbitrary “weight functions” g in ₁ Ω and g on ₂ Γ, are identical zeroes. The bounded integral from an identical zero is exactly zero, and thus, the following must be satisfied for an exact solution of u and arbitrary “weights” g , 1 g : 2

1 1 2 2 0

R ε g d ε g d

Ω Γ

=

∫

∫

Condition (1.3.4) is not satisfied by an approximate solution. If we had enough time and could gradually substitute various sets d (in modern approaches) or a (in classical approaches) into (1.3.4), then, with pre-selected weights g , ₁ g , we would obtain different values of the total ₂

“error R in comparison against zero”. For example, sets d d d_{1, 2}, ₃,K,d_m would give 6385.51, 800.75, 1263.04, , 38.51

R= − − K − . And R would be absolutely smallest, let us say, for set d₈₃₆₁₅, and it its value would be R=0.0445996. We might even come across a set that, for the given accuracy, would give RB0. This set of parameters can be then declared the best with respect to meeting condition (1.3.4) for the pre-selected g , ₁ g and the corresponding ₂

approximate solution ˆu can be used in further steps. The procedure is clear, but completely useless in practice. Even if we (i) abstracted away from the continuous possibility of changes of individual parameters d in the set _j

1, 2, 3, , , , 1, T j N N d d d d d ₋ d   =   d K K (1.3.5)

(ii) limited ourselves to certain nodal points of the number axis, (iii) eliminated technically irrational values d , (iv) managed to get over the impact of the pre-selection of _j g , ₁ g , etc., ₂

we would have such a huge number of sets (1.3.5), that even the fastest state-of-the-art computers would not find the best set in a reasonable time.

1.3 More general form of FEM

The way out of this is that after the substitution of (1.3.5) into (1.3.4) and after integration (over the elements Ω_e, Γ_e), R becomes a function of N parameters d d₁, ₂,K,d_N:

1 2 3

( , , , , _N) 0

R d d d … d = (1.3.6)

If equations (1.3.2), (1.3.3), respectively their operators L , ₁ L , are linear, then also formula ₂

(1.3.4) is linear in parameters d d₁, ₂,K,d_N. It is therefore useful to establish for them

N algebraic linear equations, which can be achieved if we write equation (1.3.4) N-times with N different weight functions ( ,g g_{1 2}) ,_j j=1, 2,K : ,N

1 2

( , , , ) 0 1, 2, ,

j N

R d d K d = j= K N (1.3.7)

The sought parameters d d₁, ₂,K,d_N, can be found from system of N equation (1.3.7), which gives the approximate solution of the problem. If operators L , ₁ L are not linear, also ₂

equations (1.3.7) are non-linear, which on one hand results in (i) well-known complications of solution algorithms (e.g. Newton-Raphson method), (ii) problem of ambiguity, etc., but, on the other hand, does not mean a principal limitation of the application of the approach. In problems of mechanics, this mathematical algorithm can be sometimes clearly mechanically interpreted. For example, already Bubnov (1910) and Galerkin (1915) considered it to be a generalised principle of virtual work, which was elaborated on by a number of other authors, best by V. Z. Vlasov in 1950 - 1960. Weight functions g are in this sense considered to be a generalised virtual “displacements” (states of deformation) of the system and L are considered to be a generalised “force” quantities. As most of the authors did not reproduce the original ideas correctly, we recommend that readers should study the original publications [10] to [14] and the literature cited in [3] to [5].

It is clear from the formal derivation of (1.3.7) that different weight functions g would have different corresponding equation systems (1.3.7) and thus different solutions d and subsequently different internal forces, stresses and deformation in the analysed structure. If the solution d was strongly dependent on the selection of g , it would not be reliable for technical applications. Bubnov, Galerkin and also Vlasov tried to clear this issue through the mechanical interpretation of g . We can briefly say: if we are not able to respect all the possible virtual displacements, let us respect at least N mutually independent and, for the given system and its connections, most characteristic displacements, i.e. functions g . Relation to other methods was scrutinised as well. For example, in mechanical problems with a potential energy, if we select the functions g to be gradually equal to all base functions of set U, and if we fulfil in advance boundary conditions L₂ =0 on Γ, we obtain equations (1.3.7) identical to the Ritz method. If we fulfil in advance conditions L₁=0 in the analysed

domain, the procedure is identical to Trefftz method, which is the historically oldest “boundary method” preceding the nowadays BEM that will be briefly described in the following paragraph.

1.4 Mathematical formulation of boundary

element method – BEM

The most concise description of BEM reads: We select such weight functions g , ₁ g ₂

and perform such per partes domain integral integrations (Gauss - Ostrogradsky theorem), so that equation (1.3.4) contains only integrals in Ω and on Γ that can be numerically calculated from the input data and so that the only unknowns are the distributions of the quantities along boundary Γ. In general, this can be achieved in problems of mechanics with boundary Γ = Γ + Γ_{p u} (stress vector p is given on Γ_p, displacement vector u on Γ_u) through the selection of

1 j, 2p j, 2u j

g =u∗ g =u∗ g = p∗ (1.4.1)

where uj

∗

is the source function for displacements and pj

∗

its “reaction” on Γ_u. The stated distributions along the boundary are parameterised the usual way: we select a finite number of elements on boundary Γ, define finite boundary elements and their nodal parameters, in total

N values for set (1.3.5). If we gradually select the source functions for individual boundary nodes to be the weight functions, we obtain such a number of algebraic equations that is just required to solve the parameters. This rather mnemonic and encapsulated overview should be elaborated at least in the following: If there is just one unknown function – e.g. the deflection of a membrane w , temperature T , torsion function F , etc.– then each boundary node has just one unknown parameter d =w T F, , etc. If there are two or three unknown functions – e.g. , , ( )u v w in a 2D (or 3D) elasticity problem – then each boundary node has two or three

parameters d, marked locally d d d , e.g. ₁, ₂, ₃ u v w in a pure deformation variant with , , reaction components p p_x, _y,p in the fixed nodes. Similarly to FEM, also other variants are _z

possible. Also notations (1.3.2), (1.3.3), (1.3.4) then represent two or three equations, so that also the number of conditions for the solution increases correspondingly as well. For the weight functions are the used what is termed fundamental functions, under special conditions the exact source functions of individual displacement components u v w . These are, in , , accordance with the general influence principle, identical with the distribution of displacement components caused by singular loads P_x =1, P_y =1 or P_z =1 acting on the analysed system in boundary nodes. The method has developed from the older method of integral equations (Boundary Integral Equation Method, BIEM) and the present-day common international name is BEM (Boundary Element Method).

1.5 Partial discretisation of the problem - Finite

Layer Method

The discretisation of the problem – i.e. the substitution of unknown functions defined over the continuum of an domain and along its boundary by a countable, even finite, set of

1.6 Impacts of current division of labour on FEM in praktice

parameters (“parameterisation of the problem”) – does not have to be complete. If we can in a certain direction, let us say in the -x direction, make a very good estimate of the character of

the course of functions f x y e.g. by means of trigonometric components, it is sufficient to ( , ) divide just the -y interval (the front arch of a prismatic folded plate, support edge of a bridge

deck, etc.) and we get elements of a “strip method” that reduces a 2D problem into a 1D task. The reduction in the dimension of the problem can also be made in advance through the following. Instead of unknown function fD of several variables in domain ΩD we introduce its projection to function f_{D s−} of fewer variables in domain Ω_{D s−} , where D is the dimension of the original domain and s is the reduced dimension – practically, s=1 or 2 in common transformations (Fourier, Laplace, Hankel and others). It is an integral transformation with different “weight” functions g , defined in the original domain _D Ω_D. After bounded integral

D s D D

f ₋ f g d

Ω

=

_∫

is introduced, the only remaining variables are those from domain Ω_{D s−} . One of the oldest technical applications deals with a layered continuum (Bufler, Nikitin-Shapiro, Falk and others) and reduces a 2D symmetrical problem or a 3D general problem into a 1D problem within the interval 0≤ ≤z H , divided into layers of thickness H , _i i=1, 2,K . The ,n

procedure is known as a finite layer method = FLM. The reduction in the dimension of type (1.5.1) is also intensively exploited for the reduction of time variable t , e.g. in viscoelasticity problems. The solution then deals with what is termed assigned elastic problem.

The reduction in the dimension of a problem from a 3D construction subsoil massif into a 2D surface problem in the footing surface is the fundamental precondition for the effectiveness of FEM programs in the field of common foundation engineering; see [8, 9].

1.6 Impacts of current division of labour on

FEM in praktice

It is a well-known fact that in the current period the extent of practically useful scientific/technical knowledge doubles within 10 years, while the half-life of scientific knowledge (replacement by new, more accurate, more economic with regard to scientific thought, and more generally valid pieces of knowledge) is about 5 years. This gives rise to the question what part of the present-day information explosion is supposed to penetrate down to the engineer-designer, what part should be understood and used actively in their design practice, what part they are supposed to know as existing in order to be able to find the details, etc. This is not a simple task as the human brain competes with the most powerful computers as far as the structure is concerned, but it dramatically lags behind in terms of the (i) speed of performed operations, (ii) scanning of information (concentration about 6 bits per second), and (iii) time over which the information is stored (some data, even most of them, are erased immediately). An erudite specialist just before retirement holds in their brain, i.e. in their operational memory, approximately 109 - 1010 bits of information, unless they develop sclerosis (through bad diet and insufficient mental gymnastics) when they are about 40 that

keeps progressing and increases the natural handicap of hundreds of thousands of neurons dying every day. Even under optimal circumstances, it is illusionary to require that the engineer-designer fully understands everything they use for their work, e.g. that they have acquired comprehensive knowledge of methods of analysis, their numerical algorithms and programs, that they know everything about the work with computer and its peripheries, scanners, printers, plotters, hardcopy generators, digitizers, etc. Similarly, they can hardly be capable of citing (by heart) even a fraction of various standards and regulations. We have to accept the fact that also the division of labour has increased dramatically and that engineering and design institutions now have specialists to tackle this issue: mathematicians-analysts, programmes, electronics engineers, operators, specialists in technical fields, sometimes even documentalists of technical standards and their amendments, etc.

What is thus left for the engineer, what burden cannot be taken from their shoulders and what cannot be done by anyone else?

There is quite a bit of it, as the everyday practice of structural engineers, production boards, site engineers and other noninterchangeable roles proves. In this text, we will focus on structural engineering, and in this field it is the structural engineer (which includes a team of structural engineers in the case of larger projects) who must (i) define the analysis model, (ii) find all the related material in applicable standards or request a corresponding survey, (iii) prepare input data for the program that will be used to analyse the model, (iv) perform the

solution at the computer terminal themselves or with the help of operators, (v) correctly interpret the results for further steps in the design process and (vi) during all phases carry

out effective checks of all input data and outputs.

The structural engineer has to safely sort all the obtained information relating to the designed structure into categories: geometrical data (lengths, angles, shapes, topology),

statical (external impulses, loads, action conditions) and physical (in general, rules for the

behaviour of substances, or Hooke’s law in the simplest terms), as they are sorted into these groups by software input interfaces. They need to have a clear idea about the conditions of

continuity and equilibrium, both in the structure and at its boundary (support conditions).

To sum up: the intrinsic task of the structural engineer is to possess the knowledge of

mechanics to the extent that is required by the selected analysis model.

As far as the calculation methods (that form the basis for the applied programs) are concerned, the structural engineer has to be familiar with their core principle to the extent relevant to their reliable application under standard circumstances. When FEM programs are used, it is necessary to know what elements are implemented, i.e. what approximations of functions are assumed on them. This must be taken into account when the density (size) of

finite elements is being chosen. In addition, also accepted assumptions concerning the

internal forces with the resulting limitations in the practical application must be known. Moreover, one must understand the style of expression used by manuals and become familiar with the terminology employed, in order to be able to communicate with the programmers when unavoidable errors in input and output have to be clarified, quickly located and corrected. One also needs to have a correct idea of the extent of application of the selected program or calculation method. With regard to complex problems that appear only occasionally in engineering practice, one must be aware of the programs and technical literature suitable for resolving the problem and must be able to study these material on one’s own.

1.6 Impacts of current division of labour on FEM in praktice

university, even though to the extent that is proportional to the (i) possibilities of the curriculum, (ii) study plan, (iii) teaching staff, and (iv) university facilities. The present era makes life-long learning in practice vitally important. And the aim of this book is to provide a part of the knowledge required.

2 Basic terms and algorithms of

finite element method

2.1 Explanation of used terminology

Element is a part of the whole that is either physically composed of the elements, or that is

divided into the elements in our theorization.

Infinitely small element, also infinitesimal, dx, dxdy , dxdydz etc. is a limit of a finite size element whose dimensions approach zero. It is used in differential and integral calculus and is commonly used in the technical practice as it has its place in a university curriculum.

Finite element is an element of finite dimensions, contrary to the infinitely small element. For

example, a rectangle a b⋅ can be divided into 4 finite elements with dimensions a 2⋅b 2, or a triangle can be split into several triangles on condition that we create the additional vertices through a kind of triangulation, etc.

A domain is a connected set of points (open, unless we consider also its boundary; closed including the points of the boundary). It can be simply or multiply connected (with openings). In technical practice, this means so-called bodies (3D) that are in fact three-dimensional in Euclidian space with coordinates , ,x y z – in fact a set of points ( , ,x y z ). And this is the way

that they are handled in dams, thick-walled blocks, soil massifs, etc. where none of the dimensions is significantly smaller. Often, however, simpler analysis models are introduced:

Two-dimensional domain (2D) of walls, plates, shells, box structures, etc., planar or spatial of

more or less complex structure, the points of which are assigned certain physical properties including those depending on thickness, i.e. sectional dimension that does not exist in a two-dimensional domain.

One dimensional domain (1D) of beams, frames, truss girders, grids, netting, etc., in general

in structures composed of what is termed beams (the beam is thus a one-dimensional model of a body that is in fact three-dimensional).

Subdomain is a set of points of the domain that have the same properties, i.e. non-zero

measure, in the corresponding dimension d =3 or 2 or 1 in spatial, planar and linear, respective three-, two-, one-dimensional domains. From this we get the shortest definition:

Finite element is a subdomain.

A connected set of finite elements, which is usually treated as what is termed substructure or segment, can also be a subdomain. This is also related to the formation of more complex finite elements, see further in the text.

Element is the English term that gave the name to the Finite Element Method.

Subelement is an element that forms a part of a larger element. For example, four triangular

subelements can form one quadrilateral element, or five tetrahedrons can form one hexahedron (called brick), etc.

2.2 Explanation of the procedure on an example deformation variant of FEM

Superelement is an element created from two or more subelements.

Substructure is a part of the structure that can be analysed separately usually with the aim to

simplify the solution of a larger structure.

Load is in FEM perceived in a generalised sense as all impacts (external and internal,

gravitational, thermal, hygroscopic) that produce internal forces and displacement of the structure. In input they are sorted into force load (common forces and moments in points or distributed over a certain 3D, 2D, or 1D domain) and deformation load (similar deformation impulses) and, quite often, thermal and similar impacts (shrinkage – analogy to cooling, etc.) are extracted from the deformation ones.

Support (support conditions) is a technical term for boundary geometric or, as the case may

be, kinematical, conditions. Basically, it means the reduction of degrees of freedom of a 0D (point), 1D (linear) or 2D (planar support) figure. Inputs strictly distinguish (i) fixed

supports in which all degrees of freedom, or more precisely deformation parameters, are a

priori nullified, (ii) support with given values of deformation parameters (if zero values are input, this type coincides with the previous one) and (iii) flexible supports.

Elastic support is the most frequent type of flexible support. It is assumed that the size of

reactions depends only on the magnitude of deformation parameters in the state when the reactions are analysed. In general, the relation may be non-linear, but the term “elastic” eliminates possible dependence on other factors, especially on the history of the loading process. That means that the relation between reactions and deformation parameters is always unambiguous.

Physically linear elastic support is a type of support where the relation between reactions and

deformation parameters is approximated by a linear law of the following type: r=k d , which

is similar to Hooke’s law. In general, the set of reactions can contain also parameters d of other nodes. Usually, however, we assume in one node a linear nature of this relation. In the simplest example with just one reaction r the matrix of various connections k is converted into what is termed a spring constant k in relation r=kd. The linkage between quantities r and d is that virtual work r on d is the full product rd (principle of net virtual work).

Element node list is a sorted list of numbers of the element nodes.

2.2 Explanation of the procedure on an

example deformation variant of FEM

The finite element method is a method based on dividing the analysed domain into subdomains, or illustratively into finite elements. In the widest meaning of the word, this name can be used for any calculation that exploits finite elements. For example, already in the years before Christ, in order to determine the area of a planar figure U, ancient mathematicians divided it into finite elements U of a simple shape (rectangles, sectors) and _i

i i

U =

∑

that is actually commonly used until now, and even for other quantities relating to planar figures, e.g. cross-sections of beams. In addition, it is sufficient that the quantity is additive in

nature, i.e. that its magnitude for U is the sum of magnitudes for U . This is for example the _i

second moment of area (moment of inertia) about the same x -axis: 2 2 i i i i J y dxdy J y dxdy J J Ω Ω = = =

∫∫

∑

∫∫

It can be easily understood that this applies to any quantity defined by the value of a bounded integral over domain U . Whatever the dimension of this domain, the theorem on the calculation of bounded integral (only one integration symbol is used in the notation here, the type of the integral depends on the dimension of domain U) whose value equals V holds for all continuous functions (this assumption can be even weakened):

i i U i U U fdU =

∑

∑

∫

∫

For example, in a one-dimensional domain 0≤ ≤x L we have: After the domain is divided into finite elements - intervals ( ,x x , _{0 1}) ( ,x x , ..., _{1 2}) (x_n−1,x_n) for x₀ =0, x_n =L in dividing points (termed “nodes”) x_j, j=1, 2,K,n−1, the mesh has 2 end-nodes and n−1 internal nodes plus n finite elements and applies to any continuous function ( )f x :

1 1 0 ( ) ( ) j j x L n j _x f x dx f x dx − = =

∑

∫

∫

Function f x can be for example the density of potential energy of internal forces in a ( ) beam: 2 2 2 2 2 2 _{( ) ( )} ( ) ( ) ( ) 1 ( ) ( ) 2 y y z z x y z k y z Q x Q x M x M x M x N x f x EA GA GA GJ EJ EJ β _β   =  + + + + +      (2.2.5)

where sectional characteristics and modules can even be – for beams of variable cross-section – functions of x . The additive nature of the energy, which is scalar, is evident: the energy accumulated in the beam as what is called deformation work of an elastic state is the sum of energies of its parts, i.e. finite elements into which the beam is divided. This applies to all, even multidimensional, elastic bodies (structures):

int int j j

Π =

∑

Here, j is the summation index. The potential energy of external forces, i.e. of the given load (index z ) and reactions, resulting usually from gravitation and other sources outside of the body, is marked Π_z and is of the same additive nature, e.g. for a general body Ω with boundary Γ:

2.2 Explanation of the procedure on an example deformation variant of FEM

Ω Γ

ext T T

d d

Π =

_∫∫∫

_∫∫

Also the singular load by concentrated forces and moments is included into the integration, i.e. elements of type

(P uk k Mk k) k

ϕ +

∑

and we sum over all the loaded points k. Using the oldest and most general principle of

virtual works (used in simple machines already by Archytas of Tarentum and Archimedes in

fourth and third century B.C.), we can derive the Lagrangean variational principle of the

minimum (for details see [5]) total potential energy of the system (body + its load) and of the

minimum of its internal and external forces. min .

int ext

Π = Π + Π = (2.2.9)

This principle means that the algebraically lowest value of Π is achieved in a real state of the body – when related to the initial state in which the zero energetic levels of Π, Πint

and Πext are defined. Following from the Clapeyron theorem, 1

2 int ext

Π = − Π in linearly elastic bodies, and therefore Π = −Πint

(there is always a decline from the initial state). If we calculate the value of Πext from any other than the real system of stresses and deformations, we always get – on condition that we meet all geometrical supporting and continuity conditions –

algebraically higher value of Π. Instead of finding the stress-state and deformation of the body from the equilibrium and continuity conditions – which are in the form of differential equations with enormous demands on the smoothness of the function (continuity up to the third derivative) – we can find the solution directly from the above-mentioned condition of the minimum, without having to transform to Euler differential equations of this variational

problem, that would lead us back to the equilibrium and continuity conditions.

The finite element method in a narrower meaning of the word is the oldest form of FEM, only

for elements Ω_e of domain Ω. Methods that introduce finite elements on the boundary, finite strips and layers (see art. 1.5) have special names. In the period of early development between 1956-1965 FEM used to be usually connected with the problem of finding the extreme of an

operator (in mechanics it is also called functional) Π. Contrary to classical variational methods that do not divide domain U and that narrow down the class of allowable functions (among which the extremal meeting the requirement of the variational problem is sought) to a linear combination

k k k

f =

∑

where base functions g are non-zero throughout almost the whole domain k U (except the set of points of zero measure in which the “coordinate axes or planes are possibly intersected”), the finite element method uses the following new means:

a) Base functions g , closely related to the division of the domain into finite elements _k

and non-zero only in the elements that contain one common node of the mesh (this set of elements is “a small compact support of function g ”). These functions have a k typical “pyramidal” character and can be named in a popular way as source functions of the sought function in terms of coefficients a , as they define the distribution of _k