2 m1 m CT = 10 T = 10 8 CT = 10 8 C 8

Meshless methods

and other computational methods compared with FEM

Sławomir Milewski

mail: slawek@L5.pk.edu.pl

Contents

Introduction

Classification criteria of computational methods Boundary value problems formulations

Domain discretization and solution approximation Methods’ review on the basis of selected criteria

Finite difference method FDM compared with FEM Numerical example –

Beam deflection analysis Meshless methods

Meshless finite difference method MFDM compared with FEM Numerical example –

FD operators generation by means of MWLS (1D and 2D cases)

Summary

Introduction

Finite element method FDM

General, the most commonly applied method

Basis of variety of commercial codes and packages

(Abaqus, Adina, Ansys, Diana, FELT, Feap, Mark, Robot, …)

Applied for most problems of physics, mathematics, engineering, and mechanics

Developed classes and types of finite elements, solid mathematical

background, results postprocesssing, methods of errors estimation

Introduction

Why we should talk about other computational methods?

historical aspects (FEM is not the oldest one…)

didactic aspects (it is easier to solve problem manually by using FDM than FEM)

practical aspects

selected applications (plates analysis, moving boundary, crack, …) available code (user one, commercial, …)

methods’ combinations (e.g. FEM + MFDM)

need for results verification by means of other method efficiency and speed of the algorithm

Need for frequent mesh modifications (adaptation)

Accuracy of solution and its derivatives (super-convergence) Final results postprocessing (methods combinations)

Recent trends in science (meshless methods)

Methods classification criteria

Boundary value problems formulations

Domain discretization base (domain, boundary, subdomain, …)

Domain and boundary discretization type (nodes, elements + nodes, …) Solution discretization type (nodal values, other degrees of freedom, …) Solution approximation type

Numerical integration type

Postprocessing type

Classification criteria of

computational methods

Local formulation

Global formulations

Ω ∂Ω

( )

u f P

u u P

u g P

= ∈ Ω

 

=

 

=  ∈ ∂Ω

 L G

L,G - differential operators of n-th i m-th orders

Ω ∂Ω

Functional

Variational equation (inequality)

I( )= 1 B( , )

u 2 u u − Lu

B( , ) u v = L v ( ) for v ∈ V B( , ) u v ≥ L v ( ) for v ∈ V

adm

Mixed formulations (e.g. local - global)

=1 v

=0 v

=0 v

( )i

Ωi

( , ) ( )

, 1,...,

i

b u l

i N

=

∈Ω =

v v

v

Variational principle satisfied in subdomains ascribed to subsequent nodes Ω

ⁱ

local FDM local MFDM

FE M

variational FDM variational MFDM

MLPG5

Finite element method (FEM)

Boundary element method (BEM) Finite difference method (FDM)

Meshless methods (MM) Meshless FDM

Weighted residuals methods Energy methods

others …

Ω ∂Ω

Domain discretization

Solution approximation

boundary methods

element methods

meshless methods

METHODS CLASSIFICATION

FINITE ELEMENT

METHOD METHOD’S

NAME

FORMULATION DISCRETIZATION BASE

DISCRETIZATION TYPE

APPROXIMATION TYPE

NUMERICAL INTEGRATION

RESULTS POSTPROCESSING

WEAK (VARIATIONAL

PRINCIPLE, FUNCTIONAL)

DOMAIN

INTERPOLATION

(SHAPE FUNCTIONS) INSIDE ELEMENT FEM + OTHERS

DOMAIN NODES

+ ELEMENTS

FINITE BOUNDARY

METHOD

INTEGRAL

EQUATION DOMAIN

BOUNDARY INTERPOLATION

ON THE BOUNDARY

(INTEGRALS) BEM + OTHERS

BOUNDARY ELEMENTS

FINITE DIFFERENCE

METHOD

STRONG

(LOCAL) DOMAIN

DIFFERENCE FORMULAS

NOT NEEDED

APPROXIMATION DOMAIN

NODES + ELEMENTS

NODES

VARIATIONAL FDM

WEAK

(VARIATIONAL) DOMAIN

DIFFERENCE FORMULAS

AMONG AND

AROUND NODES APPROXIMATION

DOMAIN NODES

MESHLESS METHOD

(MESHLESS FDM)

WEAK / STRONG

(VARIATIONAL) DOMAIN

MWLS VARIOUS

METHODS MWLS

DOMAIN NODES

RESIDUAL METHODS

(GALERKIN, COLLOCATION, …)

WEAK

(VARIATIONAL) NONE

LINEAR COMBINATION OF BASIS FUNCTIONS

ANALYTICAL INTERPOLATION NONE

ENERGY METHODS

(RITZ)

WEAK

(FUNCTIONAL) NONE

LINEAR COMBINATION OF

BASIS FUNCTIONS ANALYTICAL INTERPOLATION NONE

Finite difference method

- local version

local FDM compared to FEM

local FDM FEM

Boundary value problem

formulation

Local - Variational

( )

u f P

u u P

u g P

= ∈ Ω

 

=

 

=  ∈ ∂Ω

 L G

I( )= 1 B( , ) u 2 u u −Lu

B( , ) u v = L v ( ) for v ∈ V - Functional

Mesh generation type (rectangular, triangular, …) + modulus h

Special programs - generators

Approximation Generation of finite difference formulas

Interpolation by means of shape functions

Discrete equations generation

Collocation variational principle at element level

Integration not needed Gaussian formulas in finite element Boundary

conditions

Additional difference formulas at boundary nodes

System of equations modification

System of equations matrix

Generally non-symmetric Symmetric banded

FDM steps – mesh generation

Source: Orkisz J., „Finite Difference Method”, part III in Handbook of Computational Mechanics, ed: Kleiber, Springer, 1998

FDM steps – difference formulas generation

1D:

h h

2D:

h

h

h h

1

i− i i+1 i−1, j i+1,j

, 1 i j+

, 1 i j−

, i j

standard three-nodes star

5-nodes star

1

ui₋ uⁱ

1

ui₊

2h 2h

1

i− i i+1

1

ui₋ uⁱ

1

ui₊

five-nodes star

2 i−

2

ui₋

h

2 i+

2

ui₊

h

h

h 9-nodes

star

Generation methods:

- Formulas composition (complex formulas by means of simple formulas) - Forcing the agreement for monomials

- Interpolation and differentiation

- Indefinite coefficient method (Taylor expansion)

FDM steps – difference formulas generation – 1D examples

( ) ( )

1 1 1 1

2

' '' ' ' 2 ''' '' ' ...

2

i i i i i

i i i i i

u u u u u

u u u u u

h h

+

−

− −

− +

+

≈ → ≈ = → ≈

1 1

''

_{i i i i}

u ≈ au

₋

+ bu + cu

₊

2 1

2 1

' 0.5 '' ...

' 0.5 '' ...

i i i i

i i

i i i i

u u hu h u

u u

u u hu h u

−

+

 = − + +

 =

 

= + + +



( )

^{2 2}

0 1

0

''

_{i i}

' (

_i

) '' (0.5

_i

0.5 )

u ≈ u a b c + + + u − + ah ch + u h a + h c

2

2

2

1

2

1 a h b h c h

 =

 

−

 =

 

 =

  -indefinite coefficients method for operator type: h h

1

i− i i+1

1

ui₋ uⁱ

1

ui₊

''

_{i i}

'

_{i i} 1

u ≈ au + bu + cu

₊

2 1

' '

' 0.5 '' ...

i i

i i

i i i i

u u

u u

u

₊

u hu h u

 =

 =

 

= + + +



( )

²

0 1 0

''

_{i i}

' (

_i

) '' 0.5

_i

u ≈ u a c + + u b ch + + u h c

2

2

2

2

2

a h

b h c h

 = −

 

−

 =

 

 =

  -indefinite coefficients method for operator type: h

i i+1

i, 'i

u u u_i+1

- formulas composition:

FDM steps – difference formulas generation – 2D examples

-indefinite coefficients method for operator type:

h

h

h h

1,

i− j i+1,j

, 1 i j+

, 1 i j−

, i j

( ) ( )

2

,

''

,

''

, 1, , 1 1, , 1 ,

i j xx i j yy i j i j i j i j i j i j

u u u au

₋

bu

₋

cu

₊

du

₊

eu

∇ = + ≈ + + + +

( ) ( )

( ) ( )

( ) ( )

( ) ( )

2

1, , , ,

2

, 1 ,

, ,

2

1, , , ,

2

, 1 , , ,

, ,

' 0.5 '' ...

' 0.5 '' ...

' 0.5 '' ...

' 0.5 '' ...

i j i j x i j xx i j

i j i j y yy

i j i j

i j i j x i j xx i j

i j i j y i j yy i j

i j i j

u u h u h u

u u h u h u

u u h u h u

u u h u h u

u u

−

− +

+

 = − + +

  = − + +

 

= + + +

 

= + + +

 

 =

( ) ( ) ( ) ( ) ^{( )}

( ) ( ) ( ) ( )

2

, , , ,

0 0 0 2

2 2 2 2

, ,

2

1 1

' ' ... 1

... '' 0.5 0.5 '' 0.5 0.5 4

i j i j x i j y i j

xx i j yy i j

u u a b c d e u ha hc u hb hd

a b c d h

u h a h c u h b h d e

h

∇ = + + + + + − + + − + + 

= = = =

→  

+ + + +   = −

- formulas composition:

( ) ( )

( ) ( )

1, , 1, , 1 , , 1

2 2

, ,

1, , 1 1, , 1 ,

2

, , , 2

2 2

'' , ''

'' '' 4

i j i j i j i j i j i j

xx i j yy i j

i j i j i j i j i j

i j xx i j yy i j

u u u u u u

u u

h h

u u u u u

u u u

h

− + − +

− − + +

− + − +

≈ ≈ →

+ + + −

→ ∇ = + ≈

FDM steps – difference equations generation

( )

u f P

u u P

u g P

= ∈ Ω

 

=

 

=  ∈ ∂Ω

 L G

i i

i i

j j

i i

L u f P G u g P

=  ∈ Ω

 

 

=  ∈ ∂Ω

 

Boundary conditions

Operator works on internal nodes only (poor approximation)

Operator works on internal nodes with additional

generalized degrees of freedom

Operator works on internal nodes

and external fictitious nodes

Nodal collocation

Bending beam – 2nd order equation

q(x)

L EI

y

x

2 2

( ) ( ) , (0) 0 , ( ) 0

d y M x

f x y y L

dx = = − EI = =

( )

1 1

2

1

( ) 2 2 2

M x = qLx − qx = qx L − x mathematical formulation with mechanical model

for q = const.

2nd order problems

- analytical derivatives into difference operators

h h

1

i − i i + 1

1 1

' 2

i i

i

y y

y h

+ − −

≈

1 1

2

'' _i y ⁱ 2 y ⁱ y ⁱ

y h

− − + +

≈

EJ L

y

x

Example: mesh with 5 nodes

1

2 3 4

5

2 2

( ) ( )

(0) 0 , ( ) 0

d y M x

dx f x EI

y y L

= = −

= =

( ) ( ) ( )

0

1 2 3

2 2

2 3 4

3 2, 3 4

2 0

3 4 5

2 4

2

2 ,

2

y y y

h f x

y y y

f x y y y

h

y y y

h f x

 − +

 =

 

− +

 = →

 

 − +

 =

 

q(x)

traditional notation – for manual calculations

4 h = L

matrix notation – for computational analysis

( ) ( ) ( )

1 1

2 2 2

2 2 2

1

3 3 3

2 2 2

4 4 4

2 2 2 5 5

1 0 0 0 0

1 2 1 0

0 0

1 2 1

0 0

1 2 1

0 0

0

0 0 0 0 1

y y

h h h y f x y

y f x y

h h h

y f x y

y y

h h h

−

 

 

 

   

 −       

       

 

 

  = →   =

−

 

 

   

       

 

 

−    

       

 

 

 

^B

A

A B

Bending of cantilever beam – 2nd order equation

q

L EI

y

x

2 2

( ) ( ) , '(0) 0 , (0) 0

d y M x

f x y y

dx = = − EI = =

( ) ( )

M x = − q L − x

mathematical formulation with mechanical model

Bending of cantilever beam – 2nd order equation

- numerical FD model – version 1 (fictitious node)

1 2 3 … n-1 n

1

h L const

=n =

0

−

2 0

0 2

1

2 0 0 y y

y y h

y

−

 = → =

 

 =



x

n

= L

( )

0 1 2

1 2 1

: y 2 y y

x f x

h

− + =

( ) ( ) ( ) ( )

1

2 2

2

1

2 2 2 3

2 4

2

2 2 2 1

1

2 2 2

1 0 0 0 ... 0

2 2

0 0 ... 0 0

1 2 1

0 ... 0

... ... ... ... ... ... ...

1 2 1 ...

0 0 0

1 2 1

0 0 0

n n

n n

y

h h y

f x

y f x

h h h

y y f x

h h h f x

y

h h h

− −

−

 

 

 

 −     

     

 −     

     

  ⋅   =  

     

     

 −     

       

   

 − 

 

- numerical FD model – version 2 (improved boundary operator)

1 2 3 … n-1 n

1

h L const

=n =

−

( )

1 1 2 1

2 2

1

2 2 2

' 0

y y y f x

h h h

y

 − − + =

 

 =



x

n

= L

( ) ( ) ( ) ( )

1

2 2

2

1

2 2 2 3

2 4

2

2 2 2 1

1

2 2 2

1 0 0 0 ... 0

2 2

0 0 ... 0 0

1 2 1

0 ... 0

... ... ... ... ... ... ...

1 2 1 ...

0 0 0

1 2 1

0 0 0

n n

n n

y

h h y

f x

y f x

h h h

y y f x

h h h f x

y

h h h

− −

−

 

 

 

 −     

     

 −     

     

  ⋅   =  

     

     

 −     

       

   

 − 

 

the same system of equations as for version 1

Bending of cantilever beam – 2nd order equation

Stationary heat flow (2D)

2

''

_xx

''

_yy

T

n q

T T T f in

k

T T on

k T q on

n

 ∇ = + = − Ω

 

= ∂Ω

  ∂

 = − ∂Ω

 ∂

 Ω

Ω T

Ω T

Ω q

q

n

1 - 4

1

1 1

h

h

h h

2

1

× h

2

∇ ≈

0

( )

2

0 0 0

1 2 3 4 0 2

'' ''

4 1

xx yy

T T T

T T T T T

h

∇ = + ≈

= + + + −

„1” „0” „2”

„3”

„4”

h h

operator

value at „0”

3 m 2 m

2 m 1 m

10 C T =

T = 1 0 8 C

Stationary heat flow (2D) - example 10 C

T =

10 C T =

T = 1 0 8 C T = 1 0 8 C

Cms k J

k

k

_x

=

_y

= = 7

_o

isotropic material

( − + ) _ ^ _ ^

= m s

y J x

y x

f ( , ) 20 30 10

₂

Intensity of heat generation

inside the domain (per unit area)

3 m 2 m

2 m 1 m

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21

[ ] ^m

h = 1

mesh modulus (21 nodes)

Internal nodes (5): 8, 9, 10, 11, 14

Boundary nodes (16): 1-6, 7, 12, 13, 15-21

( ) ^{1 , 1} ( ) ^{2 , 1} ( ) ^{3 , 1} ( ) ^{4 , 1} ( ) ^{1 , 2}

Stationary heat flow (2D) - example

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0

0

0 0 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T T T T T T T T T T T T T T T T T T T T T

   

   

   

   

  

  

  

  

  

  

  

  

  

  

  

  

  ⋅

  

  

  

  

  

  

  

  

  

  

  

  

  

  

   

 

0 0 0 0 0 0 0

0 0

0 0 0 0 0 0 0 0 0 0 0

0

  

  

  

  

  

  

  

  

  

  

  

  

  

  

= 

  

  

  

  

  

  

  

  

  

  

  

  

  

  

   

equation no. 1 - node no. 1 - boundary equation no. 2 - node no. 2 - boundary equation no. 3 - node no. 3 - boundary equation no. 4 - node no. 4 - boundary equation no. 5 - node no. 5 - boundary equation no. 6 - node no. 6 - boundary equation no. 7 - node no. 7 - boundary equation no. 8 - node no. 8 - interior equation no. 9 - node no. 9 - interior equation no. 10 - node no. 10 - interior equation no. 11 - node no. 11 - interior equation no. 12 - node no. 12 - boundary equation no. 13 - node no. 13 - boundary

equation no. 15 - node no. 15 - boundary equation no. 16 - node no. 16 - boundary equation no. 17 - node no. 17 - boundary equation no. 18 - node no. 18 - boundary equation no. 19 - node no. 19 - boundary equation no. 20 - node no. 20 - boundary equation no. 21 - node no. 21 - boundary equation no. 14 - node no. 14 - interior

initial form of system of algebraic equations (21 x 21)

Stationary heat flow (2D) - example

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21

( )

²

2 9 14 7

4

8 8

,

8

h 0

T T T T T f x y

+ + + − = − k =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 1

0 0 0 0 0 0 0

4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0

0 0

0 0 0 0

−

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T T T T T T T T T T T T T T T T T T T

  

  

  

  

  

  

  

  

  

  

  

  

  

  ⋅

  

  

  

  

  

  

  

  

  

  

  

  

  

  

   

 

0 0 0 0 0

0 0

0 0 0 0 0 0 0 0

0 0 0

0

  

  

  

  

  

  

  

  

  

  

  

  

  

= 

  

  

  

  

  

  

  

  

  

  

  

  

  

  

   

Stationary heat flow (2D) –

finite difference equations

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21

( )

²

3 10 15 8 9 9 9

4 , 2

7 T T T T T f x y h

+ + + − = − k = −

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 1

0 0 0 0 0 0 0

4 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 1 4 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0

0 0 0 0 0

−

−

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0 0

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T T T T T T T T T T T T T T T T T T T

  

  

  

  

  

  

  

  

  

  

  

  

  

  ⋅

  

  

  

  

  

  

  

  

  

  

  

  

  

  

   

 

0 0 0 0 0

0 0

0 0 0 20 / 7

0

0 0 0 0 0 0

0

  

  

  

  

  

  

  

  

  

 − 

  

  

  

= 

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

Stationary heat flow (2D) –

finite difference equations