General model of wood in typical coupled tasks, Part II. – Weak solution

103

Volume LVI 10 Number 4, 2008

GENERAL MODEL OF WOOD IN TYPICAL

COUPLED TASKS

PART II. – WEAK SOLUTION

P. Koňas, E. Přemyslovská

Received: March 14, 2008

Abstract

KOŇAS, P., PŘEMYSLOVSKÁ, E.: General model of wood in typical coupled tasks, Part II. – Weak solution.

Acta univ. agric. et silvic. Mendel. Brun., 2008, LVI, No. 4, pp. 103–108

The main aim of this work is focused on weak solution of coupled physical task the microwave drying of wood with stress-strain eff ects and moisture/temperature dependency. Due to well known weak solution for separated physical fi elds without coupled eff ect, author concerns with coupled stress-strain relation coupled with moisture and temperature distribution. For scale de-pendency the subgrid upscaling method was used. Solved region is assumed to be divided into discontinual subregions according to investigated scale. This approach sugests sequential type of solution for highly coupled task. This way, very huge structures (huge according to geometry and also physics) can be solved in reasonable time and with memory consumptions. Main emphasis was putted on evaluation of structural response of the whole complex. Due to infl uence of mois-ture, temperature and time the coupled physical task of structural response is solved. Sugested aproach is of course usable not only for structural response, but for other physical fi elds, which were taken into account. Weak solution is based on slightly modifi ed Ritz-Galerkin method. The work is continuing of the previous article General model of wood in typical coupled tasks: Part I. – Phenomenological approach.

FEM, multiphysics, microwave wood drying, upscaling, homogenisation

INTRODUCTION

The task is fi nding of (T, w, p, u, v, E, B, H, J, D) ∈

H0(Ω) in the following equations.

e

t

t

0,

ρ

∂ ∇× =

∂ ∂ ∇× = +

∂ ∇ ⋅ = ∇ ⋅ =

B E

D H J

D B

(1)

where: B is the magnetic fl ux density, D is electric fl ux density, H is magnetic fi eld intensity, J is current density, ρe is electric charge density. Due to anisotro-py of wood we can itemize these variables to D = εE,

B = μH, J = σE, where ε is permittivity, μ is per mea bi-li ty and σ is electric conductivity of material.

e

t

ρ

∂ ∇ ⋅ = −

∂

J (2)

(

)

(

)

(

)

abs ext

ext

ext

T

C w p T q T T

t w

w p T w w

t p

w p T p p ,

t

ρ ∂ − ∇ ∇ − ∇ ∇ − ∇ ∇ = + −

∂

∂ _{− ∇ ∇ − ∇ ∇ − ∇ ∇ = −}

∂

∂ _{− ∇ ∇ − ∇ ∇ − ∇ ∇ = −}

∂

T

w

p

Tw Tp TT h

ww wp wT h

pw pp pT h

k k k k

k k k k

k k k k

(3)

w is mass concentration (moisture content), wext re-spective pext is moisture content respective static pressure in the surroundings, khw respective khp are

coeffi cients of convective type of fl uxes, kTw, kTp, kTT, kww, kwp, kwT, kpw, kpp, kpT are matrixes of diff usion

coeffi cients

2

ext ext

2

2 2

w w T T

t t

w w T T wT .

U

w ’ ’ ’ ’ ’ ’w

w w

˜ ˜ ˜ ˜ ˜

w T w,T

b b

2 2

vel

EG K K Ȝ

w w T T wT

u u

c c c u c

C C C C C C F

Defi nition of individual coeffi cients for eq. 4 was described in previous Part I.

2 fl 2

v 1 p

U .

t 2

K

U

U

w _{’u u ’} ’ _{’ ’}

w Ȟ Ȟ Ȟ Ȟ (5)

Where, ∇U is potential, ν is velocity of fl uid.

We should rewrite eq. 1, 2, 3, 4, 5 in the weak form. Because the weak form of 1, 2, 3 and 5 can be found in many books (e.g. BODIG, J., et al. 1982, KRIEGSMANN, 1997) we can focus on eq. 4 where

we will outline variational formulation for mixed type of elements in numerical subgrid upscaling method (Arbogast, 1998, 2000, 2002, and Korosty-shevskaya, 2006). It should be note, that we will suppose the sequential type of mentioned equa-tions. This assumption leads to simplifi cation of eq. 4, where T and w are constant in one time-step.

METHODS

The weak form of eq. 4 can be written as follows:

2

u 2 ext ext

2 2

F , w w T T , ,

t t

2 w w T T wT , 0.

U

§ w · _{’ ’ ’ ’ ’}§ _’w ·

¨ _w ¸ ¨_{© w} ¸_¹

© ¹

˜ ˜ ˜ ˜ ˜

w T w,T

b b

2 2

vel

EG K K Ȝ

w w T T wT

u u

ȟ c c c uȟ c ȟ

F C C C C C C ȟ

(6)

for all ξ ∈ H₀(Ω) and meaning of (,) as scalar product on Hilbert space. On base of mentioned simplifi ca-tion we obtained integral form. Let the funcca-tional eq. 6 is defi ned on vector space V. Let us assume the region Ω is partitioned by linear mesh Ψ_∂

1on

very fi ne scale δ₁ also we will assume that region is not fully partitioned by this fi ne mesh. Only m₁

of small regions are covered by mesh on this scale (subgrids). Functional eq. 6 is than defi ned on vec-tor subspaces V₁δ_{1, V}

2 δ_{1, …, V}

m

δ_{1 ∈ V , where V}

j

δ_{j for j =}

1, …, i are Raviart-Thomas (RT) spaces. Subspaces may not fi ll the full space V. It means that V₁δ_{1 ∪ V}

2 δ₁

∪ V₃δ_{1 ∪, …, ∪ V}

m

δ_{1 ≡ V}δ_{1 ⊆ V. Withal we declare}

men-tioned vector subspaces with bases {φδ1

V1,1, φ

δ1

V1,2, …,

φδ1 V1,n1} ∈ V1

δ_1,{φδ1

V2,1, φ

δ1

V2,2, …, φ

δ1 V2,n2} ∈ V2

δ_{1, {φ}δ1

Vm₁,1, φ

δ1

Vm₁,2, …,

φδ1

Vm₁,nm₁} ∈ Vm1

δ_{1. Complete basis φ}δ1 ≡ {φδ1

V1,1, φ

δ1

V1,2, …, φ

δ1 V1,n1,

φδ1

V2,1, φ

δ1

V2,2, …, φ

δ1 V2,n2, φ

δ1

Vm₁,1, φ

δ1

Vm₁,2, …, φ

δ1 Vm₁,nm₁} ∈ V

δ_{1 on}

vec-tor space Vδ_{1 is derived from the fi ne mesh of}

sub-grids where functions in bases are linear combina-tions of spatial direccombina-tions.

Similarly let us to partition Ω by next linear meshes Ψ_δ2, Ψ_δ2, …, Ψ_δi for diff erent scales δ1 < δ2 < … <

δi where again m2, m3, …, mi regions cover some parts of Ω on specifi c scale. Consequently similar vector subspaces can be distinguished V1δ2, V2δ2, …, Vmδ₂2 ∈ V,

V1δ3, V2δ3, …, Vmδ₂3 ∈ V, V1δi, V2δi, …, Vmδ₂i ∈ V with the same requirements:

. ,

, ,

, ,

,

, ,

,

2 2 3 2 3

3

1 2 2 2

2

2 1

2 1

2 1

V V V V

V

V V V V

V

V V V V

V

i i i

i

m m m

⊆ ≡ ∪ ∪ ∪

⊆ ≡ ∪ ∪ ∪

⊆ ≡ ∪ ∪ ∪

δ δ δ

δ

δ δ δ

δ

δ δ δ

δ

K K

K K

(7)

Also we will tie subspaces by these important rules:

Vδ_{1 ⊆ V}δ_{2 ⊆ V}δ_{3 ⊆, …, ⊆ V}δ_{i ⊆ V (8)}

δ_i is maximal scale (9)

Vδi ≡ V and V is declared on the whole Ω (10) All unknowns can be decomposed to individual scales e.g.:

u = uδ1 + uδ2 + … + uδi on some Ω

o (11)

Decomposition of unknowns to individual scales aff ects solution in sense of fi nite elements and mi-ni mi sa tion of functional eq. 6 does not provide common appearance of Ritz system.

Let us consider PDE Au = f: u ∈ V with diff erential operator A and follow common steps in solution of this task for multi-scale problem.

Functional which will be minimized has stan-dard form (Rektorýs, 1999):

F_u = (u, u)_A − 2(f, u) (12) Eq. 11 will be substituted into fi rst part of eq. 12: Lu = (u

δ1 _{+ u}δ2 _{+ … +u}δi_{, u}δ1 _{+ u}δ2 _{+ … + u}δi₎

A (13)

It can be expanded due to rules of scalar product in the following manner.

¸¸

¸ ¸

¹ ·

¨ ¨ ¨ ¨

© §

A A A

A A

A

u

i i

i i

u u

u u u

u

u u u

u u

u L

G G

G G G

G

G G G

G G

G

, , 2 ,

, 2 ,

2 ,

2 2

2

1 2

1 1

1

(14)

As usual the functional is minimized by the function:

ji

j j j

s

k k

k 1

b .

δ δ δ

ϕ =

=

∑

u_% (15)

For fi rst step we will approximate functional in subgrid on scale δ1

Finally unknown function can be by this function:

j

j j j

s

k k

k 1

a .

δ δ δ

ϕ =

=

∑

u (16)

Evaluation Lu for minimizing function

j

¸¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ © § k k j j k k j j k k j k k j k j k j k k j k k j k j k j k j k j k j s s s s s s s s b b b b b b b b b b b b u u G G G G G G G G G G G G G G G G G G G G G G G G G G

M

M

M

M

M

M

M

M

M

M

M

M

, , 2 , 2 , 2 , 2 , ~ , ~ 2 2 2 2 2 2 1 1 2 1 2 1 1 1 1 1

for j ≠ k (17)

j j j j j j j j j j j

j j

j j j j j j j j j

j j

2

1 1 1 1 2 1 2 1 s 1 s

2

2 2 2 2 s 2 s

, b 2 , b b 2 , b b

u , u 2 , b 2 , b b

G G G G G G G G G G G

G G G G G G G G G

M M

M M

M M

M M

M M

M

j j j

j j j

2

s , s bs

G

_M

G G

§ · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ . (18)

Requirement on minimisation of quadratic func-tional Fu allows evaluating a minimum of function.

Thus partial diff erentiation according to all coeffi -cients on all scales should be done.

1 i

1 1 1 1 i i i i i 1 1 1 1 i i i i

1 1 s1 s1 1 1 si si 1 1 s1 s1 1 1 si si

u u

1 b a b a , ,b a b a s b a b a , ,b a b a

L L

, , .

b δ δ δ δ δ δ δ δ b δ δ δ δ δ δ δ δ

δ δ

= = = = _{= = = =}

∂ ∂

∂ ∂

K L K K L K

K (19)

This task can be easily achieved with next relations.

a a a b u u b u u b u u j j j j k k j j k k j k k j k j k j k j k k k j k k j k k j s A s s A s A s A A A s ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § ¸¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ · ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ © § w w w w w w , , 2 , 2 0 , , 2 0 0 , ~ , ~ ~ , ~ ~ , ~ 2 1 2 1 2 2 2 1 1 1 2 1 G G G G G G G G G G G G G G G G G G G G G G G G

M

M

M

M

M

M

M

M

M

M

M

M

for j ≠ k (20)

It can be simplifi ed into the relation:

j k j k k k k j L A s b b u u G G G G G G G a R w w w 1 ~ , ~ (21) RL

Aδjδ_k is modifi ed lower triangular matrix of Ritz system.

a a a b u u b u u b u u k k k k k k j j k k j k j k k j k j k j j j k j j k j j k j s A s s A s A A s A A s ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ © § w w w w w w , 0 0 , 2 , 0 , 2 , 2 , ~ , ~ ~ , ~ ~ , ~ 2 1 2 2 2 1 2 1 1 1 2 1 G G G G G G G G G G G G G G G G G G G G G G G G

M

M

M

M

M

M

M

M

M

M

M

M

for j ≠ k (22)

k k j j j j k j U A s b b u u G G G G G G G a R w w w 1 ~ , ~ RU

Aδjδ_k is modifi ed upper triangular matrix of Ritz system.

j j j j j j j j A s b b u u G G G G G G G a R w w w 1 ~ , ~

R_Aδ

jδk

is well known matrix of Ritz system.

RESULTS AND DISCUSSION

We can evaluate eq. 19 by above mentioned rela-tionships in the following form.

i i i i i i i i i i i i i A L A L A s u U A U A A L A L A s u U A U A A L A s u U A U A A s u b b L b b L b b L b b L δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ a R a R a R a R a R a R a R a R a R a R a R a R a R a R a R + + + + = ∂ ∂ ∂ = + + + + + + + = ∂ ∂ ∂ + + + + + = ∂ ∂ ∂ + + + = ∂ ∂ ∂ 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 3 4 4 3 3 3 3 2 3 2 1 3 1 3 2 3 2 3 3 2 2 2 2 1 2 1 2 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 1 M K M M K K K K K K (24)

This complex system can be rewritten in more readable form:

j

j

j k k

j j j k k j

j

j

u 1

u _{i j 1}

U L

2

A A A A

k j 1 k 1 u s L b L b

S 2 2 .

L b δ δ δ δ δ δ δ δ δ δ δ δ − = + = ∂ ⎛ ⎞ ⎜_∂ ⎟ ⎜ ⎟ ⎜∂ ⎟ ⎜_∂ ⎟ =⎜ ⎟= + + ⎜ ⎟ ⎜ ⎟ ∂ ⎜ ⎟ ⎜_∂ ⎟ ⎝ ⎠

∑

∑

R a R a R a

M

(25)

This way we obtained numerically approximated fi rst part of eq. 12 denoted as SA on diff erential ope-ra tor A.

Minimization of functional eq. 12 is done by the relation:

1 i

1 1 1 1 i i i i i 1 1 1 1 i i i i

1 1 s1 s1 1 1 si si 1 1 s1 s1 1 1 si si

u u

1 b a b a , ,b a b a s b a b a , ,b a b a

F F

0, , 0.

b δ δ δ δ δ δ δ δ b δ δ δ δ δ δ δ δ

δ δ

= = = = = = = =

∂ ₌ ∂ ₌

∂ ∂

K L K K L K

K

(26) With including of eq. 25 the solution of the initial

problem can be reached by enumeration of a_j in the following equation:

( )

( )

( )

j j j j 1 2 A s f , f , S . f , δ δ δ ϕ ϕ ϕ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =_{⎜ ⎟} ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

M (27)

By analogy, the solution of eq. 6 with applying of

SA derivation can be rewritten.

( )

( )

( )

j j j j c 1 c 2

A B C

c s

f ,

f , S S S 2 .

f , δ δ δ ϕ ϕ ϕ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − − = _{⎜ ⎟} ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ M (28)

For diff erential operators

t C T T w w B t

A ext ext

w w ’ ’ ’ ’ ’ ’ w w T w, T b w b Ȝ K K EG c c c c , , 2 2 U

and function fc = F − (Cw·w + Cw2·w2 + CT.T + CT2·T2 +

C_wT.wT + C)

If fi nite elements with linear basis functions are used then system eq. 28 is unambiguously sol

va-ble. Solution is realized in i consequent steps of so-lution. In fi rst step eq. 28 is formed, whereas results of higher scales are unknown (in Ritz or modifi ed Ritz system). Solution on higher scales in individual nodes can be expressed by mapping of a_δ

1 or other

appropriate lower scales. From this step we obtain defi nitions in some nodes on higher scale(s) which bounds region of element on this solved scale. In the following we calculate the same eq., but on the following higher scale withal some nodes on this scale were strictly derived from previous step. This idea is repeated until the highest scale is reached.

CONCLUSIONS

Advantage of this type of solution is also that you do not need enumerate results on lower scales, but you can enumerate only results on last scale whereas results on this scale is derived from the low and lower scales. The solution is simplifi ed by this statement:

If position of node for higher scale is in some re-gion of lower scale mesh, than a_δ

j−k can be mapped directly to results on higher scale a_δ

j(aδj−k → aδj). Let each node of element on some higher scale Eδj coin-cides with node in element on lower scale Eδj−k. All contributions of higher scales a_δ

j>1 to subgrid can be

derived from consequent mapping of a_δ

1, aδ2, …, aδj−1

to required a_δ

SUMMARY

The weak solution of coupled stress-strain task with moisture/temperature dependency of material model was obtained in this project. The subgrid upscaling homogenization method for large scale hie rar chi cal structure which is typical for wood structure was used. Modifi ed Ritz-Galerkin method for simple solution was derived. Coeffi cient form of PDE suitable for nowadays numerical solvers was used (see Part I.). Suggested weak solution off ers unique and relatively accurate solution of large scale problems with dependency on low scale. The solution is very general and slight modifi cation of the approach allows solution a lot of common tasks in fi eld of Biomechanics.

SOUHRN

Obecný model dřeva v typických vázaných úlohách, Část II. – Slabé řešení

Bylo nalezeno slabé řešení sdružené napjatostní úlohy s vlhkostní/teplotní závislostí materiálového modelu. S výhodou byla použita homogenizační metoda subgrid upscaling vhodná pro hierarchické struktury velkého měřítka, jakou je např. struktura dřeva. Byla sestavena modifi kovaná Ritz-Galer-kinova metoda pro snadné použití. Rovněž byla použita koefi cientová forma obyčejné diferenciál-ní rovnice vhodná pro dnešdiferenciál-ní numerické řešiče (viz Část I.). Navrhované slabé řešediferenciál-ní nabízí jedineč-né a relativně přesjedineč-né řešení problémů na velkých měřítkách, které závisí na nižším měřítku. Řešení je velmi obecné a mírná modifi kace navrhovaného přístupu poskytuje řešení řady běžných úloh biomechaniky.

MKP, vázané fyzikální úlohy, mikrovlnné sušení dřeva, homogenizace

ACKNOWLEDGEMENT

The work is supported by project GAČR GP106/06/P363 (P526563605) – Homogenization of material properties of wood for tasks from mechanics and thermodynamics

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