IP-TA 2010 Inverse Problems: developments in theory and

IP-TA 2010 Inverse Problems: developments in theory and applications February 9-12, 2010 Warsaw, Poland

Trefftz method in solving the inverse problemsKrzysztof GrysaKrzysztof Grysa

Kielce University of Technology, Al.. 1000-lecia P.P.7, 25-314 Kielce, Polande-mail: [email protected]

1/27


Trefftz method has been known since the work of Trefftz in 1926*.Trefftz method has been known since the work of Trefftz in 1926 .

An approximate solution of a problem is a linear combination ofAn approximate solution of a problem is a linear combination offunctions that satisfy the governing differential linear equation orsuch one that is possible to be converted into such a form.

The unknown coefficients are determined from the conditions ofapproximate fulfilling the boundary and initial conditions, finallyhaving a form of a system of algebraic equations.

* E T fft Ei G t k Rit ’ h V f h P di f th 2 d I t ti l C f A li d

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Krzysztof Grysa - Trefftz method in solving the inverse problems

* E. Trefftz, Ein Gegenstuek zum Ritz’schen Verfahren. Proceedings of the 2nd International Congress of AppliedMechanics, 131–137, Zurich, 1926.


Generally, Trefftz bases fall into two broad classes, F-Trefftz basesbased on fundamental solutions and T-Trefftz bases, which areusually (but not always) obtained by separation of variables in polary ( y ) y p pand Cartesian coordinate systems.

In 2000 Ciałkowski presented two other methods one of whichIn 2000 Ciałkowski presented two other methods, one of which,based on developing function in Taylor series, is particularly simpleand effective. We will focus our attention on T-Trefftz bases.

In eighties and nineties T-complete functions have been used to findapproximate solutions of BVP, also with the use of FEM. However, noIBVP were investigated Researchers used to get rid of time variableIBVP were investigated. Researchers used to get rid of time variable,and as a result they considered the Helmholtz type equation.Therefore the first T-complete functions were found for Laplace

ti bih i d f d d tiequation, biharmonic one and for reduced wave equation(Helmholtz type eq.)

M J Ci łk ki A F k i k H t f ti d th i li ti t l i h t d ti

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M.J. Ciałkowski, A. Frąckowiak, Heat functions and their application to solving heat conduction and mechanical problems. Wyd. Politechniki Poznańskiej, Poznań, 2000. (in Polish)


During the last 10 years Trefftz method has been applied to findi t l ti f th i blapproximate solutions of the inverse problems.

Up to now the following inverse problems have been considered:- boundary value determination inverse problems,- material properties determination inverse problems,- sources determination inverse problems.

At first a “global” approach (i.e. looking for an approximate solutionin the whole domain) was applied with good results for simple

p

in the whole domain) was applied with good results for simplegeometry and initial-boundary conditions. However, a great majorityof more complex problems of mathematical modeling cannot besolved without division of the area Ω into subregions (elements) FEMsolved without division of the area Ω into subregions (elements). FEMwith Trefftz functions as trial functions is then used.

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Trefftz functions for some linear differential equations without time*

Laplace equation: 02 =∇ uLaplace equation:

2D in Ω ( ) ( ) nn zz Im,Re,1

0=∇ u

3D in Ω ( ) nqnnePr iqqn

n ≤≤−= ,...,2,1,0;cos φθ

( )θcosqnP - associated Legendre functions

2D i Ω

( )n

Reduced wave equation:

( ) ( ) ( ) ( ) ( ) θJJJ 21iθ02 =+∇ uu

2D in Ω

3D in Ω

( ) ( ) ( ) ( ) ( ) ,...,n,nθrJnrJrJ nn 21sin,cos,0 =θ

( ) ( ) n nqnnePrj iqqn ≤≤−= ,...,2,1;cos φθ

( ) ( )rJr

rj n 2/12 +=π

n

5/27Krzysztof Grysa - Trefftz method in solving the inverse problems

*A.P. Zieliński, On trial functions applied in the generalized Trefftz method. Adv. in Eng. Software, 24, 147-155, 1995.


Trefftz functions for some linear differential equations with time

Heat conduction equation: uu ∂∇ 2Heat conduction equation:

1D in Ω

tu

∂=∇

[ ]

⎭⎬⎫

⎩⎨⎧

== ∑−

, ...2,1,0!)!2(

),(2 2

nkk

txtxvn kkn

n

2D in Ω

⎭⎬

⎩⎨ −∑

=

, ,,!)!2(

),(0 kknk

n

( ) ( )⎭⎬⎫

⎩⎨⎧ +

+==== − , knnmnkntyvtxvtyxV kknm 2

1,,...,0,...,1,0),(),(,,

Beam vibration equation * : 02

2

4

4

=∂∂

+∂∂

tu

xu

1=S

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−⎥⎦⎤

⎢⎣⎡ −

−

++−⎥⎦⎤

⎢⎣⎡ −

−−⎥⎦⎤

⎢⎣⎡ −

−2

34

23

454

24234

231

nn

jnnjnnj xt

10 =S

xS =1

S ( ) ( )∑⎦⎣

=

⎦⎣⎦⎣

⎟⎟⎠

⎞⎜⎜⎝

⎛++−⎥⎦

⎤⎢⎣⎡ −

⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎥⎦

⎤⎢⎣⎡ −

−

−=

0 !454

24!234

23

1,j

n

jnnjnn

xttxStS =2

and for 3≥n


M.J. Al-Khatib, K. Grysa, A. Maciąg, The method of solving polynomials in the beam vibration problems. J. Theoret. Appl. Mech., 46, 2, 347-366, 2008.


Trefftz Methods

Trefftz Methods have not received a precise definition although thisTrefftz Methods have not received a precise definition, although thisterminology has had wide acceptance. Herrera’s definition of what ismeant by a Trefftz Method is:

Given a region of an Euclidean space or some partitions of thatregion, a „Trefftz Method” is any procedure for solving initialboundary value problems of partial differential equations or systemsboundary value problems of partial differential equations or systemsof such equations, on such region, using solutions of that differentialequation or its adjoint, defined in its subregions.

When Trefftz Method is conceptualized in this manner, it includesmany of the basic problems considered in numerical methods forpartial differential equations and becomes a fundamental concept ofp q pthat subject.

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I. Herrera, Trefftz method: A general theory. Numer. Meth. Partial Diff. Eq. 16, 561-580, 2000.


Indirect Trefftz MethodsIndirect Trefftz Methods

Consider a linear diff eq Lu=0 in Ω with values of u=u1prescribed on Γ1 and ∂u/∂n=q2 prescribed on Γ2 . Γ1 and Γ2prescribed on Γ1 and ∂u/∂n q2 prescribed on Γ2 . Γ1 and Γ2are parts of ∂Ω or are included in Ω .

( ) ( ) ( )PuaPuPuN

nn**~ uaT==≅ ∑( ) ( ) ( )

nnn

1∑= - T-complete functions

( ) ( ) ( ) ( )PPnuPqPq *~~ qaT=∂∂

=≅ *

nu

Residuals: ( ) ( ) 11*

1 0 Γ∈≠−≡ PPuPR for uaT

( ) ( ) 22*

2 0 Γ∈≠−≡ PPqPR for qaT ( ) ( ) 222 qq

1. Collocation method: residuals at the points Pi placed on Γ1 andΓ2 are forced to vanish ⇒ Ka=f ⇒ a ⇒( )Pu~

2. Least-square-method: ⇒ ⇒ Ka=f ⇒min)(21

22

21 →Γ+Γ= ∫∫

ΓΓ

dRdRF αa


⇒ a ⇒ α - weighting parameter( )Pu~


Indirect Trefftz MethodsIndirect Trefftz Methods

Consider a linear diff eq Lu=0 in Ω with values of u=u1prescribed on Γ1 and ∂u/∂n=q2 prescribed on Γ2 . Γ1 and Γ2prescribed on Γ1 and ∂u/∂n q2 prescribed on Γ2 . Γ1 and Γ2are parts of ∂Ω or are included in Ω .

3. Galerkin method formulation: 0~~)( 21 =Γ−Γ= ∫∫ dRudRqF a

and - weighting parameter

⇒ Ka=f ⇒ ( )Pu~

)(21

21 ∫∫ΓΓ

q

q~ u~−

4. Modified Trefftz formulation (T-Trefftz approach)

⇒ Ka f ⇒ ( )Pu

In this case the T-complete functions are built using fundamentalsolution for the eq Lu=0. The f.s. is a function of r(P,Qi) with P beingan integral point of Ω and Qi standing for external source placed onan integral point of Ω and Qi standing for external source placed onthe imaginary boundary surrounding the (real) boundary. Then

( ) ( ) ( ) ( )PQPuaPuPuN

in** ,~ uaT==≅ ∑ ( ) ( ) ( ) ( )PPuPqPq *

~~ qaT=∂

=≅and

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( ) ( ) ( ) ( )Qn

in1

,∑=

( ) ( ) ( ) ( )n

qq q∂and


Direct Trefftz Methods

The direct Trefftz formulation is based on the boundaryintegral equation Details for the Laplace eq: in the articleintegral equation. Details for the Laplace eq: in the articleE. Kita, N. Kamiya, Trefftz method: an overview. Adv. in Eng. Software, 24,3-12, 1995.

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„Global” approach, BVP

Consider a stationary heat conduction problem in a hollow cylinder:y p y

( ) ( ) RHzRbarzT

rT

rrT

⊂∈⊂∈=∂∂

+∂∂

+∂∂ ,0,,,01

2

2

2

2

MN

( ) bTzbT =,

∑∑==

+=≅m

mmn

nn zrwdzrhczrzrT00

),(),(),(),( θ

⎟⎞

⎜⎛ br ππ

( )HzCzaT πsin, =

Exact solution:

( ) bTrT =0,

( ) bTHrT =, ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

Hb

HaF

Hb

HrF

HzCzrT

ππ

πππ

;

;sin),(

0

0)()()()();( 00000 yIxKyKxIyxF −=

6080

100

80100

Approximate solution has been obtainedwith 8 1st kind hn and 10 2nd kind wn T-functions [ ] ⎞⎛2/ 2kk

0204060

0204060 functions [ ]

∑=

−⎟⎠⎞

⎜⎝⎛

−−

=2/

0

22

2 2)!2()!()1(),(

n

k

knkk

n zrknk

zrh

r rzz [ ]

∑ −⎟⎞

⎜⎛−2/

22)1(ln)()(

nkn

kk

k

zrarzrhzrw

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Exact Approx

a = 0.1 m, b =0.2 m , H = 0.2 m

∑=

⎟⎠

⎜⎝−

−=0

2 2)!2()!(ln),(),(

k

knn z

knkrzrhzrw


A square, 4 elements, inverse BVP

On three sides of the square the Dirichlet( )

( )⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛

21sinh

2cos ππ yx

qboundary conditions are prescribed inaccordance with the accurate solution*:

( )⎟⎠⎞

⎜⎝⎛

⎠⎝⎠⎝=

2sinh

22,π

yxT

~ ~TT ~TTUsing norms L2 and H1

the approximate and exact solutions are compared.

2

~LL TTf −=1

~HH TTf −=

T~ T2

2

L

LL T

TT −=δ

1

1

H

HH T

TT −=δ

y fL = 0,0006 δL = 0,0049

fH = 0,0084 δH = 0,0085

0 012

0.016

0.020 fH

0.012

0.016

0.020 fH

0.06

0.08

0.10

fHc)0.60

0.70

0.80

0.90

1.00

fH , H ,

0.004

0.008

0.012

fL 0.004

0.008

0.012

fL 0.02

0.04

fL

a) b)

0 00

0.10

0.20

0.30

0.40

0.50

δy

0.0 0.2 0.4 0.6 0.8 1.0

0.00

δy 0.0 0.2 0.4 0.6 0.8 1.0 0.00

δy0.0 0.2 0.4 0.6 0.8 1.0

0.00 δy

Distance from the boundary with unknown condition δy versus norms fH and fL for a) 3 b) 2 and c) 1 points with measured temperature

x 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00

Approximate solution for2 points with measuredtemperature


* M.J. Ciałkowski, A. Frąckowiak, Heat functions and their application to solving heat conduction and mechanical problems. Wyd. Politechniki Poznańskiej, Poznań, 2000. (in Polish)


FEMT f th IHCPFEMT for the IHCP:

1o FEMT with condition of continuity of temperature in the common nodes of elementsnodes of elements.

2o No temperature continuity at any point between elements.

3o N d l FEMT I t d i h fi it l t th t t i3o Nodeless FEMT. Instead, in each finite element the temperature isapproximated with the linear combination of the Trefftz functions. Theunknown coefficients of the combination are calculated from thecondition of minimizing the functional that describes the mean-squarefitting of the approximated temperature field in an element to theboundary and initial conditions.y

Moreover, the energetic regularisation is used to improve theapproximate solution, i.e. one minimizes defect of energy dissipationb t l t i l t d ti (b i lt fbetween elements or numerical entropy production, (being a result ofdiscontinuity of heat flux between elements).

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A square, 4 elements, temperature discontinuous between elementsq , , p

( ) ( ) ( )∑ ∫ ∫ ⎥⎥⎤

⎢⎢⎡

Γ−+Γ−= −+−+ ijijnn dTTdqqTQ 22 α

Objective functional:

Accuracy of the approximate solution T~

( ) ( ) ( )∑ ∫ ∫ ⎥⎦⎢⎣Γ Γ++

ijijijnn

ij ij

qqQ

( )[ ]%100

~~~

~

2/1

22

222

⋅

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

⎤⎡ ⎞⎛ ∂⎞⎛ ∂

Ω⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+−

=∫Ω

TT

dyT

yT

xT

xTTT

T

eee

δ2

⎟⎟⎟

⎠⎜⎜⎜

⎝Ω

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

+∫Ω

dyT

xTT ee

e

FEMT continuous FEMT discontinuous

∑=

=≅N

nnn yxVazrTzrT

0),(),(~),(

α=0.001 α=1 α=1000 α=0.001 α=1 α=1000

0.00 0.0042 0.0042 0.0042 0.0207 0.0141 0.0161

0.10 0.0046 0.0046 0.0047 0.0852 0.0153 0.0168 δNorm as a functionof distance for 12

T~δ

δ

0.30 0.0184 0.0184 0.0178 0.1805 0.0376 0.0333

0.50 0.3471 0.3471 0.3471 0.1856 0.0383 0.0523

0.70 0.1655 0.1656 0.3076 0.3954 0.1711 0.3990

δtrial functions. Inputdata are accurate.

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0.90 0.3426 0.3425 0.3153 0.2029 0.0753 0.3492

0.99 0.2097 0.2101 0.0958 0.3669 0.1173 0.7800


h i l i i f C *The energetic regularisation for IHCP*

In order to find an approximate solution one minimizes a heat flux jumpIn order to find an approximate solution one minimizes a heat flux jumpor defect of energy dissipation or defect of numerical entropy productionbetween elements. The terms in the functional are minimized to find

ffi i t i th f l d ibi Th dT~coefficients in the formula describing . They read:

the heat flux jump

iT

( )∑ ∫ ∫Γ

Γ−ji

t

njnie

dqqdt ~~

0

2&&

nTq∂∂

=~

~&

the defect of energy dissipation

Γji ij, 0

∑ ∫ ∫Γ

Γ⎟⎟⎠

⎞⎜⎜⎝

⎛−

ji

t

j

nj

i

nie

dTq

Tqdt ~

~

~~

0

2&&

the defect of numerical entropy

Γ ⎠⎝ji jiijTT, 0

( )∑ ∫ ∫ Γ−t

jnjinie

dTqTqdt ~ln~~ln~ 2&&

production( )∑ ∫ ∫

Γjijnjini

ij

qq, 0

*M J Ci łk ki A F k i k K G S l ti f t ti i h t d ti bl


*M.J.Ciałkowski, A. Frąckowiak, K. Grysa, Solution of a stationary inverse heat conduction problemby means of Trefftz non-continuous mthod. Int. J. Heat Mass Transfer, 50, 2170-2181, 2007.


A square three FEMT approaches the energetic regularisationA square, three FEMT approaches, the energetic regularisation

An IHCP in a square is considered*. Conditions:

( )T∂ ( ) ( )( )1i −∂ hT

( )( ) yy

y

eeyhxT −+==∂∂

1,0

( ) 02 ==∂∂ xhT

( )( ) ( )( )1

31,

sincos −+==∂

eexxxhy x

( ) iib TydT =− ,1 8,...,1=i

( )( )2

0,∂y x

Accurate solution: ( ) ( )( )yy eexxyxT −−+= sincos,

Accurate internal temperatures

Distance versus relative error for minimisation of a) heat flux, b) entropyproduction c) energy dissipation for c(ontinuous) d(iscontinuous) n(odeless) FEM

Lδbd


production, c) energy dissipation for c(ontinuous), d(iscontinuous), n(odeless) FEMA. Maciąg, Trefftz functions for some direct and inverse problems of mechanics. Politechnika Świętokrzyska, Kielce,2009 (in Polish).


Time-space finite elements

( )jjjn tyxv ,, the n-th trial Trefftz functionN( ) ( )

Nj

tyxvaTtyxTN

njjjnin

ijjjji

,...,

,,,,~

11

=

=≡ ∑=

hence the temperature in the i-th element:

( ) ( ) ikN

iki TtyxtyxT ∑= ,,,,~ ϕ

with ϕik being a linear combination of the Trefftz function.

( ) ( )k

iki yy ∑=1

,,,, ϕ

Time-space elements continuous in nodes

( ) ( )( ) 212 /,,,,~

⎥⎤

⎢⎡ −∫ ee dDtyxTtyxT ( ) ( )( )

( )22 ,,

,,,,

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=∫

∫

De

Dee

dDtyxT

yyLδ


Time-space elements discontinuous in nodesRelative error


A square, non-stationary IHCP, FEMT*

Heat transfer eq. tTT ∂∂=Δ / ( ) ( ) ( )1,01,0, ×∈yx ( )ett ,0∈Heat transfer eq. tTT ∂∂Δ / ( ) ( ) ( )1,01,0, ×∈yx ( )e,

Conditions:( ) TT 1 δ

Internal temperatures

( ) ( )yxTtyxTt

,,, 00=

=

tyhT 2)()( +

( ) ikkib TtyT =− ,,1 δyi ∈ 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, 0.9

tyx

etyhtyxT 210

),(),,( +=

==

txetxhtyxT 21)()( ++∂Th l i

tk ∈ 0.0025, 0.0050, 0.0075, 0.01

y

etxhtyxy 2

1

),(),,(=

==∂

txthtT 2)()( +∂( ) tyxetyxT 2,, ++=

The exact solution

tx

y

etxhtyxy

23

0

),(),,( +

=

==∂

* G R ś i k Diff fi i l h f i h d i


*K. Grysa, R. Leśniewska, Different finite element approaches for inverse heat conductionproblems. Inv. Probl. in Science and Eng., 18: 1, 3 — 17, 2010,


Objective functional

et( )( ) ( ) ( )( )

⎞⎛ ∂⎞⎛ ∂

+Γ−+−=Γ

∑ ∫ ∫∑ ∫

ee

e

ii

tt

i

t

ii D

i

TT

dtyhtyTdtdDyxfyxTJ

220

21

2 00

~~

,,,~),(,,~

( ) ( ) ( ) ( )

( ) ( )( )

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

+ΓΓ

∫ ∫

∑ ∫ ∫∑ ∫ ∫

ITRe

e

i

e

i

It

i

i

i

i dtxhtxyTdtdtxhtx

yTdt

22

03

02 01

~~~

,,,,,,

( ) ( )( ) +−+Γ−+−==Γ

∑ ∑∑ ∫ ∫bji xi k

ikkkkiji

ji TtyxTdTTdtδ11

2

0

2,,~~~

, ,

⎞⎛ ∂⎞⎛ ∂ tt TT ~~~~ 22

∑ ∫ ∫∑ ∫ ∫ΓΓ

Γ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂∂

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂∂

+ji

tji

ji

tji

e

ij

e

ij

dy

TyTdtd

xT

xTdt

,, 00

∫ ∫∫ ∫ ⎟⎞

⎜⎛ ∂∂⎟

⎞⎜⎛ ∂∂ t

jit

jiee TTTT

~~~~ 221111

heat flux jump, JS

∑ ∫ ∫∑ ∫ ∫ΓΓ

Γ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂∂

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂∂

+ji

j

j

i

iji

j

j

i

i ijij

dy

TTy

TT

dtdx

TTx

TT

dt,,

~~~~00

1111

∑ ∫ ∫∑ ∫ ∫ Γ⎟⎞

⎜⎛ ∂∂

+Γ⎟⎞

⎜⎛ ∂∂

+t

jit

jiee

dTT

TTdtdTT

TTdt ~l~

~l~

~l~

~l~ 22

entropy prod. jump, JE

energy diss. jump,

19/27

∑ ∫ ∫∑ ∫ ∫ΓΓ

Γ⎟⎟⎠

⎜⎜⎝ ∂

−∂

+Γ⎟⎟⎠

⎜⎜⎝ ∂

−∂

+ji

jj

ii

jij

ji

i

ijij

dTy

Ty

dtdTx

Tx

dt,,

lnlnlnln00

gy j p,JRE


Inaccurate input data

The time-spatial domain is divided into 4 elements. Themeasurements are simulated from the exact solution and disturbedwith a noise with normal distribution not greater than 5% of the exactvalue. In order to obtain good results the input data have beeng psmoothed with the use of 18 Trefftz functions.

Lδ 2Lδ

LδThe relative error versus the distance δ for accurate and inaccurate

20/27


2LδThe relative error versus the distance δb for accurate and inaccurate smoothed input data for nodeless FEMT without energetic regularisation


The approximate temperature accuracy in the nodeless FEMT for 12 Trefftz functions

21/27



The approximate temperature accuracy in the nodeless FEMT for 15 Trefftz functions

22/27



R kRemarks

Th t t bl ith ti l i ti l d t d lt The test problems with energetic regularization lead to very good results for the all three methods.

The best to apply seem to be the nodeless FEMTThe best to apply seem to be the nodeless FEMT.

The relative error does not exceed 1% even in norm for inaccurate and smoothed input data and 12 trial functions.

1Hδp

The greater number of T-functions the better results one obtains.

Smoothing the inaccurate data with the use of Trefftz functions leads to Smoothing the inaccurate data with the use of Trefftz functions leads to results comparable with those obtained with accurate input data.

In the case of a direct problem all three methods lead to good results.

23/27



Open problems:Open problems:

1. A type of time-space element depends on the type of governingequation because the length” of the time side of the elementequation, because the „length of the time side of the elementshould probably depend on the signal propagation velocity.

2. For the HC problems a „velocity” of temperature propagationp „ y p p p gshould be related to the temperature measurement accuracy.

3. Generally in FEMT big finite elements can be used. However, theiri d d h b f i l f isize depends on the number of trial functions.

4. In the places of accumulation of the investigated phenomenon theelements should be concentrated in space and time Far from suchelements should be concentrated in space and time. Far from suchplaces the time-space elements can be greater in time and space.

5. An approximate solution of a (direct, inverse) problem seems to bepp ( , ) pof better quality if the conditions and input data are formulated inthe same subspace of the space generated by the T-completefunctions

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functions.



Open problems:Open problems:

6. The nodeless FEMT seems to be an interesting method to examine interms of solution discontinuities at the edges between theterms of solution discontinuities at the edges between theelements versus the size of the elements and dimensions of thesubspace generated by T-functions.

7. In our study on T-functions application, small number of points withinput data (internal responses) led to good results. Also theincomplete data (eg lack of initial condition or the boundaryincomplete data (eg lack of initial condition or the boundaryconditions known only on a part of the boundary) lead to good(comparable with accurate) results.

8. Regularisation with the use of normal derivative jump on theborders between time-space elements seems to be interesting toinvestigate.g

9. In the case of nonhomogeneous diff equation an idea ofapproximating the right side of the eq. with the T-functions seems

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Mathematicians and engineersMathematicians and engineers

In the article of Z.C. Li, T.T. Lu, H.T. Huang, A.H.-D. Cheng, Trefftz,Collocation, and Other Boundary Methods – A Comparison. Numer.Collocation, and Other Boundary Methods A Comparison. Numer.Meth. Par.Diff. Eq, 23,93-144, 2007 I have found the followingremark:It is of interest to point out that some of the TMs are developed in theIt is of interest to point out that some of the TMs are developed in theengineering community [E. Kita, N. Kamiya, Trefftz method: an overview. Adv. inEng. Software, 24, 3-12, 1995. ], while others in the mathematical community[Z.C. Li, Combined methods for elliptic equations with singularities, interfaces[ , f p q g , fand infinities. Kluwer Academic Publisher, Dordrecht, 1998 ]. As commented inan article on the history of boundary methods [A.H.-D. Cheng, D.T. Cheng,Heritage and early developmentof boundary element. Eng. Anal. Bound. Elem.29, 268-302, 2005 ], “the developments in the two communities, the appliedmathematics and the engineering, seems to ran parallel to each other, almostdevoid of any cross citations…although many techniques have much in common,

d f tili ti i d d ”and cross-fertilization is needed.”

It would be useful to unify the studies of the two communities, andto stimulate cross-fertilization and citation.

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to stimulate cross fertilization and citation.


Thank you for your attentionThank you for your attention

Krzysztof GrysaKielce University of Technology, Al.. 1000-lecia P.P.7, 25-314 Kielce, Poland

e-mail: [email protected]

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IP-TA 2010 Inverse Problems: developments in theory and

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