Trefftz method in solving the inverse problems

AUTHOR’S COPY | AUTORENEXEMPLAR

J. Inv. Ill-Posed Problems 18 (2010), 595 – 616 DOI 10.1515 / JIIP.2010.027

© de Gruyter 2010

Trefftz method in solving the inverse problems Michał J. Ciałkowski and Krzysztof Grysa

Abstract. In this article we present different types of Trefftz functions for stationary and non-stationary linear differential equations. The Trefftz methods are then defined and briefly described. One of them, namely the LSM applied to the region divided on finite elements, is discussed. Examples of the T-functions usage in solving inverse problems and smoothing inaccurate input data are presented. Moreover, some remarks concerning nonlinear non-stationary problem of heat conduction is briefly discussed and some open problems are formulated. Keywords. Linear ill-posed problems, non-stationary inverse problems, Trefftz method. 2010 Mathematics Subject Classification. 31A25, 35J05, 35L05, 35R30, 35K05, 65N21, 74S05, 90C31, 93E24.

1

Introduction

The method known as “Trefftz method” was firstly presented in 1926, [35]. In the case of any direct or inverse problem an approximate solution is assumed to have a form of a linear combination of functions that satisfy the governing differential linear equation (without sources). Because a great majority of more complex problems of mathematical modeling cannot be solved without dividing the area into subregions (elements), such approximation is applied in each element. The unknown coefficients are then determined basing on approximate fulfilling the boundary, initial and other conditions (for instance prescribed in chosen points inside the considered body), finally having a form of a system of algebraic equations. Such approach to the approximate solving a direct or inverse problem is referred to as indirect Trefftz method, [21]. In order to apply the Trefftz method for problems described with a differential linear equation a complete system of the so-called T-functions (or T-complete functions or Trefftz base) has to be known. Generally, Trefftz bases fall into two broad classes. The F-Trefftz method constructs its basis function space by allowing many source points outside the domain Michał J. Ciałkowski has carried out this work within the grant No. 3134/B/T02/2007/33 of Polish Ministry of Higher Education.



596

M. J. Ciałkowski and K. Grysa

of interest to keep basis functions regular. For each source point the fundamental solution is adopted; therefore, this method is also called the method of fundamental solutions (MFS), [24]. The T-Trefftz bases consist of functions which fulfill identically the considered differential equation, [3, 9, 21, 36]. Here we consider only the T-Trefftz bases and as T-function we understand an element of a T-Trefftz base. Starting at the end of the 70s the T-Trefftz have been presented for both bounded and unbounded regions, for two- and three-dimensional stationary problems, [9, 36], namely for Laplace and Helmholtz equations [14, 17], biharmonic equation [13] and Stokes problem [16]. For elasticity problems T-funtions have been shown in [19, 29]. The basic sets of T-functions for stationary problems can be found in [36]. In the 80s and 90s T-complete functions have been used to find approximate solutions of boundary value problems, also with the use of FEM. In the case of non-stationary problem 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 equation, biharmonic one and for reduced wave equation (Helmholtz type eq.). However, no inverse problem was investigated. In late 90s the T-functions for non-stationary problems have been published, at first, based on the paper [33], for parabolic equations, [7, 8, 18, 32], then for wave equation [3, 6, 25], beam vibration, [1], and for plate vibration, [27]. The T-functions for the considered linear differential equation usually have been obtained by the method of variable separation. In 2000 two new methods of deriving the T-functions have been published, [3]. The one based on expanding function in Taylor series is particularly simple and effective; the second is based on the so-called inverse operators, [3, 4]. It is worth mentioning that obtaining the T-functions with the use of Taylor expansion makes it possible to estimate an error of the approximate solution (being a linear combination of the T-functions) of the considered differential equation. During the last 10 years the Trefftz method has been frequently applied to find approximate solutions of the inverse problems, [9]. Up to now the following inverse problems have been considered:

boundary/initial value determination inverse problems,

sources and forces determination inverse problems, and

material properties determination inverse problems.

At first a “global” approach was applied with good results for simple geometry and initial-boundary conditions. Next, the FEM with Trefftz functions as trial functions was investigated. Recently, the T-functions have been tested to smooth



Trefftz method in solving the inverse problems

597

the inaccurate internal data for the wave and for heat transfer equations with very good results, [11, 26]. Further research fields seem to be clearly noticeable. In the paper we present the basic sets of T-functions for stationary and nonstationary problems in bounded regions. Then, the Trefftz methods are briefly described with a special attention to the least square method in relation to nonstationary direct and inverse problems. Moreover, approximate solutions of inverse problems of heat conduction and of wave motion are shown. In conclusion, a proposition of an approach exploiting T-functions to find an approximate solution of a nonlinear non-stationary heat transfer solution is presented and some open problems are formulated.

2

Basic sets of the Trefftz functions for some differential equations

In the paper we confine our attention to the linear differential equations which model physical phenomena in bounded regions. The sets of T-functions presented below are useful in approximate solving of direct and inverse problems. Laplace equation in 2D, [3]:

Fn .x; y/ D Re

@2 @2 C 2 u D 0; @x 2 @y

.x C iy/n ; n D 0; 1; : : : ; nŠ

.x; y/ 2 R2 ; Gn .x; y/ D Im

.x C iy/n ; n > 0: nŠ (2.1)

Another, equivalent form of the functions (without factorials in denominators) is presented in [36]. In the polar system of coordinates .r; '/ the T-functions are as follows; Fn .r; '/ D r n cos n';

Gn .r; '/ D r n sin n'

(2.2)

p with r D .x 2 C y 2 /n , ' D arctan yx . In the cylindrical system of coordinates .r; z/ we have Fn .r; z/ D

Œn=2 X kD0

r 2k .1/k z n2k ; .kŠ/2 .n 2k/Š 2

Gn .r; z/ D hn .r; z/ ln r kn .r; z/


(2.3)


598


with kn .r; z/ D

Œn=2 X kD0

r 2k .1/k ak z n2k ; .kŠ/2 .n 2k/Š 2

Œx D floor.x/

and a0 D 0;

an D 1 C

1 1 C C 2 n

for n > 0:

(2.4)

Laplace equation in 3D, [3]:

@2 @2 @2 C 2 C 2 u D 0; @x 2 @y @z

.x; y; z/ 2 R3 :

In [36] the T-functions are expressed using associated Legendre functions. In [3] they are described by recurrent formulas: F00 D 1;

G00 D z:

If Rlk D Flk or Rlk D Glk , the following formulas hold for n 1 and 1 j n 1: Rn;n1 D xRn1;n1 ; Rn;j C1 D

Rn;1 D yRn1;0 ;

1 Œ.n C 1 j /Rn;j 1 xRn1;j 1 C yRn1;j : j C1

(2.5)

For FnC1;0 and GnC1;0 and n 1 we have 1 Œ2x nFn;0 .x 2 C z 2 /Fn1;0 ; n.n C 1/ 1 Œ2x.n C 1/Gn;0 .x 2 C z 2 /Gn1;0 : D .n C 2/.n C 1/

FnC1;0 D GnC1;0

Biharmonic equation in 2D, [3]:

@2 @2 C @x 2 @y 2

2 u D 0;

.x; y/ 2 R2 :


(2.6)


599


The T-functions Hj n .x; y/, n 0; j D 1; 2; 3; 4, have the following form: H1n D Fn C

y Gn1 ; 2

3 y H2n D Gn Fn1 ; 2 2

H3n D Fn ;

H4n D Gn (2.7) with Fn .x; y/ and Gn .x; y/ given by formulas (2.1); F1 .x; y/ D G1 .x; y/ D 0. Heat conduction equation in 1D, [18, 33]: @u @2 u ; D 2 @x @t

.x; t/ 2 .0; T / ; R1 :

The T-functions (also called heat polynomials) read vn .x; t / D

Œn=2 X kD0

x n2k t k : .n 2k/ŠkŠ

(2.8)

In the polar system of coordinates .r; t / the T-complete set of functions consists of two sets of functions: n r 2n2k k X t 2 and un .r; t / D gn .r; t / ln r C qn .r; t / (2.9) gn .r; t / D 2 Œ.n kŠ k/Š kD0 with qn .r; t / D

Pn

kD0

ank . 2r /2n2k t k Œ.nk/Š2 kŠ

and an defined by formula (2.4).

Heat conduction equation in 2D, [3]: 2 @ @u @2 uD ; .x; t/ 2 .0; T /; R2 : C @x 2 @y 2 @t Here Vm .x; y; t/ D vnk .x; t /vk .y; t / n D 0; 1; : : : I k D 0; : : : ; nI mD

n.n C 1/ C k; 2

(2.10)

and vn .u; t / is given by (2.8). In the cylindrical system of coordinates .r; z; t / the T-complete set of functions consists of two sets of functions: V1m .r; z; t / D gnk .r; t /vk .z; t /;

V2m .r; z; t/ D unk .r; t /vk .z; t /

with n D 0; 1; : : : I k D 0; : : : ; n and m D

n.nC1/ 2

C k.


(2.11)


600


Heat conduction equation in 3D, [3]:

@2 @u @2 @2 uD ; C C 2 2 2 @x @y @z @t

.x; t/ 2 .0; T /; R3 I

Vnkl .x; y; z; t/ D vnkl .x; t /vkl .y; t /vl .z; t /; n D 0; 1; : : : I k D 0; : : : ; nI l D 0; : : : ; k;

(2.12)

and vn .u; t / is given by (2.8). Wave equation in 2D, [25]:

@2 @2 u @2 u D C ; @x 2 @y 2 @t 2

.x; y; t/ 2 .0; T /:

Substitution z D i t transforms the Laplace equation in 3D for the wave equation in 2D. However, such approach leads to a necessity of using the symbolic calculation in order to obtain T-functions for the wave equation. Another approach, based on the variable separation method, leads to the following recurrent formulas for the T-functions (wave polynomials) Pmn0 .x; y; t/, Pmn1 .x; y; t/, Qmn0 .x; y; t/ and Qmn1 .x; y; t/: P000 D 1;

Q000 D 0;

and for n D 1; 2; : : : , k D 0; 1; : : : ; n 1 P.nk/k0 D .xQ.nk1/k0 C yQ.nk/.k1/0 n C tQ.nk2/k1 C tQ.nk/.k2/1 /; 1 P.nk1/k1 D .xQ.nk2/k1 C yQ.nk1/.k1/1 C tQ.nk1/k0 /; n (2.13) 1 Q.nk/k0 D .xP.nk1/k0 C yP.nk/.k1/0 n C tP.nk2/k1 C tP.nk/.k2/1 /; Q.nk1/k1 D

1 .xP.nk2/k1 C yP.nk1/.k1/1 C tP.nk1/k0 /: n

If any subscript is negative, the value of the function with such a subscript is equal to zero.




601

Wave equation in 3D, [27]:

@2 @2 u @2 @2 u D C C ; @x 2 @y 2 @z 2 @t 2

.x; t/ 2 .0; T / :

The T-functions P.nklm/klm .x; y; z; t/ and Q.nklm/klm .x; y; z; t/ for n D 1; 2; : : : , k D 0; 1; : : : ; n, l D 0; 1; : : : ; n and m D 0; 1 and for P0000 D 1, Q0000 D 0 may be obtained from the following recurrent formulas, [27]: 1 P.nkl/kl0 D .xQ.nkl1/kl0 C yQ.nkl/.k1/l0 n C zQ.nkl/k.l1/0 C tQ.nkl2/kl1 C tQ.nkl/.k2/l1 C tQ.nkl/k.l2/1 /; 1 P.nkl1/kl1 D .xQ.nkl2/kl1 C yQ.nkl1/.k1/l1 n C yQ.nkl1/k.l1/1 C tQ.nkl1/kl0 /; Q.nkl/kl0 D

Q.nkl1/kl1 D

1 .xP.nkl1/kl0 C yP.nkl/.k1/l0 C zP.nkl/k.l1/0 n C tP.nkl2/kl1 C tP.nkl/.k2/l1 C tP.nkl/k.l2/1 /; 1 .xP.nkl2/kl1 C yP.nkl1/.k1/l1 n C yP.nkl1/k.l1/1 C tP.nkl1/kl0 /:

Beam vibration equation, [1]:

(2.14)

@4 @2 C 2 u .x; t/ D 0; @x 4 @t

.x; t/ 2 .0; 1/ .0; T / :

Consider the Taylor series for a solution of the equation:

u .x; t/ D u C

du d 2 u dNu C C C C RN C1 1Š 2Š NŠ



602


where d n u D . @u x C @u t/.n/ and RN C1 is the rest of the series, tending to @x @t zero with N tending to infinity. In order to eliminate the derivatives with respect to variable x of the order higher than four, the following form of the governing equation will be used: 2v @4v u v @ u .1/ D ; @x 4v @t 2v

v D 1; 2; : : : :

Taylor series may be then written as follows: ˇ ˇ ˇ ˇ x2 @2 u ˇˇ @u ˇˇ @2 u ˇˇ @u ˇˇ C xC tC xt u .x; t/ D u .0; 0/ C @x ˇ.0;0/ @t ˇ.0;0/ @x 2 ˇ.0;0/ 2Š @x@t ˇ.0;0/ ˇ ˇ ˇ ˇ @2 u ˇˇ t2 x3 x2t @3 u ˇˇ xt 2 @3 u ˇˇ @3 u ˇˇ C C C C @t 2 ˇ.0;0/ 2Š @x 3 ˇ.0;0/ 3Š @x 2 @t ˇ.0;0/ 2Š @x@t 2 ˇ.0;0/ 2Š ˇ ˇ ˇ @3 u ˇˇ t3 @2 u ˇˇ x4 x3t @4 u ˇˇ C C C ˇ ˇ ˇ @t 3 .0;0/ 3Š @t 2 .0;0/ 4Š @x 3 @t .0;0/ 3Š ˇ ˇ ˇ x2t 2 xt 3 @4 u ˇˇ t4 @4 u ˇˇ @4 u ˇˇ C C C @x 2 @t 2 ˇ.0;0/ 2Š2Š @x@t 3 ˇ.0;0/ 3Š @t 4 ˇ.0;0/ 4Š ˇ ˇ @3 u ˇˇ x5 @3 u ˇˇ x4t C : C @t 2 @x ˇ.0;0/ 5Š @t 3 ˇ.0;0/ 4Š Sums of coefficients accompanying the same derivatives stand for the T-functions. Finally, the T-functions for the beam vibration equation for n D 1; 2; : : : read:

Sn .x; t/ D

n3Œ n2 3 4 2 X j D0

n2 n2 .1/j t n3Œ 4 32j x 4Œ 4 nC5C4j : n 3 n2 3 2j Š 4 n2 n C 5 C 4j Š 4 4 (2.15)

Plate vibration equation, [27]:

@4 @4 @4 @2 u .x; y; t / D 0: C 2 C C @x 4 @x 2 @y 2 @y 4 @t 2




603

The T-functions Pkl .x; y; t / and Qkl .x; y; t / for n D 1; 2; : : : ; and for P00 D 1, Q00 D t may be obtained from the following recurrent formulas, [27]: Pn1 D yP.n1/0 ;

Qn1 D yQ.n1/0 ;

Pn.n1/ D xP.n1/.n1/ ; P.nC2/0 D

Qn.n1/ D xQ.n1/.n1/ ;

2x.n C 1/P.nC1/0 x 2 Pn0 4t 2 P.n1/0 2tQ.n2/0 ; .n C 1/.n C 2/

2x.n C 1/Q.n1/0 x 2 Q.n2/0 4t 2 Q.n4/0 C 2tPn0 ; .n C 1/.n C 2/ (2.16) .n k C 1/Pn.k1/ xP.n1/.k1/ C yP.n1/k ; D kC1 1 k n 1;

Qn0 D Pn.kC1/

Qn.kC1/ D

.n k C 1/Qn.k1/ xQ.n1/.k1/ C yQ.n1/k ; kC1 1 k n 1:

Other sets of T-functions for stationary problems are mentioned and discussed in [36] and in the monograph [9].

3

Trefftz methods

Trefftz methods have not received a precise definition, although this terminology has had wide acceptance. Herrera’s definition of what is meant by a Trefftz method is, [15]: Given a region of an Euclidean space or some partitions of that region, a ‘Trefftz Method’ is any procedure for solving initial boundary value problems of partial differential equations or systems of such equations, on such region, using solutions of that differential equation or its adjoint, defined in its subregions. When Trefftz method is conceptualized in this manner, it includes many of the basic problems considered in numerical methods for partial differential equations and becomes a fundamental concept of that subject. For the inverse problems the definition has to be widened for situation in which the prescribed conditions are known in a set of points or on a line/surface inside the considered body and the other conditions are only partly known. The Trefftz methods are generally divided into two groups: direct methods and indirect methods. In the direct method the governing equation is to be fulfilled in a weak formulation with trial function being a linear combination of T-functions. In the indirect method the solution itself is approximated as a linear combination of the T-functions.



604


Among the indirect methods the most popular ones seem to be the least square method (LSM) and collocation method. Different formulation of the indirect methods are presented in [21]. The LSM seems also to be useful when dealing with the inverse problems. Let us consider a problem formulated as follows: Lu D f

in .0; te /;

u D g1 on SD .0; te /; @u D g2 on SN .0; te /; @n

p p

@u C ˛u D g3 on SR .0; te /; @n u D g4 on Sint Tint uDh

on for t D 0;

where Lu D f is a parabolic linear differential equation, SD [ SN [ SR S D @, Sint stands for a set of points inside the considered region, Tint .0; te / is a set of moments of time and the functions gi , i D 1; 2; 3; 4 and h are of proper class of differentiability in the domains in which they are determined. In order to solve in an approximate way such a problem let us consider the following objective functional: Z Z 2 .Lv f / d dt C w1 .v g1 /2 dSdt I .v/ D .0;te /

Z

p

C w2 SN .0;te /

Z C w3 Z

SR x.0;te /

@v g2 @n

SD .0;te /

2

dSdt

@v p C ˛v g3 @n 2

(3.1)

2 dSdt Z

.v g4 / dSdt C w5

C w4 Sint Tint

.v h/2 d :

Here wi , i D 1; 2; 3; 4; 5, stand for constant positive weighting parameters which are to preserve the numerical equivalence between the terms, and v 2 V . The problem may be formulated as follows: find a function u 2 V such that I .u/ D min I .v/ : 8v2V

Such approach leads to the LSM.


(3.2)


605


Let V be a space generated by a finite number of T-functions for the equation Lu D 0. If v is a linear combination of the Trefftz functions for the homogeneous equation and v part is known (the function v part stands for a particular solution of the equation Lu D f ), the first term in the equation (3.1) vanishes. If f D 0 then the function v part is equal to zero. Usually the weighting parameters are equal to one. Moreover, the objective functional may be completed with term or terms regularizing the solution, [5, 10]. The advantage of the LSM are that the stiffness matrix in the system of algebraic equations is always symmetric and positive definite (or semi-definite) and that even complicated constraints of the solution can be easily incorporated with the numerical method. However, in the case of global approach the disadvantage of LSM is that the condition number will increase significantly, compared with FEM, [23]. When the region has to be divided into subregions in order to use FEM one introduces the time-space elements, [12, 28]. In the case of stationary problem the spatial elements are used. In both cases the FEM with T-functions (FEMT) is convenient to be applied. At least three kinds of FEMT may be used: 1. FEMT with condition of continuity of temperature in the common nodes of elements. 2. FEMT with no temperature continuity at any point between elements. 3. Nodeless FEMT. Instead, in each finite element the temperature is approximated with a linear combination of the T-functions. Hence, depending of the kind of FEMT, the T-functions are used as trial functions or the approximate solution being a linear combination on the T-functions is constructed in each element separately, [11]. The elements may be greater than in the classic FEM because the approximate solution fulfill the governing equation identically. Moreover, the number of T-functions used to approximate the solution in an element can be smaller than in the classic FEM. Usually on the interior boundaries between elements one considers continuity of the approximate solution and its normal derivative. Then the objective functional is completed with terms Z

C

2

.v v / dSdt .0;te /

Z and .0;te /

@v @v C @n @n

2 dSdt

accompanied with positive weight constants. Here stands for the interior boundaries, v C and v (nC and n / denote values of the function v (normal vector n/ on both sides of , respectively.



606


Independently of the method used to build the approximate solution with T-functions, the equation (3.2) leads to a system of algebraic equations for the coefficients of the combination. Therefore, instead of a global approach with an approximate solution being a linear combination of the T-function, it is more useful to use the FEM with T-functions. In the case of operator L being a hyperbolic one the analysis is similar with the only difference concerning initial conditions and resulting in an additional term in the objective functional.

4

Examples of the inverse problems solved using Trefftz method

Example 1. Consider a transient 2D inverse heat conduction problem (IHCP) in dimensionless coordinates in a square. On three sides of the square the heat flux is prescribed. Thermal conditions on the fourth side are unknown. Instead, inside the square on a line distant at ıb 2 .0; 0:99/ from the side with unknown temperature the internal temperature responses (ITRs) are prescribed. In the case of a transient problem the heat conduction equation is considered: T D @T =@t

.x; y/ 2 D .0; 1/ .0; 1/ ;

t 2 .0; tK /

(4.1)

with the conditions T .x; y; t/j tD0 D T0 .x; y/;

(4.2)

T .x; y; t /jxD0 D h1 .y; t / D e yC2t ; ˇ ˇ @T .x; y; t /ˇˇ D h2 .x; t / D e xC1C2t ; @y yD1 ˇ ˇ @T .x; y; t /ˇˇ D h3 .x; t / D e xC2t ; @y yD0 T .1 ıb ; yi ; tk / D Tik :

(4.3) (4.4) (4.5) (4.6)

Here Tik s stand for known ITRs in points distant at ıb 2 .0; 0:99/ from the side x D 1 of the square, i D 1; : : : ; I I k D 1; : : : ; K. In order to solve the problem the time interval is divided into subintervals. In each subinterval the domain is divided into 4 finite subdomains j , j D 1; 2; 3; 4 and in each subdomain the temperature is approximated by the linear combination of the T-functions Vm .x; y; t /;

T j .x; y; t / TQ j .x; y; t/ D

N X

j cm Vm .x; y; t /;

mD1



607


given by formula (2.10). In the case of the first time subinterval an initial condition is known. For the next subintervals the initial condition is understood as the temperature field in the subdomain j at the final moment of time in the previous subinterval. The LSM is used to minimize the inaccuracy of the approximate solution on the boundary, in the initial moment of time and on the borders between elements. This way the unknown coefficients of the combination are calculated. Moreover, on the border between elements the heat flux jumps are minimized, [11]. The objective functional reads: XZ .TQi .x; y; 0/ T0 .x; y//2 dD J D i

C

XZ i

C

XZ i

C

XZ i

C

XZ i;j

C

XZ i;j

Di

Z

tm tm1

tm1

!2

i

@TQi .x; 1; t/ h2 .x; t/ @y

i

@TQi .x; 0; t/ h3 .x; t/ @y

tm1

Z

tm tm1

TQi TQj

tm1

2

d dt C

i;j

Z

tm

d dt !2

Z

tm

d dt

i

Z

tm

2

TQi .0; y; t / h1 .y; t /

ij

I X K XX

TQj .1 ıb ; yi ; tk / Tik

j

@TQi @TQj @x @x

d dt

!2 d dt C

i

2

kD1

XZ i;j

0

te

Z ij

@TQi @TQj @y @y

!2 d dt:

The dimensionless time interval is .0; 0:01, i.e. t0 D0. The temperatures are known at points (1 ıb ; yi ) with yi 2 ¹0:1; 0:2; 0:3; 0:4; 0:6; 0:7; 0:8; 0:9º for t 2 ¹0:0025; 0:0050; 0:0075; 0:01º and ıb 2 .0; 0:99/, see Figure 1. It means that I D 8, K D 4 and m D 1; 2; 3; 4. For ıb D 0 the problem becomes a direct one. To the exact values of the ITR generated from the accurate solution we have added a random noise with the mean value equal to zero and standard deviation equal to 0:05. The noisy ITRs have been next approximated with the linear combination of the T-functions due to formula T .1 ıb ; y; t/ TQıb .y; t / D

S X

as Vs .1 ıb ; y; t/:

sD1

For calculation we have taken S D 18 T-functions.


(4.7)


608


Figure 1. The boundary conditions (4.3)–(4.5), the ITR placement and the finite elements.

Because the exact solution is known, the error of the approximate solution in the square in the i -th time subinterval can be described using the norm: "R ıL2 D

Q y; t / T .x; y; t//2 dD D .T .x; R 2 D T .x; y; t/ dD

#1=2 100%;

D D .ti1 ; ti /:

The error on the part of the boundary where the temperature is calculated in the m-th time subinterval according to the norm reads " R tm ıL2 jxD1 D

tm1

#1=2 R1 dt 0 .TQ .1; y; t / T .t; y; t//2 dy 100%: R tm R1 2 tm1 dt 0 T .1; y; t/ dy

Results obtained with the nodeless FEMT for noisy and smoothed data are presented in Table 1 and in Figure 2. For exact ITRs results obtained for ıb D 0 (a direct problem) are sometimes worse than those for ıb > 0 (an inverse problem). It may be a result of approximating the solution with a small number of T-functions; sometimes, in such a case, the results inside the region under consideration are better than those close to the boundary. Comparing errors of the unknown boundary identification for exact and smoothed noisy data one observes a slight difference between their values.



609


ıb

Noisy ITRs Smoothed ITRs Exact ITR ıL2 Œ% ıL2 jxD1 Œ% ıL2 Œ% ıL2 jxD1 Œ% ıL2 Œ% ıL2 jxD1 Œ%

0

0.3916

0.5071

0.03915 0.0795

0.0428

0.07186

0.1

0.7204

1.4844

0.04318 0.0745

0.0315

0.07938

0.2

1.2183

2.5965

0.04612 0.0746

0.0334

0.08551

0.3

1.0455

0.6235

0.04470 0.0833

0.0478

0.08107

0.4

0.9212

0.8840

0.04254 0.1012

0.0416

0.07455

0.5

0.6929

0.4371

0.04069 0.0991

0.0469

0.07197

0.6

0.9228

0.8737

0.04417 0.0937

0.0438

0.08063

0.7

1.1008

0.9198

0.04463 0.0932

0.0459

0.08005

0.8

1.1216

0.3904

0.04410 0.0911

0.0429

0.07760

0.9

0.7263

0.4261

0.04339 0.0890

0.0464

0.07696

0.99 0.4265

0.3864

0.04573 0.0884

0.0517

0.08119

Table 1. Norm ıL2 and ıL2 jxD1 as functions of ıb for noisy and smoothed ITRs.

Figure 2. Norm ıL2 jxD1 of the temperature field error for noisy and smoothed ITRs.



610


Example 2. Consider free vibrations of a square membrane D .0; 1/ .0; 1/ described by dimensionless wave equation in 2D, [9, 26], with the conditions: u .x; y; 0/ D x .x 1/ y .y 1/ ; ˇ @u ˇˇ D 0; @t ˇ.x;y;0/ u .0; y; t/ D u .x; 0; t / D u .x; 1; t/ D 0; and with the ITRs in three inner points: u .1 "; 0:25; t/ D u1 .t / ; u .1 "; 0:50; t/ D u2 .t / ; u .1 "; 0:75; t/ D u3 .t / : If " > 0 we have an inverse problem and we search for the solution in the whole domain but especially u.1; y; t /. The exact solution of the problem for u.1; t; 0/ D 0 is known, [26], and is used to generate the ITRs. The approximate solution isPapproximated in the whole square according to the formula (4.7), i.e. u w D N nD1 cn Vn with Vn s being T-functions (wave polynomials) for wave equation in 2D (presented in Section 2, formulas (2.13)). The objective functional has the following form: Z 1Z 1 Œu .x; y; 0/ x .x 1/ y .y 1/2 dxdy I D 0 0 #2 Z ƒt Z 1 Z 1Z 1" ˇ @u ˇˇ .u .0; y; t //2 dydt dxdy C C @t ˇ.x;y;0/ 0 0 0 0 Z ƒt Z 1 Z ƒt Z 1 2 .u .x; 0; t // dxdt C .u .x; 1; t//2 dxdt C 0 0 0 0 μ Z t ´ X 3 .u .1 "; 0:25k; t/ uk .t //2 dt: C 0

kD1

Figure 3 shows the exact solution (solid line u D 0) for the vibration as a function of time for the location x D 1, y D 0:5 and the approximation (dash line) by polynomials from order 0 to 7 for (a) " D 0, (b) " D 0:05, (c) " D 0:3 in the case when ITRs are undisturbed. Now let us disturb the internal response ukl according to the formula udkl D ukl .1 C kl / where kl are random numbers of normal distribution with expected value equalling zero and standard deviation equaling 0:01 (N.0; 0:01/). The disturbance kl was generated separately for each ukl using random-number genera-




(a)

611

(b)

(c) Figure 3. Exact solution versus time for point x D 1, y D 0:5 and approximation by wave polynomials from order 0 to 7 (a) " D 0; (b) " D 0:05; and (c) " D 0:3.

tor. In this case, the relative error R t ED

0

Œu.1; 0:5; t / w.1; 0:5; t /2 dt R t 2 0 Œu.1; 0:5; t / dt

!1=2

has a value from circa 2000% for " D 0 (direct problem) to circa 180 000% for " D 0:8. It means that a very small disturbance causes a very bad approximation. It is easy to improve the results by smoothing the disturbed data by a linear combination of wave polynomials in a similar way like in the Example 1 (formula (4.7). Such an idea is understandable because the membrane’s vibrations are described by are chosen so as to minimize P a wave equation. Coefficients of the combination d 2 .u .1 "I .k 1/ 0:1I .l 1/ 0:1/ u / . After the process of smooth" kl kl ing we get a similar error as in the case of the exact data, [26].

5

Remarks on nonlinear non-stationary problem of heat conduction

In technical investigation concerning heat conduction a negligence of the dependence of physical properties on the temperature T in not always possible and the heat conduction equation becomes nonlinear: .T /c.T /

@T D div..T /rT /; @t

t 2 .0; 1/; x; y; z 2 R3


(5.1)


612


where ; c and stand for density, constant-pressure specific heat and thermal conductivity, respectively. Such equation may be considered with the different boundary and initial conditions. The coefficient is frequently determined from experiment. In order to avoid its differentiation one introduces the Kirchhoff substitution: Z T .s/ ds D P .T /: #D 0 .0/ Hence @# .T / @T D @t .0/ @t

and

r# D

.T / rT .0/

and finally the equation (5.1) may be transformed to the following form: .T .#//c.T .#// @# D # .T .#// @t

or

f .#/

@# D #: @t

The inverse transformation T D P 1 .#/ is possible to be made analytically only in special cases, e.g. for linear or exponential dependence of on T . However, in many practical cases the inverse transformation may be done numerically with the use of linear interpolation, for instance using trapezium method, because values of the conductivity as a function of T is assumed to be known from experiment. In order to solve the nonlinear equation (5.2) we can apply the method of successive approximations. With T D P 1 .#/ we can apply FEMT for the equation (5.2) in which the coefficient f .#/ is replaced with the mean value of f .#/ in the considered region. The obtained solution is the first approximation. In the next one in each finite element f .#/ may be replaced with its mean value in the element and FEMT with T-functions (considered in each element in the local dimensionless system of coordinates) may be applied. Repeating the procedure to the moment in which two subsequent solutions #i1 and #i are sufficiently close in the norm, say, H 2 , we stop calculations.

6

Final conclusions

Many different algorithms based on Trefftz method have been being developed intensively during last twenty years. It is interesting that some of Trefftz methods are developed in the community of engineers, while others in the mathematical community, [23]. As recommended in [2], “the developments in the two communities, the applied mathematics and the engineering, seem to run parallel to each other, almost devoid of any cross citations . . . although many techniques have much in common, and cross-fertilization is needed.”




613

The object of this paper is to present the existing T-function bases for bounded region for stationary and non-stationary problems. Not all bases are mentioned here. Some of them, like Kupradze functions, [22], or T-functions for Helmholtz equation are presented and discussed in other papers devoted to the Trefftz methods, [20, 30, 31, 36] and many others. Two applications of T-functions for solving non-stationary inverse problems show the usefulness of the functions. Many others are reported in [9]. A conclusion that follows the presented examples (and that should be check by mathematicians) is that an approximate solution of a direct or inverse problem with the use of T-functions seems to be of better quality if the functions describing conditions and input data are formulated in the same subspace of the space generated by the Trefftz base. There are still a lot of open problems. They concern some technical problems as well as mathematical background of the Trefftz methods. For instance, when dealing with the FEM or FEMT with time-space elements a question concerning a dependence between the length of the spatial and the “length” of the time dimensions of the element appears. Probably they can depend on the investigated signal propagation velocity. In the case of the heat conduction problems the “velocity” of temperature propagation is difficult to define. Theoretically, it is equal to infinity, but practically it depends strongly on accuracy of the temperature measurements. Generally, in FEMT big finite elements can be used. However, their size depends on the number of trial functions. The kind of the dependence is not known. In the second example (and in many papers devoted to the inverse heat conduction problems) small numbers of points with input data (with ITRs) lead to good results. Also the incomplete data (e.g. lack of initial conditions, [34], or the boundary conditions known only on a part of the boundary, [27]) lead to good (comparable with the accurate) results. Many other questions arise when applying Trefftz methods. From the point of view of the engineers obtaining better results with increasing number of Tfunctions is a good justification to apply the method. However, the mathematicians are kindly invited to check some ideas of the engineers. It would be useful to unify the studies of the two communities.

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Received April 18, 2010. Author information Michał J. Ciałkowski, Poznań University of Technology, 5 Maria Skłodowska-Curie Sq., 60-965 Poznań, Poland. E-mail: [email protected] Krzysztof Grysa, Kielce University of Technology, Al. 1000-lecia Państwa Polskiego 7, 25-314 Kielce, Poland. E-mail: [email protected]


Trefftz method in solving the inverse problems

Trefftz method in solving the inverse problems

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