MULTI-SCALE PERTURBATION METHODS IN MECHANICS

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1 006/3 4 PAGES RECEIVED ACCEPTED J SKRZYPCZYK MULTI-SCALE PERTURBATION METHODS IN MECHANICS Dr Hab Eng J Skrzypczyk, Prof ul Krzywoustego 7, Gliwice, Pol, Phone: , , Fax: , jerzy skrzypczyk@polslpl Author: Born 1948 Ph D 1975, Ph Hab 1995 Professor at Silesian University of Technology, V-ce Dean of Faculty of Civil Engineering, V-ce Director of Institute of Theoretical Mechanics, Chair of Theoretical Mechanics Department Prof of Technical University of Czestochowa Lectures at Universities of Stuttgart Hannover Member of American Math Soc, Polish Soc of Theoretical Applied Mechanics, Polish Soc of Computer Methods in Mechanics, Polish Math Soc Over 100 publications from mechanics, mathematics control theory, specially interested in stochastic anf fuzzy problems ABSTRACT KEY WORDS The aim of the paper is to present a modern algebraic system with specifically defined addition multiplication operations New numbers called multi-scale perturbation numbers are introduced It is proved that the system of real numbers (R,+, ) is imbedded into the new algebraic system (R εn,, ) Some additional properties such as subtraction, inversion division can be analyzed too Elementary formulations of function extensions are introduced Some basic properties of perturbation matrices, inverse matrices solution properties of linear equations eigenvalue problems in the new algebraic system can be defined Classical multi-scale perturbation problems can be solved in the new algebraic system as easily as usual problems of applied mathematics, theoretical physics techniques Additional analytical transformations are not required perturbation numbers, multi-scale perturbation, algebraic systems, boundary element, extended functions 1 INTRODUCTION The theory of perturbations is a part of science of great theoretical practical meaning It began in with the papers of Rayleigh [10] Schrödinger [1] The first papers treated eigenvalues applications in physics, namely in acoustics The first papers were formal incomplete from a mathematical point of view The first person who considered the convergence of the expansion series of perturbation theory was Rellich [11] The papers of Rellich are fundamental for perturbation theory Now, perturbation theory has a bibliography which has thouss of positions is still in use, cf the monograph by Kato [6] Finding the exact solution values for many computational problems is not an easy task Sometimes it is easier to calculate the solutions of a closely related problem then use the knowledge from perturbation theory to determine approximate solutions of the original problem In some problems, the underlying physical system may be subjected to changes (perturbations), we may want to determine the consequent change in the solutions On other occasions, we may know an problem only approximately due to errors in observation, or we may have to feed an approximation of it to a computing device In each case we would like to know how much this error or approximation would affect the solutions of the problem, cf [1],[5],[7] Classical perturbation methods can be formulated in the following sense Consider how perturbations (small) of nominal parameter values can change the solutions of the problem considered Assume that a solution of the considered problem, say x 0 corresponds to the matrix of coefficients A The basic problem of perturbation theory is to answer how much the solution changes if matrix A takes the new value A+εB, where ε is called a one-scale small parameter B is the perturbation It is often convenient to seek the solution in the form of a series of homogeneous terms in the coefficients of the perturbation matrix B, that is, solutions of the form 006 SLOVAK UNIVERSITY OF TECHNOLOGY skrzypczykindd :7:54

2 x:=x 0 +εx 1 + x +ε 3 x 3 + (1) are sought If we restrict our considerations to the first two terms in (1) we have an one-scale perturbation method of the first order In applications of the perturbation method a serious difficulty is the necessity for a large amount of analytical calculations As a result, we obtain a set of classical problems which are usually simpler to solve numerically, cf [1],[3],[6],[7],[10-17] ALGEBRAIC SYSTEM OF MULTI-SCALE PERTURBATION NUMBERS DEFINITION 1 Define a new number called further n-scale perturbation numbers (n-pn s) as ordered (n+1)-couples of real numbers ) R n+1 The set of perturbation numbers is denoted by R εn The first element x 0 of the (n+1)-couple is called the main value, the following are the perturbation values or simply the perturbations [13]-[17] Let ζ,,,ζ 3 denote any perturbation numbers ζ:= ), := (y 0,,y n ), := (z 0,z 1,,z n ), ζ 3 := (v 0,v 1,v,,v n ), x i,y i,z i,v i R, i=0,1,,,n It is determined that the two perturbation numbers are equal: if only if y i = z i for any i=0,1,,,n In the set R εn we introduce the addition ( ) multiplication ( ) as follows: = (y 0,,y n ) (z 0,z 1,,z n ) := (y 0 +z 0 +z 1 +z,,y n +z n ) () = (y 0,,y n ) (z 0,z 1,,z n ) := (y 0 z 0,y 0 z 1 +y 1 z 0,y 0 z +y z 0,, y 0 z n +y n z 0 ) (3) THEOREM 1 The set R εn with addition ( ) multiplication ( ) defined by Eqs () (3) with the selected neutral addition element 0 εn :=(0,0,,0) neutral multiplication element 1 εn :=(1,0,0,,0) is a field Defined in such a way, the field is called the field of n- PN s The field R εn as defined in Def 1 does not contain the field of real numbers R We can show that the real numbers can be considered as some elements of the field R εn with all the classical addition multiplication formulas the neutral elements of addition multiplication, cf [],[3],[9] THEOREM The map :R R εn, (x):=(x,0,0,,0) for each x R, is called the injection of the algebraic system of real numbers R into the algebraic system R εn It is single-valued mapping preserves the corresponding algebraic operations neutral elements of the additions multiplications For further details, see [13]-[17] 3 SIMPLIFIED NOTION FOR THE PERTURBATION CALCULUS Notice, that since () is the injection, then for each perturbation number of the form (a,0,0,0,,0), a R, we can identify with a real number a We use this notice to simplify the notion for perturbation operations Denote by the n-pn (0,1,0,0,,0), by n-pn (0,0,1,0,0,,0),, by n-pn (0,0,0,,0,1) respectively Then for every n-pn ζ= ) we can write ) =,0,0,,0) (0,0,0,,0) (0,0,0,0,,0) + (0,0,,0,x n ) =,0,0,,0) (x 1,0,0,,0) (0,1,0,0,,0) (x,0,0,,0) (0,0,1,0,0,,0) (x n,0,0,,0,) (0,0,,0,1) = ) (x 1 ) (x ) (x n )= x 0 x 1 x x n if we assume for simplicity s sake that any n-pn (x k,0,0,,0) is identical with the real number x k, for every x k R, k=0,1,,,n From the multiplicity formulas, it follows that = = (0,1,0,0,0) (0,1,0,,0) = (0,0,0,,0), = = (0,0,1,,0) (0,0,1,0,0) = (0,0,0,,0), = = (0,0,0,,0,1) (0,0,0,,0,1) = (0,0,0,,0), ε i ε j = = (0,0,0,,0), in the simplified notion ε i ε j =0 for any i,j=1,,3,,n in consequence ) = x 0 1 x 1 x + n x n 4 ORDER RELATION IN THE SET OF PERTURBATION NUMBERS In the set of n-pn s, similarly as in the sets: R, set of complex numbers C 1, R n, n>1, etc, it is not possible to introduce the complete order relation Following that fact, we define the relation of partial order in the following matter 41 Partial ordering in the strong form DEFINITION For, if y 0 z 0 y 1 z 1, y z,, y n z n DEFINITION 3 For, > ε if y 0 > z 0 y 1 > z 1, y > z,, y n > z n DEFINITION 4 For, = ε if In an analogous way we define the relations ε < ε DEFINITION 5 The perturbation number ζ positive (nonnegative) if ζ> ε 0 ε (ζ 0 ε ) 14 MULTI-SCALE PERTURBATION METHODS IN MECHANICS skrzypczykindd :7:56

3 DEFINITION 6 The perturbation number ζ negative (nonpositive) if ζ < ε 0 ε (ζ ε 0 ε ) REMARK 1 In further consideration we notice that, że ζ ε 0 εn (strongly), if x 0 0 x 1 0, x 0,, x n 0 REMARK In further consideration we change the symbol = ε with the simplified = 4 Partial ordering in the weak form In further consideration we often use the partial order relation in the simpler (weaker) form DEFINITION 47 For, if y 0 z 0 y 1,,y n, z 1,,z n are arbitrary DEFINITION 48 For, if y 0 > z 0 y 1,,y n, z 1,,z n are arbitrary DEFINITION 49 For, if (or equivalently y 0 = z 0 In an analogous way we define the relation DEFINITION 410 The perturbation number ζ weakly positive (nonnegative) if ζ 0 εn (ζ 0 εn ) (notice that perturbation parts x 1 can be arbitrary) DEFINITION 411 The perturbation number ζ, is said to be weakly negative (nonpositive) if ζ 0 εn (ζ 0 εn ) (notice that perturbation parts x 1 can be arbitrary) Notice that the relations between perturbation numbers of the strong type such as ε,, = ε, > ε, < ε imply the weak relations:,,,,, respectively REMARK 3 In further consideration in place of the symbol = ε the simplified notion is used =, since it is not going to cause misundersting REMARK 4 In further consideration we say that ζ 0 ε (weakly), if x 0 0 x 1,,x n are arbitrary can we underst the notion exp(ζ), where ζ= )? Notice that we can exp exp(x), x R into a classical series which is convergent for all x R Define the new function exp ε (ζ), for ζ as Following equations (4) (5), we write We can prove the generalized convergence of the Seq (6) for every ζ We additionally have that, (exp(x)) =(exp(x),0,0,,0)=exp εn (x), which proves that the new function exp ε () is the extension of the real function exp(x) 6 EXAMPLE For a simple example, see Fig 1 Multiscale perturbations (twoscale) are introduced into the boundary conditions into the (4) (5) (6) 5 EXTENDED εn-functions n-pn value functions are defined for n-pn arguments as extensions of classic elementary trigonometric functions The properties of the εn-functions are analyzed in detail, cf [13]-[17] Let D R εn be an arbitrary subset Assume that we have a rule f εn, which assigns to each element ζ D exactly one element w of R εn Then we say that f εn is an extended function defined in D with values in R εn We will denote that function as f εn :D R εn or w = f εn (ζ) or simplified w = εn-f(ζ) To illustrate how we can construct generalizations of the usual real functions, we use a simple function We now discuss an extension of a simple exponential function exp(x), x R With polynomials rational functions, it is one of the simplest elementary functions How Fig 1 Scheme of the boundary, boundary elements boundary conditions MULTI-SCALE PERTURBATION METHODS IN MECHANICS 15 skrzypczykindd :7:59

4 Fig Temperature its perturbations, perturbation on the boundary Maximal error: Fig 3 Derivative of temperature (flux) Maximal error: boundary, see Fig3 The perturbations of the boundary are of the first kind ( ); for details about the contour, see [0] The boundary temperature is perturbed in nodes ( ) The results for the temperature the flux on the boundary with the corresponding perturbations are calculated for all the nodes, see Figs 3 7 CONCLUSIONS Calculations with the use of new multi-scale perturbation numbers lead to applications which are mathematically equivalent to I-order approximations in classical perturbation methods The advantages of the new algebraic system are as follows: we can omit all the complex analytical calculations which are typical for exping the approximate values of solutions in infinite series This works for exping unknown values - solutions as well as for perturbed coefficients of the problem; we get a great simplification of all the arithmetic calculations which appear in the analytical formulation analysis of the problem; most of the known numerical algorithms can be simply adapted for the new algebraic system without any serious difficulties With the new algebraic system we get a set of very simple useful mathematical tools which can be easily used in the analytical computational parts of the analysis of complex multi-scale perturbation problems 16 MULTI-SCALE PERTURBATION METHODS IN MECHANICS skrzypczykindd :8:00

5 REFERENCES [1] Bellman, R (1976) Introduction to Matrix Analysis, McGraw- Hill Book Company, Inc, New York, Toronto, London, 1976 [] Białynicki-Birula, A (1971) Algebra, PWN, Warsaw, 1971 [3] Gelf, IM (1971) Wykłady z algebry liniowej, PWN, Warsaw, 1971 [4] Gomuliński A - Witkowski M (1993) Mechanika budowli, kurs dla zaawansowanych, Oficyna Wyd Pol, Warsaw, 1993 [5] Kaczorek, T (1998) Wektory i macierze w automatyce i elektrotechnice, WN-T, Warsaw, 1998 [6] Kato, T (1966) Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1966 [7] Kiełbasiński, A - Schwetlick H (1998) Numerische lineare Algebra VEB Deutscher Verlag der Wissenschaften, Berlin, 1988 [8] Korn, GA - Korn TM (1983) Matematyka dla pracowników naukowych i inżynierów czi, PWN, Warsaw, 1983 [9] Lankaster, P (1978) Theory of Matrices, Academic Press, New York, London, 1978 [10] Rayleigh, L (197) The Theory of Sound, Vol I, London, 197 [11] Rellich, F (1936) Störumgstheory der Spektralzerlegung I, Mathematical Annual, 113, pp , 1936 [1] Schrödinger, E (196) Quantisierung als Eigenwertproblem, Ann Physik, 80, pp , 196 [13] Skrzypczyk, J (003) Perturbation methods - New Arithmetic Zeszyty naukowe Politechnika Ślaska, series: Budownictwo, Gliwice, 003 [14] Skrzypczyk J (004) Metody perturbacyjne - Nowa arytmetyka Zeszyty Naukowe Katedry Mechaniki Stosowanej Politechniki Śląskiej 3, Gliwice 004, pp [15] Skrzypczyk, J (004) Perturbation methods I- Algebra, Functions, Linear Equations, Eigenvalue Problems: New Algebraic Methodology In: Proceedings of International Conference on New Trends in the Statics Dynamics of Buildings, October 004, Faculty of Civil Engineering STU Bratislava, Slovakia, pp [16] Skrzypczyk, J (005) Perturbation Methods - New Algebraic Methodology In: Proceedings of CMM-005 Computer Methods in Mechanics, June 1-4, 005, Częstochowa, Pol [17] Skrzypczyk, J - Winkler A (004), Perturbation methods II- Differentiation, Integration Elements of Functional Analysis with Applications to Perturbed Wave Equations In: Proceedings of International Conference on New Trends in the Statics Dynamics of Buildings, October 004, Faculty of Civil Engineering SUT Bratislava, Slovakia, pp [18] Skrzypczyk, J - Winkler-Skalna, A (006), Sound Wave Propagation Problems in New Perturbation Methodology Proceedings of Polish Acoustic Symposium OSA006, Zakopane September 006, Pol, Archives of Acoustics 31, N3, pp , 006 [19] Skrzypczyk, J - Winkler-Skalna A (006), Sound Wave Propagation Problems in New Perturbation Methodology, Proceedings of International Conference on New Trends in the Statics Dynamics of Buildings, October 006, Faculty of Civil Engineering STU Bratislava, Slovakia, pp [0] Skrzypczyk, J- Witek, H (005) Fuzzy Boundary Element Methods: A New Perturbation Approach for Systems with Fuzzy Parameters In: Proceedings of International Conference on New Trends in the Statics Dynamics of Buildings, October 005, Faculty of Civil Engineering STU Bratislava, Slovakia, Bratislava, 005 MULTI-SCALE PERTURBATION METHODS IN MECHANICS 17 skrzypczykindd :8:0