11111 111111V#Ittijl M Jtimmin Linii 1U11 I 1 I L I M

Ryszard WALENTYNSKI

■ iii m l JTT 1111111K ■ 1 13 ' I 1] l i i i i]

11111 111111V#iTTijl M jTiMMiN linii 1U11 i 1 i l i m

2 E h (- 3 ( - 1 + V2) 181® 4E h3H (apr) (a«1) (bęj^ Jari) (br3) ( y Pq) 3 (1 + v )^ !§M| 4 E h 3 H v (ai:i) \(Y p g )

4 E h 3 (a qi

4 E h 3 H f

8 E h 3 (

4 E h 3 H '

8 E h 3 H (e

4 E h 3 (a p r ) fbr ) (Ppg) 3 (3

8 E h 3 v ( a i ; i ) (Ł (Ppg) - 3 + 3 v 2 (l + v) 8 E h 3 (ąPJ) (b^1) (pP9) ~ 8 E h 3 v (ągi) (b1*) (ppq) 3 (1 + v) + -3 + 3 v 2 2 E h 3 (a«) (a«1) (c?pq5 ( 2Bh 3 v (a ^ ) (ai3) {i9pq; 3 (1 + v) + 3 - 3 v 2

^m&RWHKTwó-, V TOUTKHWKI \ >1 , Gliwice 2003 0 5 Rysz ENTYŃSKI ENTYNSKI

APPLICATION OF COMPUTER ALGEBRA IN SYMBOLIC COMPUTATIONS AND BOUNDARY-VALUE PROBLEMS OF THE THEORY OF SHELLS

2 Eh (-3 + 5 h 2 K) (aPJ) (a1*1) (ypq) 3 (1 + v )

2 Eh (-3 + 5 h 2 K) v ( a p q ) (ai j ) (TPq) 3 (-1 + v2)

4 E h 3 H ( a p r ) ( a q i ) (b ) ( b r j ) ( T p a ) 3 (1 + v '

4 E h 3 H v ( a l j

4 E h 3 ( a qj

8 E h 3 (ap:i) (bq i) ( Ppq) 8 E h 3 v (apq) (blj) (ppq)

3 (1 + v) -3 + 3 v 2

2 E h 3 (aPJ) (aqi) (Opq) 2 E h 3 v (apq) (a^) ((9pq)

3 (1 + v ) 3 - 3 v 2

WYDAWNICTWO POLITECHNIKI ŚLĄSKIEJ GLIWICE 2003 OPINIODAWCY Prof. dr hab. inż. Piotr KONDERLA — Politechnika Wrocławska Dr hab. inż. Bogumił WRANA — Profesor Politechniki Krakowskiej CONTENTS

KOLEGIUM REDAKCYJNE REDAKTOR NACZELNY — Prof. dr hab. inż. Andrzej BUCHACZ NOMENCLATURE 13 REDAKTOR DZIAŁU — Dr inż. Marianna GLENSZCZYK SEKRETARZ REDAKCJI — Mgr Elżbieta LEŚKO 1. INTRODUCTION 23

1.1. SYMBOLIC COMPUTATION - A NEW SCIENTIFIC TREND 23

1.2. TOOLS OF TENSOR ANALYSIS ASSISTANCE 26

1.3. COMPUTER ALGEBRA IN MECHANICS, DIFFERENTIAL GEOMETRY AND REDAKCJA THE THEORY OF SHELLS 27 Mgr Feliks LIPSKI 1.4. PROGRESS TENDENCIES IN THE THEORY OF SHELLS 27

1.5. AIM AND CONTENT OF THE CONTRIBUTION 29 1.5.1. Part I: Symbolic computations 30 REDAKCJA TECHNICZNA 1.5.2. Part II: Approximation of boundary value problems by means of the Least Squares Alicja NOWACKA Method 31

1. SYMBOLIC COMPUTATIONS 33

2. BASIC RELATIONS OF THE THEORY OF SHELLS 35

2.1. GEOMETRICAL DESCRIPTION 35 ZESZYTY NAUKOWE N r kol. 1587 2.1.1. Reference surface 35 BUDOWNICTWO z. 100 2.1.2. Parallel surface 37

2.2. GEOMETRICAL RELATIONS 39 PL ISSN 0434-0779 2.2.1. Kinematic relations 39 2.2.2. Deplanation 40 2.2.3. Strains 4 0 © Copyright 2003 by 2.2.4. Influence of temperature 41 Ryszard WALENTYNSKI 2.3. CONSTITUTIVE RELATIONS 42 [email protected] 2.3.1. Stress tensor 42

Utwór w całości ani we fragmentach nie może być powielany ani rozpowszechniany za 2.3.2. Internal forces 42 pomocą urządzeń elektronicznych, mechanicznych, kopiujących, nagrywających i innych, 2.4. EQUATIONS OF EQUILIBRIUM 43 w tym również nie może być umieszczany ani rozpowszechniany w postaci cyfrowej zarówno w Internecie, jak i sieciach lokalnych bez pisemnej zgody posiadacza praw autorskich. 2.5. PHYSICAL COMPONENTS 44

3 4 C o n ten ts C ontents 5

3. EQUATIONS OF EQUILIBRIUM 45 n. BOUNDARY-VALUE PROBLEMS 71 3.1. CHANGING COVARIANT DERIVATIVES IN EQUATIONS TO (ORDINARY) PARTIAL DERIVATIVES 45 7. DESCRIPTION OF THE REFINED LEAST SQUARES METHOD 7 3 3.2. SUMMATIONS 46 7.1. CLASSICAL APPROACH 73 3.3. TRANSFORMATION TO MATHEMATICS DIFFERENTIAL EQUATIONS 47 7.2. ONE-DIMENSIONAL PROBLEMS 7 4 3.4. NONLINEAR EQUATIONS OF THE SECOND-ORDER THEORY 48 7.2.1. Functional 74 4. CONSTITUTIVE RELATIONS 49 7.2.2. Application of the Ritz method 74 4.1. DEFINITIONS 49 7.2.3. System of linear algebraic equations 76

4.2. EVALUATION 50 7.3. NONLINEAR TASKS 7 7

4.3. ELEMENTS OF SIMPLIFICATION 54 7.4. MULTIDIMENSIONAL PROBLEMS 7 8 4.3.1. Grouping of terras 5 4 7.5. FEATURES OF THE REFINED LEAST SQUARES METHOD 7 8 4.3.2. Moderated simplification 55 7.5.1. Advantages of the method 78 4.3.3. Replacement 55 7.5.2. Disadvantages of the method 79 4.3.4. Result of simplifications 56 8 . ELEMENTS OF THE IMPLEMENTATION OF THE METHOD 8 0 4.4. SATISFACTION OF THE LAST EQUATION OF EQUILIBRIUM 57 8.1. TRANSLATION INTO THE MATHEMATICS L A N G U A G E 8 0 5. DESCRIPTION OF AN ARBITRARY SHELL 58 8.1.1. Polynomial approximation 80 8.1.2. Weighted differential equations 82 5.1. GEOMETRICAL DESCRIPTION OF THE REFERENCE SURFACE 58 8.1.3. Extraction of terms from equations 82 5.2. GEOMETRICAL PROPERTIES 59 8.1.4. Weighted boundary conditions 83 5.3. KINEMATIC RELATIONS 60 8.1.5. Guessing weights 83 5.4. STRAINS 61 8.1.6. Extraction of terms from boundary conditions 84

6. FINAL RESULTS OF SYMBOLIC COMPUTATIONS 62 8.1.7. Coefficients of the system matrix 84 8.1.8. Function of integration 84 6.1. CYLINDRICAL SHELL 62 8.1.9. The matrix of the system of linear algebraic equations 85 6.2. INTERNAL FORCES 63 8.1.10. Vector of free elements 85 6.2.1. Stretching forces 63 8.2. SOME COMPUTATIONAL ASPECTS OF SOLVING THE SYSTEM OF LINEAR 6.2.2. Moments 64 ALGEBRAIC EQUATIONS 86 6.2.3. Transverse forces 64

6.3. PARTIAL DIFFERENTIAL EQUATIONS 65 9. CHIMNEY EXPOSED TO AN ANTISYMMETRICAL LOAD 8 8

6.4. VARIABLE SEPARATION 66 9.1. DESCRIPTION OF THE PROBLEM 8 8 6.4.1. Stretching forces 67 9.1.1. Numerical data of the problem 88 6.4.2. Moments 68 9.1.2. Differential equations 89 6.4.3. Transverse forces 68 9.1.3. Engineering interpretation 89

6.4.4. Ordinary differential equations 68 9.2. ONE-STEP APPROACH 9 0 6 C o n ten ts C ontents

9.3. TWO-STEP APPROACH 90 11.3. ILLUSTRATING EXAMPLES 151 9.3.1. Grouping of the boundary conditions 90 11.3.1. Illustration o f the first case 151 9.3.2. Computational experiment - basis of the two-step approach 91 11.3.2. Illustration o f the third case 152 9.4. STEP ONE-BASE SOLUTION 92 11.4. ATTEMPT OF A MATHEMATICAL EXPLANATION 153 9.4.1. Rotations 94 11.4.1. Explicit tasks 153 9.4.2. Displacements 96 11.4.2. Im plicit tasks (shells) 154 9.4.3. Stretching and shear forces 99 11.5. EXTENSION OF MATHEMATICS POSSIBILITIES 156 9.4.4. Moments 102 11.6. ESTIMATION OF THE QUALITY OF THE GLOBAL CONVERGENCE 157 9.4.5. Transverse forces 104 9.4.6. Global equilibrium 104 12. REISSNER-MINDLIN PLATE 159 9.5. STEP TWO - REFINEMENT WITHIN BOUNDARY LAYERS 105 12.1. NOTATIONS AND VALUES OF DATA 159 9.5.1. Free edge 105 12.2. PROBLEM DESCRIPTION 160

9.5.2. Fixed edge 112 12.3. APPROXIMATION 162 12.4. PHYSICAL INTERPRETATION 166 10. CYLINDRICAL SHELL SUBJECTED TO A SINUSOIDAL LOAD 11 9 12.5. SOME REMARKS ON THE ANALYSIS OF ERROR AND CONVERGENCE 166 10.1. DESCRIPTION OF THE PROBLEM 119 10.1.1. Numerical data of the problem 119 13. CONCLUSIONS 168 10.1.2. Differential equations 121 13.1. SYMBOLIC COMPUTATIONS 168 10.1.3. Boundary conditions 121 13.2. BOUNDARY VALUE PROBLEMS 169 10.2. SHORT SHELL 122 13.3. ROLE OF COMPUTER ALGEBRA AND CONTRIBUTION TO ITS DEVELOPMENT 170 10.3. ERROR AND CONVERGENCE ANALYSIS 125 10.3.1. Estimation of the global error 126 14. FURTHER DEVELOPMENTS 171 10.3.2. Estimation of the local error 130 10.3.3. A few remarks on convergence and weights 134 ACKNOWLEDGEMENTS 172

10.4. LONG SHELL-ONE-STEP APPROACH 135 APPENDIX. BRIEF REVIEW OF THE AUTHOR’S CONTRIBUTIONS 173 10.5. LONG SHELL-TWO-STEP APPROACH 136 10.5.1. Base solution 136 BIBLIOGRAPHY 176 10.5.2. Boundary-layer refinement 140 SUMMARY 195 11. PHYSICAL INTERPRETATION AND MATHEMATICAL EXPLANATION 1 4 6

11.1. PHYSICAL INTERPRETATION OF THE BOUNDARY-CONDITION PHENOMENON 147

11.2. EXPLICIT PROBLEM — AS A BASE OF MATHEMATICAL EXPLANATION 148 11.2.1. Differential equation of a straight bar 149 11.2.2. Solutions of the equation 150 Spis treści 9

3. RÓWNANIA RÓWNOWAGI 45 3.1. ZAMIANA POCHODNYCH KOWARIANTNY CH W RÓWNANIACH NA POCHODNE CZĄSTKOWE (ZWYKŁE) 45 SPIS TREŚCI 3.2. SUMOWANIA 46 3.3. TRANSFORMACJA NA RÓWNANIA RÓŻNICZKOWE MATHEMATICS 47 3.4. NIELINIOWE RÓWNANIA TEORII DRUGIEGO RZĘDU 48

OZNACZENIA 13 4. ZWIĄZKI KONSTYTUTYWNE „ 49

4.1. DEFINICJE 49 1. WPROWADZENIE 23 4.2. OBLICZENIA 50 1.1. ALGEBRA KOMPUTEROWA - NOWY TREND NAUKOWY 23 4.3. ELEMENTY UPROSZCZENIA 54 1.2. NARZĘDZIA WSPOMAGANIA ANALIZY TENSOROWEJ 26 4.3.1. Grupowanie składników 54 1.3. ALGEBRA KOMPUTEROWA W MECHANICE, GEOMETRII RÓŻNICZKOWEJ 4.3.2. Upraszczanie kontrolowane 55 I TEORII POWłOK 27 4.3.3. Podstawianie 55 1.4. TENDENCJE ROZWOJOWE W TEORII POWŁOK 27 4.3.4. Wynik uproszczeń 56 1.5. CEL I ZAKRES PRACY 29 4.4. SPEŁNIENIE OSTATNIEGO RÓWNANIA RÓWNOWAGI 57 1.5.1. Część I: 2^gadnienia symboliczne 30

1.5.2. Część II: aproksymacja zadań brzegowych z użyciem Metody Najmniejszych Kwadratów 31 5. OPIS DOWOLNEJ POWŁOKI 58 5.1. OPIS GEOMETRYCZNY POWIERZCHNI ODNIESIENIA 58 5.2. WŁASNOŚCI GEOMETRYCZNE 59 1. OBLICZENIA SYM BOLICZNE 35 5.3. ZALEŻNOŚCI KINEMATYCZNE 60 5.4. ODKSZTAŁCENIA 61 2. PODSTAWOWE ZWIĄZKI TEORII POWŁOK 35

2.1. OPIS GEOMETRYCZNY 35 6. KOŃCOWE WYNIKI OBLICZEŃ SYMBOLICZNYCH 62 2.1. 1. Powierzchnia odniesienia 35 6.1. POWŁOKA WALCOWA 62 2.1.2. Powierzchnia równoległa 37 6.2. SIŁY WEWNĘTRZNE 63 2.2. ZWIĄZKI GEOMETRYCZNE 39 6.2.1. Siły rozciągające 63 2.2. 1. Związki kinematyczne 39 6.2.2. Momenty 64 2.2.2. Deplanacja 40 6.2.3. Siły poprzeczne 64 2.2.3. Odkształcenia 40 6.3. RÓWNANIA RÓŻNICZKOWE CZĄSTKOWE 65 2.2.4. Wpływ temperatury Ą\ 6.4. ROZDZIAŁ ZMIENNYCH 66 2.3. ZWIĄZKI KONSTYTUTYWNE 42 6.4.1. Siły rozciągające 67 2.3.1. Tensor naprężenia 42 6.4.2. Momenty 68 2.3.2. Siły wewnętrzne 42 6.4.3. Siły poprzeczne 68 2.4. RÓWNANIA RÓWNOWAGI 43 6.4.4. Równania różniczkowe 68 2.5. SKŁADOWE FIZYCZNE 44

8 10 11 Spis treści Spis treści

II. ZAGADNIENIA BRZEGOWE 73 9.3. PODEJŚCIE DWUETAPOWE 90 9.3.1. Grupowanie warunków brzegowych 90 7. OPIS ROZSZERZONEJ METODY NAJMNIEJSZYCH KWADRATÓW 73 9.3.2. Eksperyment obliczeniowy - podstawy podejścia dwuetapowego 91

7.1. PODEJŚCIE KLASYCZNE 73 9.4. ROZWIĄZANIE BAZOWE 92 7.2. ZADANIA JEDNOWYMIAROWE 74 9.4.1. Obroty 94 7.2.1. Funkcjonał 74 9.4.2. Przemieszczenia 96 7.2.2. Zastosowanie metody Ritz’a 74 9.4.3. Siły rozciągające i ścinające 99 7.2.3. Układ liniowych równań algebraicznych 76 9.4.4. Momenty 102

7.3. ZADANIA NIELINIOWE 77 9.4.5. Siły poprzeczne 104

7.4. ZADANIA WIELOWYMIAROWE 78 9.4.6. Równowaga globalna 104 9.5. ETAP DRUGI - UŚCIŚLENIE W OBRĘBIE WARSTWY BRZEGOWEJ 105 7.5. CECHY ROZSZERZONEJ METODY NAJMNIEJSZYCH KWADRATÓW 78 7.5.1. Zalety metody 78 9.5.1. Krawędź swobodna 105 7.5.2. Wady metody 79 9.5.2. Krawędź utwierdzona 112

8. ELEMENTY WDROŻENIA METODY 80 10. POWŁOKA WALCOWA PODDANA OBCIĄŻENIU SINUSOIDALNEMU 119

8.1. TŁUMACZENIE NA JĘZYK MATHEMATICK 8 0 10.1. OPIS ZADANIA 119 8.1.1. Aproksymacja wielomianowa 80 10.1.1. Dane liczbowe zadania 119 8.1.2. Ważone równania różniczkowe 82 10.1.2. Równania różniczkowe 121 8.1.3. Wybieranie wyrazów z równań 82 10.1.3. Warunki brzegowe 121 8.1.4. Ważone warunki brzegowe 83 10.2. POWŁOKA KRÓTKA 122 8.1.5. Zgadywanie wag 83 10.3. ANALIZA BŁĘDU I ZBIEŻNOŚCI 125 8.1.6. Wybieranie wyrazów z warunków brzegowych 84 10.3.1. Oszacowanie błędu globalnego 126 8.1.7. Współczynniki macierzy układu 84 10.3.2. Oszacowanie błędu lokalnego 130 8.1.8. Funkcja całkowania 84 10.3.3. Kilka uwag o wagach i zbieżności 134

8.1.9. Macierz układu algebraicznych równań liniowych 85 10.4. POWŁOKA DŁUGA-PODEJŚCIE JEDNOETAPOWE 135 8.1.10. Wektor wyrazów wolnych 85 10.5. POWŁOKA DŁUGA - PODEJŚCIE DWUETAPOWE 136 8.2. PEWNE ASPEKTY OBLICZENIOWE ROZWIĄZANIA UKŁADU 10.5.1. Rozwiązanie bazowe 136 ALGEBRAICZNYCH RÓWNAŃ LINIOWYCH 86 10.5.2. Uściślenie w warstwie brzegowej 140

9. KOMIN PODDANY DZIAŁANIU OBCIĄŻENIA ANTYSYMETRYCZNEGO 88 1 1 . INTERPRETACJA FIZYCZNA I WYTŁUMACZENIE MATEMATYCZNE 146 9.1. OPIS ZADANIA 88 11.1. INTERPRETACJA FIZYCZNA ZJAWISKA WARUNKU BRZEGOWEGO 147 9.1.1. Dane liczbowe zadania 88 11.2. ZADANIE NIEUWIKŁANE — JAKO PODSTAWA WYTŁUMACZENIA 9.1.2. Równania różniczkowe 89 MATEMATYCZNEGO 148 9.1.3. Interpretacja inżynierska 89 11.2.1. Równanie różniczkowe pręta prostego 149 9.2. PODEJŚCIE JEDNOETAPOWE 90 11.2.2. Rozwiązania równania 150 12 Spis treści

11.3. PRZYKŁADY ILUSTRUJĄCE 151 11.3.1. Ilustracja przypadku pierwszego 151 11.3.2. Ilustracja przypadku trzeciego 152

11.4. PRÓBA WYTŁUMACZENIA MATEMATYCZNEGO 153 11.4.1. Zadania nieu wikłane 153 NOMENCLATURE

11.4.2. Zadania uwikłane (powłoki) 1 5 4

11.5. ROZSZERZENIE MOŻLIWOŚCI MATHEMATICA' 156

11.6 . OSZACOWANIE JAKOŚCI ZBIEŻNOŚCI GLOBALNEJ 157 1. LATIN LETTERS

12. PŁYTA REISSNER’A-MINDLIN’A 15 9

12.1. OZNACZENIA I WARTOŚCI DANYCH 159 üij — the first differential form of the reference surface, metric tensor on the reference surface 12.2. OPIS PROBLEMU 160 in the actual configuration 12.3. APROKSYMACJA 162 h,: — the first differential form of the reference 12.4. INTERPRETACJA FIZYCZNA 166 surface, metric tensor on the reference surface 12.5. KILKA UWAG O ANALIZIE BŁĘDU I ZBIEŻNOŚCI 166 in the reference configuration Aij — coefficient of the matrix of the system of alge 13. WNIOSKI 168 braic equations of the Least Squares Method 13.1. OBLICZENIA SYMBOLICZNE 168 a — determinant of the first differential form 13.2. ZAGADNIENIA BRZEGOWE 169 bij — the second differential form of the reference 13.3. ROLA ALGEBRY KOMPUTEROWEJ I WKŁAD W JEJ ROZWÓJ 170 surface, curvature tensor in the actual configu ration 14. PLANY NA PRZYSZŁOŚĆ 171 — the second differential form of the reference surface, curvature tensor in the reference con WYRAZY UZNANIA 172 figuration 5, — coefficient of the free vector of the system of al DODATEK. ZWIĘZłY PRZEGLĄD PRAC AUTORA 1 7 3 gebraic equations of the Least Squares Method

BIBLIOGRAFIA 176 lb — determinant of the second differential form dj — the third differential form of the reference STRESZCZENIE 195 surface in the actual configuration Cjj — the third differential form of the reference surface in the reference configuration d — rotation vector

13 14 Nomenclature Nomenclature 15

di — derivative of the rotation vector M iJ tensor o f m o m en ts d ‘ — contravariant component of the rotation vector miifry) := E lo M n (x ,n ) sin(ny) torsion moment in the meridian direction - in the curvilinear basis physical component d3 — component of the rotation vector normal to the mn(x, y) : M u(x, n) cos{n_y) bending moment in the meridian direction - reference surface physical component di — physical component of the rotation vector m2\(x,y) :■: E*=o M 2i(x,n) cos(ny) bending moment in the parallel direction - phys ical component d\{x,y) := Z L o (x> n) c o s (n y ) —— rotation in the meridian direction - physical c o m p o n e n t m12(x, y) : E*=o Mïlix, n) sin(ny) torsion moment in the parallel direction - phys ical component di(x, y) ■■= J2n=o %h.(x, «) sin(w y) — rotation in the parallel direction - physical com po n en t P \ P 3 contravariant components of the load vector E _ Young modulus P\(x,y) := T,n=o'p '(x-n) cos(ny) load force in the meridian direction - physical co m p o n e n t T — functional of the Least Squares Method Pl(x, y) := load force in the parallel direction - physical 0 - reference functional En=o tpT^x,ri) sin(ny) co m p o n e n t Su — covariant metric tensor in 3D space Pl(x, y) ■= E*=o^3fr «) cos(ny) load force in the direction normal to the refer g ‘j — contravariant metric tensor in 3D space ence surface - physical component g — determinant of the metric tensor Q‘ contravariant components of the vector of trans verse forces 2 h _ shell thickness transverse force in the meridian direction - H _ mean curvature ") cos(ny) physical component •w _ comparative functional qi(x, y) := En=oâ 2fr ri) sin(ny) transverse force in the parallel direction - phys A" _ Gaussian curvature ical component

AT" _ tensor of stretching (tensile) forces K residuum functional ” i î (x, y) := £ * =0 N n (x, n) co s(n y) stretching — force in the meridian direction - phys r parametrization vector of the reference surface ical component in the actual configuration

n\i(x,y) := J2n=oMi2(.x,ri) sin(n;y) shear — force in the meridian direction - physical r parametrization vector of the reference surface co m p o n e n t in the reference configuration

” 21U, y) := EÎ=o^ 2i(-*-'1) sin(ny)shear force — in the parallel direction - physical R parametrization vector of the parallel surface in co m p o n e n t the actual configuration

>l22{x,y) ■= Y ,L o ^ 22(x,n) COS(ny) — stretching force in the parallel direction - physi k parametrization vector of the parallel surface in cal component the reference configuration Nomenclature 17 16 Nomenclature

angular thermal strain r■, — vector of the curvilinear basis, tangent to the ref X Lame constant erence surface Lame constant Poisson ratio rj — unit vector of the curvilinear basis, normal to the v the second strain tensor of the reference surface reference surface Pij r j stress tensor te — external temperature coefficient of shear stress distribution under the X ti — internal temperature load of a transverse force w — displacement vector of the reference surface 3. MATHEMATICAL SYMBOLS wi — derivative of the displacement vector of the ref erence surface w displacement vector of the parallel surface = — equal to, symbol of equation w* := — symbol of definition contravariant component of the displacement ; — symbol of covariant differentiation vector in the curvilinear basis — symbol of (ordinary) partial differentiation w3 component of the displacement vector normal to the reference surface Other notations are explained in the text. Wi physical component of the displacement vector W\ (x,y) '■= T,kn=o'W'ifr«) cos(ny) displacement in the meridian direction - physi 4. FUNCTIONS OF MATHEMATICA’ AND MathTensor™ cal component w2(x, y) := E l o W 2(x, n) sin(ny) displacement in the parallel direction - physical Comprehensive information on MATHEMATICA' and its functions can be found in the system component documentation provided by S. Wolfram [229, 230] and in The MATHEMATICA' Book Online: http://documents.wolfram.com/v4/index3.html m(x,y) := Yln=0 W 3fcn) COS {ny) displacement in the direction normal to the ref erence surface- physical component Brief information about all MathTensor™ functions can be found on the following WWW page: http://smc.vnet.net/mathtensorfunc.html.

2. GREEK LETTERS a, — coefficient of thermal expansion O'; — weight of a differential equation Pk — weight of a boundary condition f t — vector of averaged deplanation jj — weight of a boundary condition y-,j — the first strain tensor of the reference surface jij — strain tensor in 3D space 6/ — Kronecker delta Ei._ a, dj+if) — thermal membrane strain Sij — antisymmetric object under interchange of any index âjj — the third strain tensor of the reference surface O znaczenia 19

d‘ — kontrawariantna składowa wektora obrotu w ba zie krzywoliniowej

d13 — składowa wektora obrotu normalna do powierz OZNACZENIA chni odniesienia di — składowa fizyczna wektora obrotu di (x, y) := E t o £>i U n) cos(n y) — obrót w kierunku południkowym - współrzędna fizyczna 1. LITERY ŁACIŃSKIE di(x, y) := YlLo AC*.«) sin(n;y) — obrót w kierunku równoleżnikowym - współ rzędna fizyczna E — moduł Young’a au pierwsza forma różniczkowa powierzchni od

niesienia, tensor metryczny powierzchni odnie T — funkcjonał Metody Najmniejszych Kwadratów sienia w konfiguracji bieżącej funkcjonał odniesienia Q - °au pierwsza forma różniczkowa powierzchni od kowariantny tensor metryczny w przestrzeni 3D niesienia, tensor metryczny powierzchni odnie gu — sienia w konfiguracji odniesienia kontrawariantny tensor metryczny w przestrzeni 3D Aij współczynnik macierzy układu równań algeb raicznych Metody Najmniejszych Kwadratów wyznacznik tensora metrycznego a wyznacznik pierwszej formy różniczkowej 2 h grubość powłoki bij druga forma różniczkowa powierzchni, tensor H krzywizna średnia krzywizny w konfiguracji bieżącej •H funkcjonał porównawczy bij druga forma różniczkowa powierzchni odniesie krzywizna Gauss’a nia, tensor krzywizny w konfiguracji odniesienia K N ‘J tensor sił rozciągających Bi współczynnik wektora wyrazów wolnych ukła du równań alg-braicznych Metody Najmniej nu(x,y)--= E != o^ii(x,n) cos(ny) siła rozciągająca w kierunku południkowym - szych Kwadratów współrzędna fizyczna b wyznacznik drugiej formy różniczkowej nn(x,y) ■■= £ * =o Mn(x,n) sin(ny) siła ścinająca w kierunku południkowym - współrzędna fizyczna cij trzecia forma różniczkowa powierzchni odnie sienia w konfiguracji bieżącej n2\{x,y) ■= Yf„=o^2i(x,n) sin(ny) siła ścinająca w kierunku równoleżnikowym - współrzędna fizyczna cij trzecia forma różniczkowa powierzchni odnie sienia w konfiguracji odniesienia n22(x,y) := J2 Lo N 2l(x, n) cos(ny) siła rozciągająca w kierunku równoleżnikowym - współrzędna fizyczna d wektor obrotu di pochodna wektora obrotu

18 20 O zn a czen ia O znaczenia 21

M‘J tensor momentów n wektor bazy krzywoliniowej, styczny do po m \\(x,y) := 0 M u (x,ri) sin{n y) moment skręcający w kierunku południkowym wierzchni odniesienia - współrzędna fizyczna rj wektor jednostkowy bazy krzywoliniowej, nor m n (x, y ) := E*=o A tnfr ń) co s(n y) moment zginający w kierunku południkowym - malny do powierzchni odniesienia współrzędna fizyczna te temperatura zewnętrzna := 2 ™2\(x, y) Z l o M \(x, n) COS(ny) moment zginający w kierunku równoleżniko ti temperatura wewnętrzna wym - współrzędna fizyczna W wektor przemieszczenia powierzchni odniesie m 22(x, y) ■■= E L o M>2(*> n) sin(«_y) moment skręcający w kierunku równoleżniko nia wym - współrzędna fizyczna Wi pochodna wektora przemieszczenia powierzch p ‘, P 3 kontrawariantne składowe wektora obciążenia ni odniesienia

P i( x ,y ) ■■= E * = o ^ i (x,n) cos(ny) siła obciążenia w kierunku południkowym - W wektor przemieszczenia powierzchni równoleg współrzędna fizyczna łej P l( x , y ) ■■= E*=o “p 2 ( x , n ) sin(«;y) siła obciążenia w kierunku równoleżnikowym - w' kontrawariantna składowa wektora przemiesz współrzędna fizyczna czenia w bazie krzywoliniowej

P 3 (x , y ) := c o s ( n y ) siła obciążenia w kierunku normalnym do po w3 składowa wektora przemieszczenia normalna do wierzchni odniesienia - współrzędna fizyczna powierzchni odniesienia Q‘ kontrawariantne składowe wektora sił poprzecz Wi składowa fizyczna wektora przemieszczenia nych W\{x,y) := \(x, n) cos (ny) przemieszczenie w kierunku równoleżnikowym qi(x,y) E ^ O .f r « ) cos(n;y) siła poprzeczna w kierunku równoleżnikowym - współrzędna fizyczna - współrzędna fizyczna W2(x,y) := T,Lo

E, — «i di+i,) 2 błonowe odkształcenie termiczne «*/ obiekt antysymetryczny przy zamianie którego kolwiek z indeksów 22 O zn a czen ia

trzeci tensor odkształcenia powierzchni odnie sienia kątowe odkształcenie termiczne A stała Lame’go P stała Lame’go Chapter 1 v współczynnik Poisson’a Pu drugi tensor odkształcenia powierzchni odnie sienia r j tensor naprężenia INTRODUCTION X współczynnik rozkładu naprężenia stycznego przy obciążeniu siłą poprzeczną i ______3. SYMBOLE MATEMATYCZNE Ten i każdy następny rozdział poprzedzony jest krótkim podsumowaniem jego treści w języku polskim i angielskim. Wprowadzenie do pracy przedstawia definicję, krótki rys rozwoju systemów algebry kom puterowej, ich zastosowania w teorii powłok i naukach pokrewnych oraz krótki przegląd tendencji roz = — równy, symbol równania wojowych w teorii powłok. Następnie przedstawiono treść pracy oraz jej tezy. := — symbol definicji This and each further chapter is preceded by a short summary of its plot in Polish and English. The ! — symbol różniczkowania kowariantnego introduction presents a definition, a short history of computer algebra systems development, their appli — symbol (zwykłego) różniczkowania cząstkowe cations in the theory of shells and related sciences and a short review of development tendencies in the go theory of shells. Next the contribution content and its thesis are presented. Pozostałe oznaczenia wyjaśniono w tekście.

4. FUNKCJE MATHEMATICA‘ ORAZ M athTensor™ 1.1. SYMBOLIC COMPUTATION - A NEW SCIENTIFIC TREND

Wyczerpujące informacje o MATHEMATICA' i jej funkcjach można znaleźć w dokumentacji sys The times, when the only tools of an engineer or scientist dealing with mathematical-numerical temu przygotowanej przez S. Wolfram’a [229, 230] oraz w The MATHEMATICA’ Book Online: notions was a book, sheet of paper, pencil and calculator, have ceased. First thick engineering http://documents.wolfram.com/v4/index3.html and logarithmic tables have gone and been replaced by calculators. The technological revolu Zwięzłe informacje o wszystkich funkcjach MathTensor™ można znaleźć na stronie WWW: tion of the nineties of the previous century resulted in the implementation of personal comput http://smc.vnet.net/mathtensorfunc.html. ers, which replaced traditional methods of computations with integrated engineering programs based on the Finite Element Approach. Nevertheless, traditional methods play an important role in the education of students to let them understand structural mechanics. Nobody open-minded fancies numerical computations with Finite Elements or Finite Differ ences methods with a pocket calculator. The knowledge that setting up formulas necessary for algorithms programming of these methods or doing single calculations can be assisted by a computer is still not very well propagated. Nowadays, most computer programmers and users are not computer science engineers. They require tools which are more user-friendly, not only in straightforward data input and readable

'The text was built within I5TjjX 2g environment. The version saved with a PDF format can be browsed with AcrobatReader http://www.adobe.com/products/acrobat/ using embedded hyperlinks (equations, sections, citations and WWW pages). Navigation is supported with left hand side table of content (Bookmarks).

23 24 Chapter 1. Introduction 1.1. Symbolic computation - a new scientific trend 25

output of results for standard computational problems, but also for analysis of not typical tasks. This list has been prepared in accordance with information given on the WWW page: On the other hand, conditions of the optimal computer code in the procedure and its precision http://www.wolfram.com/products/mathematica/tour/. Detailed explanation can be found there are simplicity of formulas and minimization of numerical operations. Therefore, almost paral- and on the WWW pages of other systems provided below. lelly to numerical programs, people started to research on systems of symbolic computations sometimes called “Computer Algebra systems”. The range of possible applications shows that the simple answer to the question “What is Com puter Algebra?” is difficult. Sometimes Computer Algebra Systems (CAS) are regarded as The brief, but accurate, definition of Computer Algebra is given on the following WWW page: “Artificial Intelligence” but that is not an exact answer. It is not a “Wizard” in any sense, but http://sal.kachinatech.c0m/A/1 /. possible capabilities and results are sometimes surprising even for system creators. The name “Computer Algebra” is somehow perplexingly similar to the name of the subject taught at me The major purpose of a Computer Algebra System (CAS) is to manipulate a for chanical and civil engineering departments called “Strength of Materials”. mula symbolically using the computer. For example, expanding, factorizing, root Some computer algebra systems are: ALAM, ALGY, AMP, Ashmedai, AUTOLEV, AX finding, or simplifying an algebraic polynomial are some of the common uses of IOM*, bernina, CAM C++ Class Libraries, CAMAL, CASA, CAYLEY, CCalc, CLAM, Co- CAS. However, many systems (...) have gone far beyond that and also offer other CoA, Computer Algebra Kit, Derive, DrMath, ESP, FELIX, FLAP, FOAM, FORM, FOR functionalities like numerical calculation, graphics, and simulations. (...) MAL, Formula ALGOL, GAMS, GAP, GiNaC, HartMath, Interpreter for symbolic manipu lation, JACAL, Kalamaris, LiE, lundin.SymbolicMath, Macaulay 2, MACSYMA, Magic Pa Expanding the information provided in this definition the contemporary systems can: per, MAGMA, MAO, Maple, MAS, MathCAD, MATHLAB, Mathomatic, MAXIMA, med- itor, Mock-Mma, MuMath, MuPAD, Nother, ORTHOCARTAN, Pari, Punimax, REDUCE, • carry out interactive calculations using notebooks Risa/Asir, SAC-1, SAC2, SACLIB, SAINT, SAML, Schoonschip, Scratchpad I, SHEEP, SIMATH, SimLab, Simple Lisp, Singular, SISYPHOS, STENSOR, SYMBAL, SymbMath, • be used as a calculator Symbolic Mathematical Laboratory, texLderiv, TMath, TRIGMAN, VAXIMA, UBASIC, • be a practical treasure of mathematical knowledge Yacas. Brief information with references on these systems can be found at the follow ing WWW addresses: http://wombat.doc.ic.ac.uk/foldoc/fo[doc.cgi?symbolic+mathematics and • perform numerical calculations to any precision http://www.sal.ps.pI/A/1/index.shtmI. It is worth to visit also Symbolic Mathematical Computa • carry out symbolic calculations to get formulas tion Information Center, http://www.symbolicnet.org . • use lists to represent collections of things Only several systems from this long list have survived. The system REDUCE recent version 3.7, http://www.zib.de/Symbolik/reduce, based on the technology from the sixties of the previous • create 2D and 3D graphics century can be found on some UNIX and Linux mainframes in large computational centers. • compute integrals and derivatives Very popular, also among Polish students, MathCAD version 11 http://www.mathcad.com is a useful program for engineering purposes. A relatively small program named Derive 5.06 • solve equations symbolically and numerically http://www.derive.com is still applied in the teaching of mathematics. • Solve differential equations The development of the Maple system launched in the beginning of the eighties in the School of • manipulate vectors and matrices Computer Science at the Faculty of Mathematics at the University of Waterloo, Waterloo, On tario, Canada, by the Symbolic Computation Group. The first commercial version was released • define one’s own functions in 1985 and the recent issue has got the name Maple 9 http://www.maplesoft.com.

• import and export data in a variety of formats Stephen Wolfram started the development of MATHEMATICS in 1986. Founded by him Wolfram • manipulate, visualize and analyze data Research, Inc. in Champaign, IL, http://www.wolfram.com issued the first version in 1988. Developers of the system established an ambitious task of creating an integrated symbolic, • use pattern matching to transform expressions numerical and graphical system, for a wide group of contemporary operating systems. The tool • serve as an effective and intuitive programming language called FrontEnd for interactive contact with the user was implemented by Theodore Gray. The crucial turn of the development was the release of the version 3 of the system. FrontEnd was • permit the usage and development add-on packages changed completely. The recent version 4.1 and 4.22 S. Wolfram [229, 230] introduced many • be used as a typesetting and publishing facility improvements, for example in symbolic and numerical computations, integration with Java, • do much more... computational speed. The system has a very good context help. Unbeatable advantages of the 2The MATHEMATICS 5 was released on June 23rd ’2003, when the book was in the final editorial stage. 26 Chapter 1. Introduction 1.3. Computer algebra in mechanics, differential geometry and the theory o f shells 27

system are constantly developed symbolic and graphical, T. Wickham-Jones[227], capabilities. There are tools based on interface called MathLink for mutual contact with Matlab environment 1.3. COMPUTER ALGEBRA IN MECHANICS, DIFFERENTIAL and building optimized C and C++ codes. GEOMETRY AND THE THEORY OF SHELLS The system is not closed. Some standard packages are appended to the distribution. Moreover on page: http://www.mathsource.com a lot of packages created by researchers and programmers There are some publications devoted to the theory of shells, mechanics and differential geometry from the entire world can be found. Some packages are accessible from their authors WWW within the extensive literature on the application of computer algebra. The first contributions pages or offered by Wolfram Research, Inc. or other commercial firms. were written using REDUCE and MACSYMA: K. Bannister [7], /. Jones [69], A. Noor [118], R. Smith and A. Palazotto [165], D. Wilkins [228], I. Lottati and I. Elishakoff [97], I. Levi and Packages and MATHEMATICS programs are created with a specific, intuitive high-level language. N. Hoff [93] and J. Bocko [26]. The philosophy of programming may be studied from the books: R. Gaylord et al. [48, 49], J. Gray [54] and R. Maeder [103, 104, 105]. An important contribution to the application of MATHEMATICS in differential geometry is due to Professor A. Gray [52, 53]. He developed many packages collected on the page Many scientific publications, school and academic manuals during several years of system de http://math.cl.uh.edurgray devoted to his achievements. Problems of differential geometry were velopment have been issued. Catalogs of some of them can be found on the following WWW also developed by W. Businger [33] and Y. Tazawa [175]. pages: http://store.wolfram.com/catalog/books/, http://library.wolfram.com and in databases by N. Beebe [15]. Among Polish publications it is worth to mention R. Grzymkowski et al. [44,45]. Most contributions were developed in recent years with the application of MATHEMATICS. Among the works devoted to mechanics, the theory of shells and computational methods are the The European scientific center is placed in Hagenberg, Austria. It is the Research Institute for following publications: G. Alfano et al. [2], N. Auciello [4], A. Banschchikova et al. [8], N. Bel- Symbolic Computation of the University of J. Kepler in Linz http://www.risc.uni-linz.ac.at. The lomo et al. [16], A. Beltzer [17,18], G. Dasgupta [41], A. Iglesias and H. Power [62], T. Hata et institute chairman is Professor Franz Winkler. Publications on computer algebra application al. [58], N. Ioakimidis [63, 64, 65, 66], M. Kaoud and J. Ari-Gur [70], A. Kaw [71], K. Knight and development can be found in The Journal of Symbolic Computation, founded by Professor [76], A. Kiselev et el. [75], M. Korayem et al. [79], R. Kutylowski and K. Myslecki [85], M. Leu Bruno Buchberger. Current Editor-in-Chief is Hoon Hong http://www4.ncsu.edu/ hong/jsc.html. et al. [92], J. Nayfeh and J. Rivieccio [116], A. Noor and C. Anderson [119, 120, 121], P. Pai and T. Ser [129], A. Papusha [132, 133], R. Raouf and A. Palazotto [142, 143], M. Skrinar 1.2. TOOLS OF TENSOR ANALYSIS ASSISTANCE [160], M. Walker et al. [225], J. Wolkowisky [231].

There are several packages devoted to tensor analysis offered or attached to the computer 1.4. PROGRESS TENDENCIES IN THE THEORY OF SHELLS algebra distributions. The package called RedTen permits tensor analysis within the old- fashion system REDUCE http://www.scar.utoronto.ca/ harper/redten.html. GRTensor, origi The theory of shells is a specific branch of Continuum Mechanics dealing with the description nally written for Maple, has also limited versions for other systems, including MATHEMATICS, http://grtensor.phy.queensu.ca/. of bodies, defined in the close neighborhood of a curved surface in three dimensional space. One of the tasks of the theory of shells is to reduce the three-dimensional task of the theory of Several tensor packages were written for MATHEMATICAL for example CARTAN - H. Soleng elasticity and plasticity in a curved space to a two-dimensional one. It uses the accomplishments [167], Tensorial - R. Cabrera [34], TTC - X. Jaen et al. [35, 67] and Ricci - J. Lee [90]. of differential geometry and tools of tensor analysis, despite some opposition against it, see The package MathTensor™ http://www.wolfram.com/products/applications/mathtensor/ is one J. Paavola and E. Salonen [128]. of the largest MATHEMATICS packages. It is offered by MathTensor Inc., (previous name Math- There are a lot of shell theories and their variants. The approaches are very different. The aim Solutions, Inc.). The authors of the package S. Christensen and L. Parker published it in 1990 of the contribution is not to develop the theory of shells, but to present the possible application [39]. The recent version issued in 1994 [137] for MATHEMATICS 2.2, was adopted for the con of computer algebra and boundary value problems connected with this theory. However, the temporary version of the system . application of computer algebra enabled us to avoid some symbolic problems, to obtain some The package dhPark D. Park [136] is a development based on MathTensor™ philosophy. It new results and to develop a new approach to boundary value problems of shells. Due to contains many improvements of kernel and FrontEnd like Greek letters in the indices. Ac methodical reasons these considerations are limited to problems common for most theories. Nevertheless, it is worth to review recent developments in the theories of shells. cording to the information on page http://smc.vnet.net/MathTensor.html by Stephen Christensen MathTensor™ will also implement these advantages. Up till now some advanced symbolic tools The classical theory of shells is based on Kirchhoff’s assumption about a normal element. It must be applied to obtain Greek letters in outputs, see 3.1. on page 45. enables us to describe a deformed surface parallel to a selected reference surface with regard to its deformation and to provide internal reduction. This assumption has been criticized many times, but is still a base of consideration and recent analysis by A. Slawianowska and J. Telega [161, 163, 162, 164], based on asymptotic expansion shows that it is correct. Libai and Sim- 1.5. Aim and content o f the contribution 29 28 C h a p ter 1. In tro d u ctio n

and L. Librescu [158], Y. Suetake et al. [168], K. Sze and L. Yao [170, 171], A. Tabiei and monds [95] introduced even an idea of “constitutive Kirchhoff’s hypothesis”. The refinement R. Tanov [173, 174], R. Valid [178], J. Vinson [179], Z. Wang [226]. A review of recent Finite of the theory of plates introduced by E. Reissner [144, 145, 146, 147, 148] and R. Mindlin [109, 110] enables us to consider shape deformations caused by transverse forces due to the Elements for shells can be found in H. Yang et al. [233]. introduction of “averaged deplanation”, C. Wozniak [232]. Another refinement is the consider There are some contributions on large strains, higher order theories and “zig-zag” theories for ation of normal stresses caused by quasi point loads, described for example by S. Lukasiewicz laminated shells, examples are A. Eckstein and Y. Ba§ar [46], H. Hassis [57], N. Huang [59], [98, 99]. U. Icardi [61], H. Jing and K. Tzeng [68]. This assumption may be neglected considering the shell as a three-dimensional body, but contri More and more contributions apply an asymptotic approach in finding an approximation for butions devoted to this assumption are limited to simple shapes, although they play an important initial-boundary value problems in shells. Examples are S. Antman et al. [3], R. Gilbert and heuristic role. Among them are J. Awrejcewicz and V. Krysko [5, 6], W. Chih-Ping and L. Jyh- K. Hackl [50], R. Gregory and F. Wan [55], T.Lewiński and J. Telega [94] H. Rutten [152]. Yeuan [37], S. Dong and P. Etitum [43], A. Leissa and J. Kang [91], K. Liew and Z. Feng [96] Optimization problems have been dealt with by M. Ostwald [127] and many others. Dynamic and H. Parisch [135]. Due to the quick development of computer hardware this trend seems to problems can be found in A. Lakis et al. [86, 87], K. Le [88], W. Soedel [166] and W. Szczęśniak be very promising. The three-dimensional approach will possibly replace 2D methods even in [169]. engineering tasks. Meshless and analytical methods are investigated worldwide in the search for an alternative Despite the introduction of the Kirchhoff’s constraint, the task of integration of stresses with approach to discrete methods, like the Finite Elements, examples are P. Krysl and T. Belytschko regard to the shell thickness is not very straightforward. Used in the theory of plates and shields [84], H. Noguchi et al. [117], A. Zielinski [235]. Symbolic theoretical problems have also and adopted in the theory of shells, constitutive relations cannot satisfy the last equation of equi been researched, T. McDevitt and J. Simmonds [108] and J. Ren [150]. the Meshless Finite librium. Usually they are acceptable, but as was noticed by E. Reissner [149] and J. Cohen [40], Differences Method and adaptive Finite Elements have been researched mainly by J. Orkisz et they sometimes lead to significant errors. Therefore the problem of the last equation of equilib al. [82, 83, 89, 125, 126] and his co-workers J. Krok [81]. rium has been investigated since the thirties of the previous century. Reviews of contributions A modem field is the theory of piezoelectric shells, N. Rogacheva [151], H. Tzou [177], and may be found in the works of H. Kraus [80], E. Sechler [159] and Z. Mazurkiewicz [107]. The micropolar elasticity J. Yen and W. Chen [234]. A separate branch is the limit load bearing most popular one is V. Novozhilov’s approach [122, 123] based on the introduction of deputy capacity of shells M. Save [155,156]. internal forces. This idea was later modified by W. Koiter [77], B. Budiansky [29], P. Naghdi [112, 113, 114, 115] and J. Sanders [154]. Another way of the solution is the application a The main fact which can be concluded from this review is a tendency in contemporary theories higher-order approximation, examples of which are contributions by A. Lurie [100, 101, 102]. of shells to consider additional influences. It results in an expansion of expression - they become Unfortunately, he did not apply tensor notation so his results lost generality. This problem was very long and complicated. also investigated by S. Bielak [21], who used an approach based on the degeneration of metric tensor formula. 1.5. AIM AND CONTENT OF THE CONTRIBUTION An important contribution to the development of the nonlinear theory of shells, both in the geo metrical and physical aspect, was delivered by the Polish scientists C. Wotniak [232], W. Olszak As has already been mentioned, a universal tool in the theory of shells is tensor analysis. Tensor and A. Sawczuk [124], W. Pietraszkiewicz et al. [106, 138,139, 140]. notation is formal and elegant but seems to be too abstract for many engineers and students. A special problem of the theory of shells is stability. Due to sensibility to local imperfection it is Although tensor notation is elegant, it is prone to error if computations are carried out by hand. developed mainly in the experimental field. Reviews on contributions can be found in L. Kolldr General formulae independent of the coordinate system, useful for lecturing purposes, have [78] A. Bornstein [27], J. Harding [56], L. Samuelson [153]. The way from theory to practice to be expanded for a concrete example and then become too long. Due to the abstract nature is in this field still long, N. Morris [111]. of this notation many publications with tensor equations must be at least suspected to contain Recent contributions concentrate attention on the formulation and implementation of approx errors in indices, not necessarily made by the author but arisen in the process of publication imation by means of numerical methods, mainly the Finite Element Method. Among them and overlooked in the proof-reading. Thus, all formulas have to be verified or developed once are A. AkozandA. Oziitok [1 ],/ Chrdscielewski [38], K. Chandrashekhar and A. Bhimaraddi, again. Nevertheless an error can be repeated due to the suggestion. Summation of tensors is [36], R. Delpak and V. Peshkam [42], R. England and J. Simmonds [47], A. Paris and G. Costello a horrible job. Therefore the theory of shells is not very popular among students of civil and [134], L Taber [172], A. Tesler etal. [176],/. Vorovich [180] andG. Voyiadjis [181]. mechanical engineering in spite of the quick development of spatial structures, which require more and better educated specialists. Many papers and books are devoted to the formulation of nonlinear theories and Finite Elements procedures for them, examples are Y. Ba§ar et al. [9, 10, 11, 12, 13, 14], M. Bernadou [19], The obtained equations of shells need to be solved and at least approximated. Approximation K. Bhaskar and T. Varadan [20], B. Brank and E. Carrera [28], N. Buechter et al. [30, 31], of shells is a difficult task because of the boundary layer phenomenon, D. Zwilinger [236]. A. Ibrahimbegovic [60], A. Palazotto et al. [130, 131, 157], M. Radwanska [141], R. Schmidt Functions near the boundary are highly oscillating and the task becomes unstable in the Lyapu 30 Chapter 1. Introduction 1.5. Aim and content o f the contribution 31

nov sense. Therefore a direct numerical approach sometimes fails, especially in the case of long 204, 205, 206, 208, 212, 213, 217, 222]. This part contains only crucial elements of symbolic cylindrical shells. The problem can be solved for a selected class of problems with the popular computations. Omitted are, for example, energy problems presented in [207, 209] as they are two-step engineering approach to the problem which is based on membrane approximation. In not essential for the subject matter. the membrane approach we neglect the bending of the shell, assuming that only stretching forces The main aim of part I of the contribution, which can be regarded as its thesis, is: do occur. According to that we shall solve the system of differential equations with an operator of range 4. Hence, only four boundary conditions may be satisfied. Having the membrane solution we can estimate the moments accompanying membrane deformation, S. Bielak [21]. Computer algebra can be used in the entire process of formulation of the problem The membrane approach enables us to find closed solutions of the problems, especially for from general relations in tensor notation to expressions ready for computations. shells of revolution [23, 25, 182, 183, 184, 185, 186, 190]. The next step of the solution is to satisfy the neglected boundary conditions near the boundaries by means of the bending theory It has been also shown that this is not an automatic task. Computer algebra helps to obtain, with hypertrigonometric series approximation. The problem of instability of direct methods verify and save the result of symbolic computation. Emphasis is put on the possible involvement in comparison with the two-step approach based on the membrane approach was discussed in of the user in the simplification process and ways of scrutinizing the results. [24, 187, 188, 189]. Unfortunately this two-step approach is not a universal tool. Membrane approach fails in cases 1.5.2. Part II: Approximation of boundary value problems by means of when bending cannot be neglected. An example is a cylindrical or conical shell subjected to ovalization loads. the Least Squares Method

This contribution gathers experience and tries to prove that the whole process of building up The developed equations should be solved or at least approximated. As mentioned before, equations and its approximation can be carried out within one computational environment. due to the instability of direct approaches by means of numerical methods and to the lack of I have chosen the computer algebra system MATHEMATICS and the MathTensor™ package to generality of analytical methods based on membrane approximation, it seemed to be the most carry out tensor analysis. MATHEMATICS as an (almost) fully integrated symbolic, numerical reasonable to find something else - a “third way”. Therefore, I chose the Least Squares Method and graphical environment can be used to deal with the entire research process from the idea, for several reasons, some of them are: through symbolic and numerical computations to the publication of results. I hope that the presented ideas will be at least impressive for other researchers and engineers how to apply 1. It is an analytical method and therefore straightforward for implementation within a com computer algebra systems in science and practice. puter algebra system. An additional aim is to develop a methodology based on computer algebra assistance. I would 2. It is very simple and for this reason the algorithms are very short. like to show how a computer armed with the powerful tool of computer algebra can be used in the entire scientific process. This is a still new and developing facility and requires a specific 3. The results of the approximation are functions, not matrices of numerical data. There is approach. It allows to ’’conserve” thoughts and ideas for further development and verification. no need for interpolation or extrapolation. Let us put away a complicated symbolic task carried out by hand on paper for, say, two weeks, we have to do the whole job once again. Doing the same with the MATHEMATICS notebook 4. As the results are functions, they can be easily transformed within the computer algebra requires only short memory refreshment. system. Most publications of the present Author, except one [224], are devoted to the theory of shells and application of computer algebra in symbolic and boundary value problems of the theory of More about the features of this method can be found in chapter 7 on page 73. plates, shields and shells. Therefore this contribution is divided into two parts. The least Squares Method is a well-known quasi-variational way of finding an approximate solution of a boundary value problem, J. Glazunow [51] and D. Zwilinger [236]. The classi cal approach consists of minimizing the functional based on algebraic, differential or integral 1.5.1. Part I: Symbolic computations equations, or on a system of equations with a set of independent functions satisfying the bound ary conditions. The first part deals with problems of symbolic computations. This is not a theory of shells but it shows how to use different tools of computer algebra effectively, in different aspects. Starting I have appended the minimized functional with terms responsible for boundary conditions. with equations of equilibrium in very general tensor notation, through the integration of stresses Therefore I have called the approach a Refined Least Squares Method. The natural abbreviation over the shell thickness, simplification of formulas and geometric and kinematic description of seems to be (RLS) but there is some conflict with the Recursive Least Squares Method used for an arbitrary shell - it finalizes with equations in terms of displacements ready for approximation. dynamic optimization and the word “refined” has also been applied by somebody else. Never Some of the problems were already published separately in [195,197,198, 200, 201,202, 203, theless, it seems to be too late to change the word “refined” for any other word, since most of 32 Chapter 1. Introduction

WWW browsers return references also to my contributions for “Refined Least Squares” input. Probably it would be better to find another description but the words “extended”, “expanded” or “generalized” in association with “Least Squares” have already been used. However, I am open for any proposal in this matter. The refinement of the method resulted in a much simpler algorithm since approximating func tions do not have to satisfy boundary conditions. However, most important according to that is the result of a computational experiment which allows to neglect some boundary conditions in the approximation process and still obtaining a nonsingular system of algebraic equations and a solution feasible in the most of the domain. It is an unexpected discovery, called boundary-condition phenomenon, that the 1 O'* order operator may be approximated with only 4 boundary conditions. The aim and thesis of this part can be formulated in the following way:

It is possible to approximate the 1 O'* order shell boundary-value problems by means Parti of the Refined Least Squares Method taking into account selected boundary condi tions. The phenomenon has a physical interpretation and can be explained math ematically. Boundary conditions neglected in the first step can be satisfied locally within the boundary layer, in the second step. SYMBOLIC COMPUTATIONS

The Refined Least Squares Method was implemented to solve boundary value problems to illustrate symbolic problems presented in Part I. Its application for boundary value problems was presented for the first time in [191] and next in [193]. It was used for example to develop numerical examples for spherical [200] and catenoidal [209] shells. It was also applied to solve shields [196], plates [192], a hyperbolic model of heat transfer [193, 194] and the theory of elasticity [199] - mainly to illustrate its possibilities to deal with non-continuous boundary conditions. The contribution presents a theoretical basis of the method, main elements of its implementation and contains examples of the application of the method to boundary problems of cylindrical shells to illustrate the boundary-condition phenomenon and propose a two-step approach. There is also carried out a discussion on global and local error analysis. Some problems were already presented in [210, 211, 212, 214, 215, 216, 223]. Chapter 11 on page 146 presents a physical interpretation of the discovered boundary-condition phenomenon and its mathematical explanation. By the way, it was shown that the base approx imation approach is not limited to problems with the boundary layer phenomenon, only. To show the wide potential application of the method the last chapter of this part presents an ex ample, published in [218], of the application of this method to medium-thick plates. There was shown that also this two-dimensional problem can be approximated neglecting some boundary conditions. The book closes with conclusions, plans for the future and the last chapter with acknowledge ments. Just before the references a brief review of the Author’s contributions is appended. Chapter 2

BASIC RELATIONS OF THE THEORY OF SHELLS

Rozważania poprzedzono zestawieniem podstawowych związków teorii powłok wykorzystanych w pra cy. Ten rozdział ma pomóc Czytelnikowi w zorientowaniu się w sposobie oznaczania oraz kontekście omawianych problemów.

The consideration is preceded by a presentation of basic relations of the theory of shells applied in the contribution. This chapter should help the Reader to get familiar with the nomenclature and context of the discussed problems.

This chapter was written on the basis of several contributions, mainly by S. Bielak [21] and P. Naghdi [113]. The riotations were unified. This was somehow enforced by the system of computer algebra as it requires a clear nomenclature. It is another advantage of the system. It makes the user apply the consequent notation and logical definition of quantities according to the computational order. Chapters and sections are pointed out where the considered problems are discussed, as well as previous publications by the author where these elements have already been presented. Einstein’s summation rule is applied consequently. Indices for two dimensional problems take the values 1 and 2. They are denoted by Latin letters in accordance with the MathTensor™ notation. The package could not, at the moment when the book was being edited, produce Greek symbols. However, many contributions apply Greek indices for two-dimensional prob lems, let us hope this disadvantage will not prevent the Reader from following the presented considerations. Basic geometrical and kinematic relations, described below, are illustrated in Figure 2.1 on the next page.

2.1. GEOMETRICAL DESCRIPTION

2.1.1. Reference surface

A three-dimensional thin curved body in the Euclid space, the shell, is described with reference to the selected surface called reference surface. Commonly it is called mid-surface, which is

35 36 Chapter 2. Basic relations o f the theory o f shells 2.1. Geometrical description 37

The vector r3 defines the surface orientation, C. Wozniak [232]. For computational convenience we introduce two other vectors denoted by mt. They are com puted as derivatives of the vector r3 with respect to the parameters x“:

»■<=£■ <2-5>

Coefficients of the second and the third differential form may be computed now:

bij-.= - r r mj, ( 2 .6 )

dj ■■= mi ■ nij. (2.7)

The Gaussian curvature, which is defined as a product of the main curvatures of the reference surface K ~ k\ k2, may be computed in various ways, P. Naghdi [113]. The most common is Rys. 2.1. Związki geometryczne i kinematyczne w powłoce the following formula:

b placed in the middle, at a distance h, between bounding surfaces. The quantity 2 h is called the K := - , (2.8) shell thickness. at The reference surface is described by the parametric equation: where b is a determinant of the matrix built of coefficients of the first differential form bij, b := det ||*y||. r:=r(x\xz), (2.1) The scalar H called mean curvature, is according to its name, the mean of the main curvatures H := and is computed from the formula: where the parameters x ‘ are curvilinear coordinates on the reference surface.

Let us introduce a definition of the covariant basis. Vectors tangent to the reference surface are H := -b k\ (2.9) computed by differentiating the vector r with respect to the parameters x ‘: It should be mentioned that the third differential form depends on two other ones and the cur Г; = r , := —dr . vatures K and H : dx‘ (2.2) After the evaluation of the vectors of the covariant basis, coefficients of the first differential Cij := 2Hbij - K a,j, (2.10) form are computed. It is done with a scalar product of the vectors r,:

Cij ■- b f bpj. (2.11) aij - n -rj. (2.3) The obtained tensor atj has the properties of a metric tensor on the reference surface. These definitions are directly connected with the equivalences (4.9) applied in point 4.3.3. on page 55. The unit vector, normal to the reference surface, is computed from the vector product:

r\ Xr2 r3 := 2.1.2. Parallel surface Va ’ (2.4) An arbitrary space point in the neighborhood of the reference surface is described by a vector: where a is a determinant of the matrix built of coefficients of the first differential form au, a := det ||a,7||. R :=r + zr-i, (2.12) 38 Chapter 2. Basic relations o f the theory o f shells 2.2. Geometrical relations 39

where z is a coordinate measured along the axis determined by the vector 7-3. This coordinate The coefficient of the metric tensor can also be computed by the inverse of the matrix built from is denoted also by x3. The set of points lying at the constant distance to the reference surface the covariant components of this tensor. If we know these components it seems to be a very (z = const) is called parallel surface. convenient way. Differentiating (2.12) on the page before with respect to x* we receive vectors of the covariant The geometrical problems presented above will be more exactly considered in the chap basis Rj. Computing the scalar products of these vectors, coefficients of the first differential ter 5.1. on page 58, but will be applied also in other sections. The problems were also considered form of the parallel surface are obtained. They are components of the metric tensor in the three- in [25, 204]. dimensional space. Applying (2.3), (2.6) on the preceding page and (2.7) on the page before these coefficients may be denoted with earlier computed objects: 2.2. GEOMETRICAL RELATIONS gij := aij + 2 zbij + z1 Cij. (2.13) 2.2.1. Kinematic relations Taking int account (2.10) on the preceding page, the components of the metric tensor may be computed from the formula: The displacement of an arbitrary point of the shell is denoted by the vector W. Let us define it as a subtract of the vectors in the ortocartesian pre-system for the reference R and actual R gij ■= an (1 - Kz1) + 2zbij (1 -H z ). (2.14) configuration.

Covariant components of the metric tensor may be computed in various ways. For example W ~ R - R , (2.20) P. Naghdi [113] proposed the following formula: This vector can be related to the reference surface introducing two new vectors, g'1 ■= VpVqJapq, (2.15) W:=w + zd. (2.21) where: where the vector w defines the displacement of the reference surface from the reference config 6ij + z [W - 2 H 8A uration f to the actual r, JV := ------(2.16) w:=r-r. (2.22) The object Si is a Kronecker delta and Z := |, where g := det ||gy||. It can be computed as a function of the coordinate z and the curvatures H and K: The vector d defines the change of the normal vector caused by deformation,

Z:= J l = \-2H Z + K f, (2.17) d:=r3- r 3. (2.23) The vectors of displacement and rotation are expanded in the covariant basis according to the S. Bielak [21] proposes the following formula: formulas:

n & (1 - z 2) ~ 2 Kb‘J { \- H z ) z g = — ^ ------. (2.18) w := w* rk + w3 r3, (2.24) where: d:= dkrk +d3 r3, (2.25)

KbiJ :=2Haij-b ij. (2.19) where the last term in the equation (2.25) is negligible in the case of small rotations. It can be proved, using relations (2.10) and (2.11) on the page before, that both formulas (2.15) Vectors Wi and present in strain tensors are derivatives of displacement and rotation vectors: and (2.18) are equivalent. 41 2.2. Geometrical relations 40 Chapter 2. Basic relations o f the theory o f shells

The two others are measures of the second and the third differential form:

0.27) Pij -=\ (bij-kl), (2.32) Kinematical relations will be considered in section 5.3. on page 60.

* < / - 5 (*«-*

d‘ := - (w1 bki + w3,) alJ. (2.28) Y,j := \ (gu ~ hi) • (2.34) The application of this relation results in a set of equations with a differential operator of the 8'A order, but it leads to additional difficulties, especially in boundary conditions. These problems, Using (2.13) on page 38, (2.34) and the first Kirchhoff assumption we receive: concerning the theory of thin plates, were discussed in [22]. Applying the ideas presented by Z Kqczkowski [72] for mid-thickness plates we introduce a Yij := Yij ~ 2 zPij + z2 &ij. (2.35) vector of “averaged deplanation” 0 , C. Wozniak [232]: In the case of geometrically linear problems (infinitesimal displacements) we can apply the relation (2.14) on page 38 and express this tensor as a function of only the tensors y,y i py, but fif := dj + (w1 bu + w3,,) afj + \ ~ , (2.29) in order to apply formulas in nonlinear theories we will use (2.35). Neglecting nonlinear terms the strain tensor can be expressed by the formulas: where mj is a component of the load moment vector. Let us assume that « \dJ\, which means that the deplanation is small enough to neglect its YijU ~ \ iri • wi + ri ‘ wd - (236) influence on stretching forces and moments. Transverse forces are computed, according to the above definition, with the following formula: Pij := - - (m, • Wj + mj ■ w, + r,- ■ d} + Tj ■ d) , (2.37) Q‘ :=X ^2 h 0 , (2.30) where p is a Kirchhoff modulus, ^ is a coefficient of shear stress distribution, assumed for a $ij ~ 2 imi ' dj + mj ‘ d>) ■ (2.38) homogeneous shell, according to E. Reissner [145] to be equal to and R. Mindlin [110] to be equal to yj. For a multi-layer shell Z. Kqczkowski 's [72] approach may be applied. 2.2.4. Influence of temperature This definition is applied in 6.2.3. on page 64. These definitions of strain tensors may be generalized taking into account the influences of temperature distortions. According to the relations derived in chapter 10 of S. Bielak [21], the 2.2.3. Strains formulas presented above will be appended with additional terms.

Describing shells we define three strain tensors of the reference surface. The first one is called the tensor of membrane strain, S. Bielak [21], and is a measure of the first differential form:

Yij := ^ (aij ~ aij) • (2.31) 42 Chapter 2. Basic relations o f the theory o f shells 2.4. Equations o f equilibrium 43

&‘j ■= ~ (m i ■ d j + n tj ■ di) - £, d j - к, bij, (2 .4 1 )

m ‘J : = / \ f l W ~zbr^ T"zdz- (2'46) -h 2.3. CONSTITUTIVE RELATIONS The components of the vector of transverse forces are received from: 2.3.1. Stress tensor

It is assumed that the component r 33 of the stress tensor is negligible in tasks concerning shells. Q‘ ■= J ( 2 - 4 7 ) Other components, if the shell is built of elastic anisotropic material, are computed from the -h following formula: The evaluation of the integrals (2.45) on the facing page and (2.46) is considered in chapter 4 on page 49. It was also published in [201, 203, 205]. t 0 == CijU yu. (2.42)

The first components of the elasticity tensor in the ortocartesian system of coordinates Cijkl 2.4. EQUATIONS OF EQUILIBRIUM should be transformed to a curvilinear one CiJU, and then the components of the deputy elasticity tensor C'JU are computed according to the formula, P. Naghdi [113]: Internal forces in a shell in equilibrium satisfy the system of equations:

N ij.i - bj Q‘ + Pj = 0, (2.48) If the shell is built of isotropic material, the deputy tensor of elasticity is described by the bij N‘J + Qj.j + P3 = 0, (2.49) relation: Mij.t - QJ = 0. (2.50)

CijU := gij gU + 2ng* gjl, (2.44) This system may be appended with terms responsible for dynamical factors, surface load mo ments and the influence of large stretching forces. where fi and X are Lame constants. Chapter 3 on page 45 is devoted to the expansion of equilibrium equations in tensor form, It is worth mentioning that in this simplest model of continuum the components C‘Jkl are func their transformation to partial differential equations and how to append them with second-order tions of the coordinate z measured along the shell thickness. nonlinear terms. This problem was also considered in [185]. There is also an equation of equilibrium called the last one: 2.3.2. Internal forces (NiJ-M kjbki)= 0. (2.51) Similarly as in geometrical relations, we reduce the analysis of the three-dimensional stress state to a two-dimensional analysis of the resultant of the stress tensor. The reduction leads to where 6y is an antisymmetrical object with zero trace. the definition of tensors of stretching forces and moments, S. Bielak [21]. This equation is (actually should be) satisfied as an equivalence, if the integrals (2.45) on the The tensor of stretching forces is defined by the formula: facing page and (2.46) are computed properly. This problem will be also considered in chap ter 4 on page 49.

-A The tensor of moments has the following definition: 4 4 ______Chapter 2. Basic relations o f the theory o f shells

2.5. PHYSICAL COMPONENTS

The quantities presented above are computed in curvilinear coordinates. For engineering pur poses physical quantities have to be calculated. According to S. Bielak [21] the formulas are as follows1: Chapter 3 components of the displacement and rotation, EQUATIONS OF EQUILIBRIUM Wi := yJa~iWl, w3 := w3, dt := yfaJid1. (2.52) components of the transverse and stretching forces,

Tematem rozważań tego rozdziału są równania równowagi. Przedstawiono sposób zamiany tych rów Q‘ faTi ■■ <7. := —r= , nij '■= J (2.53) nań w zapisie tensorowym na równania różniczkowe w formie przygotowanej do dalszych obliczeń. Cafy Va» V a" proces podzielono na 3 etapy wymagające zastosowanie różnych narzędzi obliczeniowych. Pokazano components of the moments, również możliwość rozszerzenia rozważań o przypadki nieliniowe. The topic of consideration of this chapter are equations of equilibrium. The way of changing these equations in tensor notation to ordinary partial differential equation in the form ready for further compu /a a11 /a a22 ’ m'2 M ’ (2.54) tations has been presented. The whole process is divided into 3 steps requiring different computational tools. Extension of this consideration to nonlinear problems has been dealt with, too1. components of the load vector,

Pi - yfa~iPl, Pi-= P3. (2.55) The equations of equilibrium in tensor notation should be transformed to ordinary partial differ ential equations. Three steps can be distinguished in the process. Each of them is different and Caution! Do not sum with regard to repeating indices in these formulas. requires application of different tools. Procedures are similar for each equation of equilibrium The indexing rule for physical components of moments in (2.54) was introduced after S. Bielak of the system (2.48), (2.49) and (2.50) on page 43, so we will focus our attention on the first [21] to provide a coincidence of positive senses and indices of the vectors of forces with the one. pseudovectors of moments , see Figure 2.2.

3.1. CHANGING COVARIANT DERIVATIVES IN EQUATIONS TO (ORDINARY) PARTIAL DERIVATIVES

The first equation of shell equilibrium takes the following form:

N ij- i- W Q‘ +P1 = 0. (3.1)

The left-hand side of the equation can be denoted by the following MathTensor™ definition.

In[l] : = CD [n [ui, uj], li] -q[ui] b[li, uj] +p[uj] O u t[l] = Ni 3; i +Pj - (bi1) (Q1) MathTensor™ was written for MATHEMATICS 2. Therefore all symbols in it are denoted by Latin Fig. 2.2. Positive senses of forces and moments letters. To receive Greek letters the output form has to be redefined. For example the symbols Rys. 2.2. Dodatnie znaki sił i momentów 1 The considered problems were also published in [212,213]. 'Physical components will be denoted with Script fonts. 45 47 46 Chapter 3. Equations o f equilibrium 3 .3 . Transformation to MATHEMATICS differential equations of the affine connection Af f ineG have got the standard output form G. This redefines it to r. The first equation of equilibrium in tensor notation renders two partial differential equations. The metric tensor on the reference surface is equal to the first differential form gy = ay. In[2] := Form at[AffineG [a_]] := PrettyForm [r, a] The same can be done for Kronecker delta Kdelta to be expressed by <5 or indices to be In[7] := R l[u j.] =MakeSum[rl] represented by small Greek letters. Out 17] = j (a23) (N22) (322, 2 ) + | (a22) (N 23) (a22>2) + i (a23) (N 12) (a22,i) + That command changes the covariant derivatives to ordinary partial ones. Dummy indices are | (a22) (N13) 1) +

In[3] := CDtoOD [%] (a12) (N13) (a2i,x) (a23) (N11) ( a n <2) + — (a13) (N12) ( a n ;2) + Out[3] = (r^pq) (Nqp) + ( r pp<,) (Nq j) + Np3,p + P 3 - (bp3) (QP) — (a13) (N21) ( a n /2) + — (a11) (N23) ( a n , 2) + — (a 13) (N11) (al lr l ) t The space is metric, so the affine connection can be expressed in terms of a metric tensor. This is done with the Af f in eT o M etric function. The MathTensor™ function Tsimplify will j (a11) (N13) (au,i) + N13,i + N23>2 + P3 - (bx3 ) (Q1) - (b23) (Q2) simplify the expression taking into account all symmetries of the involved tensors.

In [4]:= Tsim plify[AffineToM etric[%]] 3.3. TRANSFORMATION TO MATHEMATICS DIFFERENTIAL Out [ 4] = ’ | (gP3) (Nqr) (gpq,r ) (gPq) (Nr j ) (gpq,r ) + EQUATIONS \ (gpj) (Nqr) (gpr,q) - i (gp j) (Nqr) (gqr,p) +

Np3,p + P 3 - (bp3) (Qp) Finally the equation can be transformed into an “ordinary” partial differential equation. First the components of the tensor have to be represented as “normal” functions of two variables. Both MATHEMATICS and MathTensor™ have advanced algebraic simplification tools, but some times the use of an automatic function does not leads to the simplest form. Using more complex In 18] := n [l, 1] := n ll[x, y] ; instruments better results can be obtained. The set of commands below carries out the follow n [l, 2] := nl2 [x, y]; n [2, 1] :=n21[x, y ]; ing operations. It is easy to find that the expression above contains 3 terms with Nqr. Therefore n [2,2] :=n22[x, y ]; (»andso on*) using the Collect function terms can be collected in the expression with respect to it and then Next ordinary differentiation has to be “turned on”, which means that the derivatives can be simplified term by term with Simplify. It is done by the so called mapping Map function in evaluated. the infix form /@. The result is saved for further use. In [9]:= On[EvaluateODFlag] In [5 ]:= r l = Sim plify/@ C ollect[%, n[u2, u3]] The first differential equation takes the form: Out [5]= i (gP*5) (Nr j ) (gpq,r ) (gp3) (Nqr) (gpq,r + gpr,q - gqr,P) + In [10] := eqn [1] = Rl [1] Np’,p + P 3 - (bp3) (QP) Out [10]= pi [x, y] - bmll [x, y] ql [x, y] - + It results in the following formula, which contains only “ordinary” partial derivatives. bml2 [x, y] q2 [x, y] + n21 (0'1) [x, y ] + i n22 [x, y] (agl2 [x, y] a22(0'1) [x, y] + ag ll [x, y]

N pj,P + \ gpq gpq,r N rj + \ gpj (gpq,r + gpr,q - gqr,p) N qr - bpJ Qp + P> = 0. (3.2) (2 a l 2 (0' 1) [x, y] - a 2 2 (l,0) [x, y ] ) ) + j nl2 [x, y] (agll [x, y] a l l 10'1’ [x, y] +

agl2[x, y] a22(1'0) [x, y]) + 3.2. SUMMATIONS i n il [x, y] (2 agll [x, y] a ll’1'01 [x, y] - The next step consists in the summation of the expression. The problem of the shell is two- agl2 [x, y] ( a ll<0'1> [x, y] - 4 a l2 (1'0) [x, y ]) + dimensional and the system should be informed about it before the summation process is started. ag22 [x, y] a22(1'0) [x, y ] ) +

In [6] : = D im e n s io n = 2; j n21 [x, y] (2 agll [x, y] a l l 10'1’ [x, y] +

ag22.[x, y] a22 (0'1> [x, y] + agl2 [x, y] (2 a l2 (0'1) [x, y] + a22 [x, y] We will return to this equation later, in section 6.3. on page 65. 48 Chapter 3. Equations of equilibrium

3.4. NONLINEAR EQUATIONS OF THE SECOND-ORDER THEORY

The approach can be easily extended to nonlinear problems. This is a simple example. The consideration presented here is done in accordance to ideas about taking into account large stretching forces in the theory of plates suggested by Z. Kqczkowski [72]. Chapter 4 The third equation of shell equilibrium in tensor notation is: CONSTITUTIVE RELATIONS A/'7-., - Qj = 0. (3.3)

In the second-order theory the additional moments caused by the stretching (tensile) forces acting on the normal displacements are taken into consideration. Ten rozdział przedstawia wyprowadzenie uściślonych związków konstytutywnych powłoki. Okazuje się, że w procesie obliczeń nie trzeba dokonywać żadnych dodatkowych założeń prowadzących do M iJ -» M ij + N iJ w3. (3.4) degeneracji całkowanych wyrażeń. Proponowane podejście może być zastosowane w przypadkach zadań zarówno geometrycznie jak i fizycznie liniowych i nieliniowych. Zaprezentowano wybrane ele It is implemented with the following function: menty moderowanego uproszczenia postaci wyrażeń z zastosowaniem różnorodnych narzędzi pakietu MathTensor™ oraz MATHEMATICS. Wykazano, że dla dowolnej powłoki wyprowadzone związki spełniają In [ll] : = m[ui_, uj_] :=ml [u i, u j] + n [u i, u j] w3 ostatnie równanie równowagi powłoki. The moment tensor now has the following form: This chapter shows the derivation of refined constitutive relations for shells. It occurs that in the com I n [ 1 2 ] := m [ u i , u j ] putational process there is no need to make additional assumptions leading to the degeneration the Out [12]= Mij +w3 (Nij) integrated expressions. The proposed approach can be applied in the case of both geometrically and The left-hand side (lhs) of the third equation is: physically linear and nonlinear tasks. Selected elements of moderated expression simplification with the application of various MathTensor™ and Mathematical tools are presented. It has been proved that in In [13] := CD [m [ui, u j], li] - q[uj] case of an arbitrary shell the derived relations satisfy the last equation of equilibrium for shells1. Out [13]= MiJ;i + w3 (Ni3; i) + (w3,i) (Nij) - Qj Using a similar approach as thatpresented in the previous section, the following result is ob tained:

4.1. DEFINITIONS

M p{ p - 1 M pq g'J (gpq_r - gprq - gqrp) + I M pj g * gqr,p - QJ+ One of the tasks of the theory of shells is to reduce the three-dimensional problem of the theory 1 / (3.5) +NpJ W\ P + _ w 3 ^gPq N rj g f v + gPj N ,r + gpr" _ g^ + N „ j ^ = 0 of elasticity into a two-dimensional one2. The analysis of stresses is reduced to an analysis of internal forces: stretching (tensile) forces and moments. The respective tensors are computed from the following integrals, see section 2.3.2. on page 42: Nonlinear terms are written in the second line. A r N ‘J := [ (

'The problems presented here are discussed in more details in [222]. 2This has been considered in a new MATHEMATICA' session.

49 50 Chapter 4. Constitutive relations 4.2. Evaluation 51

h In[5] := n [ui., uj-] [k_] := Simplify/@ Tsimplify[ M IJ := J (srj - z b rj) Tn Z dz. (4.2) AbsorbKdelta[ -h integ/gExpand[ N orm al[ The square root in these formulas is the following function, compare formula (2.17) on page 38: S e r i e s [ Z (K d e lta [1 3 , u j] - z b [1 3 , u j ] ) ta u [ u i , u 3 ] , { z , 0 , k}]]]]] Z '~ \ ha ~ x ~ 2 H z + K?. (4.3) This complex multi-function does the following: the considered integrand - I n [ l ] := Z := 1 - 2H z + Kz2; (Kdelta[13, uj] - zb[13, uj]) tau[ui, u3] The stress tensor for isotropic material can be derived from the formula, see section 2.3.1. on is expanded into a power series with Series and Normal. The result is algebraically ex page 42: panded with Expand and then integrated term by term (using the mapping procedure /@) with a predefined function of integration integ3 : h ^ :=E ( r ^ 2 Sij 8pq + ~ g‘p g]q) 7pq- (4.4) In [6]:= i n t e g [x_] := j xdlz -h In[2]:= tau[ui_, uj_] :=E^------M etricg[ui, uj] M etricg[ul, u2] + This approach is necessary as MATHEMATICS is a program and only a program. If the argument

—-— M etricg[ui, ul] M etricg[uj, u2]) gammastar[11, 12] of the function is complicated, it takes a lot of time to deal with it. Therefore it is usually sensible 1 + v J to divide the task into a set of simpler problems. This approach speeds up the computations. A strain tensor in 3D space can be expressed in terms of the strain tensor of the reference Mapping is a very useful tool in this process. surface, see section 2.2.3. on page 40: After integration the Kronecker delta is absorbed by AbsorbKdelta and simplified by the already mentioned functions Tsimplify and Simplify. The parameter k defines the preci Jij ■= 7ij - 2 zptj + z2 dij. (4.5) sion of computations; for engineering purposes it is sufficient if the calculations are performed with a precision of up to the third power of the shell thickness. Then the parameter k in the In[3]:= gammastar [li_ , lj_] := function should take the value 2. This commands computes the tensor of stretching forces with a gamma [ li, 1 j] - 2 z rho [ li, 1 j] + z2 th eta [ li, 1 j] required precision. The result is saved for further use. The output transformed into a traditional Contravariant components of the metric tensor 3-D shell can be computed from the formula formula is presented on the next page. which is a function of the variable z measured along the normal to the reference surface, see formula (2.18) on page 38: In [7]:= noriginal [ui-, u j.] = n [ui, uj] [2] It should be mentioned here that all MATHEMATICS inputs and outputs, as well as parts of the notebooks in this contribution have been transformed to ET]gX2e files using built-in com .. a‘j (1 - K z 2) - 2 (2 H afJ - W) (1 - H z) z g'J = — • (4.6) mands TeXForm[expr] and TeXSave [notebook] and next text-processed and com piled using the I5TgX2e package called The MATHEMATICS Virtual Font Package by J.-P. Kuska In[4]: = M e t r i c g [ u i - , u j _ ] := http://phong.informatik.uni-leip2ig.de/ kuska/. The package is distributed with the MATHEMATICS a [ui, uj] (l-K z 2)-2 (2Ha [ui, uj] - b [ui, u j]) (l-H z )z ) system and available separately on the author’s homepage. Z2“ This remark is provided here since this chapter contains very complex outputs. Text-processing of such expressions was possible only thanks to the mentioned facilities. 4.2. EVALUATION 3 The consideration can be expanded to multi-layer shells. In the case of the shell with several layers the integration should be performed within each layer so the function i n t e g would be The integrals (4.1) on the preceding page and (4.2) appear to be simple but after substituting into them the functions (4.3), (4.4), (4.5) and (4.6) they become a bit more complicated, although by integ [x-] := J xdlz applying the abilities of the system they can be computed with arbitrary precision with respect i [ i ) to the expansion of the shell thickness 2 h. This can be achieved by means of the function: This approach will permit to consider advanced nonlinear problems including tasks with large strains and phys ical nonlinearities. 52 Chapter 4. Constitutive relations 4.2. Evaluation

The result of computations of the tensor of stretching forces is: 4 E h 3 H (5 + 6h2 K) (ap3) (aqi) ( Y P q ) 0utl8]= ------ÏE1T77)------+ 4 E h3 H (5 + 6 h 2 K) V (apq) (ai 3 ) ( Y p q ) ( vU - ^ h I3 “ 5 h% K) a" aqi ypq 2E/i (3 -5 h 2 K) v apq ypq 15 (-1 + v2) 3 (1 + v) + 3 ( 1 - v2) + 2 E h3 (-1 + 3 h2 K) (apr) (aq i) (brj ) (YPq) , 4 E A3 // apr aqi brj ypq 4 E h3 H va« ari brj ypq 3 (1 + v) 3 (1 + v) + 3 (1 - v2) + 2 E h 3 (~ l + 3 h 2 K) v (apq) (ar l ) (b ^ ) ( Y p q ) 3 (-1 + v 2) + 4E h 3 H v a‘J bpq ypq _ 4E h3 v ar‘ brj bM ypq _ 4 E h3 aqi brJ bpr ypq 4 Eh3 (5+ 6h2 (H2 -2 K )) v (aiJ) (bpq) ( Y p q ) | 3 (l - v2) 3(1- v2) 3 (1 + v) + 15 (-1 + v2)

+ 4 E h3 H aql bpJ ypq AEh3 H apJ bqi ypq 4 E h3 apr brJ bqi ypq 4 E h 5 H v (ar l ) (br J) (bpq) ( Y p q ) _ 4E h 5H (aqi) (br3) (bpr) ( Y p q ) : 3 (1 + v) + 3 (1 + v) 3 (1 + v) + -5 + 5 v2 5 (1 + v) ^ 8 E h3 bPJ bqi ypq _ A Eh3 v a* brJ bri ypq 4 E h3 H v apq biJ ypq 4 E h3 (5 + 6 h2 (H2 - 2 K) ) (aq l) (bp3) ( Y p q ) | 3 (1 + v) 3 (l - v2) + 3 (l - v2) + 15 (1 + v) 4 E h3 (5 + 6 h2 (H2 - 2 K) ) (ap3) (bq i) (Ypq) f 8E h3 vbpqbijypq i 8EA3// apj aqi ppq 8E h3 H v apq aiJ ppq 15 (1 + V) 3 ( l - v 2) + 3 (1 + v) + 3 (1 - v2) + 4Eh5H (apr) (br*) (bqi) ( Y P q ) _ 8 E hs (br 3) (bPr ) (bq l) ( Y P q ) ; ^ 4E h3 apr aq‘ brJ ppq ^ 4 E h3 v apq ari brj ppq 8 E h3 v aiJ bpq ppq 5 (1 + v) 5 (1 + V)

3 (1 + v) + 3 (1 - v2) 3 (1 - v 2) + 32 E h 5 H (bpj) (bqi) ( Y p q ) 4 E h 5 H v (apq) (br3) (br l ) ( Y p q ) | 8 E h3 aqi bp> p pq 8 E h3 api bqi ppq 8E/i3v apq W ppq 5 (1 + v) + -5 + 5 v2 3 (1 + v) 3 (1 + v) 3 ( 1 - v2) + 8 E h5 v (brJ) (bpq) (br i ) ( Y p q ) - 5 + 5 v2 , 2E h3 apj aqidpq 2E/i3v apq aij dpq 4 E h 3 (5 + 6 h2 (H2 - 2 K) ) v (apq) (b1*) (YPq) | 3 (1 + v) + 3 (1 - v2) + 0 \h )- 15 (-1 + v2)

32 E h5 H v (bpq) (bi j ) ( Y Pq ) + 4 E h 3 (-1 + 3 h2 K) (ap3) (aq i) (ppq) The tensor of moments is defined by the following formula, 5 - 5 v2 3 (1 + v) 4 E h 3 ( - l t 3 h 2 K ) v ( a pq) (al j ) (ppq) _ 8 E h5 H (apr) (aq i) (br j ) (ppq) | 3 (-1 + v2) 5 (1 + v)

_ j E h3 H apj a qi y pq 4 E h3 H v apq a ‘j y pq 8 E h5 H v (apq) (ar l ) (br J) (ppq) + 8 E h 5 H v (a13) (bpq) (ppq) | 3 (1 + v) 3 ( 1 -v 2) + -5 + 5 v2 -5 + 5 v2 2 E h3 apr aqi b j y pq 2 E h3 v apq ari brj y pg 8 E h 5 v (ari) (brj) (bpq) (ppq) + 8E h 5 (aql) (br3) (bpi:) (ppq) 5 - 5 v2 5 (1 + v) 3 (1 + v) 3 (1 - v2) + 8Eh5H (aql) (bP3) (Ppq) _ 8 E h s H (aP3) (bqi) (ppq) | 4E h3vaijbpqypq 4E h3aqibp jypq 5 (1 + v) 5 (1 + v) (4.8) 3 (l - v2) + 3 (1 + v) + 8 E h 5 (apr) (brj) (bql) (Ppq) _ 16 E h 5 (bpj) (bqi) (ppq) | 4 E h3 apJ bq‘ y pq 4 E h3 v apq biJ y pq 5 (1 + v) 5 (1 + v) + 3 (1 + v) + 3 (1 - v2) + 8 Eh5 v (apq) (brJ) (brl) (ppq) + 8 E h5 H v (apq)(bi3 )(ppq) | 5-5 v2 -5 + 5 v2 _ 4 E h3apJaqippq AEh3 vapq aiJppq n /i5N 16 E h 5 v (bpq) (b13) (pPq) _ 4 E hs H (a ^ ) (aq l) ((9pq)| 3(1+v) 3 (l -v2) + ° ( h >- -5 + 5 v2 5 (1 + v)

4 E h5 H v (apq) (a1^) (flpq) _ 2 E h 5 (apr) (aq i) (br J) (<9pq): Computations can, of course, be carried out with a higher precision. For example, the result of -5 + 5 v2 5 ( 1 + v) evaluating the tensor of moments with a precision of the fifth power of shell thickness is: 2 E h 5 v (apq) (ar l ) (br j ) ((5pq) + 4 E h5v (aiJ ) (bpq) (Opq)| -5 + 5 v 2 5 - 5 v2 In [8] ;= m [ui, u j] [4] 4 E h 5 (aq i) (bpJ) ((9pq) 4 E h 5 (ap j) (bq i) (<9pq) 4 E h5 v (apq) (b13) ((9pq) The result of evaluation is presented on the facing page. 5 (1 + v) 5 (1 + v) 5 - 5 v 55 54______Chapter 4. Constitutive relations 4.3. Elements o f simplification

4.3. ELEMENTS OF SIMPLIFICATION They are saved with separate names. The function Part, presented here in the form I n i selects the n,h term from the expression.

The result can be used directly in further computations but is rather long. Nevertheless, using In [10] : = Iiy = % I I I ; n 0 = % H2H ; n „ = % 1 3 ! ; a set of MATHEMATICS and MathTensor™ tools it can be presented in a shorter form. Among them we can find: 4.3.2. Moderated simplification • Dum [ % ] to find pairs of dummy indices Automatic simplification does not necessarily render the simplest form. The system can be • Expand [ % ] for the expansion of the expression helped in this process by the user. An example of enforcing the required behavior is given below. The terms are grouped and each group is simplified. For example: • Absorb[%, a] and AbsorbKdelta to lower and raise the indices In[ll]:= Simplify/@ % • Canonicalize to find the canonical form of tensor expression 2 E h (-1 + h 2 K) v (a13) (ypp) 4 E h3 H v (a13) (bpq) (ypq) Out [11]= ------_l+v2 3 - 3 v2 • Tsimplify for tensor simplification 4 E h 3 (bp1) (bq3) (Ypq) 4 E h3 v (bpq) (bi3) (Ypq^ 3 (1 + v ) 3 - 3 v 2 • Simplify for algebraic simplification. 4 E h 3 H ( b p 1 ) (yp3) 2Eh (-1+h2K) (Y1J) 3 (1 + v) ” 1 + v 4.3.1. Grouping of terms In[12] := S i m p l i f y [% I I ] ] + % 1 6 ]] ] + S im p lify [% I2 J + % 14]) ] + Analyzing the original expression of stretching forces (4.7) on page 52 we find that it contains S i m p l i f y [% I 3 ] | + % I 5 J ] all three strain tensors y;j, pi; and i?y. 4 E h3 v (bpq) (H (a13) + b 13) ( Ypq) 4 E h 3 (bp1) ( (bq3) (Ypq) +H (Yp3) ) U 3 (-1 + v2) 3 (1+v)

Let us group the terms which accompany each tensor: 2 E h ( - 1 + h 2 K) (v (a 13) (Ypp) - ( - 1 + v ) (Y13)) - 1 + v 2 In [9]9]: := = CoJC ollect [n o rig in al [u i, u j], {gamma [11, 12], rho [11, 12], theta [11, 12] } ] 2 E h (-3 + 5 h2 K) (ap j) (aq l) 2 E h (-3 + 5 h2 K) v (apq) (a13) Out 191- ( - 3 (1 + v ) 3 (-1 + v2) 4.3.3. Replacement 4 E h3 H (apr) (aqi) (br3) 4E h3H v(apq) (ari) (br3) 3 (1 + v) + 3 - 3 v2 Replacement is a very useful tool to control the simplification process. The well known identi 4 E h 3 H v ( a 13) (bpq) 4 E h 3 v ( a r i ) (br j ) (bpq) ties (see (2.10) and (2.11) on page 37): 3 - 3 v2 -3 + 3 v2 4Eh3 (aqi) (br3) (bpr) 4 E h3 H ( aqi ) (bp3) 4Eh3H(aPi) (bqi) 3 (1 + v ) 3 (1 + v) 3 (1 + v ) bpq bjP - 2 H bqj K aq\ 4 Eh3 (apr) (br3) (bql) 8 Eh3 (bp3) (bqi) 4Eh3v(apq) (br3) (br bp‘ bjp = 2 HbiJ - K d j, 3 (1 + v) 3 (1+ v) -3+ 3V2 4 E h3 H v (apq) (bij) 8E h3v(bpq) (b13)' ") (Vpq)H are applied with the following functions: 3 - 3 v2 3 - 3 v2 „ 2 E h v (a13) (Ypp) 10Eh3Kv (a13) (Ypp) 8 E h3 H v (b13) (YPP) O u t [12] - — + ------* — + ------—— ------*— + '8Eh3H(api) (aqi) 8Eh3Hv(apq) (a13) 4Eh3 (apr) (aqi) (br3) -1 + v 2 3(-l + v 2 ) 3 - 3 v 2 ^ 3 (1 + v) 3 - 3 v2 3 (1 + v) 4 E h 3 v (bp1) (bJ>3) (Y,q) 4 E h3 H v (a13) (bpq) (YM ) 4 E h 3 v (apq) (ari ) (br 3) 8 E h 3 v ( a iJ ) (bpq) -3 + 3 v 2 + 3 - 3 v 2 3 - 3 v2 + -3 + 3 v2 4 E h 3 (bp1) (bq3) (Ypq) + 8 E h 3 v (bpq) (b13) (Ypq) : 8 E h3 (aq i) (bp3) 8 E h 3 (ap3 ) (bqi) 8 E h3 v (apq) (bi3 ) \ 3 (1 + v ) + 3 - 3 v 2 ( Ppq ) + 3 (1 + v) 3 (1 + v) + -3 + 3 v2 ) 4 E h 3 v (bpq) (b13) ( Ypq) 8 E h 3 H (bp3) (ypl) 4 E h3 H (bp1) (yp3) r 2 E h 3 (ap3) (aqi) 2 E h3 v (apq) (a13) \ -3 + 3 v 2 + 3 (1 + v ) + 3 (1 + v ) k 3 (1 + v) + 3 - 3 v2 / (<9pq) 4 E h 3 (bpq) (bP3) t ^ 1) 2 E h (y13) _ 10 E h 3 K (y13) 3 (1 + v ) + 1 + v 3 (1 + v ) 4.4. Satisfaction o f the last equation o f equilibrium 57 22______Chapter 4. Constitutive relations

This is a formula for the tensor of moments. It is received by a similar procedure. In [13]:- % /.{b [ll, 12] b [u j,u l] 2 Hb [12, U j] - K a[12, u j], = 4 E h3 (y a‘j bpq ypq + ( 1-v ) b„‘ ypl) | b [ll, ui] b[uj, ul] -» 3 ( l - v 2) 2 H b[ui, u j] - K a[ui, uj] } 2 E h3 (2 H S j - V ) (v aqi y„p + (1 - v) Y>‘) 1 = 2 E h v (a13) (YpP) _ 10 E h3 K V (a13) (ypP) 8 E h3 H v (b13) (ypP) -1 + v2 + 3 (-1 + v2) + 3 - 3 v2 + 3 (1 - v2) 4 E h 3 v ( -K (a 13) + 2 H (b13)) (yqq) 4 E h3 H v (a13) (bpq) (yPq) 4E h3 (va,7p / + (l - v) p”) -3 + 3 v 2 + 3 - 3 v2 + 3 (1 - v 2) 1 ’’ 4 E h 3 (bp1) (bq3) (yP'!) 8 E h3 v (bpq) (b13) (yPq) 3 (1 + v) + 3 - 3 v2 + It should be added that the formulas (4.10) and (4.11) are valid in the geometrically nonlinear 4 E h 3 v (bpq) (b13) (ypq) _ 8 E h 3 H (bp3) (yP1) 4 E h3H (bp1) (yPJ) approach if geometrical properties of the shell are computed in the actual configuration. -3 + 3 v2 + 3 (1 + v) + 3 (1 + v) + 2 E h (yij ) _ 10 E h3 K (y13) _ 4 E h 3 (yql) (2 H (bq3) - K (<5q3)) 1 + v 3 (1 + v) 3 (1 + v) 4.4. SATISFACTION OF THE LAST EQUATION OF EQUILIBRIUM

MATHEMATICS simplification tools do not always find the simplest form. Replacement can be very useful in such cases, for example: The next step is to check if the obtained simplified results satisfy the last equation of equilib rium, which has the following form and should be satisfied as an identity. Out [13]= 2 E h ( + h2 K) (v(a13) (ypP) - (-1+v) (y13)) -1 + v2 4 E h 3 (ypq) (v (bpq) (H (a13) + b13) - ( — 1 + v) (bp1 ) (bq3 + H (<5q3) ) ) epq (Npq - b p Mqr) = 0. (4.12) 3 ( - 1 + v2) ln[14]:= The check is carried out by means of the following function, which shows that it is satis - 1 + v2 1-V2 fied. A totally antisymmetric object epq is denoted by EpsDown[ll, 12]. The function aa. (-1 + v) -» -aa (1 - v ), aa. (-1 + h2 K) -* -aa (1 - h2 K) } contains a lot of simplification tools like tensor simplification Tsimplify, canonicalization Out [14]= 2Eh (1 ~h2K) (YpP) + (1 -v ) (y13)) : Canonicalize, absorption Absorb and AbsorbKdelta and expansion Expand. More 1 - V2 over the well-known identity: 4 E h 3 (yPq) (y (bpq) (H (a13) + b13) + ( 1 - v ) (bp1) (bq3 + H ( V ) ) ) 3 ( 1 - v 2) bpqbrp = 2Hbqr- K a qr, (4.13)

4.3.4. Result of simplifications is applied with b[ 11, 12] b[13, ul] -♦ 2Hb[12, 13] -Ka[12, 13], a.

At the end the following result is obtained: In[15]:= T s i m p l i f y [ A b s o r b [ Canonicalize[ A b s o r b [ _ 2 E h (1 - h2 K) (v aij ypp + (1 - v) y'-') N ,J := AbsorbKdelta[ 1 - v 2 E x p a n d [ 4 E h3 ypq (v bpq (H a‘j + W) + (1 - v) bp‘ ( V + H

Covariant components in MathTensor™ are denoted with negative integer numbers. Thus, the condition N e g ln te g e rQ [ i] restricts the definition to covariant components.

In [2] := r i [i_] /; NeglntegerQ [i] := ri[i] =Sx[.i]r Chapter 5

DESCRIPTION OF AN ARBITRARY SHELL

W tym rozdziale przedstawiono wybrane elementy opisu geometrycznego i związków kinematycznych, oraz zależności pomiędzy przemieszczeniami i odkształceniami powłoki o dowolnym kształcie. Szcze gólną uwagę zwrócono na możliwość popełnienia błędu przy wyprowadzaniu zależności kinematycz nych w wypadku zastosowania niewłaściwych narzędzi obliczeniowych algebry komputerowej.

This chapter discusses selected elements of geometrical description, kinematic equations and strain- displacement relations in a shell of an arbitrary shape. Special emphasis is focused on a possible error in the derivation of kinematic relations due to the application of wrong computational tools of computer algebra.1.

The problems presented in the previous two sections are a general consideration. The next step Fig. 5.1. Catenoide is to carry out more detailed calculations for a concrete shell, which is performed in a new Rys. 5.1. Katenoida MATHEMATICS session.

5.1. GEOMETRICAL DESCRIPTION OF THE REFERENCE 5.2. GEOMETRICAL PROPERTIES SURFACE

The first differential form o f the reference surface is defined by the following scalar product: A surface in 3D space can be parameterized with two variables x f. This is an example of semi geodesic parameterization [25] of the catenoide shown in Figure 5.1 on the next page. aij '•= r, ■ rj. (5 .2 ) In [1] :•= r := { Cos [x [2] ] V s 02 + x [ l ] 2 , In[3] := a[i_, j_] /; NeglntegerQ [i] S&NeglntegerQ [ j] := S i n [ x [ 2 ] ] \Js02 + x [l]2, s0 ArcSinh[— }; So a[i, j] = S im plify[ri[i] ,r i[j]] The components of the covariant curvilinear basis are computed as a derivative of the vector r It is sensible to collect these coefficients in the matrix. with respect to the parameter x f. In [4] : = aLowerMatrix := aLowerMatrix = Table [a [-i, - j], {i, 2), {j, 2}] n ■= d* r. (5 .1 ) This is a definition of the determinant a of this matrix.

1 Some problems considered in this chapter were also discussed in [213, 223]. In [5] := D e t a := D e t a = S im plify[D et[Table[a[-i,-j], {i, 2), {j, 2}]]]

The third vector of the curvilinear basis can now be computed, it is normal to the mid-surface. It is obtained from the formula: 58 60 Chapter 5. Description of an arbitrary shell 5.4. Strains 61

Derivatives of the displacement and rotation vectors are objects of rank equal to one. ri x r 2 rj := — (5.3) y a w 'j : = 8^ w, ( 5 . 6 ) In [6] := ri [-3] := ri [-3] = Simplify [ rl t"1] x rl i P owe r Expand [ VD etaJ The normal vector is necessary to calculate the second and the third differential form. In this dt := d, d. (5.7) example these computations are omitted because the process is very similar to the derivation of In [12] := wi [i-]/; NeglntegerQ [i] :=w i[i] = Sim plify [ax[_i]W] the first differential form. The tensor a-,j is a metric tensor on the reference surface, so its contravariant components can In [13] := d i[i.]/; NeglntegerQ[i] := d i[i] = Sim plify [3„[-ijd] be computed from the inversion of the aLowerMatrix. The physical components of the displacement and rotation components can be computed from the following formulae: In [7] := aUpperMatrix := aUpperMatrix =

Sim plify[Inverse[aLow erM atrix]] wi ■■= w' V^i/> (5-8) Contravariant coefficients a‘J are elements of this matrix and can be computed basing on the following definition. Here the condition PosIntegerQ[i] restricts the definition to con travariant components since contravariant indices are denoted with positive integer numbers in di := d‘ (5-9) MathTensor™. This leads to the following definition: In[8]: = a[i_, j_] /; PosIntegerQ [i] SSPosIntegerQ [ j ] := K 'i[x[l], x[2]] a [i, jl = aUpperMatrix [[i, j]] In[14] :■= ww[i_] :=w w [i] = PowerExpand[Va[-i, -i]J

dj. [x [l], x[2] ] ln[15]:= dd[i_] :=dd[i] = 5.3. KINEMATIC RELATIONS PowerExpand[Va[-i, -i]J

The displacement and rotation vectors can be decomposed in the covariant basis: 5.4. STRAINS

w ■■= wk rk + w3 #*3, (5.4) The generalized formula for the first strain tensor (containing terms responsible for temperature distortions) is: d := dk rk + d3 r3. (5.5) yij ~ i (ri ■ wj + rj ■ wi) - e Oij + (nonlinear terms). (5.10) The last term in the rotation vector is negligible in linear problems. Attention is focused by limiting further considerations to a geometrically linear theory. It is denoted by: In [9] := w:=w = ww[l] r i[-l] + ww[2] ri[-2 ] + ww[3] ri[-3 ] In[16] := gamma [i_, j.] /; NeglntegerQ [i] SSNeglntegerQ [j ] :=

In[10] ;= d := d = dd [ 1 ] r i[-l] +dd[2] ri[-2 ] (*+dd[3] [ri[-3 ]]*) gamma[i, j] = It has to be emphasized that MakeSum [ ] cannot be used in those definitions because it results i ( r i [i] .w i [ j] + r i[ j] .w i [ i] ) - 6 [x [1], x[2] ] a [i, j] in an error, for example the command: The other two strain tensors of the reference surface can be computed basing on similar defini In [ll] := w := w = MakeSum [ww[ui] r i[ li] ] +ww[3] ri[-3 ] tions. will produce a wrong result. It is probably not a bug in the package but is caused by further definitions which express dis placements with their physical components. Nevertheless it is a good example of the need to be critical concerning computer-assisted results; they need careful scrutiny. 63 6.2. Internal forces

Chapter 6

FINAL RESULTS OF SYMBOLIC COMPUTATIONS

Celem tego rozdziału jest podsumowanie dotychczas wykonanych obliczeń symbolicznych. Podstawie nie zależności kinematycznych do związków odkształcenia-przemieszczenia, a tych z kolei do związ ków konstytutywnych i dalej do równań równowagi daje końcowe wyniki w formie zależności wszystkich Fig. 6.1. Cylinder wielkości w funkcji składowych przemieszczenia i obrotu. Przedstawiono jako przykład wyniki końcowe dla powłoki walcowej, również z uwagi na rozważania następnej części pracy. Ze względu na symetrię Rys. 6.1. Walec obrotową rozważanej powłoki podano również związki i równania po rozdzieleniu zmiennych. Końco wym wynikiem tej części pracy są uporządkowane równania różniczkowe pięcioparametrycznej teorii powłok wyrażone w przemieszczniach. Tym sposobem pokazano, jak można wykorzystać system alge The boundary value problems of this shell will be approximated in the next part of the contri bry komputerowej na każdym etapie obliczeń symbolicznych. bution. Results of symbolic computations for this shell ready for numerical computations are

The aim of this chapter is to summarize the symbolic computations carried out so far. The substitution presented, below. of kinematic relations into strain-displacement equations, next to constitutive relations, and further to equations of equilibrium results in final results in terms of components of displacement and rotation. 6.2. INTERNAL FORCES As an example, results concerning a cylindrical shell are presented, also for the sake of consideration in the next part of the contribution. Due to the symmetry of revolution of cylindrical shell relations and equations after the separation of variables are presented, too. The final results of this part of the The formulas for the stretching forces and moments are presented in chapter 4 on page 49. Hav contribution are adequately arranged differential equations of a five-parametric theory of shells in terms ing already computed the kinematic relations and strain tensors, internal forces can be expressed of displacements. In this way it has been shown how computer algebra can be used at each step of in terms of displacements. symbolic computations. The results of computations of the physical components of internal forces are presented in terms of the physical components of displacements in a form returned by the computer algebra system.

6.2.1. Stretching forces 6.1. CYLINDRICAL SHELL Meridian stretching force

Let us consider a cylindrical shell, see Figure 6.1 on the facing page, parameterized with the In [17]:= nrefph[-l, -1] following vector: 2Ehe[x,y] 2Eh3x[x,y] 2EhVM/3[x,y] 2 E h vivj0' 11 [x, y ] Out [17] _ l + v + 3 (-l+ v 2)sc + (-l+v2)s0 + s0 - v2 s0

r := {s0 cos (y), sa sin (y), x), (6.1) 2 E h 3 rf|1,01 [X/ y] _ 2 E h t n 1,0> [X/ y] 3 ( -1 + V2) So -1+ V2 where the variable x e (xa, xb) is measured along the cylinder meridian, variable y e (0, 2 n) is Meridian shear force measured along the cylinder parallel and s0 is the cylinder radius. In [18]:= nrefph[-l,-2] E h ivi°'11 [x, y] E h3 d2 1,0) [x, y] E h W2 "0) [x, y ]

62 64 Chapter 6. Final results o f symbolic computations 6.3. Partial differential equations 65

Parallel shear force Parallel transverse force I n [19] : = nrefph[-2, -1] In[26]:= q p h [ - 2 ] Out [19]= ? h_3 dl_ + E h ( h 2 + 3 s I) Wi°-1) [ X , y ] E hwil, 0) [x, y] „ 5Ehrf2[x,y] 5 E h w2 [x, y] SEhivj0'1* [x, y] 3 (1 + v) s 2 3 (1 + v) s3 1 + v U 6 ( 1 + v ) + 6 (1 + V) So + 6 (1 + V) So Parallel stretching force

In [20]:= nrefph[-2, -2] 6.3. PARTIAL DIFFERENTIAL EQUATIONS 2E h e[x,y] 2Eh] K[x, y] 2 Eh (h2 + 3 sg) w3[x, y] o u tr n u j- _1 + V ~ 3 ( —1 + v2) s0 + 3 ( -1 + v2 ) s3 Substituting internal forces into the equations of equilibrium, obtained in section 3.3. on 2 E h 3 d2 °'11 [x, y ] 2 E h (h 2 +3 s2) IV20' 11[x, y] 2 E h v wj1'01 [x, y ]page 47, we receive them expressed in terms of displacements: 3 (-1+v2) s2 3 (-1+ v2) s3 -l+v2~ In[27]:= e q n r e f [ 1 ] _ . , , E h 3 rfi°'2) [x, y] E h (h2 + 3 sg) w{°'2) [x, y] 6.2.2. Moments Out [27]= Pl [x, y] + 3 (1T V) s3 + ------3 (l+ v )s < ------+

Meridian torsion moment 2 E h e (1' 0) [ X/ y] 2 E h 3 K<1'°> [ X/ y] 2 E h v » > l1'° ) [x, y] -1+v + 3(-l + v2)sc + (-l+ v 2)s 0

In[21]:= mrefph[-l,-1] E h W2 lrl) [x, y] 2 E h3 rfi2'0) [x, y ] 2 E h«/i!'°] [x, y] Cut [21] E h 3 ^ 01> lx' y] Eh3^ 1'01 [x, y] | Eh3W21,01 [x , y] s0 — v s0 3(-1 + v 2 ) S o -1 +v2 3 (1 + v) So 3 (1 + v) + 3 (1 + v) So In[28]:= e q n r e f [ 2 ] Meridian bending moment „ W , » , . Pi tx ' y] 5Ehrf2[x, y] 5Ehw2 [x, y] 2 E h e 1°'1) [x, y] U s0 6 (1+v) s£ 6 (l + v) s3 + ( —l + v) s2 In [22]:- mrefpht-1, -2] 2 E h3 K(0-11 [x, y] E h (4 h2 + (17 - 5 v) sg) ^ 0,1> [x, y] Out [22]= 2E h 3etx' Vi + 2Eh3K[x, y] + 2 Eh3 v d ^ ’^ [x, y] | 3(-l+v2)s| + 6(-l+v2)s| 3 s0 - 3 v s0 3 - 3 v 3 s0 - 3 v2 s0 2 E h3 dz°'2) [x, y] 2 E h (h2 + 3 sg) W2 °'2) [x, y] E hug1-11 [x, y] 2E h 3i/|1'01 [x, y] 2 E h 3Ki|1'tl [x,y] 3 (-1 + v2) sj 3 ( - l + v2)sj> " (-1 + v) s2 3 - 3 v2 + 3 (-1 + v2) s0 Parallel bending moment Eh3i/f'D) [x, y] + ------Ehivi2'01 [x, y] 3 (1 + V) S2 So + V So

I n [23]:= mrefph[-2, -1] In[29]:= e q n r e f [ 3 ] . _____ „ , 2 E h e [x, y] 2 E h3 k [x,y ]2 E h (h2 + 3 sg) w3 [x,y ] 2 E h3 K [x, y] 2 E h 3 w3[x, y] 2 E h3 rf2°-1> [x, y] Out [29]= p3 [x, y] + — ------( —1 + v ) s0 3 (-1 + v2) s2 3 ( - 1 + v 2) s’ 3 (-1 + v) 3 (-1 + v2) s| 3 (-1 + v2) s0

E h (-4 h2 + 55 (-1 + v) sg)sg)< d2 0,1) [x, y] 2 E h 3 1V20,1) [x, y] 2E h 3v j|1,01 [x, y] 6 (-1 + V 2 ) s3 3 (-1 + v2) s| 3 (-1+ v2) Parallel torsion moment E h ( - 4 h 2 + (-1 7 + 5 v) sg) 1V20,1) [x, y] + 6 (-1 + v2) s< + In[24]:= mrefph[-2,-2] 5 E h 1V3°'2> [x, y] 5 E h rfî1,0> [x, y ] 2 E h v ivî1'01 [x, y] 5 E h w j2'01 [x, y] Eh3^°'l,[x/y] Eh3WlM I[x,y] Eh3^ 0,[x(y] 6 ( l + v)s| 6 (1 + v) (- l+ v 2)s0 6 (1+v) 0ut^ = "3Ti'“v)80- + 3 (i + v) s 2 + J 7 I 7 ------In[30]:= e q n r e f [ 4 ] 5Ehrfi [x, y] E h3 rfj°'2> [x, y] E h 3 tn°'Z> [x, y] 6.2.3. Transverse forces U 6 (1+v) + 3(l + v)s2 + 3 (1 + v) s3 2Eh3e(1'°> [x, y] + ————————— 2E h3x(1'0|[x, y] + Meridian transverse force 3s0-3vs0 3-3v

5 E h w^1'0' [x, y ] E h 3^ 141 [ x ,y ] I n [25]:= q p h [ - l j — 1 ■■■- + ————— —— — — + 6 (1 + v) 3 s0 - 3 V So Out [25,- 5 E t ^ [X' yl + ?-^-h-K,.3.1 °> [X/ y] 6 (1 + v) 6 (1 + v) 2 E h 3 di2,0) [x, y] + 2 E h 3 w{2,0) [x, y] 3 - 3 v2 + 3 (-1 + v2) So 67 66 Chapter 6. Final results of symbolic computations 6.4. Variable separation

In[31]: = e q n r e f [ 5 ] 5ЕЬ), 2 E h 3 W2 °'21 [x, у ] E h3 d i"1* [x, у] E h3 d^'°^ [x, у ] E h3 wl2,0) [x, y] /1=1 /1=1 3 (-1 + v2) 3 (-1 + v) s2 + 3 (1 + v) sQ 3 (1 + v) s2 к к It can now be stated that it is possible to proceed from very general equations to very spe q2(x, у) ■= Q2(x, n) sin (ny), p2(x, y) := ^ V2(x, ri) sin(ny), cific ones which are ready for numerical computations. The next section will deal with these /2=1 /1=1 problems. к к w2(x, у) := ^ 2 ^ 2(х, ri) sin (ny), d2(x, y) := ^ D2(x, n) sin(ny).

/1= 1 /1=1 6.4. VARIABLE SEPARATION According to this semi-analytical approach we obtain the following definitions of one dimensional functions for each parameter n of series expansion It is well known that in the case of shells of revolution all functions (stress, forces, moments and loads) can be expanded into trigonometric series with respect to the variable y measured along the parallel. Thus, it becomes possible to reduce the two-dimensional problem to k one 6.4.1. Stretching forces dimensional ones. Each considered quantity can be expanded into the sum of sine and cosine series in the general case. To focus our attention, we will assume in further considerations that Meridian normal force the task has a plane symmetry with respect to the plane defined by the meridians y = 0 and y - it. This assumption does not lose the generality of consideration. 2EG(x,n)h 2EK(x, n)h3 2E D x’(x,ri)h3 According to that the following functions can be expressed by cosine series: «■(“« -— r^— 3Fvn:-3-(i-i5)r+ 2E^Vi'(x, n)h 2E nvfW2(x,ri)h 2E v'W 3(x,ri)h 1 - v 2 + ( l - v 2) s 0 ( l - v 2)s<,

n\\{x,y) := ^ 2 Mi(*. n) cos(ny), n22(x, y) := 5 3 N-nix, ri) cos(ny), Meridian shear force n = 0 n= 0

k k , ED 2'(x,n)h3 E n W fa t ih E 'W2 (x,ri)h tttn(x, ;y) := ^ 2 Mn(x, ri) cos(ny), m2l(x, y) := ^ M 2 \(x, ri) cos (ny), n=0 /i=0 n )= - 3 ( i - ) s , - (i + y )i. + TTT (6-5) k k Parallel shear force <7i(*. y) ■- Y 2 Q\(x, n) cos(n y), Pi (x, y) :=

Meridian torsion moment 2 E n G(x, ri)h 2 E n 7C(x, ri) h3 C2(n) ~ (1 - v ) s 02 ~ 3 (1 - v2) s03 +

M „ (,.») = ? " ^ “>■h- - E ">">*’ (69, E (4 h2 n 2 + 5 (1 - v) So2) T)2(x, ri) h 3(1+»)*. 3 (1 + y) 3(1 + »)*' 6 ( 1 - v2) s«4 Parallel torsion moment E P 2"(x, ri) h3 _ E n 'Wi'jx, ri) h (6.15) 3 (1 + v) s02 (1 - v) s02 M EnDi(x, n) h3 E D 2'(x, n)h3 En W ,(x, ri)h3 (6.10) E ^4 h2 n 2 + (12 n 2 - 5 (1 + v)) s 2) fW2(x, ri)h E

6 ( 1 - V2) S05 So A t f c „) := - 3 (1 - v) 3 (l - v2) 2 E &(x, n) h 2 E 'Kix, n) h3 2E h3n D 2(x, n) 2E h 3 n'W 1 (x,n) 2E h3 'W3(x,n) (6U ) (l-»)*'+3(l-»ï)l?+ 3 ( 1 - v2) s0 3 (1 - v2) j02 + 3 (1 - v2) j02 ‘ 5 E D\(x,ri)h Em (4h2 + 5 (1 - v) sa2) P 2(x, ri) h | + 6(1 + v) + 6 (l - v2) s03 6.4.3. Transverse forces 2 E v 'Wipe, ri) h En ((5 v - 17) s„2 - 4 h2) W 2(x, ri) h | (6.16) Meridian transverse force + (1 - v2) s0 6 ( 1 - v2) s04 E ^4 h2 + (12 + 5 n2 (1 - v)) s02) 'WiCx, ri) h . 5EhD i(x,n) 5E h'W ^x.ri) Qi(x, ri) := —— -----— (6.12) 6 (l - v2) i„4 6 (1 + v) 6 (1 + v) v ’ + 5E W M k Parallel transverse force 6 (1 + v)

5EhD2(x,n) _ 5Eh "W2(x, ri) 5E hn'W ^x.n) 2 6(l + v) 6(1 + v) S0 6 (l+ v )j0 2 E &{x,ri)h3 2E'K'(x,ri)h3 E ( 2 h 2 n 2 + 5 y 2) D i ( x , n ) h | e4(n) := 3 (l-v )5 „ + 3 (1 - v) 6 (1 + V) So2 2 E D i "(x, ri) h3 E n D2 (x, ri) h3 _ E n2 'W^x, ri) h3 ___ 6.4.4. Ordinary differential equations (6.17) + 3 (1 - v2) + 3 (1 - v) s0 3 (1 + v) s03 Finally, according to series expansion we receive n systems of ordinary differential equations. 2 E “W\ "(x, ri) h3 5 E tW3\x, ri)h _ n Each system consists of 5 equations. 3 ( 1 - v2) So 6 (1 +v) 70 Chapter 6. Final results o f symbolic computations

_ 2E n fK(x,n)h i E n £ > ,'(* , n) h3 " " 3(1 - v) j 02 - -3(V-v)^ + E (4 h2 n2 + 5 (1 - v) s„2) D 2(x, n)h E D 2"(x, n) h3 6 (1 - v2) s„3 + 3 (1 + v )s0 E (4 h2 n2 + 5 (1 - v ) s„2) n)h E ' W / U n) h3 ( 6 ' 18) 6 ( 1 - V2) 3 (1 + V) So2 E n (4 h2 n2 + 5 (1 - v) s02) ri) h 4------= 0 6 (l — V2) s04

It is easy to find that these equations are of order 2 with respect to all unknowns: 'VV'i, 'Wj, 'Wi, T)\ and D 2, so the differential operator of the problem is of the order equal to 10. In the further consideration we will deal with problems defined for a concrete value of n. Thus, Part II for the sake of shortness, all functions will be denoted, neglecting the parameter n, for example, 'Wiix, n) will be written rW i(x), and so on. BOUNDARY-VALUE PROBLEMS Chapter 7

DESCRIPTION OF THE REFINED LEAST SQUARES METHOD

W tym miejscu przedstawiono zasadnicze podstawy teoretyczne Rozszerzonej Metody Najmniejszych Kwadratów dla zadań jednowymiarowych oraz jej elementy dla zadań dwuwymiarowych. Rozszerzenie metody polega na uwzględnieniu warunków brzegowych w funkcjonale. Podano podstawowe cechy metody.

A crucial theoretical basis of the Refined Least Squares Method for one-dimensional tasks and its elements to two-dimensional problems is presented. The refinement of the method consists in taking into account boundary conditions in the functional. Basic features of the method are discussed.

7.1. CLASSICAL APPROACH

i The Least Squares Method is a well-known (7. Glazunow [51] and D. Zwilinger [236]) quasi- variational way of finding an approximate solution of a boundary value problem. The classical approach consists in minimizing the functional (7.1) based on algebraic, differential or integral equations, or on a system of equations e/ — 0 in the domain H with a set of independent functions which must satisfy the boundary conditions.

( W M O (7.1)

where an are weights or weighting functions. They play an important formal role. If we consider a physical problem which is described by a system of equations, each equation can have a different physical dimension. To obtain physical consistency of the functional weights we must set up formally appropriate physical dimensions.

'The crucial elements of the method will be presented for one- and two-dimensional problems. The algorithm can be easily expanded for multidimensional problems.

I 74 C h a p ter 7. Description o f the Refined Least Squares Method 7.2. One-dimensional problems 75

The main advantage of the method is its simplicity. If we know the equations describing the be improved by taking into account more independent u, (*) functions. Setting the indices in this considered problem, building the functional is straightforward. The disadvantage of the clas manner we avoid construction of the matrix from the scratch in the next step of approximation. sical approach is that it is not very easy to find a set of independent functions satisfying the It can be explained by the following example. If the considered problem is described by 2 boundary conditions, especially for multidimensional problems with discontinuous boundary functions yitO and y2(x), we can predict the approximate solution by means of the following conditions. This disadvantage does not occur in the refined approach. linear combination of 3 functions (p '■= 2 and n := 2), first:

7.2. ONE-DIMENSIONAL PROBLEMS ?!(*) := C0 u0(x) + C2 Ui(x) + C4 u2(x)

7.2.1. Functional y2(x) := C, u0(x) + Cj u\{x) + C5 u2(x)

Let us consider a one-dimensional problem described in the interval x e (xa,xb) with a system The indexing procedure enables us to append each function with new terms. If p ■■= 3, we of n equations en = 0 (differential, algebraic, integral or their combination) and m boundary receive: bk = 0 conditions. The following functional can be constructed for them: ;yi(x) = C0 u0(x) + C2 Mi W + C4 u2(x) + C6 u3(x)

T (a(e,)2 j dx + ^2(J3kbkf, (7.2) ■m / *=i y2(x) := Ci u0(x) + C3 ui(x) + C5 u2(x) + C7 u3(x) where an and pk are weights or weighting functions. As already mentioned, the weights must As we see, the approximating combinations are built-up, not rebuilt from the scratch. In result have physical dimensions to satisfy formally the physical consistency. For that reason they must also the matrix of the system of linear equations can be built up. It speeds up the computations be present in the formula (7.2). since already computed terms can be saved and used again. The implementation of this idea is From the mathematical point of view, weights play an important role in the method, since they presented in the next chapter, see section 8.2. on page 86. can be used to moderate the approximation path. This feature will be discussed further on. According to the notation (7.3) on the facing page, any weighted linear differential equation The crucial difference from the classical approach is that the functional is supplemented with ak ek can be expressed in the following form: terms responsible for the boundary conditions. Therefore, the approximating functions do not have to satisfy the boundary or initial conditions. They must be linearly independent only. n(p+l)-l ockek{x) = Fk(x)+ Y2 Q Kik(.x), (7.4) The Refined Least Squares Method consists in minimizing this functional. If the solution is i=0 exact, the value of the functional is equal to zero, otherwise it is positive. For example, if we consider one of two differential equation e\ = 0 of two functions (n := 2):

7.2.2. Application of the Ritz method e, := y"(x) + y'2(x) + f t(x)

Minimization is accomplished by means of the Ritz method. According to it, the approximation and predict the solution as combinations of 3 independent functions (p := 2): of any n unknown function yk (x) in the considered equations can be predicted in form of a linear combination of p independent functions w, (*) (the best stability is obtained with monic Chebyshev Tn polynomials). yiW := C0 u0(x) + C2 «iW + C4 u2(x)

p yk(x) -= ^ ' C(„j+k-i) Uj jx). (7.3) y2(x) := Ci u0(x) + C3 u,(x) + C5 u2(x) /=1 we obtain: Special indexing of unknown coefficients Q is used here to permit the building up of a matrix of the system of linear algebraic equations. It is well known that the quality of approximation can e\ f\(x) + Co uo"(x) + Ci uq’C*) + C2 \x"(x) + C3 Ui'(jc) + C4 u2”(x) + C5 U2 (x) 77 76 Chapter 7. Description o f the Refined Least Squares Method 7.3. Nonlinear tasks

which means that in the considered case Ft(x) := /,(*), K0] := u0"(jc), Kn := u0'(jr), This equation can be rearranged in the following way: #21 := U| "(jc), #3| := Ui'(x), # 4i := u2"W, #51 := u2'W- Simultaneously we can check that the summation parameter in (7.4) on the page before takes the values i := 0 ,1,..., n(p + 1) - 1 (n{p + 1) - 1 = 2(2 + 1) — 1 = 5). The launching value of i is set to 0 since Chebyshev polyno n (p + l)-l ( * r n m \ xj t n m mials Ti(x) used in the implemented algorithm (see next chapter, section 8.1 .1 . on page 80) are £ c ‘ / Y , K> ix) Ku (x) dx + J 2 Lik Uk + / Kj‘ F‘(x) dx+J 2 Lik Gk = °- (7-9) numbered starting with i = 0 . ,=o \ l i m J i /= i t-i Similar considerations concern boundary conditions pk bk'- Differentiating (7.6) on the preceding page for each unknown decision variable C„ a system of n(p + 1) linear algebraic equations is obtained: n(p+l)-l Pk bk = Gk + 5 3 C, Lik. (7.5) /=0 AijCi= -Bj. (7.10)

Fast procedures of extracting the coefficients Fk from (7.5) and #,* from (7.4) on the preceding The coefficients of the matrix A,,- of the system (7.10) can be computed according to equation page are presented in the next chapter, see section 8.1 .3 . on page 82. (7.9) with the following formula:

7.2.3. System of linear algebraic equations xi n m Aij I 5 3 Ku № Kjt M dx + Y ^L * Ljk- (7-11) " /=1 k=\ Minimization of any functional by means of the Ritz method is equivalent to the condition: The matrix is symmetrical. It results from the multiplication of the coefficients Ku (x) #;, (x) d T n and Lik Ljk in its definition. If these coefficients are real numbers or functions, matrix Atj is also t e r 0' a 6 ) a positive definite one. Taking into account the definition of the functional (7.2) on page 74 and — 2 /(Z) According to equation (7.9) the coefficients of free vectors Bj of the system (7.10) are computed we receive from (7.6): from:

n m 2 Bj := / 5 3 K* « F‘dx + 5 3 Gk- (7‘12) / ( Ê ^ + 2 Ê ^ A A = ». (7.7) “ /=1 k=1 ■to xa Since differentiation of both sides of (7.4) on the preceding page yields: Computations of integrals in (7.12) and especially (7.11) belongs to the mosttime consuming element of the method. A special procedure for subdividing the task of integration is proposed in section 8.1.7. on page 84. A function designed by the Author for quick integration is proposed £(«/«/) ^ in section 8.1.8. on page 84. ~ d c ~ ~ Kjl’ and the similar operation on (7.5) results in: 7.3. NONLINEAR TASKS dpk bk _ T dCj ~ Jk’ It is also possible to develop a procedure based on the Least Squares approach for nonlinear problems. In the such case the approach is called Generalized Least Squares Method, compare we receive from (7.7) the following equation (dividing its both sides by 2): J. Glazunow [51]. This approach requires well known techniques based on the recursive solution of a linear prob lem and will not be discussed here, as it will not introduce anything new. Xb / n / n (p + 1)—1 \ \ m / n (p + 1 )-1 \ / ^ 5 3 Kj, (x) (x) + C< Ku W j J dx + Ljk + Y , Q J = 0. (7.8) 78 Chapter 7. Description o f the Refined Least Squares Method 7.5. Features of the Refined Least Squares Method 79

1A . MULTIDIMENSIONAL PROBLEMS important is that the results of approximation are also functions. Thus, there is no need of any interpolation or extrapolation. This is yet another element of a very straightforward For two-dimensional problems, defined in the domain Cl with two types of boundary conditions: implementation. It simplifies the so called “postprocessing”. defined along the boundaries T: bk and in separate points c„ the functional has the following form: • Portability of results The obtained functions may be easily differentiated, integrated and plotted within the computer algebra system. T : = J (E e^j dn+J b^2dr + £ (* cj) 2• c u 3 ) • Straightforward estimation of the error of approximation error. There are local and In the case of more dimensions of a considered hyperspace the functional will be a bit more global ways to estimate the error of approximation. The obtained functions substituted complicated. into the equations allow to estimate the quality of their approximation. The square root of the considered functional is a generalization of the L2-norm. The value of this square Setting up a system of algebraic equations for multidimensional problems is similar and will root close to zero informs about a good global convergence. There are two other equiv not be presented here. alent criteria for error analysis. These problems are discussed in 10.3.2. on page 130 and 11.6. on page 157 Error analysis has been also discussed in 12.5. on page 166. This section shows that the method enables us to detect a “false” convergence. 7.5. FEATURES OF THE REFINED LEAST SQUARES METHOD • Boundary-condition phenomenon. Computational experiments with weights in the According to achieved experience we can find many advantages and some disadvantages of this functional have allowed to discover a very interesting feature of the refined method. Un method. der certain circumstances, some or all boundary conditions may be neglected and the approximation will still satisfy differential equations and some boundary conditions. The result is close to the exact solution on most of the domain excluding the boundary layer. 7.5.1. Advantages of the method This has made it possible to develop a two step approach to boundary-value problems. This phenomenon, called boundary-condition phenomenon, is illustrated by examples in Among the advantages we can point out: the chapters 9 on page 88 and 10 on page 119. A physical and mathematical explanations of the phenomenon can be found, see chapter 11 on page 146. • Generality. The method is general. It does not matter if the problem is described by alge braic, differential or integral equations, the algorithm is always the same. The boundary conditions are approximated together with the equations and in the case of multidimen 7.5.2. Disadvantages of the method sional problems they do not have to be continuous, and as has already been shown in [192, 193, 194, 196, 199], it is easy to approximate problems with steep boundary or Despite advantages the method has also some disadvantages, mainly caused by computer hard initial conditions. ware limitations.

• Simplicity. The method is very simple. Implementation within the MATHEMATICS system • Duration of computations Analytical integration is required to avoid problems with is straightforward, see chapter 8 on page 80. Taking into account the boundary conditions polynomial instability. These computations require a lot of time, especially for high- in the functional makes the algorithm even simpler. degree polynomials. A remedy for this is the definition of my own integration functions, see 8.1.8. on page 84 and a special indexing procedure that allows to use already com • Flexibility. One of the most important features of the method are weights. They make it puted coefficients of the algebraic system, see 8.2. on page 86. possible to moderate the approximation path. The increase of one of the weights enforces quicker approximation of this equation or condition on the cost of others. For example, if • Limitations of the analytical solution of algebraic equations. Despite of analyti we set very large weight coefficients in boundary conditions we obtain an approximation cal computations of these integrals, analytical solution of systems of linear algebraic asymptotically convergent with the result of the classical approach. Experience shows equations is limited to relatively small matrices. It is somehow connected with the previ that this is not reasonable and the weights should be generally set to allow a uniform ous feature but also with the amount of computer memory. Nevertheless, the solution of convergence of equations and boundary conditions. Setting up weights seems to be a the systems of linear equations may be computed with an arbitrary numerical precision separate scientific problem. and next “rationalized”, see 10.5.2. on page 140. • The method is analytical. The method does not generally warrant an exact solution, but only an analytical, global approximation of the functions satisfying the equations. Most The disadvantages will be soon overcome by the technological progress in computer hardware and software. 8.1. Translation into the MATHEMATICS language 81

■ In general, when we cannot assume any symmetry properties of the function 'W i (x) w e define them by:

In [1] : = W d x . ] := PolyDegree 2 (X - bx) Chapter 8 c [ 5 i ] M o n ic C h e b y s h e v T [ i, — — — ------l ] ; 1=0

In [2]:= 'W 'j t x . ] := PolyDegree 2 (x - bx) ,V ELEMENTS OF THE IMPLEMENTATION OF THE yc [5 i + 1] MonicChebyshevT[i, — -—— 1J ; METHOD 1=0 In [31:= W 3 [ x J :=

PolyDegree 2 ( X - b x ) , , y c [5 i + 2] MonicChebyshevT[i, —— —— 1J; 1=0 W tym rozdziale przedstawiono elementy wdrożenia metody w systemie M athem atics do zadania brze gowego powłok walcowych rozpatrywanych w dalszych rozdziałach. Do aproksymacji zastosowano In [4]:= ® i [x_] := moniczne wielomiany Czebyszewa, co pozwoliło na znaczne zwiększenie stopnia wielomianu aprosy- PolyDegree 2 (x - bx) , y c [5 i + 3] MonicChebyshevT[i, — —— 1J ; mującego. Całki potrzebne do wyznaczenia współczynników układu równań obliczane są analitycznie. 1=0 Oba te elementy pozwalają na uniknięcie kłopotów ze stabilnością wielomianów. Pokazano autorskie In [5] := Z>2 [X - ] := sposoby przyśpieszania analitycznego obliczania całek oraz układu równań algebraicznych przy po PolyDegree 2 (x - bx) . większaniu stopnia wielomianu aproksymującego. y c [5 i + 4] MonicChebyshevT [i, ------— ------1 J I 1=0 The elements of the implementation of the method within the M athematica r system are presented in this chapter for the boundary-value problem of cylindrical shells, which are considered in the next two chap w h ere ax stands for xa, bx fo r xb, c [k] fo r Ck, PolyDegree is an integer variable defining ters. Monic Chebyshev polynomials are applied for approximation, which makes it possible to enlarge a degree of an approximating polynomial, and monic Chebyshev polynomials are Chebyshev the degree of approximating polynomials considerably. Integrals needed to evaluate the coefficients polynomials Tn(x) divided by 2'1-1: of the system of equations are computed analytically. Both these elements make it possible to avoid ChebyshevT[n, x] In [6] := MonicChebyshevT [n_, x_] := ------— j------difficulties with polynomial stability. Some author tips for speeding up analytical computations of inte grals are shown, as well as a system of algebraic equations in the case of enlarging the degree of an Monic Chebyshev polynomials provide better polynomial stability than Tn(x) functions, see approximating polynomial. remark on page 135.

■ As the Chebyshev polynomials Tn(x) are symmetrical when n is even and antisymmetrical if n is odd, so if, for example, the function 'W i (x) is antisymmetrical with regard to the plane To focus our attention, let us analyze the implementation of the method for a concrete problem. x — the system can be informed about it by the following definition: To do that let us consider the equations and other relations, concerning a cylindrical shell, de veloped in the part of the contribution devoted to symbolic problems, see chapter 6 on page 62. In [7] : = TVi[x_] := We will discuss only the crucial points of the implementation. PolyDegree 2 ( x - b x ) 'y' c [5 i] Moni cCheby shevT [2 i + 1, —— —— l] I 1=0 8.1. TRANSLATION INTO THE MATHEMATICA' LANGUAGE ■ Similarly, if the function 'W 2 (x) is symmetrical with regard to this plane, we can inform the system by:

8.1.1. Polynomial approximation In [81:= -W 2 [ x - ] := PolyDegree 2 ( X - bx) , y c [ 5 i + 1 ] M o n ic C h e b y s h e v T [ 2 i, — — ------1 J ; Each one of the unknown functions: 'Wi (x), , (x) and D2 (x) of the system i=0 of differential equations (6.14), (6.15), (6.16), (6.17) and (6.18) on pages 69-70 can be approx imated with a linear combination of monic Chebyshev polynomials, R. Burden et al. [32]. The degree of the approximating polynomial is equal to 2 P o ly D e g re e in case of symmetrical function and 2 P o ly D e g re e + 1 in case of antisymmetrical function, of course.

It is not difficult to notice that the real domain (xa, xb) is transformed to the interval (-1,1). 80 8.1. Translation into the MATHEMATICS language 83 82 Chapter 8. Elements of the implementation o f the method

8.1.4. Weighted boundary conditions 8.1.2. Weighted differential equations Similarly each weighted boundary condition for the task is saved into the variable. Let us ■ Each differential equation is multiplied by a weight and saved into the variable: consider the case when the edge xb is fixed and the edge xa is free.

In[9]:= D ifferentialEquation[1] := ■ Then on the fixed edge we provide information that displacement and the rotation component D ifferentialEquation[1] =- should be equal to zero: h ;

In[10] : = D ifferentialEquation[2] := In [16] : = BoundaryCondition [ 1 ] := r 2 BoundaryCondition[1] = Di [b x ]E ; D ifferentialEquation[2] = In [17]:= BoundaryCondition [2] := In [ll] := D ifferentialEquation [3] := BoundaryCondition [2] = Z>2 [ b x ] E ; D ifferentialEquation[3] = In [18]:= BoundaryCondition [3] :=

In[12]:= D ifferentialEquation[4] := BoundaryCondition [3] = 'U 'i[bx]E ;

D ifferentialEquation[4] = In [19]:= B o u n d a r y C o n d i t i o n [ 4 ] := BoundaryCondition[4] = IV 2 [ b x ] E ; In [13] : = D ifferentialEquation [5] := r 5 In[20]:= BoundaryCondition[5] := D ifferentialEquation[5] = h BoundaryCondition[5] = TV3 [bx]E ; ■ For the free edge we assume that forces and moments should be equal to zero: 8.1.3. Extraction of terms from equations In[21] := BoundaryCondition[6] := N n [ a x ] According to this approximation the weighted differential equation akek and boundary condition B o u n d a r y C o n d it io n [ 6 ] = ------— — ; pn bk can be rewritten in the following form, see section 7.2.2. on page 74: In [22]:= BoundaryCondition [7] := N12 [ a x ] 5 (p + \)-\ BoundaryCondition[7] = ; ak ek (x) = Fk (x) + QK-kM, (8.1) h (=0 In [23]:= B o u n d a r y C o n d i t i o n [ 8 ] := M u [ a x ] BoundaryCondition[8] = ------5 (p + l)-l h

Pk bk = G k + 5 3 Lik, (8.2) In [24] := B o u n d a r y C o n d it io n [ 9 ] := ;=o M12 [ a x ] BoundaryCondition[9] = ------—;------where p is an assumed degree of the polynomial. hJ In [25] := B o u n d a r y C o n d it io n [ 1 0 ] := ■ Free terms Fk are extracted from (8.1) by means of the following functions: Û 1 [ a x ] BoundaryCondition[10] h 3 In[14] := D if f EqnFreeTerm [k_] := D if f EqnFreeTerm [k] = Other combinations of boundary conditions are also possible, of course. E x p a n d [ - D i f f e r e n t i a l E q u a t i o n [ k ] / . c [_ ] -» 0 ] ; The functions Kik(x) can also be found using a very similar procedure. Standard MATHEMATICS functions might be used but this one is faster. 8.1.5. Guessing weights

In[15] := D iffEqnC oefficient [i_, k_] := D iffEqnC oefficient [i, k] = It is worth mentioning here how the weights have been guessed. The word ’’guess” is appro Expand[DifferentialEquation[k]+ priate here because there is no rule for it. The weights have been chosen by the following D iffEqnFreeTerm [k]/.c[i] -*l/.c [_ ] -»0] speculation. Displacements in differential equations are multiplied by E ft or Eh3, to equal ize “minimization chances”. Boundary conditions in displacements were multiplied by E and 84 Chapter 8. Elements o f the implementation of the method 8.1. Translation into the MATHEMATICS language 85

forces and differential equations were divided by h or h3, respectively. Guessing weights is ■ Function integl is defined for computing a definite integral of polynomials. There are pre generally not a very easy task and should be researched more deeply in future. defined formulas for the integration of monomials and dealing with sums, which accelerates integration of the polynomials.

8.1.6. Extraction of terms from boundary conditions In [29] : = in teg l [z-Plus] := xxintegl/@ z; In [30]:= in te g l [z_] := xxin teg l [ z ] ;

■ The following commands are applied to extract Lik (jc) and G* from the boundary conditions In[31 ] := x x i n t e g l [z _ ] := i n t e g x [z, a x , b x ] ; (8.2) on page 82. a ( - a x 1+n + b x 1+n) In[32]:= integx[a_x“- , ax_, bx.] := ------/; FreeQ[a, x] ; In[26]: = BoundCondFreeTerm[i_] := BoundCondFreeTerm[i] = 1 + n Expand[-BoundaryCondition[i]/ . c[_] -* 0] ; b x 1* “ - a x 1+n In [33] := integx [x , ax., bx_] := ------; 1 + n In [27] : = BoundCondCoef f ic ie n t [k_, i_] := In[34]:= integx[a_, ax., bx.] := a (bx - ax)/; FreeQ [a, x] ; BoundCondCoefficient[k, i] = bx Expand[BoundaryCondition[i]+ In [35] := integx[z_, sue., bx.] := J z d x ; BoundCondFreeTerm [i]/.c[k] -*l/.c [_ ] -»0] ax

8.1.7. Coefficients of the system matrix 8.1.9. The matrix of the system of linear algebraic equations As x e (xa, xb), coefficients of the symmetric matrix can be computed by means of the following formula, see section 7.2.3. on page 76: ■ The matrix of the system of linear algebraic equations consists of the matrix coefficients, of course. The matrix is symmetrical so only the lower triangle should be saved.

Xb / 5 \ 10 In [36] : = System M atrix := System M atrix = K *(x) Kjk (X) U + £ U k L jk . (8.3) / Table[Table[ r \*=i / *=i M atrixC oefficient[j, i] , o, j} ] , { j, 0, NumberOfEqns } ] ■ This is done with: ■ where, for our task: In[28]: = M atrixC oefficient [i_, j.] := M atrixC oefficient[i, j] = M atrixC oefficient[j, i] = In [37] := NumberOfEqns = 5 (PolyDegree + 1) - 1 Expand [in te g l [Expand [

5 y^DiffEqnCoef fic ie n t [i, k] D if fEqnCoeff icien t [j, k]]] + 8.1.10. Vector of free elements k=l E x p a n d [ Elements of the free vector can be obtained, see section 7.2.3. on page 76, from: 10 yBoundCondCoefficient[i, k] k.l BoundCondCoef fic ie n t [j, k]]j

8.1.8. Function of integration

MATHEMATICS has got very sophisticated built-in procedures of integration within the function I n te g r a te . Due to its complexity it does not always make computations very fast. It is reasonable to define an own function for concrete problems. 86 Chapter 8. Elements of the implementation o f the method 8.2. Some computational aspects o f solving the system o f linear algebraic equations 87 ■ This is achieved with: System matrix In[38]: = FreeVecCoefficient [i_] :=

FreeVecCoefficient[i] = Expand[integl [Expand[

5 y D iffEqnC oefficient [i, k] D if fEqnFreeTerm [k] ]] k-l QJ + E x p a n d [ > 10 BoundCondCoefficient[i, k] k = l BoundCondFreeTerm[k ] ] ] 29 30 31 32 33 34 35 36 ■ These coefficients are used to build a vector of free terms. 37 38 39 40 41 42 43 44 45

In [39] : = FreeVector := FreeVector = Fig. 8.1. System of linear algebraic equations in the Cholesky-Banachiewicz method (detailed descrip

Table[FreeVecCoefficient[i], tion in the text) 8.1. { i, 0, NumberOfEqns}] Rys. Układ równań algebraicznych liniowych dla metody Cholesky’ego-Banachiewicza (szczegóło wy opis w tekście)

8.2. SOME COMPUTATIONAL ASPECTS OF SOLVING THE Due to matrix symmetry we can save some computer memory asking the computer to remember SYSTEM OF LINEAR ALGEBRAIC EQUATIONS only the lower triangle of the matrix, see Figure 8.1. The Cholesky-Banachiewicz procedure deals only with terms on and below the matrix diagonal. As the system matrix Aiy- of the Refined Least Squares Method is symmetrical and positive Approximation may be improved by adding a term to the approximating functions, elements definite the Cholesky-Banachiewicz method of solving the linear system of equations was im in white boxes in Figure 8.1. It results in an enlargement of the matrix but already computed plemented. This was done also for many other reasons, which will be discussed below. coefficients do not have to be evaluated once again, elements in hatched boxes in Figure 8.1. This is the only part of the procedure which must sometimes be carried out numerically. Nev The Cholesky-Banachiewicz algorithm is stable, it does not require pivoting during factoriza ertheless, the results of the approximation are functions. tion of the matrix. The factorization is performed row by row. Therefore, if we have already factorized the matrix in the previous step, we can append it with new terms and factorize only ■ The MATHEMATICS code of the procedure is presented. The first argument of the procedure those. All this saves computation time. is a system matrix, the second a free vector and the third a required starting precision. The If we deal with multi-dimensional tasks, we have to implement a special indexing procedure. It output of the procedure is a solution of the system of linear algebraic equations. is relatively simple and will not be discussed here. I n [40] := Cholesky [aaJjist, bbJiist, prec_] := M o d u le [ { a = a a , b = b b , i , j , n = L e n g t h [ b b ] } , ■ The solution of the Cholesky-Banachiewicz procedure is saved into the function whose ar D o[{Print [i] , gument is the required working precision.

In [41]:= s o l u t i o n l [n _ ] := a [ [ i , i l = E x p a n d [ N[a Hi, i j , prec] - ^ a [[i, k l2], solutionl[n] = Cholesky[System Matrix, FreeVector, n] b W = Expand^111 ~E ^ aŒi- M hlH Ji 1 a Hi, ij J' ■ Next the solution is transformed into a substitution list.

Do [a Œ j, IJ = Expandf— *11 ~ k]1 a 1 L aŒi, il J' In[42]:= s o l u t i o n [ n _ ] := { j, i + 1, n }]}, {i, n}]; solution[n] = T a b l e [ c [ i ] -» s o l u t i o n l [ n ] H i + 1 1 , Do[{Print[i],b Œil = Expand[b gl11 l]1 b ŒjI1l) L a Hi, il J ‘ { i, 0, NumberOfEqns) ] {i, n, 1, -1 }]; {a, b}] It is worth mentioning that the procedure is not limited to positive definite matrices. This was already discussed in [210, 211]. 9.1. Description o f the problem 89

Chapter 9

CHIMNEY EXPOSED TO AN ANTISYMMETRICAL LOAD

W tym rozdziale na przykładzie komina obciążonego antysymetrycznym parciem wiatru, przedstaw iono podstawy zastosowania podejścia dwuetapowego. W pierwszym etapie rozwiązano zadanie z pominięciem sześciu warunków brzegowych. Otrzymane (ścisłe) rozwiązanie, nazwane bazowym, jest inżyniersko poprawne na większości obszaru z wyjątkiem warstw brzegowych. W drugim etapie po prawiono rozwiązanie bazowe w obrębie warstwy brzegowej biorąc pod uwagę wszystkie warunki brze gowe.

This chapter presents the basis of a two-step approach on the example of a chimney loaded with antisymmetrical wind pressure. The task has been solved with negligence of six boundary conditions. The received (exact) solution is feasible from the engineering point of view in most of the domain, excluding boundary layers. In the second step the base solution was refined within the boundary layer taking into account all boundary conditions.

Fig. 9.1. Chimney, dimensions in [m]

9.1. DESCRIPTION OF THE PROBLEM Rys. 9.1. Komin, wymiary [m]

9.1.1. Numerical data of the problem 9.1.2. Differential equations

Let us consider a cylindrical cantilever shell, made of steel (E := 2 • 108, v := ^), length / := 40 The problem is described by a set of 5 differential equations (6.14), (6.15), (6.16), (6.17) and m, thickness 2 h := 20 mm and cylinder radius s0 := 2 m. Its undeformed reference surface is (6.18) on pages 69-70, with the parameter n := 1. presented in Figure 9.1 on the facing page. It is exposed to the load p3 := P3 cosfy), P3 := 1 ™ normal to the cylinder mid-surface, p\ = 0, p2 - 0. 9.1.3. Engineering interpretation The variable y e < 0, lit) is measured along the parallel of the cylinder, and x e {xu x2) along the meridian. The cylinder is fixed on the edge xt = -20 m and it is free (not supported) on the From the engineering point of view the problem can be considered as a static scheme of a steel edge x2 = 20 m. chimney loaded with an anti-symmetric component of the wind. The load is symmetrical with regard to the vertical plane parallel to the wind direction and antisymmetrical with regard to the plane perpendicular to the wind direction, see Figure 9.2 on the next page.

88 9.3. Two-step approach 91 90 Chapter 9. Chimney exposed to an antisymmetrical load

and two components of stretching forces on the free edge,

bz — (.Xb) 0, ^ 2) bn := N 12 (xb) = 0.

This group of 4 boundary conditions will be called in our consideration boundary conditions of the first type. The other 6 boundary conditions can be satisfied applying the bending theory in the limited domain called boundary layer. These are rotations and normal displacement on the fixed edge:

b5 ■■= D\ (xa) ~ 0, b6 :=D2 (xa) = 0, (9.3)

h := 'Ws (Xa) = o,

and bending moment, torsion moment and transverse force on the free edge:

Fig. 9.2. Distribution of the load over the parallel cross-section bs := M,2 (xb) = 0, Rys. 9.2. Rozkład obciążenia w przekroju równoleżnikowym b9 ~ M n (xb) = 0, (9.4) bio := <2i t e ) = 0- 9.2. ONE-STEP APPROACH This group of 6 boundary conditions will be called in our consideration boundary conditions of This problem is described by the 10"1 order differential operator so it needs 10 boundary con the second type. ditions. A boundary layer phenomenon occurs in the solution of the problem. In the boundary layer the functions are highly oscillating. The task is unstable in the Lyapunov sense if we try to 9.3.2. Computational experiment - basis of the two-step approach approximate it in one step, especially in the case of long thin cylindrical shells. As such a prob lem is also discussed in the next chapter 10 on page 119, the analysis of a one-step approach is One of the most important features of the Refined Least Squares Method are the weights, by omitted in this chapter. which a better or worse satisfaction of a selected equation or boundary condition can be en forced. The idea of the base solution presented below has been developed by experimenting 9.3. TWO-STEP APPROACH with weights. Knowing the boundary layer phenomenon connected with the satisfaction of the boundary con ditions of the second type I have tried to apply smaller and smaller numbers for weights standing 9.3.1. Grouping of the boundary conditions by them. Frankly speaking it did not help much. Full of desperation I have done what seems not to be very wise: I have put zeros! It has been an equivalent of neglecting the conditions of We can distinguish two groups of conditions. The consideration of grouping is based here on the the second type. In some numerical methods it results in lack of some equations or a singular famous two-step engineering approach - membrane approximation. In the membrane approach matrix. It has been a great surprise that a nonsingular system has been received. Moreover the we neglect the bending of the shell, assuming that only stretching forces do occur. According to obtained approximation is as stable as the membrane approach but it is much better. First of that, we shall solve the system of differential equations with an operator of the order equal to 4. all, we do not have to assume the negligence of moments. It makes it possible to approximate a Hence, only four boundary conditions may be satisfied. The boundary conditions in membrane wider range of tasks. The next one is a better convergence than in the one-step approach. approximation are: two components of displacements on the fixed edge, It is an unexpected situation but it can be interpreted physically, see 11.1. on page 147. It has been found that it is good enough to apply polynomials of a much lower degree than in the one- bx ■= *Wx (xa) = 0, step approach to obtain quite good results, as the base approximation is free of boundary layer (9.1) b2 ■■=W2 (xa) = 0, 93 92 Chapter 9. Chimney exposed to an antisymmetrical load 9.4. Step one - base solution phenomenon. This discovery becomes the basis of the development of the two-step approach. ■ We are taking into account the whole domain, so that the parameters describing the position It is somehow similar to the two-step approach based on membrane approximation but free of of the boundaries take the values. its disadvantages and therefore more general. In [11]:= a x = 2 0 ; The first step of the Refined Least Squares Method consists in computing the base solution sat isfying only the essential boundary conditions. This solution is feasible in most of the problem domain except the boundary layer. Some boundary-layer conditions are neglected. They are adjusted in the second step, where we consider only the boundary layer.

9.4. STEP ONE - BASE SOLUTION

It has been found that it is possible to solve the system (6.14), (6.15), (6.16), (6.17) and (6.18) on pages 69-70 exactly, in the considered case of the cylindrical shell loaded with antisym metrical wind pressure, applying the Refined Least Squares Method, taking into account only the essential boundary conditions (9.1) and (9.2) on the preceding page and neglecting the boundary-layer (9.3) and (9.4) on the page before. They were simply multiplied by zero. The deformed shape of the shell is presented in Figure 9.3 on the facing page.

■ The appropriate set of boundary conditions is presented below.

t -' 1 7 — BoundaryCondition [1 ] := BoundaryCondition [1] = X>i [bx] E 0 ;

In [2]: = BoundaryCondition [2] :=

BoundaryCondition[2] = C2 [bx] E 0;

I n [31:= BoundaryCondition [3] := BoundaryCondition [3] = TVj. [bx] E;

I n [4]:= BoundaryCondition [ 4 ] :=

BoundaryCondition [4] = [bx] E;

In [5] : = B o u n d a r y C o n d i t i o n [ 5 ] := BoundaryCondition[5] = *W3 [ b x ] E 0 ;

In [6]: = BoundaryCondition[6] :=

BoundaryCondition [6] = —11 h^ax^.. ln[7] : = BoundaryCondition [ 7 ] :=

BoundaryCondition [7] = ^ 12 t8*] ; h In [8]:= BoundaryCondition[8] :=

BoundaryCondition [8] = Ml1 0 ; h In [9]: = BoundaryCondition [9] :=

B o u n d a r y C o n d i t i o n [ 9 ] = —h3 0 ; In [10] : = BoundaryCondition [10] := Fig. 9.3. Deformed chimney, 100 times exaggerated BoundaryCondition[10] = 0; Rys. 9.3. Deformacja komina, skala skażona 100 razy 9.4. Step one - base solution 95 94 Chapter 9. Chimney exposed to an antisymmetrical load

The solution for rotation in the parallel direction f)2(x) is the function presented below as a The solution is feasible in most of the domain except the boundary layers, the sizes of which MATHEMATICS output. are limited to about 1200 mm from each edge (shown in gray in Figures 9.4 9.5 on the facing page, 9.6 on page 96,9.7 on page 97,9.8 on page 98, 9.9 on page 99,9.10 on page 100, 9.11 on In [13] : = d2 [x_] = Expand [02 [x ]/. solu tion [co] ] page 101, 9.12 on page 102, 9.13 on page 103, 9.14 on page 103, 9.15 on page 104. The 913310 918200808534200999617871251481 actual functions are highly oscillating but they quickly decay to the base solution, which is Out [13] = 114162 91525882 94 6052 6396109777680000 smooth and has no oscillations. Hence, it is practically impossible to satisfy all the boundary 127408341697x 127408341697x2 conditions in one step, as the problem becomes ill-conditioned in the Lyapunov sense and is 16987590 67 86833300 6795036271473332000 slowly convergent. The base solution is not a membrane because the moments and shear forces are not zero functions. Note that the negligence of some boundary conditions usually results in The diagram of this function is shown in Figure 9.5: a singular system. Here we have received a well-conditioned non-singular problem.

r m m -, 9.4.1. Rotations m 0.03 The solution for rotation in the meridian direction D\(x) is the function (9.5). 0.025

& 0.02 o / s 145875892187309029455347404987 T3 (T\ / ______W ~ 244627829045233265891823326000000' •o 0.015 58709275859518574758520364987 x (9.5) 48925565 80904665317836466520000000 + E 0.01 1061697000 x2 17694950 x3 1698759067868333 1698759067868333' 0.005

The diagram of this function is presented in Figure 9.4: 0

Fig. 9.5. Base solution: function of the rotation in the parallel direction T>i versus the coordinate mea sured along the chimney meridian Rys. 9.5. Rozwiązanie bazowe: funkcja obrotu w kierunku równoleżnikowym ł >2 względem współrzędnej mierzonej wzdłuż południka komina

x[m]

Fig. 9.4. Base solution: function of the rotation in the meridian direction D \ versus the coordinate mea sured along the chimney meridian Rys. 9.4. Rozwiązanie bazowe: funkcja obrotu w kierunku południkowym D \ względem współrzędnej mierzonej wzdłuż południka komina 96 Chapter 9. Chimney exposed to an antisymmetrical load 9.4. Step one - base solution 97

9.4.2. Displacements The solution for displacement in the parallel direction 'IV2OO is the function presented below as The solution for displacement in the meridian direction 'W i (x) is the function presented below a M athem atics output.

as a M athem atics output. In [15]:= w2[x_] = Expand[1^2 [x ]/. solution [ 00] ]

In [14] : = wl [x_] = Expand [TVi [x ]/. so lu tio n [oo] ] 183180773135170611198823854 64 9 Out[15]- 2 4 462782904523326589182332600000+ „ r,_, 143062836007370424506933884987 Out [14] = ------+ 146242918720647997804230541541x 122313914522 616632 945911663000000 244 62782 90452332658 91823326000000+ 61522332039457179706933884987 x 5834224 932 6179606409637228433x2 2446278290452332658918233260000000~ 9785113161809330635672933040000000” 2123394000 x2 35389900x3 353899000 x3 8847475x4 1698759067868333 + 1698759067868333 1698759067868333 + 3397518135736666

The diagram of this function is shown in Figure 9.6. The diagram of this function is shown in Figure 9.7: O 'TV i [mm] 0 IV 2 [mm]

15 0) U 60 W) u ■a -o00 p q S W u ■>< 8 1 0 E [L,

x[m] -2 0 -1 0 0 10 20 x[m]

Fig. 9.6. Base solution: function of the displacement in the meridian direction rW\ versus the coordinate Fig. 9.7. Base solution: function of the displacement in the parallel direction "1^2 versus the coordinate measured along the chimney meridian measured along the chimney meridian Rys. 9.6. Rozwiązanie bazowe: funkcja przemieszczenia w kierunku południkowym 'Wi wzglądem Rys. 9.7. Rozwiązanie bazowe: funkcja przemieszczenia w kierunku równoleżnikowym rU/2M wzglądem współrzędnej mierzonej wzdłuż południka komina współrzędnej mierzonej wzdłuż południka komina 98 Chapter 9. Chimney exposed to an antisymmetrical load 9.4. Step one - base solution 99

The solution for displacement in the direction normal to the reference surface *tV3(jc) is the 9.4.3. Stretching and shear forces function presented below as a MATHEMATICS output.

In [16]:= w3 [x_] = Expand [TV3 [x ]/. so lu tio n [o o ] ] The base approximation in the considered case can be called a quasi-membrane solution. For the membrane approach we receive for the stretching force in the meridian direction: 1101433024236505894 637 93395695668329 Out [16] = 14 677 6697427139959535093 995600000000 145875983395023489405883861541x M i« :=^(*-20)2. ( 9 . 6 ) 24462782 90452332 6589182332 6000000 58709184 651804114807 983908433x2 The quasi-membrane (base) approximation is close to the membrane solution, but not the same, 9785113161809330635672933040000000~ the stretching force in the meridian direction M i W is different: 353899000x3 8847475x4

1698759067868333 + 3397518135736666 In[17]:= nll[x_] = Expand [//u [x ]/. solution [ 00] ]

The diagram of this function is shown in Figure 9.8: 169872 935596000000 16987293559600000 x Out[17] 1698759067868333 1698759067868333+ 424682338990000 x2 IV 3 [mm] 1698759067868333

The diagram of this function is shown in Figure 9.9:

« kN o " u t — ] 60 m wT3 •a 9X 300 u o 60 60 T3w ■a 200 W •o0) X E x[m] 100 Fig. 9.8. Base solution: function of displacement in the direction normal to the shell rW3 versus the coordinate measured along the chimney meridian Rys. 9.8. Rozwiązanie bazowe: funkcja przemieszczenia w kierunku normalnym do powłoki 'W 3 wzglę dem współrzędnej mierzonej wzdłuż południka komina Fig. 9.9. Base solution: function of the stretching force in the meridian direction N \\ versus the coordi nate measured along the chimney meridian Rys. 9.9. Rozwiązanie bazowe: funkcja siły rozciągającej w kierunku południkowym N u względem współrzędnej mierzonej wzdłuż południka komina 100 Chapter 9. Chimney exposed to an antisymmetrical load 9.4. Step one - base solution 101

In the case of other functions it is very similar. For the membrane solution the shear force in the This is a formula in MATHEMATICS notation for the stretching force in the parallel N 22{x) ac meridian direction N\2(x) would be: cording to the definition (6.7) on page 67: 2 E h 3 n D 2 [x] 2 E h (3 s 2 + h 2 ) [x ] In [19]:= N 2 2 [x_] := Nn(x) ~ 20-x, (9.7) 3 s 2 ( 1 - v 2 ) 3 s 3 ( 1 - v 2 ) 2 E h (3 s 2 + h 2) TV3 [x ] 2 E h v W ; [ x ] whereas here we have a slightly different function received for the base approximation N\2(x). 3 s 3 ( 1 - v 2 ) 1 -v 2

In [18] : = n l 2 [ x _ ] = E x p a n d [//12 [ x ] / . s o l u t i o n [oo] ] For the membrane solution we will receive the constant value N 22(x) := -2. Here we see a small variation of N 22{x) from this value. 61154985974334492683330 6115498597433449268333x Out [18]= ------3057822381212239054 989 6115644762424478109978 In [20] : = E x p a n d [N22 [x ] / . s o l u t i o n [00] ]

371070244 90192879644501827538506 Out [20] = The diagram of this function is shown in Figure 9.10: 185514737295597 87335539367838873 140000000000 x 3500000000 x2 » kN N 12 [ — ] 5096277203604 999 50 96277203604999 m 40 The diagram of the function Ń 22(x) is shown in Figure 9.11:

30 ° kN 1) N 22 [ —m ] 00 0 20 s ■o

10 tu)

Fig. 9.10. Base solution: function of the shear force in the meridian direction Ń\2 versus the coordinate measured along the chimney meridian Rys. 9.10. Rozwiązanie bazowe: funkcja sity ścinającej w kierunku południkowym N\i względem współrzędnej mierzonej wzdłuż południka komina x[m] Fig. 9.11. Base solution: function of the stretching force in the parallel direction N22 versus the coordi nate measured along the chimney meridian Rys. 9.11. Rozwiązanie bazowe: funkcja siły rozciągającej w kierunku równoleżnikowym Ń22 względem współrzędnej mierzonej wzdłuż południka komina 102 Chapter 9. Chimney exposed to an antisymmetrical load 9.4. Step one - base solution 103

9.4.4. Moments In [22] : = m i l [x_] = E x p a n d [ M u [x ] / . s o l u t i o n [00] ] 7841519509164 67094 43 78415195091646709443x Out [22] = Another feature of the base solution is the presence of the moments and transverse forces which 101927412707074 6351663000 20385482541414 927033260000 do not occur in the membrane approximation. r Nm The solution of the bending moment at the meridian direction Mn{x) is the function presented M u L——Jm below as a MATHEMATICS output.

In[21]: = ml2[x_] = Expand [M\2 [ x ] / • s o l u t i o n [o o ] ]

-177 89918006563859830445063508084188519 Out[21]= ------5194 41264427 6740453951022 994 8844400 594238166660x 29711908333 x2 + 1698759067868333 ” 3397518135736666

The diagram of this function is shown in Figure 9.12:

0 x[m] -2 Fig. 9.13. Base solution: function of the torsion moment in the meridian direction Mu versus the coor dinate measured along the chimney meridian -4

Fig. 9.12. Base solution: function of the bending moment in the meridian direction At 12 versus the coor dinate measured along the chimney meridian

9.4.5. Transverse forces 9.5. STEP TWO - REFINEMENT WITHIN BOUNDARY LAYERS

Moments are usually associated with transverse forces. The solution for transverse forces in the Having a base solution we can approximate it near the free layers. To obtain it we used poly meridian direction <3i(x) is the function presented below as a MATHEMATICS output. nomials of degree 35. The coefficients of the system of algebraic equations were computed ex

In[24] : = ql [x_] = Expand[Qi [x]/ . so lution [co] ] actly and the system was solved with precision up to 256 digits by the Cholesky-Banachiewicz method. 1461649910288416450 146164991028841645x Out [24]= ------3057822381212239054989 61156447 62424478109978 The diagram of this function is shown in Figure 9.15: 9.5.1. Free edge

0 N The free edge layer domain is an interval (x3, x2) with x3 = 188/10. QA-]m In [26]:= a x = 2 0 ;

ln[27]:= b x = H p 0.8 This is the full set of boundary conditions for this case:

'Om 0.6 In [28]:= BoundaryCondition[l] := W BoundaryCondition [1] = (2>i [ax] -d l[ax ]) E ; 'Oo H 0.4 E In [29]:= B o u n d a r y C o n d it i o n [2 ] := BoundaryCondition [2] = (O2 [ax] -d2[ax]) E ; 0.2 In [30]:= BoundaryCondition [3] := BoundaryCondition [3] = ('W'i [ax] - wl [ax] ) E; -2 0 -1 0 0 10 20 x[m] In[31] := BoundaryCondition[4] := Fig. 9.15. Base solution: function of the transverse force in the meridian direction Q{ versus the coordi BoundaryCondition [4] = ("Wj [ax] -w 2[ax]) E; nate measured along the chimney meridian In[32]:= BoundaryCondition[5] := Rys. 9.15. Rozwiązanie bazowe: funkcja siły poprzecznej w kierunku południkowym Ói względem BoundaryCondition [5] = (% [ax] -w3[ax]) E; współrzędnej mierzonej wzdłuż południka komina In [33]:= BoundaryCondition [ 6 ] := A/ 11 [b x ] BoundaryCondition[ 6 ] = ; 9.4.6. Global equilibrium n

In [34]:= BoundaryCondition[7] := If the transverse force <3i(x) is added to the shear force N n (x) (presented on page 100) obtained „ ril ^ 12 [b x ] from the base solution, we receive the formula for the shear force of membrane solution N l2(x) BoundaryCondition[7] = ------—------/ n (9.7) on page 100. In [35]:= B o u n d a r y C o n d it i o n [ 8 ] := I n [25]: = %24 + %18 A in [b x ] BoundaryCondition [ 8 ] = —; ; Out [25]= 20 - x n J

It proves that the global equilibrium of horizontal forces holds true. By that means it is also In [36]:= BoundaryCondition [9] := shown that the results of the Refined Least Squares Method can be checked by known methods [b x ] of verification (see the last paragraph of section 12.5. on page 167). BoundaryCondition [9] = ----- —j-----;

In [37]: = BoundaryCondition[10] := Qi [b x ] B o u n d a r y C o n d it i o n [1 0 ] = — —^— ; 106 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers 107

Satisfaction of all the boundary conditions within the free edge layer results in approximate 'W 1 [mm] functions, the diagrams of which are presented in Figures: 9.16 and 9.17 - below, 9.18 1.3393 and 9.19 on the facing page, 9.20 and 9.21 on page 108, 9.22 and 9.23 on page 109, 9.24 and 9.25 on page 110, 9.26 and 9.27 on page 111. 1.33925 ^ mm,

1.33915 -----Bas ie k -----Rei ined B 1.3391 Lm m m f— H 18.8 19 19.2 19.4 19.6 19.8 20 x[m] Fig. 9.18. Refinement within the boundary layer at the free edge: function of displacement in the meridian direction 'TV 1 versus the coordinate measured along the chimney meridian Rys. 9.18. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja przemieszczenia w kierunku południkowym 'W\ względem współrzędnej mierzonej wzdłuż południka komina x[m] Fig. 9.16. Refinement within the boundary layer at the free edge: function of rotation in the meridian direction 0 \ versus the coordinate measured along the chimney meridian 'W 2 [mm] Rys. 9.16. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja obrotu w kierunku południkowym D\ względem współrzędnej mierzonej wzdłuż południka komina

^ rmm, £>2 -----m 0.000525 -----Base L 0.00052 — RefinedJT aou 0.000515 W 0.00051 Pm x[m] 0.000505 Fig. 9.19. Refinement within the boundary layer at the free edge: function of displacement in the parallel direction versus the coordinate measured along the chimney meridian 0.0005 Rys. 9.19. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja przemieszcze 18.8 19 19.2 19.4 19.6 19.8 20 nia w kierunku równoleżnikowym 'W względem współrzędnej mierzonej wzdłuż południka x[m] 2 komina Fig. 9.17. Refinement within the boundary layer at the free edge: function of rotation in the parallel direction T>2 versus the coordinate measured along the chimney meridian Rys. 9.17. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja obrotu w kierunku równoleżnikowym £ > 2 względem współrzędnej mierzonej wzdłuż południka komina 108 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers 109

kN N 12 [^Tm ] 1.2

1 -----Base 1> — Refined 00 0.8 oou 0.6 IL, 0.4

0.2

0 18.8 19 19.2 19.4 19.6 19.8 20 x[m] x[m] Fig. 9.20. Refinement within the boundary layer at the free edge: function of displacement in the direction Fig. 9.22. Refinement within the boundary layer at the free edge: function of shear force in the meridian normal to the shell "W-} versus the coordinate measured along the chimney meridian direction Nn versus the coordinate measured along the chimney meridian Rys. 9.20. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja przemieszczenia Rys. 9.22. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja siły ścinającej w kierunku normalnym do powłoki względem współrzędnej mierzonej wzdłuż południka w kierunku południkowym N\i względem współrzędnej mierzonej wzdłuż południka komina komina

kN kN. N 22 [ - ] N m 0.35 0.3 -----Base 0.25 00u 0.2 0.15 0.1 0.05 0 18.8 19 19.2 19.4 19.6 19.8 20 x[m] x[m\ Fig. 9.23. Refinement within the boundary layer at the free edge: function of stretching force in the Fig. 9.21. Refinement within the boundary layer in the free edge: function of stretching force in the parallel direction N22 versus the coordinate measured along the chimney meridian meridian direction N\ i versus the coordinate measured along the chimney meridian Rys. 9.23. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja siły rozciągającej Rys. 9.21. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja siły rozciągającej w kierunku równoleżnikowym N22 względem współrzędnej mierzonej wzdłuż południka w kierunku południkowym N\ i względem współrzędnej mierzonej wzdłuż południka komina komina 110 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers 111

Ma a 12 [ rN----- m] n m m 0.07 0 0.06 -0.0025 0.05 00

- 0.01 0.02 - - - - 1 iase 0.01 —...... T'efined | \ -0.0125 0 1 V -0.015 18.8 19 19.2 19.4 19.6 19.8 20 x[m\ Fig. 9.24. Refinement within the boundary layer at the free edge: function of bending moment in the Fig. 9.26. Refinement within the boundary layer at the free edge: function of torsion moment in the meridian direction M \2 versus the coordinate measured along the chimney meridian parallel direction M22 versus the coordinate measured along the chimney meridian Rys. 9.24. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja momentu zgina Rys. 9.26. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja momentu skręcają jącego w kierunku południkowym M \2 względem współrzędnej mierzonej wzdłuż południka cego w kierunku równoleżnikowym M22 względem współrzędnej mierzonej wzdłuż południka komina komina

M\A u l rN — m T] Q l[^m ]

X) S u a u-

18.8 19 19.2 19.4 19.6 19.8 20 x[m] x[m] Fig. 9.25. Refinement within the boundary layer at the free edge: function of torsion moment in the Fig. 9.27. Refinement within the boundary layer at the free edge: function of transverse force in the meridian direction Mu versus the coordinate measured along the chimney meridian meridian direction Q\ versus the coordinate measured along the chimney meridian Rys. 9.25. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja momentu skręca Rys. 9.27. Uściślenie w obrębie warstwy brzegowej przy brzegu swobodnym: funkcja siły poprzecznej jącego w kierunku południkowym M\ \ względem współrzędnej mierzonej wzdłuż południka w kierunku południkowym Qi względem współrzędnej mierzonej wzdłuż południka komina komina 113 112 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers

9.5.2. Fixed edge

The fixed edge layer domain is an interval jc4) with x4 = -188/10.

I n [38]:= b x = - 2 0 ;

In [39] := a x = 10 This is the full set of 10 boundary conditions for this case:

In[40] BoundaryCondition[1 ] := BoundaryCondition[1] = [bx]E;

In [41]:- BoundaryCondition [2] := BoundaryCondition [2] = Z)2 [ b x ] E ; -20 -19.8 -19.6 -19.4 -19.2 -19 -18.8 In [42]:- B o u n d a r y C o n d it i o n [3 ] := x[m\ BoundaryCondition[3] = rW1 [ b x ] E ; Fig. 9.28. Refinement within the boundary layer at the fixed edge: function of rotation in the meridian In[43] :- BoundaryCondition[4] := direction 0 \ versus the coordinate measured along the chimney meridian BoundaryCondition [4] ='M'2 [ b x ] E ; Rys. 9.28. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja obrotu w kierun ku południkowym T>\ względem współrzędnej mierzonej wzdłuż południka komina In [44] := B o u n d a r y C o n d i t i o n [5 ] := BoundaryCondition[5] = “W jtbx]E;

In[45]:= BoundaryCondition[ 6 ] := N u [ax] - nil[ax] BoundaryCondition[ 6 ] = h

In[46] := BoundaryCondition[7] := N12 [a x ] - n l 2 [ a x ] BoundaryCondition[7] = h

In[47]:= BoundaryCondition[ 8 ] := M u [a x ] - m i l [ a x ] BoundaryCondition[ 8 ] = h 3

In[48]:= BoundaryCondition[9] ::

A< 12 [a x ] - m l2 [ a x ] BoundaryCondition[9] = h 3

In[49] :- BoundaryCondition[10] := x[m] <2i [ax] -ql[ax] BoundaryCondition[10] Fig. 9.29. Refinement within the boundary layer at the fixed edge: function of rotation in the parallel h 3 direction T>2 versus the coordinate measured along the chimney meridian Satisfaction of all the boundary conditions on the fixed edge layer results in the functions, the Rys. 9.29. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja obrotu w kierun diagrams of which are presented in: Figures 9.28 and 9.29 on the next page, 9.30 and 9.31 on ku równoleżnikowym Di względem współrzędnej mierzonej wzdłuż południka komina page 114, 9.32 and 9.33 on page 115, 9.34 and 9.35 on page 116, 9.36 and 9.37 on page 117, 9.38 and 9.39 on page 118. 114 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers 115

'W 3 [mm]

1> 00 ■o to -a X

-20 -19.8 -19.6 -19.4 -19.2 -19 -18.! x[m] x[m] Fig. 9.30. Refinement within the boundary layer at the fixed edge: function of displacement in the merid Fig. 9.32. Refinement within the boundary layer at the fixed edge: function of displacement in the direc ian direction 'Wi versus the coordinate measured along the chimney meridian tion normal to the shell 'Ws versus the coordinate measured along the chimney meridian Rys. 9.30. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja przemieszczenia Rys. 9.32. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja przemieszczenia w kierunku południkowym 'Wi względem współrzędnej mierzonej wzdłuż południka komina w kierunku normalnym do powłoki ^ 3 względem współrzędnej mierzonej wzdłuż południka komina

'W 2 [mm] kN N „ [— ] m 400

395 -----Base U .... . u ----- Refinecl i 00 3 390 -o

380

x[m] -20 -19.8 -19.6 -19.4 -19.2 -19 -18.8 x[m] Fig. 9.31. Refinement within the boundary layer at the fixed edge: function of displacement in the parallel direction rW2 versus the coordinate measured along the chimney meridian Fig. 9.33. Refinement within the boundary layer at the fixed edge: function of stretching force in the Rys. 9.31. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja przemieszcze meridian direction N\ \ versus the coordinate measured along the chimney meridian nia w kierunku równoleżnikowym W 2 względem współrzędnej mierzonej wzdłuż południka Rys. 9.33. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja siły rozciągającej komina w kierunku południkowym N\ 1 względem współrzędnej mierzonej wzdłuż południka komina 116 Chapter 9. Chimney exposed to an antisymmetrical load 9.5. Step two - refinement within boundary layers 117

N n Em ^ ]

-20 ■19.8 -19.6 -19.4 -19.2 -19 -18.8 -20 -19.8 -19.6 -19.4 -19.2 -19 -18.8 x[m] x[m\ Fig. 9.34. Refinement within the boundary layer at the fixed edge: function of shear force in the meridian Fig. 9.36. Refinement within the boundary layer at the fixed edge: function of bending moment in the direction Nn versus the coordinate measured along the chimney meridian meridian direction Af 1 2 versus the coordinate measured along the chimney meridian Rys. 9.34. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja siły ścinającej Rys. 9.36. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja momentu zgi w kierunku południkowym N\i względem współrzędnej mierzonej wzdłuż południka komina nającego w kierunku południkowym Ati2 względem współrzędnej mierzonej wzdłuż południka komina

kN N 22 [— ] m 120

100 -----Base L 80 J ----- Refined 1 60 •au E 40 20 1 0 1 1

— I.----- . . -20 -19.8 -19.6 -19.4 -19.2 -19 -18.S x[m\ -20 -19.8 -19.6 -19.4 -19.2 -19 -18.8 x[m] Fig. 9.35. Refinement within the boundary layer at the fixed edge: function of stretching force in the parallel direction Nn versus the coordinate measured along the chimney meridian Fig. 9.37. Refinement within the boundary layer at the fixed edge: function of torsion moment in the Rys. 9.35. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja siły rozciąga meridian direction M\ 1 versus the coordinate measured along the chimney meridian jącej w kierunku równoleżnikowym N22 względem współrzędnej mierzonej wzdłuż południka Rys. 9.37. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja momentu komina skręcającego w kierunku południkowym M\\ względem współrzędnej mierzonej wzdłuż południka komina 118 Chapter 9. Chimney exposed to an antisymmetrical load

m 0 ------2 ----- Base U Chapter 10 &> -4 -o № -6 CYLINDRICAL SHELL SUBJECTED TO A SINUSOIDAL

-12 i______-20 -19.8 -19.6 -19.4 -19.2 -19 -18.: x[m] Przedstawiony w tym rozdziale przykład ilustruje zastosowanie Metody Najmniejszych Kwadratów do znalezienia rozwiązania przybliżonego powłoki walcowej poddanej obciążeniu sinusoidalnemu wzglę Fig. 9.38. Refinement within the boundary layer at the fixed edge: function of torsion moment in the dem równoleżnika. Podejście jednoetapowe zbiega się szybko dla powłoki krótkiej. Ten przykład jest parallel direction M22 versus the coordinate measured along the chimney meridian okazją do przedyskutowania dodatkowych narzędzi analizy błędu jakie daje metoda. Przedstawiono Rys. 9.38. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja momentu sposoby oceny zarówno globalnego jak i lokalnego błędu aproksymacji. Następnie dokonano porówna skręcającego w kierunku równoleżnikowym M22 względem współrzędnej mierzonej wzdłuż nia wyników dla podejścia bezpośredniego - jednoetapowego z dwuetapowym dla powłoki długiej. Tym południka komina samym wykazano, te przybliżenie bazowe daje poprawne wyniki dla prawie całej dziedziny z wyjątkiem warstwy brzegowej.

The example presented in this chapter illustrates the application of the Least Squares Method to search 0 1 [— ] for the approximate solution of a cylindrical shell subjected to a sinusoidal load with respect to the m parallel. The one-step approach converges quickly in the case of a short shell. This example is an occasion to discuss additional tools of error analysis given by this method. Ways are shown of evaluating -----Base L both global and local approximation errors. Next, the one-step approach has been compared with the two-step one for a long shell. By these means, it was proved that the base approximation leads to correct

10.1. DESCRIPTION OF THE PROBLEM

-20 -19.8 -19.6 -19.4 -19.2 -19 -18.: 10.1.1. Numerical data of the problem x[m] Fig. 9.39. Refinement within the boundary layer at the fixed edge: function of transverse force in the Let us consider two cylindrical shells made of steel (Young modulus E := 2 • 108, Poisson ratio meridian direction <3i versus the coordinate measured along the chimney meridian v — i ) with length of / := 4 m (short) and I := 60 m (long), the thickness 2h := 10 mm and Rys. 9.39. Uściślenie w obrębie warstwy brzegowej przy brzegu utwierdzonym: funkcja siły poprzecznej cylinder radius s0 := 2 m. The shapes of their undeformed reference surfaces are presented in w kierunku południkowym <2i wzgłędem współrzędnej mierzonej wzdłuż południka komina Figures 10.1 and 10.2 on the next page, respectively. They are subjected to a periodical load, ‘The load is described by a cosine function. The diagram of the cosine function is a sinusoid shifted in phase. Thus, the title of the chapter contains the word “sinusoidal”.

119 120 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.1. Description of the problem 121

with regard to the parallel direction, which is normal to the reference surface p3 := !P3 cos(5 y), P3 "— 1 normal to the cylinder mid-surface, tangent load components p\ := 0, p2 ■- 0.

Fig. 10.1. Short cylindrical shell - reference surface, dimensions in [m] Fig. 10.3. Sinusoidal load of the cylinder (pipe) Rys. 10.1. Powloką walcowa krótka - powierzchnia odniesienia, wymiary w [m] Rys. 10.3. Obciążenie snusoidalne walca (rury)

10.1.2. Differential equations

The problem is described by a system of five ordinary differential equations of the second-order (6.14), (6.15), (6.16), (6.17) and (6.18), see 6.4.4. on pages 69-70, with the parameter n ~ 5.

10.1.3. Boundary conditions

2 Similarly to the previous example the system of 5 second-order differential equations (6.14), 1 0 (6.15), (6.16), (6.17) and (6.18) on pages 69-70 requires 10 boundary conditions. -1 -2 In the considered case we have four boundary conditions of the first type: The boundary conditions of the first type in our problem are: Fig. 10.2. Long cylindrical shell - reference surface, from the plane of symmetry x = 0 to the fixed end x = 30, dimensions in [m] bi = 'Wl (Xa) = 0, Rys. 10.2. Powloką walcowa długa - powierzchnia odniesienia, od płaszczyzny symetrii x —Odo końca b2 = 'W2(xa) = o, zamocowanego x — 30, wymiary w [m] (10.1) b3 =

b4 = W 2(xb) = 0 . The variable y e < 0,2n) is measured along the parallel of the cylinder, and x e (xa, xb) along the meridian. The cylinder is fixed at both edges xa := - | and xb := |. They are essential to receive a non-singular system of linear algebraic equations. These boundary conditions are the same as those used in the membrane approach. Here it needs From the engineering point of view the problem can be considered to be a static scheme of to be stated that the membrane approach is not correct for the considered task as the cylindrical a steel pipe, fixed on both ends, loaded with a fifth periodic component of the wind. The distribution of load is shown in Figure 10.3 on the facing page. shell is significantly bent by this type of load, so the moments and transverse forces cannot be neglected. 122 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.2. S h o rt sh ell 123

The boundary conditions of the second type in our problem are: The approximation has been carried out with polynomials whose degrees attain the value 36 for even functions and 37 for odd ones. These degrees of polynomials were necessary to satisfy b5 - T) 1 (xa) = 0, the condition (10.17) on page 129 for the residuum, which are introduced further on within the b6 = 'Di (xa) = 0, discussion about error estimation, see section 10.3. on page 125. bi = (xa) == 0, h = £>i (xb) = 0, bg = ©2 (Xb) = 0, bio =

10.2. SHORT SHELL

The approximation for a short shell (/ = 4 m) is not difficult. Its deformed shape is presented in Figure 10.4. Figures 10.5, 10.6 on the next page, 10.7 and 10.8 on page 124, and 10.9 on page 125 show diagrams of some physical internal forces concerning the considered case.

-2 -1 ; x[m] Fig. 10.5. Approximation of a short cylindrical shell: function of the stretching force in the meridian direction N\ i versus the coordinate measured along the cylinder meridian Rys. 10.5. Aproksymacja powłoki walcowej krótkiej: funkcja siły rozciągającej w kierunku południko wym N\ i względem współrzędnej mierzonej wzdłuż południka walca

Meridian Shear Force Ni2 [—m ]

Fig. 10.6. Approximation of a short cylindrical shell: function of the shear force in the meridian direction Fig. 10.4. Short cylindrical shell: deformation (exaggeration 5000 times), dimensions in [m] N\2 versus the coordinate measured along the cylinder meridian Rys. 10.6. Aproksymacja powłoki walcowej krótkiej: funkcja siły ścinającej w kierunku południkowym Rys. 10.4. Powloką walcowa krótka: deformacja (skala skażona 5000 razy), wymiary w [m] N\2 względem współrzędnej mierzonej wzdłuż południka walca 124 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.3. Error and convergence analysis 125

Meridian Bending Moment M\i [—m ] m Meridian Transverse Force âj [ —m ]

-10

-2 0

- 3 0

" 2 _1 0 1 2 x[m] Fig. 10.7. Approximation of a short cylindrical shell: function of the bending moment in the meridian Fig. \0.9. Approximation of a short cylindrical shell: function of the transverse force in the meridian direction M \i versus the coordinate measured along the cylinder meridian direction (2i versus the coordinate measured along the cylinder meridian Rys. 10.7. Aproksymacja powłoki walcowej krótkiej: funkcja momentu zginającego w kierunku południ Rys. 10.9. Aproksymacja powłoki walcowej krótkiej: funkcja siły poprzecznej w kierunku południkowym kowym At i2 względem współrzędnej mierzonej wzdłuż południka walca <2, względem współrzędnej mierzonej wzdłuż południka walca

Meridian Torsion Moment Л1ц Г——1 10.3. ERROR AND CONVERGENCE ANALYSIS m 3 The considered task of a short shell subjected to a sinusoidal load is an occasion to discuss some 2 aspects of error and convergence analysis.

1 As already mentioned, the aim of the Refined Least Squares Method is the approximation of a set of equations with boundary conditions. It is achieved by the minimization of the functional: 0

-1 (ai

Adding both sides of (10.7) on the facing page and (10.10) on the preceding page we receive. It can be shown that if we use an appropriate degree of the approximating polynomial, which satisfies the global criterion of an error, the approximate solution approaches the actual solution despite the presumed weights. This will be shown in the end of this analysis. [ ( e (f' to)2) dx+ ji (g*)2 = / ( e w ) ) ) * + e (^ ) ) ) • xa{ \/=i / /=i - <* \/=i / *=i 10.3.1. Estimation of the global error (10.11) In each considered weighted equation ar( et (x) we can distinguish a free term Ft (x) independent The functional Q on the left-hand side of this equation depends only on known quantities Ft (x) of functions yi (jc) and a part that depends on them Z(yi (x) ). and Gk, and can be computed exactly and a’priori.

a, et (x) := F, (x) + £(y,(x) ) = 0 . (10.4) I f n \ m 0 := / E (F‘ t o ) 2 ) dx + Y 2 {Gk)2 * o. (10.12) “ V /=1 / k= 1 The free term F/ (x) may be zero. In linear problems £(yt (x)) is a linear combination of yt (x) Xa functions and/or their derivatives (or integrals). Q can be called a reference functional. The satisfaction of the equation (10.4) is equivalent to: The right-hand side functional 7Y depends on unknown functions yk which should be approxi mated. Ft (x) = -£ (yi(x)). (10.5) This yields that also the following equation should be satisfied, /Xb / n 2\ m 2 (x))) dx + Y , \K(yk (x,))J > 0. (10.13) xar \;=i / *=i (F,(x))2 = (-£()-, (x)))2, (10.6) This functional 'H can be called a comparative one. and finally we can state that also the equilibrium holds true after the integration of both sides within the considered domain x e (x„, xb). Q — 0 when both all Ft (x) = 0 and all Gk = 0, only. It yields a trivial solution of the equations: yk (.x) = 0. Therefore, in practical cases, at least one of Ft (x) or Gk must attain non-zero values. According to that § > 0, and subsequently 'H > 0. Equations (10.4) and boundary (initial) conditions (10.8) on the facing page are usually approximated, not exact. Therefore both sides /(!>,«/)*=/ E ^ f y « ) ) ^ dx. (10.7) are not equal. Nevertheless, we can define the global criterion for the estimation of the quality of the approxi Similar considerations may concern boundary or initial conditions. In each weighted boundary mate solution. Let us introduce the residuum functional H that is defined as follows: condition we can distinguish a part 'K(yk (xit)), which depends on functions, and the free term Gk which does not depend on unknown functions. The free term may be equal to zero. K:= (10.14)

Pk bk ■= Gk + T<{yk (x.t) ) — 0, (10.8) and it should satisfy conditions: and "R « "H, which is equivalent to Q ^ rH. where xs ■= xa or xs := x^. Additionally, we can easily prove that always the value of the functional T « Q when K « Q. which yields, Substituting (10.4) and (10.8) on the preceding page into (10.3) on page 125 we receive:

Gk= ~'K(yk(xs)), (10.9) 7 ( " \ m / E (F< to + £{yi (x) ))2 \d x + Y ,(G k + K(yk (x,) ))2. (10.15) which results, after similar consideration, in the following equation: xa/ V /=i / *=i

n m 2 From that we can find that Q may be computed by setting all ^(x) = 0 in the functional T. E(G*)2= E ( ^ * f e ) ) ) - (10.10) i=i *=i 128 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.3. Error and convergence analysis 129

— 100 @

Fig. 10.10. Convergence of the square root of the functional T, equation (10.3), versus the degree of the polynomial approximating the task Fig. 10.12. Convergence of the condition of the error criterion (10.19) versus degree of the polynomial approximating the task Rys. 10.10. Zbieżność pierwiastka kwadratowego funkcjonału T, równanie (10.3), ze względu na stopień wielomianu aproksymującego zadanie Rys. 10.12. Zbieżność warunku na kryterium błędu (10.19) ze względu na stopień wielomianu aproksy mującego zadanie

| 1 0 0 Since functionals T , Q and Tl are computed as sums of squares and integrals of squares - it is reasonable to compare the square roots of these functionals. Therefore we can define another global criterion as:

\Jt p « \[T o- (10.16)

Figure 10.10 on the preceding page presents the convergence of the square root of the func tional with respect to the degree of the approximating polynomial degree p. This and further functionals are computed for the considered problem of a short cylindrical shell subjected to a sinusoidal load. The second criterion can be expressed in the form:

ypR « ^[q. (10.17) Fig. 10.11. Convergence of the condition of the error criterion (10.18) versus the degree of the polyno mial approximating the task It can also be denoted as: Rys. 10.11. Zbieżność warunku na kryterium błędu (10.18) ze względu na stopień wielomianu aproksy mującego zadanie rR (10.18) p->lim oo \]l yr = °’ On the other hand any minimization of the functional, also a very poor one, yields T < Q. It or leads to the conclusion about equivalence of the criteria. Let us minimize the functional with, for example, constant functions yk(x) ak and evaluate the value of the functional To < § and next with a set of p independent functions yk (x) ■- £ f =| C(n/+*_u u, (x). If the value of the (10.19) functional Tp « T o , we consequently have that also K « Q.

where p is the degree of the polynomial applied in the approximation. 10.3. Error and convergence analysis 131 130 Chapter 10. Cylindrical shell subjected to a sinusoidal load e sa isfa i n f e fi s diffe en ia e a i n Figure 10.11 on page 128 presents the convergence condition (10.18) on the page before with respect to the degree of the approximating polynomial degree p. 0 .0 0 0 0 4 Figure 10.12 on the preceding page presents the convergence condition (10.19) on the page 0.0000 before with respect to the degree of the approximating polynomial degree p. Figures 10.10 on page 128, 10.11 on page 128 and 10.12 on the preceding page show that 0.00002 a admissible error is attained for the degree of the polynomial equal to 24 in the case of even functions and 25 in the case of odd functions. The values on these diagrams have been computed for the approximating polynomials whose degrees attained values p := 0,4, 8,12,16, 20, 24. 0.00001

k ( W W V W v w _ -wvW W Vrtlli 10.3.2. Estimation of the local error -1 1 ' x[m]

A global estimation informs us about the error obtained under the assumed weights in the Fig. 10.13. Absolute value of the residuum of the equation (10.20) versus the coordinate measured along equations and boundary conditions. We may be interested in satisfaction of equations in certain the cylinder meridian points of the considered domain. Therefore, we must estimate the local error of the equations or Rys. 10.13. Bezwzględna wartość reziduum równania (10.20) względem współrzędnej mierzonej wzdłuż boundary conditions as analysis of the residuum for a selected equation or boundary condition. południka walca The consideration will be carried our for the first differential equation of the system of equations on pages 69-70, of the problem discussed in this chapter: This case is also encountered here since for the considered task P\{x, n) := 0, 7C(x) := 0 and S(x) := 0 in equation (10.6) on page 126 and we have the equation with a free term equal to 0 _ _ 2 E G'(x) h _ 2 E 7 C (x ) h3 _ E n2 D {(x) h3 (homogenous equation). h 3 n2 E D i [x ] h n2 E (h2 + 3 s2) TVi[x] Out[l] = £' n 1-v 3(1- v2) s0 3(l+v)s03 + 3 ( 1 + v ) s3 3 (1+ v) s£

e h3 n 2 Oi [x] E h (3ag + h2) 11 r i * 3 s3 (1 + v) 3sJ (1 + v) ai ei (x) := ^ 2 £,, ( 10.21) 2Ehe:txJ.. 2Eh3x:ixi + Ehn^ [x]. 2Bh.v.r» w . 1-V 3 So (1 -v 2) s0 (1 — v) sQ (1 — V2 ) i=i 2 E h 3 D'i [x] 2E h'W '1'[x] where £, := X, (yi (x)) is a function of yt (x) or its derivatives. 3sQ (1-v2) + 1-v2 For example for the first differential equation in the considered case we have the following The residuum of the equation can be computed by substituting into it the approximated functions vector: (for the degree of the polynomial p := 36 in even functions and 37 in odd ones). Figure 10.13 on the next page presents the absolute value of the residuum of the equation (10.20), We can find In[2] ;= FirstD ifferentialEqnTerm s = that the equation is satisfied exactly in selected points, only. We find also that the maximum N [ E v a l u a t e [ absolute error is attained in the points x — ± 2 m. , E h 3 n2 £>i [x] 2Eh 3 Z>i'[x] Eh (3 s 2 + h 2 ) n2 ■Wj. [x ] 3 s 3 (1 + v) , ~ 3 s 0 (1 - v2) , ~ 3 s 4 (1 + v) The approximation quality may be locally estimated by comparing both sides of the equations 2 E h ' W ' i [x] E h n TVj [x] 2 E h v ' T 3 [ x ] 1 (10.6) or (10.9) on page 126, but it fails when F; (x) or Gk, respectively, are equal to zero. Also 1-v2 ' s0 (1-v) s0 (1-v2)' in the neighborhood of zero points of the functions F/ (x) the residuum may be relatively large. /.solution[256]/.x -> 2]] In the zero points of these functions we encounter singularities of a relative error. 132 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.3. Error and convergence analysis 133

We have substituted the approximated solution and computed the values in the point x — 2. Out [2] = {-6.75849 x 10'13, 0.249769, -1.43116 x 10‘6, 32.2743, -32.7816, 0.257591} The absolute value of the sum of this vector components returns an absolute (dis)satisfaction of the first differential equation in the point x = 2, compare Figure 10.13 on the page before.

I n [3] : = Satisfaction = Abs[ FirstDifferentialEqnTerm si] i=i Out[3]= 0.0000420803 For an approximate solution the equation is not satisfied exactly. We can compare this residuum with a norm computed for components of the sum (10.21)2 on the page before, which can be regarded as components of the vector £ = Li, • • •, -£„]. It can be, for example, /2-norm:

-2 x[m] E ^ 2- (10.22) /=1 Fig. 10.14.12-norm for the equation (10.20) versus the coordinate measured along the cylinder meridian

Rys. 10.14. Norma / 2 równania (10.20) wzglądem współrzędnej mierzonej wzdłuż południka walca I n [4]:= L 2N orm = У FirstDifferentialEqnTerm si 2 \ Out[4]= 4 6 .0 0 4 3 Relative error of the first differentia] equation If the condition is satisfied:

E* « (10.23)

the equation is well approximated. Satisfaction In [5] : = L 2N orm Out 15] = 9.14704 x 10“7 A similar consideration may be carried out for other equations and boundary (initial) conditions. The consideration may concern any point of the domain. Figure 10.14 on the facing page presents a diagram of the /2-norm computed within the domain and Figure 10.15 on the next Fig. 10.15 .Relative error of the equation (10.20) versus the coordinate measured along the cylinder page shows the relation of the absolute error to the /2-norm. meridian The approach fails in points (and their close neighborhood), where all _£, in the equation ap Rys. 10.15. Błąd względny równania (10.20) względem współrzędnej mierzonej wzdłuż południka walca proach 0. In such cases the norm approaches zero, too, and the error becomes relatively large. This problem is usually encountered in tasks with symmetries - near points of symmetry. In such points the equations are satisfied automatically due to their symmetries and .£, = 0. It ХЪ 4 dx « occurs in the point x = 0 in the considered equation. E* u dx. (10.24) xb - X a J Xb —- x a J/I I Therefore it is reasonable to compare the average error with an average norm within the domain. The “globalized” condition is: The local analysis is useful in the evaluation of weights. A comparison of the average or local 2In the considered case this is the equation (10.20) on page 130 error may enable us to find which equations are satisfied better or worse and implement adaptive weights (or weight functions) estimation. Such problems will be researched in the future. 134 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.4. Long shell - one-step approach 135

10.3.3. A few remarks on convergence and weights The parameter k takes the values: 1, 0.01, 0.001, 0.0005 and finally 0.00025. Figure 10.16 on the preceding page shows the convergence of the displacement component rWi(0) with respect One of the very important problems of the method are weights standing by equations and to the degree of the polynomial used for the approximation p. boundary or initial conditions. As has already been mentioned, they can be used to moder ate the convergence path. They were guessed according to the rule mentioned in point 8.1.5. on This example shows that the convergence paths are different. However when the polynomial page 83. degree attains the value 24 all paths converge to the common value. We have already noticed that this degree of the polynomial is sufficient to satisfy the conditions of the global error discussed Nevertheless, it is worth stating that any choice of the weights leads to the same result. The in the previous section 10.3.1. on page 126. problem is the choice of the appropriate degree of the approximating polynomial. By the way, we can propose another way of analyzing the convergence. We can try different The influence of weights on the convergence of the task has been analyzed. The problem was weights for concrete equations and boundary conditions. If tasks converge to a common solution approximated several times with constant weights standing by differential equations and differ it means that we have obtained an appropriate degree of the approximating polynomial. ent weights by boundary conditions. The problem of weights presumption will be investigated in future since weights play an impor tant role in the method and possibly they influence precision of the solution and consequently the numerical stability.

10.4. LONG SHELL - ONE-STEP APPROACH

If the problem of a long cylindrical shell had been approximated with a full set of boundary con ditions, then difficulties would have been experienced with the convergence and stability. This is a high-order differential operator and therefore the problem is ill-conditioned. The degrees of the approximating polynomials have to be quite large numbers and the working precision of the computations has to be large. Despite of this the loss of the precision is considerable. The deformation from the plane of symmetry x — 0 to the fixed end x = 30 (exaggeration 500 times) is shown in Figure 10.17 on the next page, other results of this approximation are presented in Figures 10.18(a) and 10.19(a) on page 137, 10.20(a) and 10.21(a) on page 138, and 10.22(a) on page 139. The figures to the the right of them present results of the “base” Polynomial Degree approximation discussed in the next section. The approximation has been carried out with polynomials whose degrees attained the value 92 Fig. 10.16. Convergence of the value 'W3(0) versus the degree of the approximating polynomial, for for even functions and 93 for odd ones. Such high degrees of polynomials were necessary to different miltipliers k of the weights of the boundary conditions satisfy the conditions (10.17) on page 129 for the residuum. Rys. 10.16. Zbieżność wartości 'WjiO) ze względu na stopień wielomianu aproksymującego, dla różnych On the other hand, such a high degree of approximating polynomials is only possible with mnożników k wag warunków brzegowych the application of monic Chebyshev polynomials, see the definition on page 81. Moreover it requires obligatory analytical (symbolic) integration during computations of the coefficients of The boundary conditions were multiplied by the coefficient k, as follows: the system matrix and the solution of this system of linear algebraic equations with a starting In [6] := BoundaryCondition [ 1 ] := B o u n d a r y C o n d it i o n [ 1] = k î > i [ b x ] E ; precision equal to 512 digits. The loss of precision during the computations was larger then 200. Fortunately MATHEMATICS can compute numbers with an arbitrary precision. Moreover, it In[7]:= BoundaryCondition [2] := BoundaryCondition [2] = k2 [b x ] E ; )2 truncates false digits and returns the result with an actual precision. In[8] : = BoundaryCondition[3] := BoundaryCondition[3] = k 1 V i [b x ] E ; The final high precision of the coefficients of high degree polynomials is necessary to avoid

In [9]:= BoundaryCondition [4] := BoundaryCondition [4] =k'W 2 [b x ] E ; problems with their instability. Dealing with high-order polynomials requires special tech niques. The handling of these technical problems will be presented in further contributions. In[10]:= BoundaryCondition[5]:=BoundaryCondition[5] =k'W 3 [ b x ] E ; 137 136 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.5. Long shell - two-step approach

(a) Long shell - one-step approach (b) Long shell - “base” approximation Powloką długa - podejście jednoetapowe Powłoka długa - przybliżenie „bazowe” Fig. 10.18. Comparison of one step and base approximations: function of the stretching force in the meridian direction N\ i versus the coordinate measured along the cylinder meridian x[m] Rys. 10.18. Porównanie przybliżeń jednoetapowego i bazowego: funkcja siły rozciągającej w kierunku Fig. 10.17. Base approximation of the long cylindrical shell: deformation (displacements scale exagger południkowym Nu względem współrzędnej mierzonej wzdłuż południka walca x[m\ ated 500 times), dimensions in [m] Rys. 10.17. Bazowa aproksymacja powłoki walcowej długiej: deformacja (skala przemieszczeń skażona 500 razy), wymiary w [m]

10.5. LONG SHELL - TWO-STEP APPROACH Meridian Shear Force N\j [—] Meridian Shear Force N\i [ — ]

10.5.1. Base solution

Similarly to the previous example, the system (6.14), (6.15), (6.16), (6.17) and (6.18) on pages 69-70 is approximated by means of the Refined Least Squares Method, taking into ac count only the essential boundary conditions (10.1) on page 121 and neglecting the boundary- layer ones (10.2) on page 122. As has already been mentioned, they are simply multiplied by zero, for example on the fixed edge:

In [ll] := BoundaryCondition[5] := BoundaryC ondition [5] = Z>i [ax] E 0 ; (a) Long shell - one-step approach (b) Long shell - “base” approximation In[12]:= BoundaryCondition[ 6 ] := Powłoka długa - podejście jednoetapowe Powłoka długa - przybliżenie „bazowe” BoundaryCondition[ 6 ] = Z>2 [ a x ] E 0 ; Fig. 10.19. Comparison of one-step and base approximations: function of the shear force in the meridian In[13]:= BoundaryCondition [7] := direction JVj2 versus the coordinate measured along the cylinder meridian x[m] BoundaryCondition[7] = W 3 [ a x ] E 0 ; Rys. 10.19. Porównanie przybliżeń jednoetapowego i bazowego: funkcja siły ścinającej w kierunku Opposite to the previous example, see chapter 9 on page 88, the system can be approxi południkowym W12 względem współrzędnej mierzonej wzdłuż południka walca x[m] mated, not solved exactly. Nevertheless, the base approximation is free from a boundary layer phenomenon and can be approximated with a polynomial of a smaller degree. 138 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.5. Long shell - two-step approach 139

Meridian Transverse Force ö i [-

1000

500

-500

-30 -20 -1 0 10 20 30 -30 -20 -1 0 20 30 (a) Long shell - one-step approach (b) Long shell - “base” approximation (a) Long shell - one-step approach (b) Long shell - “base” approximation Powłoka długa - podejście jednoetapowe Powłoka długa - przybliżenie „bazowe” Powłoka długa - podejście jednoetapowe Powłoka długa - przybliżenie „bazowe” Fig. 10.22. Comparison of one-step and base approximations: function of transverse force at the merid Fig. 10.20. Comparison of one-step and base approximations: function of the torsion moment in the ian direction Oi versus coordinate measured along the cylinder meridian x[m\ meridian direction Mu versus the coordinate measured along the cylinder meridian x[m\ Rys. 10.22. Porównanie przybliżeń jednoetapowego i bazowego: funkcja siły poprzecznej w kierunku Rys. 10.20. Porównanie przybliżeń jednoetapowego i bazowego: funkcja momentu skręcającego w kie południkowym Q\ wzglądem współrzędnej mierzonej wzdłuż południka walca x[m] runku południkowym M\\ względem współrzędnej mierzonej wzdłuż południka walca x[m\

The results of this approximation are shown in figures 10.18(b) and 10.19(b) on page 137, 10.20(b) and 10.21(b) on the preceding page and 10.22(b). The approximation was carried out with polynomials whose degrees attained the value 20 for even functions and 21 for odd ones. These degrees of polynomials were necessary to satisfy the conditions of the residuum (10.17) on page 129. It is worth denoting that these degrees of polynomials are much smaller than in the case of a one-step approach (respectively 92 and 93) and even smaller than for a short shell (respectively 36 and 37). i Comparing the diagrams on both sides of Figures 10.18 and 10.19 on page 137, 10.20 on the preceding page and 10.21, and 10.22 we find that the “base” approximation is feasible in most of the domain except the boundary layers, the sizes of which are limited to about 1.2 m from each edge. The actual functions are highly oscillating but they quickly fade to the base solution, which is smooth. Hence, it is difficult or even practically impossible to satisfy all the boundary conditions in one step, as the problem becomes ill-conditioned in the Lyapunov sense and is slowly convergent. The base solution is not a membrane because the moments and shear forces are not zero functions. Note that neglecting some boundary conditions in other methods usually (a) Long shell - one-step approach (b) Long shell - “base” approximation results in a singular system. Here a well-conditioned and non-singular problem is obtained. Powłoka długa - podejście jednoetapowe Powłoka długa - przybliżenie „bazowe” Fig. 10.21. Comparison of one-step and base approximations: function of the bending nwment in the meridian direction M n versus the coordinate measured along the cylinder meridian x[m\ Rys. 10.21. Porównanie przybliżeń jednoetapowego i bazowego: funkcja momentu zginającego w kie runku południkowym M n względem współrzędnej mierzonej wzdłuż południka walca x[m\ 10.5. Long shell - two-step approach 141 140 Chapter 10. Cylindrical shell subjected to a sinusoidal load

In [28] := BoundaryCondition [6] := 10.5.2. Boundary-layer refinement N u [bx] - rdl BoundaryCondition[6] = The base solution allows to refine the approximation at the boundary layers. In [29] := BoundaryCondition [7] := We are taking into account a limited domain, so the parameters describing the position of the Afi2 tbx] - rd2 boundaries take the values: BoundaryCondition[7] = In[30] :- BoundaryCondition[8] := I n [14]:= a x = - 3 0 ; Ain [bx] - rd3 BoundaryCondition[8] = In [15] :- bx = -288/10; h 5 ; As no symmetry can be expected, the approximation polynomials should contain both odd and I n [31]:= BoundaryCondition [9] := Mi2 [bx] - rd4 even functions. BoundaryCondition[9] = h3 PolyDegree In [16]:= *Wi [x_] := c [5 i] MonicChebyshevT[i/ ——— - l ] ; In [32]:= BoundaryCondition [10] := 1=0 ax - bx £2i [bx] - rd5 BoundaryCondition[10] In [17]:- -w-ztx-] := h3 ' PolyDegree Satisfying all boundary conditions on the fixed edge layer results in refined functions whose di c[5 i + 1] MonicChebyshevT[i, —----- —-----l ] ; t i a x - b x J agrams are shown in figures 10.23, 10.24 and 10.25 on the following page, 10.26 and 10.27 on page 143, 10.28, and 10.29 on page 144 and 10.30 on page 145, compared with the base so A full set of boundary conditions is now taken into account; for the fixed edge the boundary lution. It can be observed that taking into account additional boundary conditions results in a conditions are expressed by displacements. small decrease of displacements. It can be interpreted physically as a local increase of the shell

In[18]:= BoundaryCondition [1] := stiffness due to more constraints in the boundary conditions. The obtained approximation error is admissible for engineering purposes. BoundaryCondition [1] = 2>i [ax] E;

In[19] : = BoundaryCondition[2] := Displacement Component "W3 [mm] BoundaryCondition[2] = D2 [ a x ] E ;

In [20] :- BoundaryCondition [3] := B oundaryC ondition [3] = 'VV'itax] E;

In [21]:- BoundaryCondition [4] := BoundaryCondition [4] =TV2[ax]E ;

I n [22] : = BoundaryCondition[5] := BoundaryCondition[5] ='W3[ax]E; The boundary conditions on the “artificial” edge x = 28.8 are expressed by forces. It is assumed that the forces are equal to those obtained from the base approximation. To avoid a further loss of precision we can change the numerical values of base approximation by means of a R a tio n a liz e function to “exact” fractions. It is a kind of cheating, but ensures symbolic computations of the coefficients of the system of algebraic equations. Moreover the calculations x[m] take less time. Fig. 10.23. Boundary layer refinement of the “base" approximation: function of the displacement in the In [23] :- rdl = Rationalize[nll[bx], ; direction normal to the shell 'W^ versus the coordinate measured along the cylinder meridian ln[24] :- rd2 = R ationalize[nl2[bx], io-pr*ci*io“t“12tten ], Rys. 10.23. Uściślenie przybliżenia „bazowego" w warstwie brzegowej: funkcja przemieszczenia w kie runku normalnym do powłoki 'W3 względem współrzędnej mierzonej wzdłuż południka walca I n [25] : = rd3 = R ationalize [mil [bx], io-*™=i>ion[mii[b,]]j .

In [26] : = rd4 = Rationalize [ml2[bx], .

In[27] :- r d 5 = R a t i o n a l i z e [ q l [ b x ] , i o - ',™=i-i °” Cqi[bx] 1 ] . 142 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.5. Long shell - two-step approach 143

-0.05

- 0.1

-0.15

- 0.2

-0.25

-30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 -30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 x[m] x[m] Fig. 10.24. Boundary layer refinement of the “base " approximation: function of the rotation in the merid ian direction D\ versus the coordinate measured along the cylinder meridian Fig. 10.26. Boundary layer refinement of the “base” approximation: function of the stretching force in the meridian direction N\ i versus the coordinate measured along the cylinder meridian Rys. 10.24. Uściślenie przybliżenia „bazowego ” w warstwie brzegowej: funkcja obrotu w kierunku południkowym T>\ względem współrzędnej mierzonej wzdłuż południka walca Rys. 10.26. Uściślenie przybliżenia „ bazowego " w warstwie brzegowej: funkcja siły rozciągającej w kie runku południkowym N\ i względem współrzędnej mierzonej wzdłuż południka walca

Rotation Component £>2 [----- ] kN m Meridian Shear Force N i2 [— ] m 0.6 14 \ \ 0.5 \ 13 -----I Jase 0.4 ----- Fdefined 12 0.3 11 0.2 10 0.1 9

-30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 x [m\ Fig. 10.25. Boundary layer refinement of the “base” approximation: function of the rotation in the par allel direction versus the coordinate measured along the cylinder meridian Fig. 10.27. Boundary layer refinement of the “base” approximation: function of the shear force in the meridian direction Nn versus the coordinate measured along the cylinder meridian Rys. 10.25. Uściślenie przybliżenia „bazowego“ w warstwie brzegowej: funkcja obrotu w kierunku rów noleżnikowym D2 względem współrzędnej mierzonej wzdłuż południka walca Rys. 10.27. Uściślenie przybliżenia „bazowego” w warstwie brzegowej: funkcja siły ścinającej w kie runku południkowym N\2 względem współrzędnej mierzonej wzdłuż południka walca 144 Chapter 10. Cylindrical shell subjected to a sinusoidal load 10.5. Long shell - two-step approach 145

Meridian Transverse Force (2i [— ] m 0 1000 I - - - B:ise L -20 800 * — Kenned g| 600 1 -40 400 \ -60 200 \

0 . _____ -30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 -30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 x[m] x[m] Fig. 10.28. Boundary layer refinement of the “base” approximation: function of the bending moment in Fig. 10.30. Boundary layer refinement of the “base" approximation: function of transverse force at the the meridian direction M n versus the coordinate measured along the cylinder meridian meridian direction <3i versus coordinate measured along the cylinder meridian Rys. 10.28. Uściślenie przybliżenia „bazowego" w warstwie brzegowej: funkcja momentu zginającego Rys. 10.30. Uściślenie przybliżenia „bazowego " w warstwie brzegowej: funkcja siły poprzecznej w kie w kierunku południkowym M n względem współrzędnej mierzonej wzdłuż południka walca runku południkowym <2i względem współrzędnej mierzonej wzdłuż południka walca

The approximation has been carried out with polynomials whose degrees attained the value 25 to satisfy the conditions for the residuum (10.17) on page 129. The contribution [223] discusses also some other aspects of convergence and error analysis concerning the task considered in this chapter. -2 Resuming the computational experiment presented in the last two chapters we can now state that it is possible to neglect some boundary conditions approximating the boundary value problem -4 of long cylindrical shells. The approximate solution is feasible in most of the domain excluding the boundary layer. -6 The observed boundary-condition phenomenon should be interpreted physically and explained mathematically. This is done in the next chapter. -8

-30 -29.8 -29.6 -29.4 -29.2 -29 -28.8 x[m] Fig. 10.29. Boundary layer refinement of the “base" approximation: function of the torsion moment in the meridian direction M\\ versus the coordinate measured along the cylinder meridian Rys. 10.29. Uściślenie przybliżenia „bazowego ” w warstwie brzegowej: funkcja momentu skręcającego w kierunku południkowym Mn względem współrzędnej mierzonej wzdłuż południka walca 11.1. Physical interpretation of the boundary-condition phenomenon 147

11.1. PHYSICAL INTERPRETATION OF THE BOUNDARY- CONDITION PHENOMENON

The phenomenon connected with boundary conditions, presented in the previous two chap Chapter 11 ters: 9 on page 88 and 10 on page 119, which enables us to approximate the boundary-value problem taking into account a limited number of boundary conditions, has the following physi cal interpretation. PHYSICAL INTERPRETATION AND MATHEMATICAL Let us compute the integrals for stretching forces N iJ and moments M EXPLANATION h I— N iJ := f (6/ - zb j) rndz, (11.1) -h W tym rozdziale przedstawiono interpretację fizyczną i dokonano próby znalezienia wytłumaczenia zjawiska warunku brzegowego od strony matematycznej, gdyż żaden eksperyment nie jest dowo M ij := J ^ { 5j -zbj) ^'zdz, (11.2) dem. Analizuje się zadanie zginania prostego pręta na sprężystym podłożu poddanego działaniu dużych sił osiowych i opisanego równaniem różniczkowym czwartego rzędu. Pokazano, że poszukując -h rozwiązania bazowego zadania nie można pominąć warunków brzegowych odpowiadających najniższej with a precision of integrand expansion about the point of the variable z — 0 of the order equal pochodnej poszukiwanej funkcji, która występuje w równaniu różniczkowym. Ponadto stwierdzono, że to 1. To distinguish these results from those obtained more precisely they will be emphasized zarówno zjawisko jak i metoda dwuetapowa nie ograniczają się do zadań z warstwą brzegową oraz by a bar over the symbol. to, że zastosowanie metody umożliwia rozszerzenie możliwości systemu Mathematics w znajdowaniu rozwiązań przybliżonych zadań brzegowych. Then we obtain that the tensor N iJ is symmetrical and depends only on the first strain tensor:

This chapter presents a physical interpretation and attempt to explain the boundary-condition . u 2 E h (v aij y / + ( 1 - v) y j) , ,, phenomenon from the mathematical point of view, since no experiment is a proof. The task of bending a N ,J := i - L-L + 0(h3), (11.3) straight bar, subjected to large axial forces, on elastic subsoil and described by a differential equation of 1 - V2 v 7 the fourth order. It has been shown that looking for a base solution the boundary conditions correspond and the tensor of moments is equal to zero, precisely: ing to the lowest derivative of the function in the differential equation must not be neglected. Moreover, it has been found that both the phenomenon and the method are not limited to tasks with a boundary M iJ - 0 + 0 (h3) . (11.4) layer and that application of the method permits an extension of the Mathematics system capabilities in searching approximations of boundary-value problems. It yields also that transverse forces are equal to zero too, due to the equation of equilibrium:

Despite the fact that the result of computational experiments presented in the previous two Mij.%i - Qj = 0. (11.5) examples, that the boundary-value problem can be approximated taking into account limited number of boundary conditions, holds true (see chapters 9 on page 88 and 10 on page 119), it This is called a membrane approximation and may be used for engineering purposes in many would be good to try to understand the boundary-condition phenomenon. It was not a simple task and took a lot of time. cases of shells. It fails in cases when bending cannot be neglected. If we neglect moments and transverse forces, the system of equations of equilibrium is reduced Probably that is not all that can be said about the explanation but at least an attempt of un derstanding. This explanation is not a formal mathematical proof, which remains to be done. Nevertheless, it must be emphasized that MATHEMATICS helped me, as a professional civil en gineer, understand the crucial aspects of the phenomenon. + pj = 0, (11.6)

bijŃij + P3 = 0 . (11.7)

146 149 148 Chapter 11. Physical interpretation and mathematical explanation 11.2. Explicit problem — as a base of mathematical explanation

The membrane approach is internally statically determinate. The tensor equation (11.6) on the 11.2.1. Differential equation of a straight bar page before yields 2 differential equations and the equation (11.7) on the preceding page is an To do that, let us consider a straight bar supported on the elastic Winkler subsoil and subjected algebraic one. It means that the number of unknown functions (Nn, N 12 — N 2' and N 22), which are internal forces, is equal to the number of equations of equilibrium. The membrane to a large axial force and a transverse, distributed load. The length of the bar is I, so the problem approach can be sometimes solved exactly, especially for shells of revolution. Examples are domain is x e (0,1). presented in [23, 182, 183, 184, 185, 186, 188, 189, 191] since these problems can be reduced Taking into account the influence of a large axial force - the moments M{x) can be expressed to very simple equations [190]. by the following equation: Substitution of the first strain tensor in terms of displacements, (11.8): M(x) := - EJ(x)y"(x) + n(x)y(x), (11.11)

7ij := ^ (n ■ wj + rj ■ Wi) - £, aiJt (11.8) where, EJ(x) is a function of the bending stiffness, y(x) is a function of normal displacements and n(x) is a function of axial forces. into the constitutive relation (11.4) on the preceding page and then into equations of equilibrium (11.6) and (11.7) on the page before results in 3 differential equations of the order 2 with respect In[l] := M [x_] := - E J [x ] y" [ x ] + fl[x ] J /[x ] to displacements IV i (x), 'Wj, (x) and one algebraic equation (of the order 0) with respect to Transverse forces Q(x) are related to moments with the following equation of equilibrium nV3 (x). So the order of the differential operator is equal to 4 and can satisfy only boundary conditions of the first type (see sections 9.3.1. on page 90 and 10.1.3. on page 121): Q(x) = M \x) + p(x), (11.12) two displacements on fixed edge: where fi(x) is a function of load moments bi ■■=

In [3] := Q' [x] + />[x] - *[x] y [ x ] ; b3 N\ i (xt) — 0, where k{x) is a function of the subsoil elasticity and p(x) is a normal load. (11.10) b4 := N n (xb) = 0. Substituting (11.11) into (11.12) and then into (11.13), we obtain the following differential equation of the fourth order. Other conditions can be satisfied only by chance. Nevertheless, the consideration of the bound ary conditions of the first type is necessary and enough to satisfy the condition of general sta In [4] := e [1 ] = bility. The influence of other boundary conditions in shells is usually limited to the boundary Sim plify/gC ollect [Expand[%], {m/[x] , w' [ x ] , w" [x ], iv<3) [x ], iv<4) [x] }] Out [4]= p [x] + 2 rf [x] w' [x] + // [x] + k^x] (-*[x]+ n" [ x ] ) + (n[x]- E J " [ x ] ) w" [x] - layer. Therefore neglecting the additional (second type) boundary conditions during the approx 2 E J ' [x] tv<3) [ x ] - E J [ x ] w(,) [ x ] imation of the 1 O'* order operator does not result in any singularity of the system matrix. which can be rewritten in the following form: Besides the physical interpretation the problem should be explained mathematically. To do that let us consider a rather simpler problem. w(x) (k (x) - ri' (x)) — 2 ri (x) W (x) + (EJ" (*) -n (x )) W‘ U)+ 2 EJ' (x) u/3) (*) + +EJ (x) w<4) (jc) = p(x)+ p! (x). 11.2. EXPLICIT PROBLEM — AS A BASE OF MATHEMATICAL EXPLANATION If k{x) ■■=%— const, n(x) := N = const and EJ(x) := 8J = const, the equation takes a simpler form Equations of shells are an implicit system of differential equations of five functions. To find a mathematical explanation of the phenomenon, let us consider a physical problem which is (11.15) described by an explicit differential equation of one function.

It is obvious that the differential equation of an order equal to 4 should satisfy 4 boundary or • Case 4 initial conditions. Let us consider the case when both edges of the bar are fixed. In this case the And finally, if both K—O and N = 0, and &SF t 0: function of normal displacements should satisfy: y(0) = y(l) := 0 and y'(0) — y’( l ) 0.

y(x)-.= Ci+ xC2 +x1C 3 + ^ 4- ^ . (11.19) 11.2.2. Solutions of the equation In this case no boundary condition can be neglected applying the Refined Least Squares It is possible to find an exact solution of the equation (11.15) on the page before. To focus our Method. attention, let us consider this equation with the following right-hand side: p(x)+n’( x ) sin(x). Depending on the value of the constants 7C, N and 8 J we distinguish four specific cases: 11.3. ILLUSTRATING EXAMPLES

• Case 1 The consideration will be illustrated by two examples of two specific cases. In both cases If % ± 0, N ± 0 and + 0: I := 167T, sox e <0, 16tt). The Refined Least Squares Method is more general than, for example, asymptotic approaches. There is no need to lower the degree of the differential operator and assume a small parameter. ( 2 . ( H-y/N2-i£JK\ ^ It will be shown within the examples. y(x) := e V ) c, + e V ^ ) c2+ , ,_____Ni , . l (11.16) , I N+'JH'1-A&.TK \ I №-Va/2-»&7*’ 1 • , v + ex{ -^ r ~ ) c3 + e c4-—S ln W 11.3.1. Illustration of the first case S J + K + W Let the equation (11.15) on page 149 have the following form: It can be checked that in this case all boundary conditions may be neglected applying the Refined Least Squares approach. The obtained base solution is equivalent to the particular - y(x) + 2y'\x) - yw(x) = (jc- 15^) (x - 13;r) (x-3n) (x-n). (11.20) integral of the equation (11.16). • Case 2 Note that none of the coefficients of this differential equation can be regarded as a small param eter. If 7f * 0 ,^ = 0 and £J-*0: The exact solution of the problem is:

y(x) := e~a~0x V ^ c , + e(l~i)x^ C 1+ y(x) := - 72 - 1240tt2 - 585?r4 + 96;r (4 + 9n2) x+ + V & c, + e"+i)x c. - Sin(x) ° 1 '17) - 2 (12+ 155 tt2) x2 + 32 jtx3 - x 4+ SJ+'K' 72+ 1240 n2 + 585 n4 In this case all boundary conditions may be neglected, too. + e? (e32,r + 32e16,r7r- l) (11.21) • Case3 • ^ (e16* (16tt- 1 - x) + e2(S,r+x) (1 + 167r-x) + e2j: (x- D + e32* (1 + x)) + If 7C = 0, JV £ 0 and &ST t 0: + 96tt (4 + 97i2) (e2(8,r+JC) (I6 n - x) + e32* x - e2x x +e16* (x - 167r))^.

y{x) := C, + x C2 + f C3 + e-xJ& C t - (11.18) Neglecting all boundary conditions, we receive a base solution, which is, in this case, equal to o J + N its particular integral: In this case only two boundary conditions can be neglected. They are y'(0) = y’(l) := 0.

yb(x) := - 72 - 1240TT2 - 585n4 + 96tt (4 + 9tt2) x+ (11.22) - 2 (12 + 155tt2) x2 + 32nx3 - x 4. 152 Chapter 11. Physical interpretation and mathematical explanation 11.4. Attempt of a mathematical explanation 153

As it is a polynomial, the Refined Least Squares algorithm finds its exact closed form. It only Neglecting two boundary conditions and taking into account two others we receive the base so requires an exact solution of the system of linear algebraic equations, which is possible, in lution yb(x), which is not equal to the particular integral of the equation (11.23) on the preceding reasonable time, in the case of small systems. page as it actually satisfies equation (11.23) on the facing page (exactly) and two boundary con Taking into account all boundary conditions it is impossible to find an exact solution by means ditions (yb(0) — yb( 16 7i) := 0). of polynomial approximation. The degree of the polynomial exceeds 40 to receive a satisfac tory approximation, whereas the base approximation which is feasible in most of the domain -An (90 - 810tt2 + 6983 tt4) x (6 -3 1 0 ^ + 585^) x2 excluding the boundary layer requires a polynomial degree equal to 4. yb(x) ■■=------—------+ ------j ------+ 15 4 (11.25) The diagrams of base and exact solutions are presented in Figure 11.1. , , (-3 + 155^2)x 4 47TX5 x6 + 8?r (l - 9j?) x^ + ------— -!■------+ 12 5 60 «/(x) 10-4 Figure 11.2 shows the results of the approximation of the exact solution (Exact in the leg x end) with consideration of all (Approx) and neglecting two boundary conditions (Base) and a polynomial degree equal to four in both cases of approximation. It can be seen that the exact solution “oscillates” around the base solution. If all boundary conditions are taken into ac count, the convergence is rather poor. In this case satisfactory approximation is obtained for the polynomial degree equal to 46! Moreover we have found that the approach is not limited to problems with a boundary layer but is also applicable if the influence of the boundary conditions propagates over the whole domain of the problem.

yOO IQ'6

Fig. 11.1. Exact and base solutions of the equation ( 11.20) ^ 4 TT 6 00 n 1( n 12 n 1 ' n — £TV2 A Rys. 11.1. Dokładne i bazowe rozwiązania równania (11.20) y s V - 4 ”

— E xact 11.3.2. Illustration of the third case —tn — Base /1 — A p p ro )c l Z 4 Let the equation (11.15) on page 149 be: O- /] — JLÜ- 2 t/"(x) + j/4)(x) = (x - 15 n) (x - 13 n) (x - 3 n) (x - n). (11.23) - i ? h The exact solution of the problem is: Fig. 11.2. Exact, base and approximate solutions of the equation (11.23) -4 n (90 -810TT2 + 6983 n4) x (6 - 310^ + 585 ;r4) x2 y(x) :=------^------>— + S------L— + Rys. 11.2. Dokładne, bazowe i przybliżone rozwiązanie równania (11.23)

+ 8* (1 - 9/r2) x3 + (d L L 1557r2) + V ’ 12 5 60 11.4. ATTEMPT OF A MATHEMATICAL EXPLANATION - 4 n (9 0 - 8107T2 + 6983;r4) fcos (8 V L r)-c o s (V2 (8jt-x)) ) • (1L24) 11.4.1. Explicit tasks 8 V 2 tt c o it (8 V2^]1 - 1

OO This consideration makes it possible to try to explain the mathematical mechanism of the 240 Tt cos ^ 8 V2tt^1 -1 5 V2 sin | phenomenon. In the first and second cases the differential equation can be presented in the 155 11.4. Attempt o f a mathematical explanation 154 Chapter 11. Physical interpretation and mathematical explanation form: 2 E h (1 - h2 K) (v a‘j y / + (1 - v) y j) , , Nu"{x) - £ J y w(x) + p(x) + m'(x) ~ 1 - v 2 + y(x) = k • (11 26) 4 Eh1 y™ (v bpq (H aiJ + bij) + (1 - v) bp‘ ( V + H V ) ) The equation looks like an algebraic one, now. Although the functions y"{x) and / 4)(jc) depend + 3 (1 - v 2) + on y(x), they may be formally regarded as independent ones. Therefore, all boundary conditions 8 E h3 (v aij bpq pw + (1 - v) V P") (11.31) may be neglected, as an algebraic equation does not require any. Nevertheless, the base solu 3 (1 - v2) + tion is not a result of the approximation of the algebraic equation but satisfies the differential equation and its boundary conditions only by chance. 4 E h3 (2 H V - V ) (v a9‘ Pp” + ( ! “ >') Pq0 In the third case the equation can be presented in the form: + M T ^ ) + 2Etf (v ^V + q - , ) » 0 „, x £>Jym(x) - p(x) - m'(x) 3 (1 - v1) ' ’ y (x) = ------—. (11.27) N can formally, taking into account the definition (11.3) on page 147, be denoted as: If we integrate both sides twice, we obtain a form similar to (11.26). NiJ := N ij + N ij, (11.32) ,lM __ n n , II & Jy{A\x) - p(x) - m'(x) , , y(x) — C1 +XC2 + / / ------—------dxdx. (11.28) // where N ‘J is defined by (11.3) on page 147 and N ,J is: It is now obvious that two boundary (initial) conditions must be satisfied in this case; for this boundary value problem they are y(0) — y(l) ■■= 0. 2E h3K (v d> ypp + (1 - v) yJ) In the fourth case we can provide the following transformation: _ 1-v2 + 4 Eh3 r q (vbpq (H a‘j + b‘j) + (1 - v) bp‘ [bqj + H V ) ) (1U9) + ; ^ —+ 3 (1 -v 2) 8E/z3 (v aij bpq pPi + (1 - v) bpl ppj) (11.33) Then no boundary condition, from among the four, may be neglected because the equation takes 3 (1 - v2) + a form similar to (11.26) or (11.28) after four times integration of the equation on both sides: 4 Eh3 (2 H V - V ) (v a9‘P / + (1 -v) p«‘) + + y(x) = Cl +xC2 +x2 C3 +x3 C4 + //// dxdx dxdx. (11.30) 2 Eh3 (v aiJ dpp + (1 - v)

to the already discussed explicit problem. The part responsible for membrane behavior is ex tracted here, similarly to the extraction of the lowest derivative in the explicit problem, compare equations (11.26) or (11.27) on page 154. x ( 167T + x)\ w + 2 y 'W +y4)w = (JC_ i5>r) (*_ 13*) (x- 3 tt) (.x - k ). (11.36) 64 it2 J This explains the boundary-condition phenomenon from the mathematical point of view. Since the membrane approach, which is the simplest and usually physically acceptable approximation Figure 11.3 shows the result of approximating the equation. A satisfactory base approximation of the shell boundary-value problems, requires only 4 boundary conditions in the considered (neglecting of boundary conditions) was obtained with a polynomial degree equal to 10 and an tasks, other conditions can be neglected in search for a base solution. “exact” one was received with a polynomial degree equal to 32. It is impossible to find a closed Nevertheless, the base solution of the Refined Least Squares Method finds neither the membrane solution in this case, as the differential equation (11.36) has polynomial coefficients. approximation in the case of shell problems nor the solution of algebraic equation in the case of equation (11.20) on page 151. The base solution approximates the problem defined by the *(x) 10"5 10,h order differential operator or the differential equation, but, of course, with negligence of boundary conditions of the second type. The base solution which is free from influences of some or sometimes every boundary condition enables us to find an approximation which is close to the particular integral of the problem far from the boundary layer, where the influences of boundary conditions of the second type are negligible or in some cases finds a function is found for which the exact solution oscillates around it. x 11.5. EXTENSION OF MATHEMATICS POSSIBILITIES

It is remarkable that the Refined Least Squares approach extends the MATHEMATICS possibilities due to dealing with differential equations. The function DSolve for the solving of differential Fig. 11.3. “Exact" and base solutions of the equation (11.36) equations fails in the case of a high-order differential equation with non-constant coefficients. For example: Rys. 11.3. „Dokładne" i bazowe rozwiązania równania (11.36)

r / X ( - 1 6 n + X) > „ ... , In [5]:= DSolve [(l - ■■ ^ ^ ------)y[x] + 2y"[x] + y(4)[x] = 0,y[x],x] 11.6. ESTIMATION OF THE QUALITY OF THE GLOBAL Out [5]= DSolve[(l - —^ - 7 - 5— ^) y [X1 + 2 y" [x] + y (4) [x] = 0, y [x ], x] L 64 7TZ ' J CONVERGENCE The numerical approach NDSolve works well with initial-value problems,

In [6] := NDSolve[{y(4) [x] + y" [x] + y [x] = 1, y [0] = 1, y/ [0] = 1, Let the last example be an occasion to present again the way of estimating the quality of the y " [ 0 ] ==2 , y '" [ 0 ] ==2 },y[x], {x, 0 , 1 }] global convergence. This can be achieved by substituting the solution into the functional (7.2) Out [6]= {{y[x]-> InterpolatingFunction [{{0., 1.}), <>] [x])} on page 74. If the functional attains a value close to zero, it means that the approximation is but sometimes fails for boundary-value problems. acceptable. The question is what “close to zero” means was already explained in section 10.3.1. on In[7] NDSolve[{y(4> [x] + y" [x] +y[x] = 1, y[0] = 1, y[l] = 1, page 126. The recipe for it is: if the value of the functional root \JTip) is considerably smaller y ' [ 0 ] = 2 , y (3> [ l ] = 2 },y[x], {x, 0 , 1 }] Out [7]= NDSolve [{y[x] +y"[x] + y(4)[x] = 1, y [0] = 1, than the value of the functional for p = 0, where p is an approximating polynomial degree, then a satisfactory approximation is attained. Figures 11.4 and 11.5 on the following page show the y[l] = 1/ y'[0] = 2, y'[l] =2}, y[x], {x, 0, 1}] convergence of the functional (its root) with respect to the polynomial degree. In the case of The Refined Least Squares algorithm finds an approximate solution in such cases. Let us con base approximation is much quicker than in the case when all boundary conditions are taken sider a similar equation with the same boundary conditions and the domain as in previous ex into consideration. amples. 158 Chapter 11. Physical interpretation and mathematical explanation

Chapter 12

REISSNER-MINDLIN PLATE

W tym rozdziale przedstawiono zastosowanie Metody Najmniejszych Kwadratów do płyt średniej grubo ści opisanych związkami konstytutywnymi Reissner’a-Mindlin’a. Jako, że w danym przypadku występuje zjawisko warstwy brzegowej zastąpiono dwa warunki brzegowe jednym. Wykazano, że zadanie jest wte Fig. 11.4. Base approximation: convergence of the square root of the functional versus the degree of the approximating polynomial dy szybciej zbieżne do wyniku akceptowalnego z punktu widzenia inżynierskiego. Tym samym wska zano na szersze pole zastosowania proponowanego podejścia dwuetapowego. Proponowana metoda Rys. 11.4. Przybliżenie bazowe: zbieżność pierwiastka kwadratowego funkcjonału ze względu na stopień wielomianu aproksymującego pozwala wiarygodnie ocenić błąd aproksymacji i umożliwia wykrycie zjawiska pozornej zbieżności. Te zagadnienia zostały zwięźle przedyskutowane na końcu tego rozdziału.

This chapter presents the application of the Least Squares Method in mid-thickness plates described by Reissner-Mindlin constitutive relations. As in the considered case we also encounter a boundary layer y fr phenomenon, so two boundary conditions were replaced by one. The task proved to be quickly conver 300000 gent with results acceptable from the engineering point of view. Thus, a wider field of the application of the proposed two-step approach has been indicated. The proposed method permits to reliably estimate 250000 an error of the approximation and makes it possible to detect a false convergence. These problems have 200000 been briefly discusses in the end of this chapter.

150000

100000 This chapter shows the results, already presented in [218], of the attempt to apply this method to the two-dimensional boundary value problem of medium-thickness plates described by the 50000 Reissner-Mindlin constitutive relations. The boundary layer phenomenon occurs and for that reason the approximation has a poor convergence. Therefore the idea of neglecting some bound ary conditions seems to be reasonable. 5 10 15 20 25'

Fig. 11.5. Approximation of an “exact” solution: convergence of the square root of the functional versus 12.1. NOTATIONS AND VALUES OF DATA the degree of the approximating polynomial Rys. 11.5. Przebliżenie rozwiązania „dokładnego zbieżność pierwiastka kwadratowego funkcjonału ze The equations in this chapter are formulated according to the generic theory of plates, względu na stopień wielomianu aproksymującego Z. Kqczkowski [72], so let us introduce the notations valid only in this chapter. Mxx — bending moment in the direction x Myy — bending moment in the direction y Mxy — torsion moment

159 160 Chapter 12. Reissner-Mindlin plate 12.2. Problem description ______161_

Qx — transverse force in the direction x e.:= ^ + ^+<7 = 0, (12.1) Qy — transverse force in in the direction y ox oy Vx — substitute transverse force in the direction x dMxx _ dMxy Vy — substitute transverse force in the direction y := ~— + -r-z ~Qx = 0, ( 12.2) ox oy w — normal displacement (fix — rotation in the direction x e^:=d-j f + d-jf~Qy = °- (12-3) ipy — rotation in the direction y Reissner-Mindlin plates are described by the following constitutive relations: E — Young modulus (E 2 ■ 105 MPa) v — Poisson ratio (v := 0.3) I t f . a f * ’ (12.4, h — plate thickness (h := 0.50 m) 12 (1 -v2) \dx dy ) q — uniformly distributed load (q := 1.0 kPa) M •= Ehi ( d_*l + vd_?±\ (125) Other notations are provided in the further text. 12(l-v2) \ dx +vdy ) ’ ( )

m - E h 3 fd tp x d

Let us consider a square plate fixed on 2 perpendicular, neighboring edges x = 0, y = 0 and ^ 5Eh (dw \ free on two others: x — a, y = a, a := 5 m, see Figure 12.1. Qx ~ 12(1 +v) (^7 +iPx) '

5 Eh f dw \ ^ :=120TV ) { ^ + ^)- (12-8) Substitution of the constitutive relations into the equations of equilibrium results in a set of three differential equations of order two in terms of normal displacements w and rotations

b{ -.= w — 0, b2 ■= (fix = 0, b3 := ipy = 0, (12.9)

free boundary x = a:

Z>4 := Mxx — 0, b5 :=Mxy = 0, b6 := Qx = 0, (12.10) Fig. 12.1. Sketch of the plate fixed boundary y = 0: Rys. 12.1. Szkic płyty b7 :=w — 0, b% '■= ipx = 0, bg ■■= ipy == 0, (12.11) The considered problem of Reissner-Mindlin, E. Reissner [145], and R. Mindlin [110] - compare Z. Kączkowski [72], plate (medium thickness) is described by the following system of equations free boundary y - a: of equilibrium: b\o := Myy — 0, b n Mxy — 0, b n Qy — 0. (12.12) 12.3. Approximation 162 Chapter 12. Reissner-Mindlin plate

12.3. APPROXIMATION

The approximation is based on minimizing the following functional:

a a 3 a 6 a 12 T ■= J J '^ j ei dxdy + J '^ T b i dx + J ^ f c , dy. (12.13) 0 0 1=1 0 i=1 0 1=7 The results of the approximation with natural boundary conditions are presented on the left of Figures: 12.2 and 12.3 on the facing page, 12.4 and 12.5 on page 164. The approximation was obtained with polynomials of a degree equal to 18. The convergence decreases if the plate is thinner. It is easy to explain when we look at Figure 12.4(a) on page 164. The function of moments is steep near the free edges. Therefore substitute boundary conditions (12.14) and (12.15) were applied instead of natural (a) Natural boundary conditions (b) Substitute boundary conditions ones (12.10) on the page before and (12.12) on the preceding page, respectively: Naturalne warunki brzegowe Zastępcze warunki brzegowe free boundary x — a : Fig. 12.2. Normal displacement w\m] /5 A /f Rys. 12.2. Przemieszczenie normalne w[m\ b4 :=Mxx = 0, bs :=Vx = Qx+— ^ = 0, *6:=0 = 0, (12.14) dy free boundary y — a:

b\0 := Myy = 0, bn :=Vy = Qy+ - ^ = 0, bl2 := 0 = 0. (12.15)

The substitute transverse force, compare Kqczkowski [72], used in the theory of thin plates has been applied, so that two boundary conditions were substituted by one. The differential operator of the sixth order is approximated with 10 boundary conditions. It should be remarked that it is impossible to proceed with, for example, the Finite Differences Method. The results of approximation with substitute boundary conditions are presented on the right of Figures: 12.2 and 12.3 on the next page, 12.4 and 12.5 on page 164. The convergence is much better (compare the left and the right side of Figure 12.3. on page 165); the presented results of forces were received here with polynomials of degree 12. Comparing both approximations, with natural and substitute boundary conditions, we can ob serve a very good coincidence in the case of functions of normal displacements and bending moments. The functions of torsion moments and transverse forces are comparative in most of the domain (a) Natural boundary conditions (b) Substitute boundary conditions Naturalne warunki brzegowe Zastępcze warunki brzegowe except the boundary layer. This is shown in Figure 12.7 on page 165, where the function of the torsion moment is presented for y = 2.5 m (the cross-section in the middle of the plate). Fig. 123. Bending moment Mxx Figure 12.7 on page 165 shows a comparison of the torsion moment M^ [^p] in the cross- section y = 2.5 m for natural (NBC) and substitute (DBC) boundary conditions. Rys. 12.3. Moment zginający Ma [^-p] 164 Chapter 12. Reissner-Mindlin plate 12.3. Approximation ______165

(j j) M

(a) Natural boundary conditions (b) Substitute boundary conditions Naturalne warunki brzegowe Zastępcze warunki brzegowe

(a) Natural boundary conditions (b) Substitute boundary conditions Fig. 12.6. Convergence of the normal displacement in the center of the plate, w (|, |) versus the degree Naturalne warunki brzegowe Zastępcze warunki brzegowe of approximating polynomial Rys. 12.6. Zbieżność przemieszczenia normalnego w środku płyty, w (|, |) ze względu na stopień wielo Fig. 12.4. Torsion moment Mxy mianu aproksymującego Rys. 12.4. Moment skręcający Mxy

- 0.2 -0.4

- 0.6

- 0.8

-1

- 1.2 x[m] (a) Natural boundary conditions (b) Substitute boundary conditions Fig. 12.7. Comparison of the torsion moment in the cross-section y == 2.5 m (in the middle of the Naturalne warunki brzegowe Zastępcze warunki brzegowe span of the plate) Rys. 12.7. Porównanie momentu skręcającego Mxy w przekroju poprzecznym y — 2.5 m (w połowie Fig. 12.5. Transverse force Qx [^] rozpiętości płyty) Rys. 12.5. Siła poprzeczna Qx [^] 166 Chapter 12. Reissner-Mindlin plate 12.5. Some remarks on the analysis of error and convergence 167

12.4. PHYSICAL INTERPRETATION convergence of the functional square root with respect to the degree of the polynomial. It can be seen that its value is relatively small, in comparison with Q, in the case if the degree of the Similarly as in the case of shells this phenomenon has a physical interpretation. The problem polynomial is larger or equal to 18. Therefore we may accept the approximation with a polyno of plates is internally statically indeterminate (we have 3 equations and 5 unknown forces). On mial of this degree. Figure 12.6(a) on page 165 shows that satisfactory convergence is attained the other hand the condition of rotation parallelly to the fixed edge is satisfied automatically by for the displacement in the center of the plate for p = 12. However, in this case T < Q, only. the condition of normal displacement. This condition may also be neglected. By analogy, the Thus, we should continue the approximation and stop it when T « Q. corresponding condition on another edge can be neglected, too. The natural boundary condition can be replaced by a substitute transverse force analogically to the theory of thin plates and VF due to the fact that this artificial condition does not enforce satisfaction of the natural boundary conditions, neither moment of torsion, nor transverse force. The method “chooses” the function which is most suitable to minimize the functional. It results in approximation, which has a better convergence as the approximation is free of the boundary layer phenomenon.

12.5. SOME REMARKS ON THE ANALYSIS OF ERROR AND CONVERGENCE

The criterion of the estimation of the global error for two dimensional problems is very similar to that presented in section 10.3. on page 125. The difference is that we have to use double integrals computing Q and 'H functionals. The formulas are similar but a bit longer and will not be presented here.

The computation of § is simple in the considered case. It has been said, compare remark on Fig. 12.8. Convergence of the square root of the functional versus the degree of the polynomial approxi page 127, that Q can be computed from the formula for the functional IT setting all searched mating the task functions to be zero. By the way, the value of the functional T for the polynomial degree Rys. 12.8. Zbieżność pierwiastka kwadratowego funkcjonału ze względu na stopień wielomianu aproksy- p — 0 is equal to Q. Presuming constant functions of approximation and substituting them into mującego zadanie equations (12.1) on page 161, (12.2) and (12.3) on page 161 we obtain zero for all derivatives present in them. Therefore the algorithm has got no information what to do with the equations According to all that we can state again that we have an additional reliable tool for convergence since no decision parameters are present in them. The only actual data concern the satisfaction and error analysis by means of the Least Squares Method. The convergence of results may lead of these boundary conditions that are expressed in displacements. Since they should be zero on to wrong conclusions about finishing the approximation. We meet this problem in this example. the fixed boundary the algorithm returns zero both for displacements and rotation in every point Note, that for degrees of the polynomial equal to 4, 5 and 6 we have in the case of natural of the domain. boundary conditions a so called “false convergence”. Nevertheless the value of the functional If we look at the equations (12.1) on page 161, (12.2) and (12.3) on page 161, and boundary con T is large enough to recognize this phenomenon and makes us continue the approximation with ditions (12.9), (12.10), (12.11) and (12.12) on page 161 we find that the only non-homogenous a higher degree of the polynomial. equation is (12.1), since q := 1.0 kPa. Of course, we can use other known methods of error analysis, like local and global analysis of It can be easily found that for the considered task setting all functions equal to zero in the equilibrium (see the remark on page 104 in section 9.4.6.), energy balance and so on. formula for the functional !F (12.13) on page 162 we receive:

u a Vq = q dx dy - 5. (12.16) JJ 0 0

The functional T was computed for natural boundary conditions and for every approximat ing polynomial of the degree p = 0,1,..., 20. Figure 12.8 on the facing page presents the 13.2. Boundary value problems 169

13.2. BOUNDARY VALUE PROBLEMS

1. Some approach to the Least Squares Method was implemented in the computer algebra Chapter 13 system and applied to the boundary-value problems. The refinement of the method is based on the idea that boundary conditions are considered within the minimized func tional. This idea resulted in a not only simpler implementation of the algorithm but also CONCLUSIONS in a better convergence of the approximation. 2. Difficulties with the stability of high-degree polynomials degree stability were overcome. This was possible thanks to the analytical integration and arbitrary precision provided by computer the algebra system and the use of monic Chebyshev polynomials for approxi 13.1. SYMBOLIC COMPUTATIONS mation. 3. It has been found that it is possible to approximate high-order boundary-value problems 1. The entire process of the problem formulation from general relations in tensor notation to by means of the Refined Least Squares Method with consideration of selected boundary expressions ready for computations may be computer assisted (aided). conditions. The task becomes becomes well-posed and its solution is quickly convergent. 2. Each step of computation requires different tools. The contribution presents crucial ele 4. The obtained solution, called a “base” one, is feasible in most of the domain except the ments concerning the application of computer algebra in different aspects. boundary layer. It had also been shown that in some cases the exact solution oscillates 3. It is possible to evaluate integrals in constitutive relations without any assumptions lead around the base solution. ing to the degeneration of expressions. An effective procedure has been shown to evaluate 5. The observed phenomenon, consisting in neglecting some boundary conditions, has its the expressions for tensors of stretching forces and moments. The obtained equations are physical interpretation and has been explained mathematically. more general and describe the problem better. The constitutive relations satisfy the last equation of equilibrium. By its means one inconsistency in the theory of shells has been 6. Boundary conditions neglected in the first step can be satisfied locally within the boundary overcome. layer, in the second step.

4. Besides standard simplification tools the MATHEMATICS system has a range of other func 7. The comparison of the method with other approaches has not been the aim of this contri tions to handle symbolic expressions. It is shown how to use these tools effectively to bution. Most contemporary, so called computer methods, like Finite Element, Boundary obtain a additional simplification meeting the needs of the user. Element, Finite Differences or some meshless approaches base on local approximation. It makes them relatively quick but sensitive, for example, to instability connected with steep 5. Tensor notation enables us to describe reliably and effectively all phenomena in mechan functions. It is worth mentioning that the proposed approach is a global one. Therefore it ics and physics. It is especially useful to describe problems in a curved space. However, requires high order approximation, especially in problems described by steep functions. computations carried outby hand are prone to error and boring. The presented problems In such cases computations are relatively slow. However, the method is not sensitive of shells may be used as, at least, an inspiration how to apply computer algebra in solving to instabilities including the Lyapunov one, and connected with considerations of steep similar or even more advanced tasks in other sciences. functions. Some asymptotic approaches applied to the problem with a boundary layer require the reduction of the order of the considered functional. 6. The applied system is very powerful. Nevertheless, the results of symbolic computation have to be carefully scrutinized. Computer algebra must be regarded as a tool not a 8. The Refined Least Squares Method enables us to consider a problem described by a high wizard. Full responsibility for the obtained results is up to the user. Therefore special order operator with a limited set of boundary or initial conditions, an still obtain feasi emphasis is put on the scrutiny of results. ble approximation. Therefore the proposed method can be used not only as a tool for verification of the other methods but can be an effective alternative for them. 7. MATHEMATICS language and input is intuitive. Its symbolic output is readable and legible. It can be easily text-processed. The computer algebra system has been applied to produce 9. Moreover, it has been shown that the method has additional reliable tools of both global formulas and most of graphics in this book. Processing of results is important technical and local error and convergence analysis. Therefore one can detect a “false” convergence matter in scientific work. phenomenon. Tools for the analysis of the error used in methods based on local approxi mation are also available. 168 170 Chapter 13. Conclusions

10. The results of the method are functions. They are portable and can be used elsewhere. There is no need to interpolate or extrapolate the results. Differentiation, integration or other algebraic operations are straightforward. The method is in that sense analytical. In most cases the result is approximate but there is a chance to get an exact (closed) solution. Chapter 14

13.3. ROLE OF COMPUTER ALGEBRA AND CONTRIBUTION TO ITS DEVELOPMENT FURTHER DEVELOPMENTS

These accomplishments were possible with the assistance of computer algebra in the entire process of symbolic and numerical computations. On the other hand this work is a contribution of the Author to the development of the system, dissemination and its application. Some of There are several problems that seem to be open and worth to be researched in future. Some of them are: them have been already mentioned in the text. Let us collect them together.

1. Despite the more than 40 years of the development of computer algebra systems the 1. The most important matter seems to be the research of weight presumption. This problem knowledge about their possibilities is still small or at least not sufficient. The contribution occurs in many computational methods but in the Refined Least Squares Method it seems presents the application of the MATHEMATICS system to advanced problems of computa to be of crucial significance. Weights decide about the convergence path. The proper tional mechanics and engineering. Advanced, non-standard tools have been presented choice of weights may speed up computations and improve numerical stability. showing wide possibilities of the system. The applications to symbolic and numerical 2. It would be interesting to find out if the observed phenomenon connected with neglecting problems of the theory of shells, continuum mechanics and other tasks presented here boundary (initial) conditions can be possible in other methods, especially the Finite Ele and in other publications can be (let us hope) an inspiration for the use of computer alge ment one. It seems to be possible, since the FEM can be formulated basing on the Least bra systems in other and possibly even more advanced tasks. Squares Method, and the boundary conditions in it are appended to the stiffness matrix. 2. The methods of approximate (numerical) solutions of differential equations are well im 3. The Refined Least Squares algorithm requires further development. The notebooks have plemented within the MATHEMATICS system concerning initial-value problems. The class of boundary-value problems possible to approximate within the MATHEMATICS system is to be transformed into computational packages. still very limited. The implementation of the Refined Least Squares Method enables us to 4. The computational speed, especially concerning multidimensional problems, is not fully solve a wider range of such problems. satisfactory. Special emphasis has to be put on integration in the process of building up the system matrix. Fortunately, Wolfram Research, Inc. developed recently a system called 3. The developed Refined Least Squares Method was applied to ill-conditioned problems gridMATHEMATICS that permits to employ computers connected in the net and spread the of the theory of shells. The difficulties with stability are due to the boundary layer task over them. It concerns most the task of integration within the method. phenomenon. There are many problems within computational physics and applied math ematics that have a boundary layer. Therefore the proposed two-step approach neglecting 5. Since the method seems to be universal and as already mentioned, similar phenomena are selected boundary (initial) conditions in the first step has wide potential applications. observed also for other tasks, it should be investigated if the two-step approach can be 4. The tricks and tips proposed in this book to speed up symbolic and numerical computa applied for such problems. tions may be applied for other purposes within MATHEMATICS environment. 6. There are many other symbolic tasks within the theory of shells and Continuum Mechan ics that should be translated into MATHEMATICS and MathTensor™ language and solved 5. The Author became a fellow of the Mathematica Inner Circle for the contribution in with their assistance. Among them there are geometrical and physical nonlinearities. development and dissemination of the system. He was the first Polish grantee of Wolfram Research, Inc. Visiting Scholar program. A valuable appreciation of the Author’s work 7. There are many fields in Civil Engineering (practice and education) that can be computer were several grants for participation at international conferences abroad, taking the chair algebra assisted. I plan to develop and to disseminate these possibilities. Among them at their sessions and an invited lecture in the Research Institute for Symbolic Computation there are dynamics of structures and temporarily set back the application of catenoidal in Linz. shells for cooling towers and chimneys.

171 ACKNOWLEDGEMENTS APPENDIX: BRIEF REVIEW OF THE AUTHOR’S CONTRIBUTIONS

Wolfram Research, Inc. has supported the Author with a Visiting Scholar Grant in 1998. Many of the symbolic problems presented in the contribution were developed during the Author’s stay Most of the Author’s contributions have already been mentioned in the text, since this book at the Wolfram Research, Inc. headquarters in Champaign, IL, USA. The firm supported the Author with a free upgrade of the system MATHEMATICS 4.1 in 2001. summarizes recent developments in my research and publications. This appendix collects all publications together in a bit more historical context. The idea to append the functional of the Least Squares Method with terms responsible for boundary conditions appeared during the Author’s stay at the University of Glamorgan in Wales, My first articles [22, 23, 24, 25] were prepared together with Professor S. Bielak, who was the UK in 1996. This visit was possible thanks to the Individual Mobility Grant awarded within promotor of my PhD thesis. the TEMPUS program of the Europeean Union. Thanks to this grant the Author had some Contribution [22] discusses problems of substitute transverse force introduced by Kirchhoff, spare time and opportunity to explore libraries. This activity was also continued within further compare Z. Kqczkowki [72]. This force replaces the transverse force and the torsion moment. mobility grants at other European Universities. The Author is very grateful to all Deans of It results that neither the transverse force, nor the torsion moment attain a zero value at the the Civil Engineering Faculty in the years 1996-2000 and host universities for this valuable free edge of the plate. I returned to the problem of substitute transverse force in [216] and in opportunity. chapter 12 on page 159. There it has been shown that deputy transverse force may be applied The valuable financial support has been provided by HM Rector of the Silesian University of in the analysis of the Reissner-Mindlin model of plates. Technology with a grant No BW/RGH-17/RB-0/2003. A catenoidal shell is described in [23]. This kind of shell is interesting for many reasons and may be used for engineering purposes. It seems to be possible to use this shell for chimneys and cooling towers. This problem will be researched in future. The shape of a catenoide is used in chapter 5 on page 58 to show the elements of the geometrical description of a shell. This shape of a shell was also used in the numerical example considered in [209]. Semigeodesic parametrization of a shell of revolution is presented in [25]. This kind of parametrization is very practical. Curvilinear coordinates measured along the meridian are dis tributed uniformly. Moreover, in many cases, the coefficient of differential forms of the surface are specified by polynomials. Consequently, differential equations contain also polynomial co efficients. This kind of parametrization was used in numerical examples presented in [200] - where a spherical shell is discussed, and in [209] - where a catenoidal shell is invesigated and also in this book in chapter 5 on page 58. One of the tasks of my PhD program was the analytical solution of a sloped cantilever shell of revolution. It is connected with the engineering problem of cooling towers and chimneys subjected to the influences of mining exploitation. I used the two-step approach based on the membrane approximation. Several papers were devoted to this problem [182, 183, 184]. Another task of my PhD thesis was the analysis of the same problem by means of numeri cal approach. I experienced difficulties with the convergence and stability of numerical ap proximation. The problem was discussed in my PhD thesis and [189] and also published in [24,187, 188].

172 173 174 Appendix: Brief review o f the Author’s contributions Appendix: Brief review of the Author’s contributions 175

Doing research for my PhD thesis I started to use the computer algebra system. This aid helped definite matrices. This problem should be researched in more details in future, since it permits to me to find that the system of differential equations describing the membrane problem of any extend the possibilities of this algorithm. The Refined Least Squares Method was also presented shell of revolution can be reduced to a simple linear equation. This equation has a closed in [214, 215] and next in [216]. solution for many shapes of the meridian. My experience has been collected in [185, 186, 189, 190]. The application of MATHEMATICS to numerical and symbolic problems in the theory of shells was the topic of an invited lecture at the international conference on Symbolic and Numerical Just after my PhD exams I started research on the application of the Least Squares Method con Scientific Computation at the Research Institute for Symbolic Computation at the University of cerning boundary- and initial-boundary value problems. The method was implemented within Linz in Austria [212], see also http://www.risc.uni-linz.ac.at/conferences/snsc01/. It resulted in the MATHEMATICS system. It was an attempt to find a “third way” between the analytical ap an article [223] presenting both experience in symbolic computation-and the application of the proach based on membrane approximation and numerical methods, which are sensitive to ill- Least Squares approach in problems of shells. conditioned tasks. The first experiences were presented in [191,192]. I showed that the method can be “refined” with consideration of boundary or initial conditions in the functional. The works [220, 221] present the application of the Refined Least Squares Method to initial problems unstable in the Lyapunov sense. There it was shown that the method is not sensitive Contribution [193] presents the basic features of the Refined Least Squares Method. It has been to that kind of instability. shown that the approach with boundary conditions appended to the minimized functional is well convergent. The example, also repeated in [194], presents an initial-boundary value task The mathematical interpretation of the boundary layer phenomenon observed in the Refined of a hyperbolic-parabolic model of heat transfer with discontinuous steep initial conditions. Least Squares Method, presented in chapter 11 on page 146, is discussed in some more details There it was shown that the method is applicable to initial-value problems. The work [194] in [219]. discuses implementation problems of the method within the computer algebra environment. Symbolic computations connected with equations of equilibrium discussed in chapter 3 on The paper [196] presents the application of the method to the problem of a shield subjected to page 45 were earlier presented in [213]. discontinuous load on the boundary. The application of the Refined Least Squares Method to Reissner-Mindlin plates, presented in Simultaneously to research on the Least Squares Method I started investigations on the applica chapter 12 on page 159, was published in [218]. tion of MathTensor™in Continuum Mechanics and the Theory of Shells. My first papers were devoted to the integration of constitutive relations [195]. Some experience with the application The problem of a moderated simplification of complex tensor expressions connected with the of MathTensor™in the theory of shells is presented also in [197]. In [199] MathTensor™ was evaluation of refined constitutive relations, discussed in chapter 4 on page 49 was also discussed applied to scale an elastic half-space to the finite curved domain and to develop displacement in [217, 222]. equations in such a curved space. These equations were approximated by means of the Least There is also a contribution written together with my former MSc student M. Kogut [224] de Squares Method. voted to the numerical analysis of the dynamical behavior of a thin-walled structure of a blast I had a valuable opportunity to develop my research with the support of the Wolfram Research, compensating structure. This was a non-standard engineering task connected with the erection Inc. at their headquarters in Champaign, IL, USA within a Visiting Scholar grant. I realized of General Motors (Opel) factory in Gliwice. Due to insurance company requirements it had the project entitled Refined Constitutive Equations in Shells. The project was devoted to the to be proved that the wall will collapse under pressure not greater than 0.75 kPa. Usually an solution of symbolic problems of shells and the application of the obtained results. It led to engineer estimates the load-bearing capacity of a structure from the opposite direction. several publications. The first one was [198]. The paper [200] presents the application of the obtained constitutive relations to a spherical shell. Next this problem was presented in [201]. The development of the application of MATHEMATICS and MathTensor™ was, continued in [202, 203, 205, 206, 208]. The geometrical description of shells with computer algebra assistance, discussed in chap ter 5 on page 58, was presented in [205]. According to a procedure similar to that presented in chapter 4 on page 49 the refinement of strain energy was computed. This was published in [207] and [209]. The example discussed in chapter 9 on page 88 was presented in [210, 211] to show a boundary layer phenomenon observed in shells approximated by means of the Refined Least Squares Method together with a proposal of the two-step approach. It was also shown that the Cholesky- Banachiewicz algorithm, see section 8.2. on page 86, is not limited to systems with positive Bibliography 111

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Heildelberg, 2003. Research Institute for Symbolic Computation, Springer Verlag. ISBN 3-540-40554-2. APPLICATION OF COMPUTER ALGEBRA IN SYMBOLIC COMPUTATIONS AND BOUNDARY-VALUE 224. R. A. Walentyński and M. Kogut. Statical and dynamical computations of thin walled PROBLEMS OF THE THEORY OF SHELLS element of the blast compensating wall using finite element method/Obliczenia staty czne i dynamiczne metodą elementów skończonych cienkościennego elementu obudowy Summary kompensującej wybuch. Journal of Silesian University of Technology, Civil Engineering, 1404(85/1998): 115-124, 1998. PL ISSN 0434-0779. The contribution consists of two parts. The first part deals with the application of the computer algebra system MATHEMATICS and the package for tensor analysis MathTensor™ in symbolic computations in the 225. M. Walker, T. Reiss, S. Adali, and P. M. Weaver. Application of MATHEMATICS to the theory of shells. The second part is devoted to finding an approximate solution of shell boundary value optimal design of composite shells for improved buckling strength. Engineering Compu problems. The contribution is neither a theory of shells nor does it aspire to be a theory of approximation. tations, 15(2):260-267, 1998. The problem related to shells, being an object of interest - the description of bodies in a curved space, is the background of considerations: 1) the possibility of showing various tools of computer assistance in 226. Z.-W. Wang. The geometrically nonlinear theory of anisotropic sandwich shells faced symbolic computations, 2) a proposal of the application of some approach to the Least Squares Method with laminated composites. The Quarterly Journal of Mechanics and Applied Mathemat in search for approximate solutions applying computer algebra. ics, 50(3):349-378, August 1997. The consideration in the first part is preceded by a presentation of basic relations of the considered theory 227. T. Wickham-Jones. MATHEMATICS Graphics: Techniques and Applications. TELOS, of shells. One approach to the theory of shells has been explored, but ways of the application of computer Santa Clara, 1994. algebra tools presented in the contribution may be further developed in other theories. 228. D. J. Wilkins, Jr. Application of a symbolic algebra manipulation language for composite The aim of this part of the contribution is to present an effective application of computer algebra system structure analysis. Computer and Structures, 3:801-807, 1973. capabilities of formulating equations and their adaptation for further numerical computations. It has been shown that computer algebra is not “a wizard box” for an automatic derivation of expressions, but 229. S. Wolfram. The MATHEMATICS Book. Cambridge University Press and Wolfram Re an assistant device, which helps a conscious research worker to obtain desired and reliable results. search, Inc., New York and Champaign, fourth edition, 1999. Ways of the application of advanced tools of symbolic computations have been presented, which permit 230. Wolfram Research. MATHEMATICS 4.0 Standard Add-on Packages: the Official Guide to to receive final expressions in the desired and possibly simplest form. Moreover, it was presented that Over a Thousand Additional Functions for Use with MATHEMATICS 4. Wolfram Media, there is no need and that it is not advisable to neglect “ad hoc” any terms in expressions. Thanks to Champaign, 1999. computer algebra applications it has been possible to determine the constitutive relations of shells, which satisfy the last equation of equilibrium (2.51). 231. J. H. Wolkowisky. Shooting the buckled plate. In Keranen et al. [74], pages 507-515. The reliability of results is an important aspect of this part of the contribution. Thus, special emphasis 232. C. Woźniak. Nonlinear Theory of Shells/Nieliniowa Teoria Powłok. Państwowe is put on the scrutiny of the obtained formulas. The computer algebra system is a program and only a Wydawnictwo Naukowe, Warszawa, 1966. program. Although it is an advanced technological product, the entire responsibility for the results is up to the user. 233. H. T. Y. Yang, A. Masud, and R. K. Kapania. A survey of recent shell finite elements. International Journal for Numerical Methods in Engineering, pages 101-126, January The attention of the contribution is focused reasonably on linear problems and shells of simple shapes, as 2000. its aim is to present a rational application of the tool, the computer algebra system. However, it has been pointed out how it is possible to extend the range of consideration to nonlinear tasks. Such approaches to 234. J.-T. Yeh and W.-H. Chen. Shell elements with drilling degree of freedoms based on the tasks for computer algebra have been shown, that the computations might be completed successfully micropolar elasticity theory. International Journal of Numerical Methods in Engineering, and as quickly as possible. pages 1145-1158, April 1993. The results of symbolic computations presented in the contribution are differential equations in terms of 235. A. P. Zieliński. Boundary Series Method Applied to Plates and Shells with Curvilinear displacements and other relations for a cylindrical shell. Contour/Metoda Szeregów Brzegowych w Zastosowaniu do Płyt i Powłok o Krzywolinio The approximation of equations derived in the first part is considered in the next part of the contribution. wym Konturze. Monografia; v. 98. Politechnika Krakowska im. Tadeusza Kościuszki, An application of some approach to the Least Squares Method to search an approximate solution of the Kraków, 1990. differential equation system is presented. In this approach to the method the functional, which is an object of minimization, is appended with terms taking into account boundary conditions. This makes 236. D. Zwillinger. Handbook of Differential Equations. Academic Press, Inc., New York, it possible to simplify considerably the implementation of the method algorithm within the computer second edition, 1989. algebra environment and permits to approximate multidimensional tasks with a discontinuous boundary. Approximations of such tasks have been shown in several other contributions of the Author. 196 S u m m a ry

Within the elements of implementation selected solutions of the application of advanced tools of the ZASTOSOWANIE ALGEBRY KOMPUTEROWEJ system have been presented, which permits to speed up the computational process, in particular the W OBLICZENIACH SYMBOLICZNYCH integration of polynomial expressions and the construction of a system matrix of algebraic equations and I ZAGADNIENIACH BRZEGOWYCH TEORII POWŁOK its decomposition. In boundary-value problems of shells, especially the considered long cylindrical shells the phenomenon Streszczenie of a boundary layer occurs, so that direct a approach to the problem by means of methods of numerical approximation slowly converges with an actual result. The application of the two step approach based on Praca składa się z dwóch części. W części pierwszej przedstawiono zastosowanie systemu algebry membrane approximation fails, due to the bending flexibility of cylindrical shells in the parallel direction. komputerowej MATHEMATICA' oraz pakietu analizy tensorowej MatliTensor™ do obliczeń symbolicznych w teorii powłok. Część druga dotyczy znajdowania rozwiązania przybliżonego zagadnienia brzegowego These difficulties have been overcome applying the proposed approach of the Least Squares Method. powłok. Praca nie jest teorią powłok, ani też nie aspiruje do miana teorii aproksymacji. Zagadnienia By means of computational experiment a boundary-condition phenomenon has been discovered - that związane z powłokami, z uwagi na przedmiot zainteresowań - opis ciał w zakrzywionej przestrzeni, is connected not only with the method, but also with the character of the differential equations - which stanowią kanwę rozważań z uwagi na: 1) możliwość pokazania różnorodnych narzędzi wspomagania permits to approximate described the problem with a tenth-order operator taking into account only four komputerowego w zakresie obliczeń symbolicznych, 2) propozycję zastosowania pewnego ujęcia Me boundary conditions applied in the membrane approach. The obtained approximation has been called tody Najmniejszych Kwadratów do znajdowania rozwiązań przybliżonych z wykorzystaniem algebry base solution and is feasible in most of the domain, excluding the boundary layer. komputerowej. This discovery becomes the basis of the two-step approach proposal, presented by examples of two tasks Rozważania części pierwszej poprzedzono zestawieniem podstawowych zależności omawianej teorii related to cylindrical shells. This approximation is as stable as that obtained by means of the method powłok. Omówiono jedno z podejść do terii powłok, jednak przedstawione w pracy sposoby zasto based on membrane approximation and simultaneously free from the already mentioned disadvantages of sowania narzędzi algebry komputerowej mogą być wykorzystane w innych teoriach. this approach. In the first step the task is approximated taking into account selected boundary conditions. Neglected boundary conditions are satisfied locally in the second step. Celem tej części pracy jest przedstawienie możliwości efektywnego wykorzystania systemu algebry komputerowej w formułowaniu równań oraz ich przygotowaniu do dalszych obliczeń numerycznych. Additional methods of global and local error evaluation are shown, They allow, among others, to find a Pokazano, że algebra komputerowa nie jest „czarodziejską skrzynką” do automatycznego uzyskiwania phenomenon of false convergence. An effective parameter of error evaluation is the value of the mini wyrażeń ale asystentem pomagającym świadomemu badaczowi na uzyskanie żądanych i wiarygodnych mized functional. wyników. The Least Squares Method is a global approach. Its results are functions and in this context it is an Przedstawiono tu sposoby wykorzystania szeregu zaawansowanych narzędzi obliczeń symbolicznych, analytical approach. Therefore, there is no problem with differentiation and integration, the problem które pozwalają na uzyskanie końcowych wyrażeń w żądanej i możliwie najprostszej postaci. Ponadto of interpolation and extrapolation does not occur, either. The estimation of a global and local error of pokazano, że nie trzeba, ba nawet nie należy pomijać „ad hoc” jakichkolwiek wyrazów w wyrażeniach. approximation is straightforward. Thanks to that the approach is free from disadvantages of discrete Dzięki wykorzystaniu algebry komputerowej udało się wyznaczyć związki konstytutywne powłok speł methods and in times of computer algebra development may become a tool of scrutiny and moreover niające ostatnie równanie równowagi (2.51). an alternative to numerical methods. The most important feature is the discovered possibility of a two- step solution of tasks with a boundary layer. Thus, it may be applied in other tasks of mechanics and Ważnym aspektem tej części pracy jest problem wiarygodności wyników. Dlatego też, szczególną uwagę poświęca się weryfikacji otrzymanych wzorów. System algebry komputerowej jest programem i tylko mathematical physics, where similar phenomena occur. Opposite to asymptotic approaches the method programem i mimo swojego zaawansowania technicznego całą odpowiedzialność za wyniki ponosi jego requires neither the setting up of assumptions of small parameter nor to lower the order of the differential operator. Thus, it may be applied to the problems of a wider class. użytkownik. The contribution shows a wide spectrum of computer algebra applications starting with formulating W pracy uwaga została skupiona celowo na zagadnieniach liniowych i powłokach o prostych kształtach, equations of the problem, through numerical computations, to the publication of results . The computer gdyż jej celem jest przedstawienie racjonalnego wykorzystania narzędzia, jakim jest system algebry algebra system has been employed in the production of most contribution graphics and formulas. The komputerowej. Wskazano jednakże, w jaki sposób można rozszerzyć zakres rozważań na przypadki za processing of results is a crucial technical element connected with scientific work. dań nieliniowych. Pokazano sposoby takiego formułowania zadań dla systemu algebry komputerowej, aby obliczenia zakończyły się powodzeniem i przebiegały możliwie jak najszybciej. The Author hopes that the presented considerations might become at least an inspiration to apply modern Wynikiem obliczeń symbolicznych przedstawionych w pracy są przemieszczeniowe równania różnicz tools of computer assistance in processes of formulating and solving complex problems of mechanics and kowe i inne związki dla powłoki walcowej. physics. Aproksymacja równań otrzymanych w części pierwszej jest przedmiotem rozważań następnej części opracowania. Przedstawiono tu zastosowanie pewnego ujęcia Metody Najmniejszych Kwadratów do znajdowania rozwiązania przybliżonego układu równań różniczkowych. W tym ujęciu metody funkcjo nał, będący przedmiotem minimalizowania, uzupełniono o wyrazy uwzględniające warunki brzegowe. Pozwala to na znaczne uproszczenie wdrożenia algorytmu metody w obrębie środowiska algebry kompu terowej oraz umożliwia aproksymację zadań wielowymiarowych z nieciągłym brzegiem. Aproksymacje tego typu zadań pokazano w kilku innych pracach Autora. 198 Streszczenie

W ramach przedstawionych elementów wdrożenia metody przedstawiono wybrane rozwiązania wyko rzystania zaawansowanych narzędzi systemu algebry komputerowej pozwalające na przyśpieszenie pro cesu obliczeniowego, w szczególności całkowania wyrażeń wielomianowych i budowy macierzy układu równań algebraicznych i jej dekompozycji. W zagadnieniu brzegowym powłok, a w szczególności rozważanych długich powłok walcowych, wy stępuje zjawisko warstwy brzegowej, które sprawia, że podejście bezpośrednie do problemu z użyciem metod aproksymacji numerycznej wolno zbiega się do poprawnego wyniku. Zastosowanie podejścia dwuetapowego opartego na przybliżeniu błonowym okazuje się zawodne z uwagi na podatność powłok walcowych na zginanie w kierunku równoleżnikowym. Trudności te udało się pokonać z zastosowaniem proponowanego ujęcia Metody Najmniejszych Kwadra tów. W wyniku eksperymentu obliczeniowego odkryto zjawisko warunku brzegowego - jak się okazuje związane nie tylko z metodą, ale i charakterem równań różniczkowych - pozwalające na aproksymację problemu opisanego operatorem różniczkowym dziesiątego rzędu z uwzględnieniem jedynie czterech warunków brzegowych - stosowanych w podejściu błonowym. Uzyskana aproksymacja nazwana roz wiązaniem bazowym jest poprawna na większości dziedziny problemu z wyjątkiem warstwy brzegowej. To odkrycie stało się podstawą zaproponowania podejścia dwuetapowego, przedstawionego na przy kładach dwóch zadań dotyczących powłok walcowych. Przybliżenie to jest stabilne i szybko zbieżne tak jak to otrzymywane w metodzie opartej na przybliżeniu błonowym i jednocześnie wolne od już wspomnianych wad tego podejścia. W pierwszym etapie zadanie jest przybliżane z wybranymi warun kami brzegowymi. Pominięte warunki brzegowe są spełniane lokalnie w drugim kroku. W pracy przedstawiono interpretację fizyczną zaobserwowanego zjawiska obliczeniowego oraz doko nano próby jego wyjaśnienia od strony matematycznej. Przy okazji wskazano na potencjalnie szerokie pole zastosowań metody. Pokazano dodatkowe metody oceny błędu lokalnego i globalnego aproksymacji pozwalające między innymi wykryć zjawisko pozornej zbieżności. Efektywnym parametrem oceny błędu jest wartość mini malizowanego funkcjonału. Metoda Najmniejszych Kwadratów jest globalnym podejściem do problemu. Wynikiem rozwiązania są funkcje i w tym sensie jest metodą analityczną. Dlatego też nie ma kłopotu z różniczkowaniem lub cał kowaniem, nie występuje też problem interpolacji i ekstrapolacji. Łatwo też ocenić lokalny i globalny błąd aproksymacji. Dzięki temu podejście jest wolne od wad metod dyskretnych i w dobie rozwoju sys temów algebry komputerowej może stać się narzędziem weryfikacyjnym, a nawet alternatywą dla metod numerycznych. Najistotniejszą cechą jest jednak dostrzeżona możliwość rozwiązywania dwuetapowe go zadań z warstwą brzegową. Może zatem znaleźć zastosowanie w innych zagadnieniach mechaniki i fizyki matematycznej, w których występują podobne zjawiska. W przeciwieństwie do podejść asymp totycznych metoda nie wymaga stawiania założeń o małym parametrze oraz obniżania rzędu operatora różniczkowego. W związku z tym może być wykorzystana do aproksymacji szerszej klasy zadań. Praca pokazuje szerokie spektrum wykorzystania systemu algebry komputerowej od formułowania rów nań problemu, poprzez obliczenia numeryczne, do publikacji wyników. System algebry komputerowej został wykorzystany do przygotowania większości grafiki występującej w pracy oraz wzorów. Opraco wanie wyników do publikacji stanowi istotny, techniczny element związany z pracą naukową. Autor żywi nadzieję, że przedstawione w pracy rozważania staną się przynajmniej inspirujące do wyko rzystania nowoczesnych narzędzi wspomagania komputerowego w procesie formułowania oraz rozwią zywania złożonych problemów mechaniki i fizyki. Książki Wydawnictwa można nabyć w księgarniach

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