Instituto Tecnológico de Aeronáutica FINITE ELEMENTS I. Class notes AE-245
|
|
|
- Blaise Jackson
- 6 years ago
- Views:
Transcription
1 Insttuto Tecnológco de Aeronáutca FIITE ELEMETS I Class notes AE-5
2 Insttuto Tecnológco de Aeronáutca 5. Isoparametrc Elements AE-5
3 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton What are the condtons to ensure that as the mesh s refned the Galerkn appromate soluton converges to the eact soluton? AE-5
4 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Shape functons must be Smooth at least C on each element nteror. Contnuous across element boundares Complete AE-5
5 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Condtons and guarantee fnte jumps across element nterfaces. n n n n AE-5 5
6 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Shape functons that satsf condtons and are of class C 0 Euler-Bernoull beams requre hgher order of dervatves. Thus shape functons must be of class C C on the element nteror and C across boundares. AE-5 6
7 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Integrands wth dervatves of order m: C m on the element nteror and C m- across boundares. AE-5 7
8 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Completeness requres that the element nterpolaton functon s capable of eactl representng an arbtrar lnear polnomal when the nodal degrees of freedom are assgned values n accordance wth t. AE-5 8
9 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Let u h m d where d u h AE-5 9
10 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness D stuaton: shape functons are complete f mples d c0 c c c u h c0 c c cz z AE-5 0
11 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Argument: as the mesh s refned the eact soluton and ts dervatves approach constant values over the elements. In elastct completeness means that the element can represent all rgd bod motons and constant stran states. AE-5
12 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Blnear quadrlateral element Straght edge four node quadrlateral mappng AE-5
13 Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Straght edge four node quadrlateral
14 Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Assume blnear epansons β β β β α α α α 0 0
15 Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons
16 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons: propertes Shape functons of class C wthn elements. s alwas smooth n and but t ma be dscontnuous n and. Shape functons of class C 0 boundares. across element AE-5 6
17 Insttuto Tecnológco de Aeronáutca AE-5 7 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons: propertes Completeness: c c c c c c c c c c c c d u h
18 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Isoparametrc element Defnton m u h m d same shape functons The blnear quadrlateral element s soparametrc AE-5 8
19 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Isoparametrc element If the mappng s one-to-one and f ts Jacoban s postve for ever then the convergence condton s met. Isoparametrc elements automatcall satsf the three basc convergence condtons. Isoparametrc elements can assume convenent shapes for practcal analss. AE-5 9
20 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular element Constant stress/stran element. Coalescence of two nodes. AE-5 0
21 Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Trangular element m
22 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular element Eercses Sketch the shape functons for the trangular element n the doman. Compute the Jacoban determnant at 0 for the element shown AE-5
23 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element ζ AE-5
24 Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Trlnear heahedral element ζ α ζ α ζ α α ζ α α α α ζ α α α α α α α α
25 Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Trlnear heahedral element 8 ζ ζ ζ ζ
26 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element Degenerated wedge coalescence AE-5 6
27 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element Degenerated tetrahedral coalescence AE-5 7
28 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Shape functons of order hgher than lnear More accurate representatons Boundares ma be curved Computatonall more epensve AE-5 8
29 Insttuto Tecnológco de Aeronáutca AE-5 9 ISOPARAMETRIC ELEMETS Hgher order elements Lagrange polnomals m m m j j j m j j j l L L L L th term omtted
30 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Lagrange polnomals l j δ j m-noded element n one dmenson: l AE-5 0
31 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Eamples -noded one dmensonal element: -noded one dmensonal element: 0 AE-5
32 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Lagrange elements: hgher order two- and three- dmensonal elements can be obtaned b takng products of Lagrange polnomals AE-5
33 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes D quadrlateral element wth 5 to 9 nodes Add ffth node to sde -: 5 AE-5
34 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes 5 l l ow shape functons and must be modfed to be zero at node AE-5
35 Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Varable number of nodes
36 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes ode 9 present Bubble functon: 9-8 do not vansh at node 9. Hence 9 j j 9 j AE-5 6
37 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Serendpt faml of quadrlateral elements AE-5 7
38 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Serendpt quadrlaterals m m O O m m AE-5 8
39 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Lagrange faml of quadrlateral elements AE-5 9
40 Insttuto Tecnológco de Aeronáutca AE-5 0 ISOPARAMETRIC ELEMETS Element famles Lagrange quadrlaterals m m m m m m m m m m m m O O
41 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Standard trangular elements AE-5
42 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Trangles O m m m L O m O m m AE-5
43 Insttuto Tecnológco de Aeronáutca Integraton of ISOPARAMETRIC ELEMETS umercal ntegraton Ω f dω In one dmenson b a f d AE-5
44 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS umercal ntegraton umercal ntegraton methods Trapezodal rule nd order accurate Smpson s rule th order accurate ewton-cotes formulas Gaussan quadrature fewer ponts AE-5
45 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature b a d α f α f... α n f f n R n α weghts samplng ponts AE-5 5
46 Insttuto Tecnológco de Aeronáutca AE-5 6 ISOPARAMETRIC ELEMETS Gaussan quadrature... 0 p f β β β ψ where... n p n n l f l f ψ n n l L L L L
47 Insttuto Tecnológco de Aeronáutca AE-5 7 ISOPARAMETRIC ELEMETS Gaussan quadrature At the samplng ponts f p f... 0 ψ β β β ψ Integraton of f elds 0 b a n b a b a d p d l f d f β
48 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS b Gaussan quadrature Requre that: p d 0 for 0... n a p s of order n. Hence ntegrals of polnomals of order n to n are zero. Samplng weghts: α b l d a AE-5 8
49 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Change of varables Gaussan quadrature b a f d f d Advantage: standardze results AE-5 9
50 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature n r α r smmetrcall postoned ± ± α 5 ± ± ± ± ± ± ± AE-5 50
51 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature Two and three dmensons: f r s drds α f r s ds α α j j f r s j AE-5 5
52 Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Dervatves of shape functons Construct stffness matrces Recover strans and stresses Compute and Chan rule
53 Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Dervatves of shape functons Chan rule The Jacoban matr must be computed frst!
54 Insttuto Tecnológco de Aeronáutca Jacoban matr ISOPARAMETRIC ELEMETS Dervatves of shape functons m m m m m m AE-5 5
55 Insttuto Tecnológco de Aeronáutca AE-5 55 ISOPARAMETRIC ELEMETS Dervatves of shape functons Chan rule det J
56 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular and tetrahedral elements Elements that model comple geometres Facltate mesh transtons Avod dstorted quadrlaterals/heahedra Area/volume coordnates AE-5 56
57 Insttuto Tecnológco de Aeronáutca AE-5 57 ISOPARAMETRIC ELEMETS Trangular elements Area coordnates 0 r s s r t s r s s r r s r 0 t constant
58 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular elements Area coordnates s r t AE-5 58
59 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular elements Eample: 6 node trangle rr ss tt rs rt st AE-5 59
60 Insttuto Tecnológco de Aeronáutca AE-5 60 ISOPARAMETRIC ELEMETS Tetrahedral elements Volume coordnates r s t t s r u t s r t t s r s t s r r t s r
61 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular and tetrahedral elements umercal Integraton n r / s / t / α 0.5 / /6 /6 /6 fdω m Ω α f det J /6 /6 / /5 /5 / /6 / /5 /5 /6 / / /5 /5 /6 /6 7/96 5/96 5/96 /5 /5 /5 5/ AE-5 6
62 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Specal shape functons Elements wth specal propertes Infnte elements Modelng of sngulartes cracks AE-5 6
63 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Specal shape functons One dmensonal eample: 0 / 0 a L u u u u a L AE-5 6
64 Insttuto Tecnológco de Aeronáutca Obtan stran: ISOPARAMETRIC ELEMETS Specal shape functons 0 / 0 a L a L L a a L If s a purel quadratc functon of then a L/ L AE-5 6
65 Insttuto Tecnológco de Aeronáutca Obtan stran: ISOPARAMETRIC ELEMETS Specal shape functons 0 / 0 L/ L du du d α α 0 α ε d d / d L L / α 0 L lm 0 ε Sngulart modelng AE-5 65
66 Insttuto Tecnológco de Aeronáutca Membrane FE code AE-5 66
67 Insttuto Tecnológco de Aeronáutca Membrane FE code Man routnes nput_data mechancal renum crout_dec crout_sol AE-5 67
68 Insttuto Tecnológco de Aeronáutca Membrane FE code Routne nput_data Purpose: generate mesh nput materal allocate memor fnd band wdth Features: user defned geometr dependant AE-5 68
69 Insttuto Tecnológco de Aeronáutca Membrane FE code Routne mechancal Purpose: generate element stffness matr and global stffness matr Features: Gaussan numercal ntegraton bquadratc and 6-node trangular elements AE-5 69
70 Insttuto Tecnológco de Aeronáutca Membrane FE code Routne renum Purpose: Appl essental boundar condtons Features: Zeroes rows and columns AE-5 70
71 Insttuto Tecnológco de Aeronáutca Membrane FE code Routne crout_dec Purpose: Decompose banded matr K L U upper trangular matr lower trangular matr Features: In-place decomposton smmetrc matr AE-5 7
72 Insttuto Tecnológco de Aeronáutca Membrane FE code Routne crout_sol Purpose: Solve sstem Kq f L U q f Features: Two step procedure AE-5 7
73 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate radus.0 m thckness.0 mm E 08 MPa ν 0. AE-5 7
74 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate u 6.E-0 5.5E-0.8E-0.50E-0.6E-0.75E E E E E-0 -.6E E-0 -.8E-0-5.5E-0-6.E v 6.E-0 5.5E-0.8E-0.50E-0.6E-0.75E E E E E-0 -.6E E-0 -.8E-0-5.5E-0-6.E AE-5 7
75 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate u 6.6E-0 6.0E E-0 5.E-0.87E-0.E-0.99E-0.5E-0.0E-0.66E-0.E-0.77E-0.E E-0.E v -5.8E E- -.6E- -.6E- -.7E- -.6E- -.8E- -.5E- -.90E- -5.5E E- -6.5E E- -7.6E- -8.9E AE-5 75
76 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate Radus.0 m Thckness.0 mm E 08 MPa ν 0. AE-5 76
77 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate u 6.95E E00.97E00.97E00.98E00.99E00 9.9E E00-9.9E E E E E E E v.06e00 9.0E E E-0.55E-0.0E-0.5E E00 -.5E-0 -.0E E E E-0-9.0E E AE-5 77
78 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate u 8.5E00 7.0E E00.87E00.65E00.E00.E E00 -.E00 -.E E E E00-7.0E00-8.5E v.06e00 9.0E E E-0.55E-0.0E-0.5E E00 -.5E-0 -.0E E E E-0-9.0E E AE-5 78
79 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate u.0e0 8.6E00 7.0E E00.E00.88E00.E E00 -.E E00 -.E E00-7.0E00-8.6E00 -.0E v.8e00.0e00 8.0E-0 6.7E-0 5.0E-0.6E-0.68E E E-0 -.6E-0-5.0E-0-6.7E-0-8.0E-0 -.0E00 -.8E AE-5 79
80 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate m P m m thckness.0 mm E 08 MPa ν 0. AE-5 80
81 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate u v AE-5 8
82 Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate u v AE-5 8
Plate Theories for Classical and Laminated plates Weak Formulation and Element Calculations
Plate heores for Classcal and Lamnated plates Weak Formulaton and Element Calculatons PM Mohte Department of Aerospace Engneerng Indan Insttute of echnolog Kanpur EQIP School on Computatonal Methods n
C PLANE ELASTICITY PROBLEM FORMULATIONS
C M.. Tamn, CSMLab, UTM Corse Content: A ITRODUCTIO AD OVERVIEW mercal method and Compter-Aded Engneerng; Phscal problems; Mathematcal models; Fnte element method. B REVIEW OF -D FORMULATIOS Elements and
MEMBRANE ELEMENT WITH NORMAL ROTATIONS
9. MEMBRANE ELEMENT WITH NORMAL ROTATIONS Rotatons Mst Be Compatble Between Beam, Membrane and Shell Elements 9. INTRODUCTION { XE "Membrane Element" }The comple natre of most bldngs and other cvl engneerng
Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D
Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean
One Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
PLATE BENDING ELEMENTS
8. PLATE BENING ELEMENTS Plate Bendng s a Smple Etenson of Beam Theor 8. INTROUCTION { XE "Plate Bendng Elements" }Before 960, plates and slabs were modeled usng a grd of beam elements for man cvl engneerng
CIVL 8/7117 Chapter 10 - Isoparametric Formulation 42/56
CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 4/56 Newton-Cotes Example Usng the Newton-Cotes method wth = ntervals (n = 3 samplng ponts), evaluate the ntegrals: x x cos dx 3 x x dx 3 x x dx 4.3333333
Module 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
C PLANE ELASTICITY PROBLEM FORMULATIONS
C M.. Tamn, CSMLab, UTM Corse Content: A ITRODUCTIO AD OVERVIEW mercal method and Compter-Aded Engneerng; Phscal problems; Mathematcal models; Fnte element method. B REVIEW OF -D FORMULATIOS Elements and
Numerical Methods. ME Mechanical Lab I. Mechanical Engineering ME Lab I
5 9 Mechancal Engneerng -.30 ME Lab I ME.30 Mechancal Lab I Numercal Methods Volt Sne Seres.5 0.5 SIN(X) 0 3 7 5 9 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0-0.5 Normalzed Squared Functon - 0.07
Numerical integration in more dimensions part 2. Remo Minero
Numerca ntegraton n more dmensons part Remo Mnero Outne The roe of a mappng functon n mutdmensona ntegraton Gauss approach n more dmensons and quadrature rues Crtca anass of acceptabt of a gven quadrature
PART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
ME 501A Seminar in Engineering Analysis Page 1
umercal Solutons of oundary-value Problems n Os ovember 7, 7 umercal Solutons of oundary- Value Problems n Os Larry aretto Mechancal ngneerng 5 Semnar n ngneerng nalyss ovember 7, 7 Outlne Revew stff equaton
Lecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.
![( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.](/thumbs/84/90601477.jpg "( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.")
3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( , then there ests a polynomal
Finite Difference Method
7/0/07 Instructor r. Ramond Rump (9) 747 698 rcrump@utep.edu EE 337 Computatonal Electromagnetcs (CEM) Lecture #0 Fnte erence Method Lecture 0 These notes ma contan coprghted materal obtaned under ar use
Report on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
2.29 Numerical Fluid Mechanics
REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton
Chapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods
Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary
APPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
Finite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
Numerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
Georgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
Inductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
Professor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
Solutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1
Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton
Error Analysis of Heat Conduction Partial Differential Equations using Galerkin s Finite Element Method
Indan Journal of Scence and Technolog, Vol 9(36), DOI: 0.7485/jst/06/v936/058, September 06 ISS (Prnt) : 0974-6846 ISS (Onlne) : 0974-5645 Error Analss of Heat Conducton Partal Dfferental Equatons usng
Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
Why Monte Carlo Integration? Introduction to Monte Carlo Method. Continuous Probability. Continuous Probability
Introducton to Monte Carlo Method Kad Bouatouch IRISA Emal: kad@rsa.fr Wh Monte Carlo Integraton? To generate realstc lookng mages, we need to solve ntegrals of or hgher dmenson Pel flterng and lens smulaton
Interconnect Modeling
Interconnect Modelng Modelng of Interconnects Interconnect R, C and computaton Interconnect models umped RC model Dstrbuted crcut models Hgher-order waveform n dstrbuted RC trees Accuracy and fdelty Prepared
More metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Finite Element Method - Jacques-Hervé SAIAC
COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC FINITE ELEMENT METHOD Jacques-Hervé SAIAC Départment de Mathématques, Conservatore Natonal des Arts et Méters, Pars,
Causal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
Lecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
e i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
3 Basic boundary value problems for analytic function in the upper half plane
3 Basc boundary value problems for analytc functon n the upper half plane 3. Posson representaton formulas for the half plane Let f be an analytc functon of z throughout the half plane Imz > 0, contnuous
Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
Chapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
Pattern Classification
Pattern Classfcaton All materals n these sldes ere taken from Pattern Classfcaton (nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wley & Sons, 000 th the permsson of the authors and the publsher
PHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
ONE DIMENSIONAL TRIANGULAR FIN EXPERIMENT. Technical Advisor: Dr. D.C. Look, Jr. Version: 11/03/00
ONE IMENSIONAL TRIANGULAR FIN EXPERIMENT Techncal Advsor: r..c. Look, Jr. Verson: /3/ 7. GENERAL OJECTIVES a) To understand a one-dmensonal epermental appromaton. b) To understand the art of epermental
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
The Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
Generative classification models
CS 675 Intro to Machne Learnng Lecture Generatve classfcaton models Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Data: D { d, d,.., dn} d, Classfcaton represents a dscrete class value Goal: learn
Calculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
MPY308. Clinical Engineering & Computational Mechanics
MPY38 Clncal Engneerng & Computatonal Mechancs Recommended Readng: John D Anderson Jr, Computatonal Flud Dynamcs- he Bascs and Applcatons, McGraw-Hll, ISBN -73-4 Brown, Smallwood, Barber, Lawford, Hose,
NUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST
Appled Computer Scence, vol. 13, no. 4, pp. 56 64 do: 10.23743/acs-2017-29 Submtted: 2017-10-30 Revsed: 2017-11-15 Accepted: 2017-12-06 Abaqus Fnte Elements, Plane Stress, Orthotropc Materal Bartosz KAWECKI
Tensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
Finite Element Solution for Heat Transfer Flow of. a Third Order Fluid between Parallel Plates
Adv. Studes Theor. Phs., Vol. 5,, no. 3, 7 - Fnte Element Soluton for Heat Transfer Flow of a Thrd Order Flud between Parallel Plates R. Mahmood a, M. Sajd a, and A. Nadeem b a Theoretcal Plasma Phscs
SL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
Constitutive Modelling of Superplastic AA-5083
TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy
In this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
PLAXIS 3D TUNNEL. Scientific Manual. version 2
PLAXIS 3D UNNEL Scentfc Manual verson 2 ABLE OF CONENS ABLE OF CONENS 1 Introducton...1-1 2 Deformaton theory...2-1 2.1 Basc equatons of contnuum deformaton...2-1 2.2 Fnte element dscretsaton...2-2 2.3
CHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
Exercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
INVERSE HEAT CONDUCTION IN ANISOTROPIC AND FUNCTIONALLY GRADED MEDIA BY FINITE INTEGRATION METHOD
IVERSE HEA CODCIO I AISOROPIC AD FCIOALLY GRADED MEDIA BY FIIE IEGRAIO MEHOD Jn, J; WE, P he Author(s) 8 Open Access hs artcle s lcensed under a Creatve Commons Attrbuton 4. Internatonal Lcense, whch permts
Errors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
Affine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
Module 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I
5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can
E91: Dynamics. E91: Dynamics. Numerical Integration & State Space Representation
E91: Dnamcs Numercal Integraton & State Space Representaton The Algorthm In steps of Δ f ( new ) f ( old ) df ( d old ) Δ Numercal Integraton of ODEs d d f() h Δ Intal value problem: Gven the ntal state
Modelli Clamfim Integrali Multipli 7 ottobre 2015
CLAMFIM Bologna Modell 1 @ Clamfm Integral Multpl 7 ottobre 2015 professor Danele Rtell danele.rtell@unbo.t 1/30? roduct of σ-algebras Let (Ω 1, A 1, µ 1 ), (Ω 2, A 2, µ 2 ) two measure spaces. ut Ω :=
ACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
Modal Identification of the Elastic Properties in Composite Sandwich Structures
Modal Identfcaton of the Elastc Propertes n Composte Sandwch Structures M. Matter Th. Gmür J. Cugnon and A. Schorderet School of Engneerng (STI) Ecole poltechnque fédérale f de Lausanne (EPFL) Swterland
A new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
Chapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
Relaxation Methods for Iterative Solution to Linear Systems of Equations
Relaxaton Methods for Iteratve Soluton to Lnear Systems of Equatons Gerald Recktenwald Portland State Unversty Mechancal Engneerng Department gerry@pdx.edu Overvew Techncal topcs Basc Concepts Statonary
Implicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
Chapter 2 Transformations and Expectations. , and define f
Revew for the prevous lecture Defnton: support set of a ranom varable, the monotone functon; Theorem: How to obtan a cf, pf (or pmf) of functons of a ranom varable; Eamples: several eamples Chapter Transformatons
Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
Poisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
CALCU THIRD EDITION. Monfy J. Strauss. Texas Tech University. Gerald L Bradley. Claremont McKenna College. Karl J. Smith. Santa Rosa Junior College
CALCU THIRD EDITION Monfy J. Strauss Texas Tech Unversty Gerald L Bradley Claremont McKenna College Karl J. Smth Santa Rosa Junor College Prentce Hall, Upper Saddle Rver, New Jersey 07458 Preface x 1 Functons
Non-linear Canonical Correlation Analysis Using a RBF Network
ESANN' proceedngs - European Smposum on Artfcal Neural Networks Bruges (Belgum), 4-6 Aprl, d-sde publ., ISBN -97--, pp. 57-5 Non-lnear Canoncal Correlaton Analss Usng a RBF Network Sukhbnder Kumar, Elane
Formal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
Lecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
Numerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method
Appled Mathematcs, 6, 7, 5-4 Publshed Onlne Jul 6 n ScRes. http://www.scrp.org/journal/am http://.do.org/.436/am.6.77 umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton
(5.1.1) v Here, u and v are the displacements parallel to x and y directions respectively.
Lecture : Constant Stran rangle he trangular elements wth dfferent numbers of nodes are used for solvng two dmensonal sold members. he lnear trangular element was the frst tpe of element developed for
Grid Generation around a Cylinder by Complex Potential Functions
Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around
The equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
New Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
Implementation of B-Spline Based Finite Element Analysis. for the Boundary-Value Problem. Andrew J. Marker. Master of Science
Implementaton of B-Splne Based Fnte Element Analyss for the Boundary-Value Problem Andrew J. Marker A project submtted to the faculty of Brgham Young Unversty n partal fulfllment of the requrements for
STATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
Lecture 3: Dual problems and Kernels
Lecture 3: Dual problems and Kernels C4B Machne Learnng Hlary 211 A. Zsserman Prmal and dual forms Lnear separablty revsted Feature mappng Kernels for SVMs Kernel trck requrements radal bass functons SVM
Rigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
Appendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
Chapter 10. Numerical Solution Methods for Engineering Analysis
Appled Engneerng Analss - sldes or class teachng* Chapter Numercal Soluton Methods or Engneerng Analss * Based on the tetbook on Appled Engneerng Analss, b Ta-Ran Hsu, publshed b John Wle & Sons, 8 (ISBN
8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
Appendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
Numerical Differentiation
Part 5 Capter 19 Numercal Derentaton PowerPonts organzed by Dr. Mcael R. Gustason II, Duke Unversty Revsed by Pro. Jang, CAU All mages copyrgt Te McGraw-Hll Companes, Inc. Permsson requred or reproducton
MAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
), it produces a response (output function g (x)
Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans
Modeling and Simulation of a Hexapod Machine Tool for the Dynamic Stability Analysis of Milling Processes. C. Henninger, P.
Smpack User Meetng 27 Modelng and Smulaton of a Heapod Machne Tool for the Dynamc Stablty Analyss of Mllng Processes C. Hennnger, P. Eberhard Insttute of Engneerng project funded by the DFG wthn the framework
Ballot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,