MPY308. Clinical Engineering & Computational Mechanics

Transcription

1 MPY38 Clncal Engneerng & Computatonal Mechancs Recommended Readng: John D Anderson Jr, Computatonal Flud Dynamcs- he Bascs and Applcatons, McGraw-Hll, ISBN Brown, Smallwood, Barber, Lawford, Hose, Medcal Physcs and Bomedcal Engneerng, Insttute of Physcs Publshng, ISBN (pb) MJ Fagan, Fnte Element Analyss: heory and Practce, Longman Scentfc & echncal, ISBN

2 Motvaton Many physcal phenomena are descrbed by dfferental equatons he equatons are often too dffcult to solve analytcally, partcularly for comple geometres or boundary condtons Solutons can be sought by the applcaton of numercal methods

3 Module Overvew hs module descrbes some of the numercal technques that can be used to eplore physcal systems, wth llustratons from heat conducton, bomechancs and boflud mechancs Fnte dfferences wll be ntroduced from frst prncples, and smple code wll be developed to solve lnear partal dfferental equatons he theoretcal bass of fnte element methods wll be brefly dscussed, and the lectures wll be supported by laboratory sessons n whch the student wll apply commercal codes to nvestgate problems n the medcal sector

4 Outlne Syllabus Introducton, obectves, hstory and development, commercal eplotaton, typcal applcatons of numercal methods n bomechancs and boflud mechancs Introducton to MALAB, matr manpulaton Presentaton of results Eercses One-dmensonal heat conducton Soluton by fnte dfference and fnte element schemes Etenson to D and 3D Shape functons Galerkn weghted resduals Laplaces equaton Numercal soluton and practcal applcatons Matr stffness methods and applcatons to truss problems Symmetry and antsymmetry Fnte element formulaton for D bar Prncple of vrtual dsplacements Introducton to ANSYS Pre-processng, soluton and post-processng phases Practcal aspects of fnte element analyss Fnte element formulaton for a D contnuum Completeness and compatblty Patch test Parametrc element formulatons Jacoban matr Computatonal mplementaton Numercal ntegraton

5 Methodology In order to construct effectve numercal schemes to solve the equatons we need to understand the fundamental characterstcs of the equatons We shall learn how to reduce the equatons to a set of smultaneous equatons hese can readly be wrtten n matr algebra

6 Software - Matlab Modern packages lke MALAB have a whole host of routnes for effcent manpulaton of matrces hey also feature brllant graphcs routnes for dsplayng the results so that we can understand them! We shall apply numercal schemes that we have constructed n MALAB to solve problems relevant to the actvtes of a medcal physcst

7 Software - ANSYS We shall learn how to use ANSYS, a maor commercal fnte element analyss package, to solve problems ncludng structural, flud dynamc, electrcal and thermal applcatons We wll construct models n ANSYS to study problems wth a more comple geometrcal shape

8 Ams and Obectves At the end of ths module you should: Understand the role of numercal methods n the soluton of real problems n bophyscs and boengneerng; Be able to construct smple fnte dfference and fnte element schemes for the soluton of partcular systems of dfferental equatons; Understand the assumptons n the numercal schemes and the requrements for approprate boundary condtons, consttutve equatons and numercal formulatons; Have a feel for the sze of a problem, the requrements for an approprate numercal scheme, and understand how to go about choosng or developng software Have completed a mn-proect usng MALAB, ANSYS or FLORAN to nvestgate a bomedcal system; Be aware of the requrements for comprehensve verfcaton, valdaton and reportng of numercal solutons, and recognse these requrements n the producton of a mn-proect report;

9 Structure one hour lectures/ laboratores Prvate work, eamples, assgnments, readng ~ 5 hours Assessment %age Hand In Assgnment (Matlab) % Wk 3 Assgnment (Fnte Dfferences) 5 % Wk 8 Mn-Proect Assgnment (ANSYS) 65 % Wk

10 Laplaces Equaton

11 Laplaces Equaton + + y z Apples to many physcal problems ncludng, for eample, steady-state temperature dstrbuton n a conductve body

12 Laplaces Equaton depends only on the spatal coordnates (,y,z) s a scalar It mght, for eample, be temperature Once we know the soluton for we can construct a colour map showng ts dstrbuton hs s often called a frnge or contour plot

13 Laplaces Equaton ( we say grad ) s a vector It represents a varaton of n space We mght plot t on a grd as an arrow at each pont he colour or length of the arrow mght descrbe the magntude of the vector and ts drecton wll ndcate the drecton of the gradent of What physcal quanttes mght these arrows represent? ( formally dv (grad )) s a scalar agan In Laplaces equaton ts value s zero everywhere n the soluton doman

14 Laplaces Equaton n D Boundary Boundary emperature Dstrbuton? Descrbes steady state dstrbuton of temperature along a bar

15 Not all pars of boundary condtons are useful to us We wll eplore ths problem later Laplaces Equaton n D Obvously we can solve ths very easly usng ntegraton, but we wll use t as a vehcle to develop our understandng of the numercal soluton procedures It s a second order (lnear, homogeneous) dfferental equaton, so to solve t we would epect to need two boundary condtons here are four possble boundary condtons - the value of the temperature and the gradent of the temperature at each end of the bar

16 Fnte Dfferences

17 Spatal Dscretsaton of Doman he bass of our numercal schemes wll be to dvde the doman up nto sectons and to seek the soluton at ponts n the doman We wll cause these ponts nodes In the D case we wll have: Boundary Boundary L + n he fundamental unknowns n our numercal representaton wll be the temperatures at each node, If we are to solve the problem numercally, we need to go from the dfferental equaton to a seres of smultaneous equatons How do we represent n the dscretsed doman?

18 Fnte Dfferences n D + + h How do we represent the gradent of temperature at node?

19 Dfference Epressons for Gradent Backward dfference h Forward dfference Central dfference h + h + + Whch, f any, s most accurate? + h

20 Dfference Epressons for Curvature Can you construct a dfference epresson for curvature?? + + h

21 Central Dfference Epresson for Curvature + + h h Gradent to left Gradent to rght h + + h h h + h + +

22 Dscretsed Laplaces Equaton h Contan eternal nodes need to replace these equatons h + + h + + n n n n Away from boundares (n- equatons)

23 Boundary Condtons Optons?? ) Specfy emperature eg 7 ) Specfy Gradent eg forward dfference h

24 Boundary Condtons h Can boundary gradent accuracy be mproved by central dfferencng? ) Specfy Gradent eg central dfference h [ ] h h + Central dfference wth Laplaces equaton yelds same boundary condton as forward dfference h

25 Numercal Eample o C o C/m m 4 m

26 Numercal Eample m o C/m + 3 h h h 4 5 h 3

27 Numercal Eample m C/m 3 5 o

28 Numercal Eample C/m 3 5 o m { } Soluton methods Gaussan elmnaton, Iteratve, you choose!

29 D Central Dfference Solver for Laplaces Equaton % D Central Dfference hermal Solver nn5; % otal number of nodes L4; % Length of doman hl/(nn-); % Dstance between nodes Problem varables bc; % Boundary condton flag at node : s temperature, s gradent ()8; % and the value for temperature, d for gradent bcn; % Smlarly at node n d(nn)-3; Boundary condtons

30 D Central Dfference Solver for Laplaces Equaton kzeros(nn,nn); % Intalse 'stffness' matr bzeros(nn,); % Intalse rght hand vector - 'load' vector for : nn- % Set up rows of stffness matr for all nternal nodes k(,-)-; k(,); k(,+)-; end;

31 D Central Dfference Solver for Laplaces Equaton f bc % Set up equaton at node k(,); b()(); else k(,); k(,)-; b()-h*d(); end; emperature boundary condton Gradent boundary condton note the cost of generalty

32 D Central Dfference Solver for Laplaces Equaton f bcn % Set up equaton at node n k(nn,nn); b(nn)(nn); else k(nn,nn-)-; k(nn,nn); b(nn)h*d(nn); end; emperature boundary condton Gradent boundary condton

33 D Central Dfference Solver for Laplaces Equaton f rank(k)<nn 'Error, k s sngular' end % Check that the problem s well-posed An error trap k\b; % Solve hs operaton n Matlab s perfectly adequate for our current purposes: f we have large matrces we mght reconsder!

34 D Central Dfference Solver for Laplaces Equaton plot(); % Plot results graphcally Improve ths fgure -tle -Label Aes (Unts!) -X as should be poston (m), not matr entry

35 Observatons on Matr Structure m C/m 3 5 o Sparse rdagonal Storage, specal soluton methods?

36 Laplaces Equaton n D + y Descrbes steady state dstrbuton of temperature n d space

37 Doman Dscretsaton n D y Column Row + Row Row

38 Laplacan n D,, + +, y h, y Column,,, + h, + Row + Row Row h h,,, +,, +,

39 Laplacan n D h h,,, +,, +, y Column Row + - Row Row -

40 Numercal Eample y y m y m

41 Numercal Eample y y Boundary Boundary m Boundary y Boundary grd : unusual because only one nternal node

42 Numercal Eample Boundary m y Boundary 3 5 Boundary y Grd sze h m Boundary y + 5 [ y ] Co-ordnates of nodes

43 Numercal Eample Boundary 3 5 y + 5 y + 45 y m Boundary Boundary 4 Boundary Set flags { } { } bc ; _ spec {} { } { } 5 7?????? dy ; 9?? 7?? 5?? d ;???? Compute specfed temperatures and gradents

44 D Central Dfference Solver for Laplaces Equaton % D Central Dfference hermal Solver nn3; L4; Problem varables nny3; Ly4; hly/(nny-); % Note that we are consderng only square grds nnnn*nny; for :nn % Calculate the co-ordnates of each node for :nny n+nn*(-); y(n,)h*(-); % and store n the array [y] y(n,)h*(-); end; Nodal co-ordnates end;

45 D Central Dfference Solver for Laplaces Equaton zeros(nn,); % Intalse the temperature vector _speczeros(nn,); % and the 'specfcaton' flag bc; % Specfy the condtons on the boundary for n : nn : nn-nn+ (n); _spec(n); end; bc; % and on the L boundary for n nn : nn : nn d(n)*y(n,)+5; end; bc3; % and on the y boundary for n : nn (n)5*y(n,); _spec(n); end; bc4; % and on the y L boundary for n nn-nn+ : nn dy(n)*y(n,)*y(n,)+45*y(n,); end; Defne Boundary Condtons

46 D Central Dfference Solver for Laplaces Equaton kzeros(nn,nn); % Intalse 'stffness' matr bzeros(nn,); % and 'load' vector for :nn- % Set up rows of stffness matr for all nternal nodes for :nny- n+nn*(-); k(n,n-nn)-; k(n,n-)-; k(n,n+)-; k(n,n+nn)-; k(n,n)4; end; end;

47 D Central Dfference Solver for Laplaces Equaton for n : nn : nn-nn+ % Set up equatons on boundary f bc k(n,n); k(n,n+)-; b(n)-h*d(n); end; end; for n nn : nn : nn % Set up equatons on L boundary f bc k(n,n); k(n,n-)-; b(n)h*d(n); end; end; for n : nn- % Set up equatons on y boundary f bc3 k(n,n); k(n,n+nn)-; b(n)-h*dy(n); end; end; for n nn-nn+ : nn- % Set up equatons on yl boundary f bc4 k(n,n); k(n,n-nn)-; b(n)h*dy(n); end; end; Set up gradent boundary condtons Row 9 overwrtten

48 D Central Dfference Solver for Laplaces Equaton for n : nn % Impose specfed temperatures f _spec(n) for : nn k(n,); end; k(n,n); b(n)(n); end; end; Impose known temperatures Rows 3 and 7 overwrtten

49 D Central Dfference Solver for Laplaces Equaton f rank(k)<nn % Check that the problem s well-posed 'Error, k s sngular' end k\b; % Solve An error trap When settng up the equatons some lnes were overwrtten by other lnes Does ths change the answers? Is there a more accurate alternatve?

50 D Central Dfference Solver for Laplaces Equaton % Reshape soluton vector nto a matr wth entres correspondng to % the geometrcal postons of the nodes arrayreshape(,nn,nny)'; % Plot the results % Fgure Surface plot of temperature surf(array); % Fgure Shaded nterpolated flat mage fgure, pcolor(array), shadng nterp, as equal % Compute the entres of the gradent vector [g,gy]gradent(array,h); g-g; gy-gy; % Fgure 3 Vector plot of gradent fgure, contour(array),hold on, quver(g,gy),hold off;

51 D Central Dfference Solver for Laplaces Equaton Results for grd

52 Heat Flu he heat flu across a regon of the boundary s gven by the product of the materal conductvty, the gradent normal to the boundary and the surface area of the boundary, Flu k n ds An adabatc boundary s descrbed by the zero flu condton, mplyng no heat flu across t

53 Symmetry A symmetry plane s one about whch both the geometry and the loadng are symmetrcal A symmetry plane must be an adabatc boundary (zero flu, zero gradent)

54 Antsymmetry An antsymmetry plane s one about whch the geometry s symmetrcal but the loadng s antsymmetrcal he temperature on such a boundary must be zero Use of symmetry and/or antsymmetry condtons can produce sgnfcant savngs n the sze of the computatonal model

55 heoretcal Fundamentals

56 heoretcal Fundamentals aylor seres epanson of a functon about a pont ' ( ) ( ) ( )( ) f f a + f a a + '' ( ) n ( )( ) ( )( ) f a a f a a + +! n! n + If the value of a functon and all of ts dervatves are known at a pont a then ts value at all ponts n the neghbourhood of a can be computed

57 Order of Accuracy of Fnte Dfference Appromatons runcatng at order h : Rearrangng: ( ) ( ) '( )( ) + h f + f h O(h ) f + ( ) f ( h) f ( ) + f ' + h O(h) he forward dfference scheme that we derved ntutvely s frst order accurate What s the order of accuracy of the backward dfference and of the central dfference schemes that we wrote down?

58 Order of Accuracy of Fnte Dfference Appromatons runcatng at order h 3 : Addng: Rearrangng: ''' ( )( h) f ( )( h) '' ' f f ( + h) f ( ) + f ( )( h) O! 3! ''' ( )( h) f ( )( h) '' ' f f ( h) f ( ) f ( )( h) + + O! 3! ( )( h) '' f f ( + h) + f ( h) f ( ) + + O ( ) f ( + h) + f ( ) + f ( h) '' f + h O 3 3 ( h) 4 ( h) ( h) 4 ( h) 4 and so the - - scheme s second order accurate

59 Order of Accuracy of Fnte Dfference Appromatons Need hgher accuracy? Wrte more terms! Subtractng: ''' 3 '''' ( )( h) f ( )( h) f ( )( h) '' ' f f ( + h) f ( ) + f ( )( h) O! 3! 4! ''' 3 '''' ( )( h) f ( )( h) f ( )( h) '' ' f f ( h) f ( ) f ( )( h) O! 3! 4! ( )( h) ''' ' f f ( + h) f ( h) f ( ) h + + O 3 Smlarly, wrtng f(+h) and f(-h) and subtractng: ''' 3 ' 8 f ( ) ( ) ( ) ( )( h) + h f h 4 f h ( h) 5 f + O 3 ( h) 5 ( h) 5 ( h) 5 and by elmnatng the thrd dervatve a fourth order accurate epresson for the gradent s obtaned

60 ransent Problems

61 D transent heat conducton α t α s thermal dffusvty, k/ρc What are ts unts? k s thermal conductvty (W/mK) ρ s densty (kg/m 3 ) c s specfc heat capacty (J/kgK)

62 D transent heat conducton α t One dmensonal transent heat conducton hs s a parabolc second order partal dfferental equaton Such equatons are open and lend themselves to tme marchng schemes Generally we have a set of ntal condtons, and then we follow the evoluton of the soluton varables he drecton n whch we follow the soluton does not have to be tme but t often s

63 ransent Soluton Storage One dmensonal transent heat conducton α t Store the soluton as a matr array {} + + m n m m n n,,,,,,,,,,,,,,,,, Spatal soluton at tme me Space emporal soluton at poston

64 One dscretsaton opton {} + + m n m m n n,,,,,,,,,,,,,,,,, Spatal soluton at tme emporal soluton at poston + +,,, h dt,, t + me Space Central dfference n space Forward dfference n tme

65 Dscretsaton α t Central dfference n space Forward dfference n tme,, + +,, +, h α dt α dt h ( ), +, +,, + +,

66 An Eplct Scheme α t Central dfference n space Forward dfference n tme α dt h α dt h ( ), +, +, + +, Gven the soluton at any tme, the soluton at tme + can be calculated mmedately from ths eplct equaton No matr nverson s requred t s computatonally nepensve!

67 Numercal Eample m m Bar of length 4 m, ntally at temperature zero, has one end rased n temperature by C/s for seconds, and then the temparature there s mantaned at C he temperature at the other end s mantaned at zero throughout he dffusvty s -5 m /s Compute the temperature dstrbuton at second ntervals at the ends and at three equ-spaced nternal ponts, untl 4 s have elapsed

68 Numercal Eample emperature at node Intal emperature everywhere tme t s t m s 4 m t What do you epect to happen? Do we need boundary condtons at the rght hand end? what would happen wthout?

69 Numercal Eample Intal emperature everywhere tme 5 {}

70 Numercal Eample {} ( ),,,, h dt + h dt α α tme 5 ( ),,,, α -5 m /s, dt s, h m An eplct scheme

71 Numercal Eample tme At t s, t,, ( ) , 3,, 3, (, 4,) ( ) , 4, 3, 5, {} 5,

72 Numercal Eample tme At t s,3 t,3, ( ) ,3 3,, 3, (, 4, ) ( ) ,3 4, 3, 5, {} 5,3

73 Numercal Eample tme At t 3 s,4 3t 3,4,3 3 ( ) ,4 3,3,3 3,3 (,3 4,3 ) ( ) ,4 4,3 3,3 5,3 {} ,4

74 Numercal Eample and so we go on {}

75 Numercal Eample Soluton at start, s (heatng tme) and 4s (end) {}

76 Numercal Eample Make a note of the temperature at the md-node () at t 4s Eperment wth changng the grd sze (h) and wth changng the tmestep (dt) what do you fnd? Convergence? Stablty?

77 Numercal Eample Make a note of the temperature at the md-node () at t 4s Eperment wth changng the grd sze (h) [dts] h (, 4) ????

78 Numercal Eample Make a note of the temperature at the md-node () at t 4s Eperment wth changng the grd sze (h) [dts] h (, 4) ???? 565

79 Numercal Eample Make a note of the temperature at the md-node () at t 4s Eperment wth changng the grd sze (h) [dts] h (, 4)

80 Numercal Eample Make a note of the temperature at the md-node () at t 4s Eperment wth changng the tmestep (dt) [h65m] dt (, 4) 4????

81 Eplct Scheme Stablty Lmt? α t Central dfference n space Forward dfference n tme α dt h α dt h ( ), +, +, + +, α dt What happens when >? h Go back to your tables and enter the value of α dt h What do you see? and what s the rght answer??? Accuracy?

82 Alternatve Dscretsaton? α t dt h,,,,, α + + Central dfference n space Backward dfference n tme dt h,,,, α

83 An Implct Scheme α t Central dfference n space Backward dfference n tme + h + dt α,, +,, hs looks rather lke our scheme for D Laplaces equaton ( addtonal term on man dagonal and entry on rght) rangular matr at every tmestep computatonally epensve! but are there compensatons?

84 Numercal Eample Repeat the tests performed wth the eplct scheme, completng the same tables Eperment wth changng the grd sze (h) and wth changng the tmestep (dt) what do you fnd? Convergence? Stablty? We stll need to deal wth Accuracy

85 More Dscretsaton Schemes here are very many varants on these schemes, many of whch bear the names of ther developers (eg Crank-Ncholson, ) he dfference between them s the way n whch the equatons are dscretsed We seek to mprove the accuracy and stablty of eplct schemes and the accuracy of mplct schemes For nonlnear problems we seek also to mprove the rate of convergence

86 Eplct v Implct Fast computaton at each tmestep rvally parallelsable Only condtonally stable Mght need very small tmestep Uncondtonally stable Computatonally epensve per tmestep long tmesteps stable but mght be naccurate! What spatal and temporal resoluton does the physcs demand? What type of computatonal resource s avalable? Now choose!

87 Fnte Elements

88 Prncples of Fnte Element Analyss In fnte element analyss, lke fnte dfference methods, we descrbe the soluton n terms of the values of the unknowns at dscrete ponts (NODES) n the doman Unlke fnte dfference methods, we make drect assumptons about the magntudes of the unknown varables everywhere her values at all ponts wthn each ELEMEN are assumed to be known functons of the values at the nodes of the element he (unknown) values of the soluton varables at the nodes are called the DEGREES OF FREEDOM (DOFs) We attempt to ensure that some measure of the soluton error s mnmsed over the whole doman

89 A D, DOF Element Φ Φ Node Element e Node L e How does vary over the element?

90 Lnear D, DOF Element Φ Φ Node Element e Node L e he most natural choce s a lnear varaton

91 Lnear D, DOF Element Φ Φ Node Element e Node + Node Element e Node L e L e Φ Φ Node Element e Node L e

92 Lnear D, DOF Element Φ Φ Node Element e Node + Node Element e Node L e L e ( ) L e Φ ( ) L e Φ N Φ Φ N Node Element e L e Node N Φ + N Φ N [ ]{ Φ}

93 he Shape Functons he functons N are called shape functons N descrbes how the parameter ( n ths case) vares over the element when t has a value of at node and zero at all other nodes N N Node Node Node Node L e L e

94 Methods of Weghted Resduals For feld problems the dscrete numercal equatons are usually formulated by mnmsaton of some measure of the error over the doman akng Laplaces equaton n D as an eample:

95 Methods of Weghted Resduals In each element we have assumed the value of e as a functon of ts values at the nodes e [ ] { } e Φ e N Hence wthn each element: ([ N] { } ) N e e e {} e

96 Methods of Weghted Resduals but f Laplaces equaton s to be satsfed we requre that ths functon be zero at all ponts n the doman he amount by whch the equaton s not satsfed at any pont s called the resdual In ths case, wthn each element: R N e {} e

97 Methods of Weghted Resduals We mght take one measure of the error n the soluton to be the sum of the errors over the whole doman R d elements N e {} e d A lttle thought mght persuade us that ths s not a partcularly good measure of error not least because huge + and errors wll cancel

98 Methods of Weghted Resduals More generally, and more approprately, we mght take a weghted sum of the resduals W N ( ) R d W ( ) {} elements e e d where W() s a weghtng factor

99 Galerkn s Method of Weghted Resduals In Galerkn s method we take the weght to be the dsplacement tself NR d N { } [ ] N {} elements e e e e d he order and manner n whch the terms are wrtten s chosen for convenence and consstency wth systems wth more DOFs at each node

100 Galerkn s Method of Weghted Resduals he soluton vectors are ndependent of and can be taken outsde the ntegral, and ndeed of the summaton NR d elements N [ ] N d {}{} e e Now f ths quantty s to be mnmsed, ts dfferental wth respect to each of the entres n the nodal soluton vector must be zero

101 Galerkn s Method of Weghted Resduals Dfferentatng the error measure wth respect to each of the soluton varables ( ) n turn: N [ ] N d {} {} elements e hs matr equaton represents n smultaneous equatons n the n unknowns e

102 Galerkn s Method of Weghted Resduals he second order dfferental can be elmnated by ntegraton by parts [ ] [ ] d d e e e e e e N N N N N N he frst term s a boundary condton, and wll be dealt wth separately he second term we wrte as: [ ] [ ] [ ] d k e e e B B where [ ] e e N B

103 he Stffness Matr [k] s usually called the stffness matr, because t was frst derved (usng a dfferent approach ) for structural problems For the lnear D element: [ ] [ ] L L L N B e e e e e [ ] [ ] [ ] L B B e e e e d k

104 System Stffness Matr he system stffness matr s assembled by summng over the elements [ k] ([ k] ) global elements e

105 he Boundary Condtons In the absence of heat sources or snks along the length of the bar, the heat conducton from the rght of one element must equal that nto the left of the net one, and the relevant boundary condtons for the system are those at the two ends L Φ Φ At the left hand end: [ ] Φ Φ Φ Φ Φ Φ L L L L e e e e N N From the Galerkn formulaton: L Φ Φ n n n n At the rght hand end:

106 Numercal Eample o C o C/m m 4 m

107 Numercal Eample C/m 3 5 o m [ ] L e k [ ] 3 k [ ] k [ ] 4 k

108 Numercal Eample m C/m 3 5 o [] global k [] global k

109 Numercal Eample m C/m 3 5 o here are no nternal sources or snks and so the forces on nternal nodes are zero he frst and last equatons can be replaced by the boundary condtons as for the fnte dfference method and n ths case the system of equatons s dentcal to that derved by the fnte dfference approach

110 he D Heat equaton Laplaces equaton represents the steady state dstrbuton of temperature n the absence of heat sources or snks For the solutons so far we assumed that there was no source or snk ecept at the boundary, and treated the boundary separately More generally: k + Q Workng through the same procedure as llustrated prevously, the fnte element method wll naturally yeld the full system equatons, although specal treatment wll be requred for postons at whch the temperature s fed

111 Parametrc Element Formulatons Fundamental to the fnte element method s the descrpton of the varaton of the governng parameters over the whole of the element n terms of ther values at the nodes Such descrptons can be dffcult to wrte for elements of comple shape It s attractve to produce the analytcal dervatons n an dealsed space, wth a mappng to and from real space descrbed by a Jacoban for each element hs s the bass of the parametrc element formulatons

112 A D Isoparametrc Element Node l Φ l η Node k Φ k - ξ Φ Node - Φ Node N Φ + N Φ + N Φ + N Φ [ N]{ Φ} k k l l

113 A D Isoparametrc Element N N l + k (( ξ)( η) ) : N ( ξ)( η) ( ) (( ξ)( η) ) : N ( ξ)( η) ( )

114 From Real to Parametrc Space We need to descrbe the relatonshp between the real space n whch the element s defned and the dealsed, parametersed space n whch ts shape functon s defned y η (,) ξ (-,-)

115 he Jacoban Matr η y ξ (,) J (-,-) he mappng s descrbed n terms of the Jacoban, [J] y y y y η η η ξ ξ ξ + + By the chan rule of dfferentaton : [ ] y J y y y η η ξ ξ η ξ

116 Evaluaton of Jacoban y J (-,-) η (,) ξ o evaluate the Jacoban we need to know how the and y co-ordnates are related to the ξ and η ones he most common assumpton (soparametrc element) s that the relatonshp s eactly the same as that whch descrbes the dstrbuton of the soluton parameters N + N + N + N N k k l l [ ]{ } [ ]{ } y N y + N y + N y + N y N y k k l l

117 he [B] Matr [ ] [ ] ξ η ξ ξ ξ ξ η η η η y J J N N N N N N N N k l k l 3 4 Φ Φ Φ Φ [B] he dervatves of the shape functons n parametrc space are very easy to calculate [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ξ ξ ξ ξ η η η η J B

118 he Element Stffness Matr For the D verson of Laplaces equaton, the element stffness matr s: [ ] [ ] B [ B] d dy k e Area [B] s defned n parametrc co-ordnates, and t s natural to perform the ntegral n these co-ordnates [ ] [ ] B [ B] k e - det[j] dξ dη

119 Numercal Integraton [ ] [ ] B [ B] k e - det[j] dξ dη In the general case the Jacoban, and hence the [B] matr and det[j], vary over the element he stffness matr s usually computed usng a numercal ntegraton procedure Gaussan Quadrature s commonly used for ths purpose Gaussan quadrature s accurate for a cubc polynomal he procedure becomes naccurate when the element becomes too dstorted here are many measures of dstorton

120 Stffness Matr for a Square Element y h h J (-,-) η (,) ξ h J [ ] [ ] k e Check t!

121 Global Stffness Matr Fnte Element Model Element Stffness Matrces he global stffness matr s: sparse suggests specal soluton technques Global Stffness Matr sngular requres attenton to boundary condtons before soluton

122 est Problems

123 Structural Analyss Problem y mm N/mm E MPa υ 3 Dameter 5 mm mm Stress concentraton at hole n plate?

124 Structural Analyss Problem y mm N/mm 4 3 R5 mm 5 5 mm 8 E MPa υ 3

125 Blood Flow Problem 7% (by area) Asymmetrc Stenoss 4 mm V m/s y R? Pa mm 8 mm Not to Scale Q 5 ltres/mn ρ 5 kg/mm 3 µ 4 Pas