Finite Element Method and its Applications（簡體書）
商品資訊
ISBN13：9787030421920
出版社：科學出版社
作者：Kaitai Li; Aixiang Huang; Qinghuai
出版日：2014/12/01
裝訂／頁數：平裝／16頁
規格：23.5cm*16.8cm (高/寬)
版次：1
商品簡介
目次
Contents
Chapter 1 The Structure of Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Galerkin Variational Principle and Ritz Variational Principle . . . . . . . . . . . . . . . . . . . . . 1
1.2 Galerkin Approximation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Finite Element Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Element Stiffness and Total Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
Chapter 2 Elements and Shape Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 2.1 Rectangular Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
2.2 Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
2.3 Shape Function of Three Dimensional Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Iso-parametric Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5 Curve Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 3 Procedure and Performance of Computation of Finite Element Method. . . . . . . . .60
3.1 The Procedure of Finite Element Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 One dimensional Store of Symmetric and Band Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Computation of Element Stiffness Matrix and Synthesis of Total Stiffness Matrix . . . . . . 72
3.4.2 The Computation of Element Stiffness Matrix and Element Array . . . . . . . . . . . . . . 76
3.4.3 Superposition of Elements of Total Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Direct Solution Method for Finite Element Algebraic Equations. . . . . . . . . . . . . . . . . .79
3.5.1 Deposition for Symmetric and Positive Definition Matrix . . . . . . . . . . . . . . . . . . . 80
3.5.2 Direct Solution for Algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 Other Solution Method for Finite Element Algebraic Equations . . . . . . . . . . . . . . . . . . 85
3.6.1 The Steepest Descent Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
3.6.2 Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.7 Treatment of Constraint Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7.1 Treatment of Imposed Constraint Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
3.7.2 Treatment of Periodic Constrain Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
3.7.3 Remove Periodic Constrain and Matrix Transformation . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7.4 Performance of the Method on Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.8 Calculation of Derivatives of Function . . . . . . . . . . . . . . . . . . . 98
3.9 Automatic Generation of Finite Element Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter 4 Sobolev Space . . . . . . . . . . . . . . . . . . . 104
4.1 Some Notations and Assumptions on Domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Classical Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Lp(Ω) Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 Spaces of Distr
書摘/試閱
TheStructureofFiniteElementMethod
Thefiniteelementmethodisanumericalcomputationalmethodfordifferentialequationsand
partialdifferentialequations.Inordertosolvethegeneralfieldproblembyusingfiniteelement
method,itmustpassthroughthefollowingprocesses:
1)Findthevariationalformulationassociatedwithoriginalfieldproblem.
2)Establishfiniteelementsubspace.Forexample,selecttheelementtypeandassociated
phasefunctions.
3)Establishelementstiffnessmatrix,elementcolumnandassembleglobalstiffnessmatrixfull
column.
4)Treatmentoftheboundaryconditionsandsolvingofthesystemoffiniteelementequations.
5)Comebacktotherealworld.
Inthisbook,thefirstfourprocesseswillbesystematicformulationsinthefirstchaptertill
thirdchapter.
1.1GalerkinVariationalPrincipleandRitzVariationalPrinciple
Asanexample,weconsiderthelinearellipticboundaryvalueproblemoftwodimension,
(1.1.1)
where,ΩisaconnecteddomaininR2,.Ω=Γ1∪Γ2isapiecewisesmoothboundary.Let
ndenotetheunitoutwardnormalvectorto.Ωdefinedalmosteverywhereon.Ω.p(x,y)∈
C1(Ω),p(x,y)≥p0>0,σ(x,y)∈C0(Ω)andσ(x,y)≥0.
Throughoutthischapterwemakenotation:C0(Ω)=thesetofallcontinuousfunctionin
anopensubsetinRn.Ck(Ω)=thesetoffunctionsv∈C0(Ω),whosederivativesoforderk,
existandarecontinuous;
whereα=(α1,???,αn),|α|=α1+???+αn.
Assumethatu(x,y)∈C2(Ω)satisfies(1.1.1)inΩandon.Ω,thefunctionu(x,y)iscalled
classicalsolutionofproblem(1.1.1).
Next,weconsiderweaksolutionof(1.1.1).Definethenorm
(1.1.2)
SobolevspaceH1(Ω)isaclosureofC∞(Ω),underthenorm(1.1.2)withtheinnerproduct
(1.1.3)H1(Ω)isaHilbertspacewhichiscalledoneorderSobolevspace.Let
C∞0(Ω)={v:visaninfinitedifferentiablefunctionandsupportofv.Ω},
H10(Ω)=theclosureofC∞0(Ω)underthenorm(1.1.2),
itisequivalentto
H10(Ω)={v:v∈H1(Ω),v|.Ω=0}.
Inaddition,let
C∞#(Ω)={v:v∈C∞(Ω),v|Γ1=0},
V(Ω)=closureofC∞#(Ω)underthenorm(1.1.2),
whichisequivalentto
V={v:v∈H1(Ω),v|Γ1=0}.
ItisclearthatVisaHilbertspacewithinnerproduct(1.1.3).Furthermore,
H10(Ω).V.H1(Ω).
Letusintroducebilinearfunctional
(1.1.4)
In(1.1.4),fixedu,thenB(u,v)isalinearfunctionalofv,whilevisfixed,itisalinearfunctional
ofu.Inotherwords,supposeα1,α2,β1,β2arearbitraryconstants,then
B(α1u1+α2u2,β1v1+β2v2)=α1β1B(u1,v1)+α1β2B(u1,v2)
+α2β1B(u2,v1)+α2β2B(u2,v2),.u1,u2,v1,v2∈H1(Ω).
Itisclearthat(1.1.4)satisfies
(1)Symmetry,
B(u,v)=B(v,u).(1.1.5)
(2)ThecontinuityinV×V,i.e.,thereexistsaconstantM>0,suchthat
|B(u,v)|Mu1,Ωv1,Ω,.u,v∈V.(1.1.6)
(3)CoercivenessinV,i.e.,thereexistsconstantγ>0,suchthat
B(u,u)γu21,Ω,.u∈V.(1.1.7)
Ofcourse,
isacontinuouslinearfunctionalinv.
TheGalerkinVariationalFormulationfor(1.1.1):Findu∈V,suchthat
B(u,v)=f(v),.v∈V.(1.1.8)
Asolutionusatisfying(1.1.8)iscalledaweaksolutionof(1.1.1).ThespaceViscalled
admissiblespaceortrialspace.Ontheotherhand,(1.1.8)mustbesatisfiedforeveryv∈V,
therefore,Viscalledtestfunctionspace.Iftrialandtestspaceforthevariationalproblemare
thesameHilbertV,inthiscase,Viscalledenergyspace.
OwingtotheboundaryconditiononΓ2iscontainedinthevariationalproblem(1.1.8),the
boundaryconditiononΓ2iscallednatureboundarycondition,whiletheboundarycondition
onΓ1iscalledessentialboundarycondition.
Thefollowingpropositiongivestherelationshipbetweenclassicalsolutionandweaksolution
of(1.1.1).
Proposition1.1Supposeu∈C2(Ω).Ifuisaclassicalsolutionof(1.1.1),then,uis
theweaksolutionof(1.1.1).Otherwise,ifuisaweaksolutionof(1.1.1),thenuisaclassical
solutionof(1.1.1).
ProofAssumethatu∈C2(Ω)isaclassicalsolutionof(1.1.1),.v∈V,multiplyingboth
sidesof(1.1.1)byvandintegrating
ApplyingGausstheorem
Inviewofv∈V,andusatisfyingboundaryconditionwehave
B(u,v)=f(v),.v∈V,
i.e.,usatisfies(1.1.7).
Conversely,letu∈C2(Ω)beasolutionof(1.1.8),owingtov|Γ1=0,weobtain
Substitutingaboveequityinto(1.1.8)leadsto
Bythearbitraryofv∈V,ityieldsthatuisaclassicalsolutionof(1.1.1).Theproofis
complete.
ThefollowingLax-MilgramtheoremguaranteestheexistenceoftheGarlerkinvariational
problem(1.2.8).
Theorem1.1(Lax-MilgramTheorem)LetVbeaHilbertspace,B(u,v)isabilinear
functionalinV×Vandsatisfies:
SymmetryB(u,v)=B(v,u),.u,v∈V.(1.1.9)
ContinuityThereexistsapositiveconstantMindependentof(u,v),suchthat
|B(u,v)|Muv,.u,v∈V.(1.1.10)
CoercivenessThereexistsaconstant
購物須知
大陸出版品因裝訂品質及貨運條件與台灣出版品落差甚大，除封面破損、內頁脫落等較嚴重的狀態，其餘商品將正常出貨。
特別提醒：部分書籍附贈之內容(如音頻mp3或影片dvd等)已無實體光碟提供，需以QR CODE 連結至當地網站註冊“並通過驗證程序”，方可下載使用。
無現貨庫存之簡體書，將向海外調貨：
海外有庫存之書籍，等候約45個工作天;
海外無庫存之書籍，平均作業時間約60個工作天，然不保證確定可調到貨，尚請見諒。
為了保護您的權益，「三民網路書店」提供會員七日商品鑑賞期(收到商品為起始日)。
若要辦理退貨，請在商品鑑賞期內寄回，且商品必須是全新狀態與完整包裝(商品、附件、發票、隨貨贈品等)否則恕不接受退貨。