Example: barber

Short Introduction to Finite Element Method - NTNU

TDT24 Short Introduction to Finite ElementMethodGagandeep SinghContents1 Introduction22 What Is a Differential Equation? Boundary conditions .. Solution methods .. Finite Difference .. Differential Formula .. Finite Element .. Strong Formulation: Possion s Equation .. Motivation .. Weak Formulation: Possion s Equation .. Advantages of Weak form Compared to Strong Form .. 93 Basic Steps in FE Approximating the Unknownu.. What is Approximation in FE? .. Basis Functions .. Basis Functions 2D .. 134 A Practical The Poisson Equation .. Minimization formulation .. Weak formulation .. 175 Geometric Element Integral .. Load Vector .. 196 Did we understand the difference between FE and FD?

Finite Element (FE) is a numerical method to solve arbitrary PDEs, and to acheive this objective, it is a characteristic feature of the FE approach that the PDE in ques- tion is firstreformulated into an equivalent form, and this formhas the weakform.

Tags:

  Introduction, Methods, Short, Elements, Finite, Finite element, Short introduction to finite element method

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Short Introduction to Finite Element Method - NTNU

1 TDT24 Short Introduction to Finite ElementMethodGagandeep SinghContents1 Introduction22 What Is a Differential Equation? Boundary conditions .. Solution methods .. Finite Difference .. Differential Formula .. Finite Element .. Strong Formulation: Possion s Equation .. Motivation .. Weak Formulation: Possion s Equation .. Advantages of Weak form Compared to Strong Form .. 93 Basic Steps in FE Approximating the Unknownu.. What is Approximation in FE? .. Basis Functions .. Basis Functions 2D .. 134 A Practical The Poisson Equation .. Minimization formulation .. Weak formulation .. 175 Geometric Element Integral .. Load Vector .. 196 Did we understand the difference between FE and FD?

2 207 Finite Element on GPU2111 IntroductionIn this Short report, the aim of which is to introduce the Finite Element Method , avery rigorous Method for solving partial differential equations (PDEs), I will take youthrough the following: What is a PDE? How to solve a PDE? Introduce Finite difference Method Introduce Finite Element Method . A pratical example using Laplace Equation. Fundamental difference from Finite What Is a Differential Equation?Popular science could describe a partial differential equation as a kind of mathemat-ical divination or fortue-telling. In a PDE, we always have amathematical expres-sion describing the relation between dependent and independent variables throughmultiplications and partials. Along with this expression we have given initial con-ditions and boundary requirements.

3 Solution of the PDE can then use this littleinformation, and give us all the later states of this information. This is just like afortuneteller who asks you about some information and tellsyou what s going on inyour life. Well, I do not need to tell the difference between aPDE and relation must be something like "true" and the negation of is not easy to master the theory of partial differential equations. Unlike the the-ory of ordinary differential equations, which relies on the"fundamental existenceand uniqueness theorem", there is no single theorem which iscentral to the , there are separate theories used for each of the major types of partial differ-ential equations that commonly , a partial differential equation is an equation which contains par-tial derivatives, such as the (wave) equation u t= 2u x2(1)In Eq (1), variablesxandtare independent of each other, anduis regarded as afunction of these.

4 It describes a relation involving an unknown functionuof severalindependent variables (herexandt)and its partial derivatives with respect to thosevariables. The independent variablesxandtcan also depened on some other vari-ables, in which case we must carry out the partial derivatives simple elliptic model equation is the Poisson Equation, given by u= 2u= 2u x2+ 2u y2=f, defined on R2(2)2 1 1 Color: u Height: u Figure 1: Solution of Eq (2) withf= will use this equation in the practical example and when introducing the finiteelement Method . Notice thatudepends symmetrically onxandy. So, if we choosea symmetric , the solution will be symmetric. If ( 1,1)2, the solutionuwillhave the form shown in Fig 1. Here we have usedf= Boundary conditionsIn Fig 1 I have usedu=0 on the entire boundary, that is, along the lines(x, 1) and( 1,y).

5 Using a constant value along the entire boundary is calledHomogeneousDirichlet. When we useu=g(x,y) along the entire boundary, it is called Inho-mogenous Dirichelet. It is also possible to specify a constant solutionu=const(oru=g(x,y)) along a (continous) part of the boundary. On the rest of theboundarywe can specify flux variation using so-called Neumann boundary conditions. Thisis called Mixed boundary conditions. Neumann boundary conditions specify thedirectionalderivative of u along a normal vector. We denote it asd ud n=~n u= (x,y) is given and in the end we have known values ofuat some(continous) part of the boundary and the directional derivative on another. Fig 3shows another solution of Eq (2) using Inhomogeneous boundary conditions on ( 1,1)2.

6 Boundary conditions are shown in Fig 2. There are also some otherboundary conditions, but these are the most common ones occurring in physcialmodels and simulations. When we introduce Finite Element Method , we will showhow these conditions are fundamental and reflects in the Solution MethodsDepending on the nature of the physcial problem and the corresponding mathe-matical formulation, we can use between a range of solution methods . The mostknown solution Method isfinite difference Method , which essentially replaces everyoccurrence of partials with a discrete approximation usinggrid information. Thefinite difference Method may be a good Method to use when the geometry is sim-3 Figure 2: Boundary conditions on ( 2,2)2 2 1 2 1 10123456 Color: u Height: u Figure 3: Solution of Eq (2) using boundary conditions shownin Fig 24ple and not very high accuracy is required.

7 But, when the geometry becomes morecomplex, Finite difference becomes unreasonably difficult to Finite Element Method is conceptually quite different from Finite difference whenit comes to the discretization of the PDE, but there are some similarities. It is themost commonly used Method to solve all kinds of PDEs. Matlab hasPartial Differ-ential Equation Toolboxwhich implements Finite Element with pre-specified knownPDEs. COMSOL is also based on Finite Element the next section, I will mention some few points on Finite difference, then gostraight to Finite Element Method . We will then be in a position to discuss someof the differences between Finite difference and Finite Finite DifferenceIn Eq (2), we have an operator working onu. Let us denote this operator byL.

8 Wecan then writeL= 2= 2 x2+ 2 y2(3)Then the differential equation can be written likeLu=f. If for exampleL= 2 2 +2, the PDE becomes 2u 2 u+2u=f. Finite difference methods are basedon the concept of approximating this operator by a discrete Differential FormulaLet me remind you of Taylor s formula with the residual part of the functions of twovariables:u(x+h,y+k)=nXm=01m! h x+k y mu(x,y)+rn(4)wherern=1(n+1)! h x+k y n+1u(x+ h,y+ k), 0< <1(5)In (4), I have used step lengthshandkinxandydirection, respectively. Let us useh=kin the following for simplicity. Let us now develop this series ofu(x+h,y). Idenote it byum+1,n. The rest follows from +1,n=um,n+hum,nx+h22!um,nxx+h33!um,nxxx+ O(h4)(6)The subscript inuxrefers to the partial derivative ofuwith respect tox.

9 Now, let usdevelop the series ofum 1,n,um,n+1, andum,n 1,n=um,n hum,nx+h22!um,nxx h33!um,nxxx+O(h4)um,n+1=um,n+hum,ny+h22! um,ny y+h33!um,ny y y+O(h4)um,n 1=um,n hum,ny+h22!um,ny y h33!um,ny y y+O(h4)(7)5If we supposeum,nxxis small compared toum,nxandh 0, we see from Eq (6) and thefirst equation in (7) thatum,nx=1h um+1,n um,n +O(h) Forward differenceum,nx=1h um 1,n um,n +O(h) Backward difference(8)In the same way, we getum,ny=1h um,n+1 um,n +O(h) Forward differenceum,ny=1h um,n 1 um,n +O(h) Backward difference(9)In the same way, if we supposeum,nxxxis small compared toum,nxxandh 0, we seefrom Eq (6) and the first equation in (7) thatum,nxx=1h2 um+1,n 2um,n+um 1,n +O(h2) Central difference (10)We also get a second order central difference inyfrom second and third equation inEq (7)um,ny y=1h2 um,n+1 2um,n+um,n 1 +O(h2) Central difference (11)Let us look at Poisson Eq (2).

10 It contains two separate secondderivatives, and wecan use Eq (10) and (11) to approximate these withO(h2). 2u=uxx+uy y=um,nxx+um,ny y+O(h2)(12)This expression gives us the well-known five point formula. One thing this expres-sion tells us, is that in order to use Finite difference methods like this one, we need arelatively structured grid. Otherwise, we will have difficulties defining the directionof step the next section, I will introduce Finite Element Method . Iwill start from whatis called "point of departure" for the Element Finite ElementOne remark that can be made already, is that the Finite elementmethod is mathe-matically very rigorous. It has a much stronger mathematical foundation than manyother methods (It has a more elaborate mathematical foundation than many othermethods), and particularly Finite difference.


Related search queries