Boundary Element Method (BEM)
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- Buddy Hutchinson
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1 Boudary Elemet Method BEM Zora Ilievski Wedesday 8 th Jue 006 HG 6.96 TU/e
2 Talk Overview The idea of BEM ad its advatages The D potetial problem Numerical implemetatio
3 Idea of BEM 3
4 Idea of BEM 4
5 Advatages of BEM Reductio of problem dimesio by Less data preparatio time. Easier to chage the applied mesh. Useful for problems that require re-meshig. 5
6 Advatages of BEM High Accuracy Stresses are accurate as there are o approximatios imposed o the solutio i iterior domai poits. Suitable for modelig problems of rapidly chagig stresses. 6
7 Advatages of BEM 3 Less computer time ad storage For the same level of accuracy as other methods BEM uses less umber of odes ad elemets. 7
8 Advatages of BEM 4 Filter out uwated iformatio. Iteral poits of the domai are optioal. Focus o particular iteral regio. Further reduces computer time. 8
9 Advatages of BEM. Reductio of problem dimesio by.. High Accuracy. 3. Less computer time ad storage. 4. Filter out uwated iformatio ad so focus o sectio of the domai you are iterested i. BEM is a attractive optio. 9
10 The D potetial problem Where ca BEM be applied? Two importat fuctios. Descriptio of the domai. Mappig of higher to lower dimesios. Satisfactio of the Laplace equatios ad how to deal with a sigularity. The boudary itegral equatio BIE 0
11 The D potetial problem Where ca BEM be applied? Where ay potetial problem is govered by a differetial equatio that satisfies the Laplace equatio. or ay other behavior that has a related fudametal solutio e.g. The followig ca be aalyzed with the Laplace equatio: fluid flow torsio of bars diffusio ad steady state heat coductio.
12 The D potetial problem The Laplace equatio for D + x y 0. x y Laplacia operator otetial fuctio Cartesia coordiate axis
13 The D potetial problem Two importat fuctios. The fuctio describig the property uder aalysis. e.g. heat. Ukow λ The fudametal solutio of the Laplace equatio. These are well kow 3
14 4 The D potetial problem l p r p π λ l p r p π λ Fudametal solutio of the D Laplace equatio for a cocetrated source poit at p is p p y Y x X p r + Where Descriptio of the domai
15 The D potetial problem Mappig of higher to lower dimesios Boudary of ay domai is of a dimesio less tha of the domai. I BEM the problem is moved from withi the domai to its boudary. This meas you must i this case map Area to Lie. The well kow Grees Secod Idetity is used to do this. λ λ λ λ da λ λ d A have cotiuous st ad d derivatives. ukow potetial at ay poit. kow fudametal solutio at ay poit. uit outward ormal. derivative i the directio of ormal. 5
16 The D potetial problem Satisfyig the Laplace equatio The ukow will satisfy The kow fudametal solutio satisfies λ 0 λ except the poit p where it is sigular. 0 everywhere i the solutio domai. everywhere λ p l π r p X x + Y y r p p p 6
17 The D potetial problem How to deal with the sigularity Surroud p with a small circle of radius ε the examie solutio as ε à 0 New area is A Aε New boudary is + ε λ λ da λ d A Aε + ε Withi area A Aε 0 & λ 0 λ The left had side of the equatio is ow 0 ad the right is ow 7
18 8 The D potetial problem How to deal with the sigularity + d d ε λ λ λ λ 0 The secod term must be evaluated ad to do this let α d εd r r r π λ λ. Ad use the fact that
19 9 The D potetial problem How to deal with the sigularity π π α ε ε ε π λ λ π ε. l 0 d d 0 / C Evaluated with p i the domai o the boudary Smooth surface ad outside the boudary. π θ C For coarse surfaces
20 0 The D potetial problem The boudary itegral equatio d K d K C K K λ λ Where K ad K are the kow fudametal solutios ad are equal to π θ C
21 The D potetial problem BEM ca be applied where ay potetial problem is govered by a differetial equatio that satisfies the Laplace equatio. I this case the D form. A potetial problem ca be mapped from higher to lower dimesio usig Gree s secod idetity. Show how to deal with the case of the sigularity poit. Derived the boudary itegral equatio BIE C K d K d
22 Numerical Implemetatio Dirichlet Neuma ad mixed case. Discretisatio Reductio to a form AxB
23 Numerical Implemetatio Dirichlet Neuma ad mixed case. C K d K d The ukows of the above are values o the boudary ad are Dirichlet roblem is give every poit o the boudary. Neuma roblem is give every poit o the boudary. Mixed case Either are give at poit 3
24 4 Numerical Implemetatio Discretisatio N i N i i d K d K Ukows
25 Numerical Implemetatio Discretisatio Let K i K i d K i K d i Ukows N N i Ki K i 5
26 Numerical Implemetatio i whe i N K + i δi i N K Ax Bz 6
27 Numerical Implemetatio Neuma roblem Ax c Matrix A ad vector C are kow Dirichlet roblem c Bz Matrix B ad vector C are kow Mixed case Ax Bz Ukows ad kows ca be separated i to same form as above 7
28 8 Numerical Implemetatio As each poit p i the domai is expressed i terms of the boudary values oce all boudary values are kow ANY potetial value withi the domai ca ow be foud. d K d K C
29 Boudary Elemet Method BEM THE END Book: The Boudary Elemet Method i Egieerig A.A.BECKER 9
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