Sınır elemanları metodu
Boundary elements method
- Tez No: 66440
- Danışmanlar: PROF. DR. TEOMAN KURTAY
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Makine Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 100
Özet
Sınır Elemanları Metodu, bazı üstünlükleri nedeniyle, son zamanlarda Sonlu Elemanlar Metoduna karşı güçlü bir alternatif olarak ortaya çıkmıştır. Özellikle gerilme yığılması ve alanın sonsuza uzamasının söz konusu olduğu problemler daha yüksek kesinlikte sonuçlar gerektirdiğinden, sınır elemanları etkin bir çözüm aracı olmaktadır. Sınır Elemanları Metodu'nun önemli bir özelliği, hacim yerine yüzeyin analizinin yeterli olmasıdır. Bu şekilde yapılan temsil eden elemanların oluşturduğu ağlar daha kolay ve çabuk bir şekilde ortaya çıkarılmakta, sonradan gerekli görülen değişiklikler kolayca yapılmaktadır. Sonlu Elemanlar Metodu ise ağların tanımlanmasından nispeten yavaş yöntemler nedeniyle daha çok zaman gerektirmektedir. Ayrıca bu yöntemin Bilgisayar Destekli Sistemlere uyarlanmasında da zorluklarla karşılaşılmaktadır. Bu çalışmada Sınır Elemanları Metodunun dayandığı matematiksel kavramlar anlatılmış, değişik tipteki problemlerin çözümü için öngörülen farklı yöntemler açıklanmaya çalışılmıştır. Daha sonra Laplace veya Poisson denklemleriyle kapsanan potansiyel problemleri ele alınmıştır. Bu kısımda Sınır Elemanları Metodu, ağırlıklı artanlar yöntemiyle anlatılmıştır. Bu yaklaşım, metodun diğer yöntemlerle bağlantısını sağlamakta ve sınır elemanlarının tanıtımı için kolay bir yol oluşturmaktadır. Potansiyel problemlerinin bu formulasyonu Jaswon tarafından yapılmıştır. Jaswon 1963 yılında Symm ile birlikte, sınır integral denklemlerini çözmek üzere sının küçük parçalara bölmek ve her parçada sabit yoğunluk farz etmek suretiyle bir sayısal yöntem oluşturmuştur. Ağırlıklı artanlar tekniğinin avantajı onun genelliğinden kaynaklanmaktadır. Bu genellik, metodun daha karmaşık kısmi diferansiyel denklemleri çözmek üzere genişletilebilmesine olanak sağlar. Bölümün sonunda potansiyel problemlerin çözümüne dair, Fortran dilinde yazılmış bir program anlatılmış olup, bir ısı akış problemi ele alınarak bu program vasıtasıyla çözülmüştür. Elde edilen sonuçlar, modelin basitliği de göz önüne alınarak kesin çözümlerle karşılaştırıldığında, mükemmel denilebilecek bir paralellik sağlandığı gözlemlenmiştir. VIH
Özet (Çeviri)
Boundary elements have emerged as a powerful alternative to finite elements particularly in cases where better accuracy is required due to problems such as stress concentration or where the domain extends to infinity. That's why finite elements have been proved to be inadequate or inefficient especially at these engineering applications and also Finite Element Method cumbersome to use and hence difficult to integrate in Computer Aided Engineering Systems. Finite Element Analysis is still comparatively slow process due to the need to define or redefine meshes in the piece or domain under study. The most important feature of boundary elements over finite elements is that it only requires discretization of the surface rather than the volume. Boundary element codes are easier to use with existing solid modellers and mesh generators. This advantage is particularly important for designing as the process usually involves a series of modifications which are more difficult to carry out using finite elements. Meshes can easily be generated and design changes do not require a complete remeshing. Boundary element meshes, especially three dimensional ones can easily be linked to Computer Aided Engineering Systems as the structure is defined using only the boundary. The discretization process is even simpler when using discontinuous elements, which are not admissible in finite elements. Boundary elements are an ideal tool for engineering design mainly because it is easy to generate the data required to run a problem and carry out the modifications needed to achieve an optimum design. With computer costs declining while engineers time becomes more expensive the saving in engineer's time is of primary importance. More important still, any tool that can shorten the“turn around”time taken by the analysis and design can bring forward the completion date of a project. Although better computational performance is important in Boundary Elements Method, particularly for three dimensional problems, improvements in CPU times should not come at the expense of precision and accuracy. For instance, applying coarse numerical integration techniques to Boundary Elements Method codes can result in large savings in computer codes and give reasonable results in many cases. For other cases however the solution may be of very poor accuracy or give non- convangent results. This makes such codes unreliable. IXAnother important advantage of Boundary Elements Method over Finite Element Method is when analysing problems with stress (or flux) concentration. Many such studies have now been carried out and they tend to demonstrate the high accuracy of boundary elements for problems such as re-entry corners, stress intensity problems and even fracture mechanics applications. As an illustration, finite element solutions, a photo-elastic model Boundary Elements Method applied to a pressure vessel. Results for a finite element mesh consisting of approximately 500 degrees of freedom (69 elements) and using eight nodes elements are compared against Boundary Elements Method solutions obtained using 20 elements. While the 69 elements finite element results show lack of equilibrium in the domain as well as the boundary, reasonably accurate solutions were obtained using boundary elements. It was only when using a very refined finite element mesh that the finite element results were in agreement with the boundary element and photo-elastic model solution for results obtained using 240 elements. The typical of boundary elements include torsion, diffusion, seepage, fluid flow and elastostatics. Corrosion engineers have used the method to design better cathodic protection systems for offshore structures, ships and pipelines. Many of these structures are basically three dimensional and the region of interest extends to infinity. Consequently they could not be effectively analysed before the development of boundary elements. Early attempts to use finite differences or finite elements to solve these problems met with little success. For these cases the computer model has to represent the potential field around the structure, representing the shielding effect of the structural geometry and the effect of the different materials involved. Unlike a structural model the cathodic protection model is concerned with the seawater around the structure and the interface between the seawater and the structure. Hence the use of Finite Elements Method to analyse the problem.would require the subdivision of the seawater surrounding the structures. The use of Boundary Elements Method represents the only practical solution for this problem. The advantage of the method is that only the structure needs to be defined as the Boundary Elements Method automatically cares of the field extending to infinity. The advances made in cathodic protection modelling using boundary elements are just one of the applications of the technique for systems extending to infinity. The method is nowadays extensively used in other problems with infinite or semi-finite domains such as those occurring in geomechanics, ocean engineering, foundations aerodynamics, flow through porous media and many others. BASIC CONCEPTS FUNDAMENTAL CONCEPTS Consider a very simple differential equation applying in a one-dimensional domain x, from x=0 to x=l Xd'u dx2 + A2u-b = 0 0) u is the function which governs the equation and we need to find it using a numerical technique which gives an approximate solution. X2 is a known positive constant and b is a known function of x. The solution of above equation can be found by assuming a variation for u consisting of a series of known shapes ( or functions ) multiplied by unknown coefficients. The coefficients than be found by forcing the above equation to be satisfied at a series of points. This is the basis of the collocation method and is essentially what one does when using finite differences. In finite elements instead the solution is found using the concept of distribution of error within the domain. If the equation is multiplied by a w function whose derivatives are continuous up to a required degree and arbitrary except for being continuous up to a required degree and integrate on the domain x as follows. (- - +A2u -b)w dx = 0 dx~ (2) If we integrate this equation two times we obtain; J j ic-r + (A2u - b)w f> + dx' -ll - =0 (3) du To solve this equation u or - - needs to be known at x=0 and x=l. This gives an insight into the boundary conditions required to solve the problem. The boundary conditions can be considered as follows: (4) where the derivatives of u are now defined as q and the terms with bars represent known values of the function and its derivatives. XIIf these conditions applied to the original expression and by integrating by parts two times the equation obtained is: d2u \\^w+^'ı,-b)w\dx-h~q)w\^ + (M_// dw dx (5) This equation implies that one is trying to enforce not only satisfaction of the dw differential equation in x but the two boundary conditions. The w and -7- functions dx can be seen as Lagrangian multipliers. POISSON'S EQUATION An important equation in engineering analysis is the so-called Poisson Equation which for two dimensions can be written as: d2u â2u, in Q, veya V2u = b in Q (6) where V“ ( ) = - ; - - +,, is called the Laplace operator, x 1 and x2 are the two d'x âc coordinates and b is a known function of x 1, x2. Q. is the domain on which the equation applies is assumed to be bounded by T. The outward normal to the boundary is defined as n. By multiplying the equation (6) by an arbitrary function, continuous up to the second derivative and integrating by parts two times equation of Green's theorem is obtained. j{(v2u)w-(v2w)u}dn=l(-w-u^f\dr (7) XIIAlthough this theorem is in many cases given as the starting point for many engineering applications, including boundary element formulations, it is much more illuminating to use the concept of the concept of distribution as it illustrates the degree of continuity required of the functions and the importance of the functions and the importance of the accurate treatment of the boundary conditions. In this regard if the F boundary of the Q domain is divided into two parts H and I~2 (r=rı + Y2). u = // /// r. du q = - = q in T, (8) on By applying these boundary conditions and integrating by parts two times to retrieve the original Laplacian V2u the equation obtained is: J {(V2w - b)w}da -\{q- q)wdT + j{u - u)-dT = 0 (9) APPROXIMATE SOLUTIONS In engineering practice the exact solution can only known in a few simple cases and it is hence important to see how the solution behaves when an approximation is introduced. If the function u is considered to define an approximate rather than the exact solution what can be written is u=al J. A,in - - + X'u-b*Q mx (11) dx~ The same will generally occur with the boundary conditions corresponding to this equation. The concept of an error function or residual which represents the errors occurring in the domain or on the boundary due to non-satisfaction of the above equations can than be introduced. The error function in the domain is called R. and is given by d2u ”, R = - - + £ıı - b (12) dx~ The numerical methods used in try to reduce these errors to a minimum by applying different techniques. This reduction is carried out by forcing the errors to zero at certain points, regions in a mean sense. This operation can be generally interpreted as distributing these errors. WEIGHTED RESIDUAL TECHNIQUES The solution of the boundary value problem defined by equations given before or similar sets for other problems can be attempted by choosing an approximation for the function u. Three types of method can be used then: i. If the assumed approximate solution identically satisfies all boundary conditions but not the governing equations in Q, one has a purely 'domain' method. ii. If the approximate solution satisfies the field or governing equations but not the boundary conditions one has a 'boundary' method. iii. If the assumed solution satisfies neither the field equation nor the boundary conditions, one has a 'mixed' method. XIVWEAK FORMULATIONS The fundamental integral statements of the boundary element and the finite element methods can be interpreted as a combination of a weighted residual statement and a process of integration by parts that reduces or 'weakens' the order of the continuity required for the u function. The weakness can be interpreted as due to two reasons: i. The order of u function continuity has been reduced as its derivatives are now of a lower order ( first rather than second order ). ii. Satisfaction of the natural boundary conditions is done in an approximate rather than exact manner, which reduces the accuracy of boundary values of this variable. BOUNDARY AND DOMAIN SOLUTIONS Weighted residual technique was classified into boundary, domain and mixed methods. Boundary methods were defined as those for which the assumed approximate solution satisfies the governing or field equation in such a way that the only unknowns of the problem remain on the boundary. The satisfaction of the field equation may be of its homogeneous form or a special form with a singular right hand side. In the process of double integration the derivatives of the approximate solution u has to be transferred to the weighting function w and so the conditions previously imposed on the former apply now to the latter. A boundary method can be obtained by choosing a weighting function w in either of the following two ways, i. By selecting a function w which satisfies the governing equation in its homogeneous form, or ii. By using special types of functions which satisfy those equations in a way that it is still possible to reduce the problems to the boundary only. The best known of the functions applied as right hand side of the equation in the second method are the Dirac delta functions which give simply a value at a point when integrated over the domain. XV
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