Bilgisayar destekli tasarımda sonlu eleman analizi için otomatik ağ yapısı oluşturulması

Abstract

Mekanik dizayn genellikle matematiksel olarak tanımlanması güç karmaşık şekilleri içerir. Mühendisler 1950 'ler de tüm şekli ufak elemanlara bölüp her eleman için bir denklem takımı elde etme ve bunu çözümlemeyi bulmuşlardır. Yöntem sonlu eleman yöntemi, bölünmüş parçalara da sonlu eleman denir. Sonlu elemanlardan oluşan yapıya da ağ yapısı denir. Sonlu elemanlar iki boyut da genelde üçgen veya dörtgen, üç boyutda da tetrahedra veya tuğla elemanı olabilir. Büyük veya karmaşık şekilli parçaları elemanlara bölmek ve ağ yapısını el ile üretmek imkansızdır, özellikle üç boyutda bilgisayar yöntemlerine ihtiyaç vardır, bu tezde de iki ve üç boyutlu otomatik ağ üretim teknikleri anlatılmışdır. Tez sonlu eleman yönteminin genel bir tanıtımı ile başlar. ikinci bölümde iki boyutlu otomatik dörtgensel ağ üretim tekniği anlatılır. Bu teknik şekil optimizasyonu bakımından da önemlidir. Bir sonraki bölümde katı modellemeden oluşturulan üç boyutlu otomatik ağ üretimi ile ilgili iki yöntem anlatılmışdır. Bu yöntemlerden birinin adı tekrarlanan uzaysal ayrıştırma, diğeri de üç boyutlu Delaunay üçgenlemesidir.

In this thesis, automatic finite element mesh generation techniques which integrate CAD (Computer ' Aided Design) systems are included. First, the basic concepts of finite element method was introduced. Then, two dimensional automatic quadrilateral mesh generation technique was explained. Newt, three dimensional automatic mesh generation techniques which use solid modelling system were explained. Mechanical design involves shapes that s>.rs often complex and not easily described mathematically. In the 1950s, engineers found that by breaking down the entire shape of a given structure into many smaller ones, and combining the solutions to each subdivision a designer could formulate an approximate answer to the original problem. Each subdivision came to be called a finite element and the solution method finite element method (FEM). The broken-up form is called Mesh. The elements, unlike the original form, &rB easy to describe and their responses to phenomena such as stress or heat aesily predicted. Theengineer could then solve\the original problem. FEA enables engineers to observe the responses of physical systems to certain imposed conditions when it may not be possible to find an exact solution. The quality of an engineer ing decision, therefore, depends on the quality of these approximate solut ions. Actually, FEA is adigital equivalent to continoum mechanics. It can be applied to many types of problems.that mathematicians call field problems, such as heat transfer, electric potential distribution, fluid flow and seapage through porous media. In fact, many people use FEA to analyze heat transfer in integrated circuets. The accuracy and the expense of the calculations are strongly affected by the goodness of the underlying mesh. An ideal mesh is characterized by small elements where stress gradients are small and elements which are regular or undistorted shape. After the original domain has been broken down to subdomains, the -finite element are described by simple algebric -functions usually polynomials, which are called elemental basis -functions or shape -functions. A-fter- the mesh and -functions sre established, the designer uses an FEA program to solve the approximate description o-f the problem over and over again, each time getting closer to the actual solution, and finally c on verging on it. In FEA, the quality o-f each approximate solution (the relative error in energy norm; depends on the types o-f subdivision and shape -function. The quality changes with each approximation depending on the discretization, which is the refinement of the finite element mesh and the polynomial degree. For large analysis especially those which are time dependent, it is not possible to interpret very much of a printed output. In this cases a graphic package -is certainly necessity. The graphics package should be capable of plotting Reformed gr ids «.magn i f y ıng the displacements if necessary) producing stress and strain plots and producing producing time history plots. In FEA, both the building of model '.preprocessing,1 and the interpreting of resul ts «.postprocessing^ require interactive graphics. For most mechanical parts «.especial iy large bodies). Automatic Mesh Generation is necessary, üne of the initial incentives for automating mesh generation is to reduce the time end money spend creating a mesh, which could amount to half of the total time required to perform the FEM analysis. In this thesis. Part 2 describes finite element method briefly. Part 3 includes two dimensional quadrilateral mesh generation technique. And Part 4 relates to three-dimensional mesh generation techniques Tram solid models. 2-dimensi.onal automatic mesh generation When quadrilateral elements ar& used some characteristics can sometimes be difficult to achieve. For example, adjusting the element size to accommodate changing stress gradients normally result in regions of distorted elements, called 'transition elements, which connect the finer and coaser areas of the mesh. Also, boundary curvature of the region to be meshed result in distorsion, to some degree, of the elements adjoining the boundary. Despite these drawbacks, quadrilateral elements are often preferred over triangular elements in creating a mesh. Though a triangular mesh is easier to generate, it can oe less accurate than a quadrilateral mesh in predicting performance characteristics such as stress or deflection. This is because triangular elements are constant-strain elements, while quadrilateral elements are linear-strain elements. The end result of this difference is similar to approximate the area under a curve as opposed to using trapezoids. In the 1970" s attention was focused on generating meshes in irregularly shaped domains with curved boundaries. Also, researcners began developing meshes in irregularly shaped domains which attempted local and/ or global refinement of the mesh to improve computational accuracy. Adaptation, which icludes both mesh optimization and mesh refinement, was shown to be especially importent when modelling parts with localised stress risers and point loads. - As the technology.of mesh generation began to mature literally hundreds of different methods and algorithms were developed to solve specific problems. As computers became more widely applied to the mesh generation process, the amount of time and money spent generating and checking the input data did not diminish a great deal and a skilled prof essionel was still required in many phases of the process. The 1970 s also saw the development of three-dimensional mesh generation routines. -V 3 1 - In the 1980s the use of free-form curves and surfaces has become a regular part of most CAD /CAM hardware systems, forcing the continued development o-t generation schemes -for irregular- shapes. The 60 s have also oeen characterized by the attempt to develop coplete 'design systems' which seel:: to integrate man, computer processes which have previously "been done independently. Today automatic mesh generation is understood to mean the creation o-f a we'll conditioned mesh in two or three dimensions with a minimum ca user-- interact ion. Ideally the user should need only to spcify geometric boundary information and the preferred element size along the boundary..i-d i men s i on a 'j au t oma 1 1 c mesh q en e t- a 1 1 on N o t a. b le sue: c: ess has also bee n a chieved b y three-dimensional CAD systems for the design of curves and sculptured surfaces in /the automobile, aerospace and shipbuilding industries. Although such so-called wire-frame CAD systems are useful for the design of smooth exterior surfaces (for example, automobile sheet metal panels), they ^re awkward and difficult to use tor the design of solid functional components such as automobile pistons, connecting rods, crankshafts, housing or other parts that are "usually cast, moulded o r m a c hi i. n e d / ftec en t J. y d e.. e 1 op ed solid mod e i 1 i n g = y s terns f o r the design of complex physical solids us:ing interact], e computer graphics offer the exciting possibility ot an integrated design/analysis system. Called geometric modellers, these systems build complex solids from p r i m 1 1 1 ve so 1 i ds < cubes, cy 1 ı nders, so 1 1 d patches, etc; and macro sol ids '.comb inat ion of primitives;. Nearly all current solid model ling systems ar& based internally on one or both of the following representation schemes ; Construct i ve Sol id Geometry (.CBS; and Boundary Representation ', B Rep). CSG exploits the notion of adding a n d s u b 1 1- a c 1 1 r ı g si n. pie solid building b locks ? pr imi 1 1 ves ;,ia set -union and set d 2 f f erence opera 1 3 ons. The various schemes that ha.-e been proposed tor automatic mesh generation from solid models can be devided for present purposes into three -rami lies :. o 1 um e t r i angu 1 a 1 1 on, e J emen t ex t r a,c t ] on and r&c ur- s ı. e- spa t i a 1 d ec omp os i t i on..v:ı 1