Prisitaikančiosios baigtinių elementų strategijos plokštuminiams tamprumo teorijos uždaviniams ; Adaptive finite element strategies for solution of two dimensional elasticity problems
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Lina VASILIAUSKIENĖ ADAPTIVE FINITE ELEMENT STRATEGIES FOR SOLUTION OF TWO DIMENSIONAL ELASTICITY PROBLEMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) 1268 Vilnius 2006 PDF created with pdfFactory Pro trial version www.pdffactory.comVILNIUS GEDIMINAS TECHNICAL UNIVERSITY Lina VASILIAUSKIENĖ ADAPTIVE FINITE ELEMENT STRATEGIES FOR SOLUTION OF TWO DIMENSIONAL ELASTICITY PROBLEMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius 2006 PDF created with pdfFactory Pro trial version www.pdffactory.comDoctoral dissertation was prepared at Vilnius Gediminas Technical University in 2000 – 2006. The dissertation is defended as an external work.
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Lina VASILIAUSKIENĖ ADAPTIVE FINITE ELEMENT STRATEGIES FOR SOLUTION OF TWO DIMENSIONAL ELASTICITY PROBLEMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)
Vilnius 2006
VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Lina VASILIAUSKIENĖ ADAPTIVE FINITE ELEMENT STRATEGIES FOR SOLUTION OF TWO DIMENSIONAL ELASTICITY PROBLEMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)
Vilnius 2006
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Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2000 2006. The dissertation is defended as an external work. Scientific Supervisor Prof Dr Habil Romualdas BAUYS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering 09T) The Dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Mindaugas Kazimieras LEONAVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering 09T) Members: Prof Dr Habil Juozas ATKOČIŪNAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering 09T) Prof Dr Habil Genadijus KULVIETIS Gediminas Technical (Vilnius University, Technological Sciences, Mechanical Engineering 09T) Prof Dr Habil Gintautas DZEMYDA(Institute of Mathematics and Informatics, Technological Sciences, Informatics Engineering 07T) Prof Dr Habil Feliksas IVANAUSKAS(Vilnius University, Physical Sciences, Informatics 09P) Opponents: Prof Dr Habil Rimantas KAČIANAUSKAS Gediminas (Vilnius Technical University, Technological Sciences, Mechanical Engineering 09T) Prof Dr Habil Rimantas BARAUSKAS(Kaunas University of Technology, Physical Sciences, Informatics 09P) The dissertation will be defended at the public meeting of the Council of Scientific Field of Mechanical Engineering in the Senate Hall of Vilnius Gediminas Technical University at 2 p. m. on 19 June 2006. Address: Saulėtekio al. 11, LT-10223 Vilnius-40, Lithuania Tel.: +370 5 274 49 52, +370 5 274 49 56; fax +370 5 270 01 12; e-mail doktor@adm.vtu.lt The summary of the doctoral dissertation was distributed on 19 May 2006 A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, Vilnius, Lithuania) © Lina Vasliliauskienė, 2006
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Vilnius 2006
VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS Lina VASILIAUSKIENĖ PRISITAIKANČIOSIOS BAIGTINIŲELEMENTŲ STRATEGIJOS PLOKTUMINIAMS TAMPRUMO TEORIJOS UDAVINIAMS Daktaro disertacijos santrauka Technologijos mokslai, mechanikos ininerija (09T)
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Disertacija rengta 2000 2006 metais Vilniaus Gedimino technikos universitete. Disertacija ginama eksternu. Mokslinis konsultantas prof. habil. dr. Romualdas BAUYS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos ininerija 09T). Disertacija ginama Vilniaus Gedimino technikos universiteto Mechanikos ininerijos mokslo krypties taryboje: Pirmininkas prof. habil. dr. Mindaugas Kazimieras LEONAVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija 09T). Nariai: prof. habil. dr. Juozas ATKOČIŪNAS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos ininerija 09T), prof. habil. dr. Genadijus KULVIETIS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos ininerija 09T), prof. habil. dr. Gintautas DZEMYDA(Matematikos ir informatikos institutas, technologijos mokslai, informatikos ininerija 07T), prof. habil. dr. Feliksas IVANAUSKAS(Vilniaus universitetas, fiziniai mokslai, informatika 09P). Oponentai: prof. habil. dr. Rimantas KAČIANAUSKAS Gedimino (Vilniaus technikos universitetas, technologijos mokslai, mechanikos ininerija 09T), prof. habil. dr. Rimantas BARAUSKAS(Kauno technologijos universitetas, fiziniai mokslai, informatika 09P). Disertacija bus ginama vieame Mechanikos ininerijos mokslo krypties tarybos posėd. 14 val. Vilniaus Gedimino technikosdyje 2006 m. birelio 19 universiteto senato posėdiųsalėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius-40, Lietuva. Tel.: +370 5 274 49 52, +370 5 274 49 56; faksas +370 5 270 01 12; el. patas doktor@adm.vtu.lt Disertacijos santrauka isiuntinėta 2006 m. geguės 19 d. Disertaciją galima periūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, Vilnius, Lietuva). VGTU leidyklos Technika“ 1268 mokslo literatūros knyga. © Lina Vasiliauskienė, 2006
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1. General Characteristic of the Dissertation Topicality of the problem. The advent of modern computer technologies provided a powerful tool in numerical simulations. One of the most frequently used method for the discretization of the physical domain is Finite element Method (FEM). One of the main problems in a finite element analysis is the adequacy of the finite element mesh. Since the quality of the finite element solution directly depends on the quality of meshes, the additional process to improve the quality of meshes is necessary for reliable finite element approximation. In order to perform quality-assessed numerical simulation, the adaptive finite element strategies have been developed. These strategies integrate the finite element analysis with error estimation and fully automatic mesh modification, user interaction with this process is limited by initial geometry data and possible error tolerance definition. The finite element solution, obtained during adaptive finite element strategy process, approximates quite good different engineering structures. Despite many works in this area the problem of the adequate finite element mesh is not fully solved and additional developments are needed in order to improve adaptive mesh refinement strategy process. Aim and tasks of the work to obtain methodology for quality assessed discretization to finite elements for complex geometry engineering structures by adaptive finite element strategies. To realize this purpose the following scientific tasks have been dealt: · to develop an automatic system for the generation of the initial almost optimal finite element mesh; · to analyze adaptive finite element strategies, based on global and local mesh optimality criteria and define the influence of different mesh optimality criteria to the element distribution and number in the final optimal mesh solving problems with different topology complexity; · to propose the new method for the accounting singular point influence to the adaptive finite element strategies. Scientific novelty.It is carried out in this work that the final optimal finite element mesh will be obtained with the minimum amount of resources, satisfying the global and local permissible errors of the solution at the same time. Using finite element method theory and computer technology the new strategies for the adaptive finite element analysis, consisting of these parts, are proposed: · The new strategy for the initial sub-optimal mesh generation was developed. The purpose was achieved with the new concept of control sphere and control space definition. It is proved in this work, that proposed definitions
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allow us describe more precisely regions with different topological incompatibilities. · Incorporating proposed initial sub-optimal mesh generation strategy and different global and local mesh optimality criteria to the standard adaptive finite element analysis, the new methodology was obtained, taking less iteration number until the final finite element solution is calculated and allowing user to control element number and distribution in the final optimal mesh. · The new method for the accounting singular point influence to the adaptive finite element process was proposed using singular polynomial instead of the standard quadratic in the error estimation analysis. Methodology of research includes methods for modeling mechanical systems. The analysis is done by the most popular engineering technique finite element method. For the mesh generation the Advancing Front Technique is chosen. Practical value. the proposed initial sub-optimal mesh generation Using algorithm it is possible to control element density and size in the most critical parts of our construction before the finite element analysis. Such an improvement allows us to reduce the number of iterations to get the final mesh, because even in the first iteration the problem domain is approximated more precisely in comparison with traditional approach. After the brief analysis of different optimality criteria it is concluded, that the elements number and density in the final optimal mesh straightforward depends on the choice of one or another optimality criteria. Incorporating initial sub-optimal mesh generation procedure to the standard approach and combining with different mesh optimality criteria we can influence the adaptive finite element analysis process in two ways: the use of the sub-optimal mesh allows us to reduce iteration number and the choice of the mesh optimality criteria allows us to control element distribution in the final mesh. Iteration number of the adaptive finite element analysis and element number in the final optimal mesh can be reduced using new method for the solution improvement in the singularity zones also. Defended propositions · The methodology for initial sub optimal mesh generation. · The adaptive finite element strategies with local and global mesh optimality criteria. · The postprocessed solution improvement methodology taking to the account geometrical singularity. Approval of work. esPrtaenonti s: 7th international conference “Modern building materials, structures and techniques, held on May 1618, 2001 in Vilnius, Lithuania.
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International conference of “Mathematical Modeling and Analysis held on May 31June 2, 2001 in Vilnius, Lithuania. International conference “Mechanics ’2003, held on April 34, 2003 in Kaunas, Lithuania. 8th conference of “Mathematical Modeling and Analysis held international on May 2831, 2003 in Trakai, Lithuania. 12th conference of the Association of Lithuanian Digital Mechanics, held on April 9, 2004 in Vilnius, Lithuania. International conference “Mechatronic Systems and Materials, held on October 2023, 2005 in Vilnius, Lithuania Publications: 3 accepted scientific publications and 1 other scientific publication. The scope of the scientific work. The scientific work consists of the general characteristic of the dissertation, 4 chapters, conclusions, list of literature, list of publications, list of pictures and list of tables. The total scope of the dissertation 99 pages, 57 pictures, 9 tables, 129 literature sources and 4 publications according to the dissertation theme list. 2. Literature Analysis During literature analysis the history of finite element development strategies was reviewed. A brief analysis of existing mesh generation methods was done, also. It is resumed, that the Advancing Front technique is the most suitable mesh generator for the adaptive finite element analysis. An overview of different error estimation methods is presented and the works of scientists in this field are discussed. The advantages and weakness of existing singular point estimation methods were presented. Zienkiewicz O. C., Wiberg N. E, Szabo B. A., Owen S. J., Onate E., Löhner R., Li L. Y., Yosibash Z., George P. L, Frykestig J., Frey P. J., Bugeda G., Oliver J., Th., ApelBorouchaki H., Aalto J., Stupak E., arnovskij V. and other scientists worked on improving adaptive finite element method. All of them used different methods and testing techniques, got different results and made a great job in developing finite element method. 3. Adaptive Finite Element Strategies With the help of reliable error estimation procedures available today, a suitable element size distribution can be predicted and new mesh is constructed using automatic mesh generators. Prediction of finite element size is based on the results of a previous stage of analysis. In this manner, subsequent meshes of
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better quality are designed adaptively. After a few trials, solutions with an accuracy corresponding to a user specified tolerance are obtained. This process is called adaptive mesh generation algorithm. It uses intermediate results for mesh modification in a such way, that the final mesh is in some sense optimal. Iterative process is controlled by an error tolerance. The classical adaptive algorithm starts from the coarse user defined mesh and specifies an error tolerance. After that FE-solution is computed on a given mesh. Then the resulting error distribution is evaluated and compared with the tolerance if the error tolerance is not met, a new mesh is generated based on the error distribution. Analysis is terminated if the estimated error is less than or equal to the tolerance. Adaptive finite element analysis consists of two main parts: error estimation and new mesh generation. In this work we turn our attention to post processed error estimates that are based upon information obtained during the solution process. The accuracy of a FE-calculation will be increased by using h-adaptive grid refinement strategy when more elements of the same type are taken. The essence of the post processed error estimator is to replace the exact solution with a post processed solution of higher quality:eu»eu=u*-uh, whereeuthe point-wise estimated error. Using the improved solution we is have an estimation over all elements in the domainW: nel nel e2=åei2=åòeuTLeudWi, i=1i=1 whereWiis an element domain andnelis the total number of elements. The absolute error defined by an energy norm is not convenient for use in practical computations. The dimensionless forms are favored and are customarily expressed as: h=nåelhi2,hi=e uh2+e2, (2) i i=1 whereuhdenotes the norm of the FE solution itself andh andhi are the relative global and relative element error, respectively.
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The problem inh-adaptivity for finite element methods may be formulated as follows: construct a finite element mesh with as few degrees of freedom, as possible, such as that: h£hTOL, wherehTOLis the maximum permissible error. This is a non-linear minimization problem, which may be solved, approximately in an iterative process. The mesh is redefined using an adaptive scheme to predict the distribution of the mesh sizeh. New element size is calculated according formula: wherehi an old mesh element size,xi element refinement parameter, based on some mesh optimality criterion,p degree of shape functions polynomials. 4. Sub-Optimal Initial Mesh Generation The efficiency of the adaptive finite element strategies straightforward depends on the number of iterations until optimal mesh is constructed. Traditionally mesh generation process starts with a coarse user-constructed background mesh and mesh sizing function is defined manually by giving its value for each element of the background mesh and is judged only from element shape without any information about boundary or loading conditions. During the finite element meshing process the target element size at the new point is defined commonly from a linear interpolation. After that a corresponding solution is computed and the discretization error estimate analysis is performed so as to redefine mesh sizing function from a posteriori error estimate. This process is the most time consuming and requires a lot of steps for the convergence to the desired accuracy. In order to simplify this process the more perfect initial mesh should be taken for the first finite element analysis. The best mesh for a given finite element analysis problem can be defined as compromise between the need for accurate results ant the desire for the computational efficiency. The compromise between accuracy and efficiency is usually achieved by grading the mesh, but in this place we have one problem the solution gradients are unknown in the first iteration step of the adaptivity
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Fig 1.Extended control space definition according points A and B
a b Fig 2.Initial meshes with the same element number: a sub optimal initial mesh; b uniform initial mesh strategies and tools of estimating mesh density requirements is needed before the automatically graded meshes can be produced. In order to overcome this difficulty, the extended control sphere and control space notion are proposed and implemented in this section. In contrast to the standard control space definition, the mesh size function is assigned to the most critical region of the given domainW. This improvement allows us reflect more precisely all topological and physical domain features to the resulting finite element mesh and gives us more flexibility in mesh gradation control. For the mesh size determination around each critical region we propose to use the general control sphere concept: the set of spheres n n ìíS(g(t ,)R)=ånjiS(Pi, R)üý,g(t)=åjiPi,R=å(1,2)ikh (5) ' îi=0þi=0i=0 is said to be the control sphere along the curvegt, if the centers of all spheres lies on this boundary curve. There radiusR defines the spacing between any
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