MESHING USING P-METHOD TWORZENIE MODELI DYSKRETNYCH ZA POMOCĄ MODELI TYPU P

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MARTA ŻAKOWSKA MESHING USING P-METHOD TWORZENIE MODELI DYSKRETNYCH ZA POMOCĄ MODELI TYPU P Abstract Accuracy and effcency of analyss carred out thanks to the FEM method manly depends on the qualty of

1 MARTA ŻAKOWSKA MESHING USING P-METHOD TWORZENIE MODELI DYSKRETNYCH ZA POMOCĄ MODELI TYPU P Abstract Accuracy and effcency of analyss carred out thanks to the FEM method manly depends on the qualty of dscrete model. That s why there are many tools used n practce to create dscrete models. The most often are used tools usng models of h- and p-elements. The characterstc feature of dscrete models usng h-elements s approxmaton of contnuous feld by lnear functons, or second order polynomals. In case of dscrete models usng p-elements approxmaton s carred out usng hgher order polynomals, whch allows automatc and good mesh adaptaton on these models. In the artcle there were presented examples of creatng dscrete models usng h- and p-elements and llustrated dfferences between dscrete models created usng both knds of methods. Addtonally, there was presented mesh adaptaton process on dscrete models made by p-elements. Keywords: p- and h-verson of the fnte element method mesh generaton, p-method of fnte element analyss Streszczene Efektywność dokładność analz prowadzonych z zastosowanem MES zależy główne od jakośc modelu dyskretnego. Dlatego też jest wele narzędz służących do tworzena model dyskretnych stosowanych w praktyce, z których najczęścej używane są narzędza wykorzystujące modele elementów typu h oraz typu p. Cechą charakterystyczną model dyskretnych wykorzystujących elementy typu h jest to, że aproksymują cągłe pole funkcjam lnowym bądź też welomanam stopna drugego. W przypadku model dyskretnych wykorzystujących elementy typu p aproksymacja przeprowadzana jest z zastosowanem welomanów wyższego rzędu, co pozwala na automatyczną dobrą adaptację satk na tych modelach. W nnejszym artykule przedstawono przykłady tworzena model dyskretnych z użycem elementów typu h p oraz pokazano różnce występujące w modelach dyskretnych wykonanych za pomocą tych metod. Pokazano równeż proces adaptacj satk na modelach dyskretnych wykonanych elementam typu p. Słowa kluczowe: MES, elementy typu h typu p Marta Żakowska, IV year student, Insttute of Appled Informatcs, Cracow Unversty of Technology.

2 Introducton Competton and costs of creatng a new product force desgn engneers to analyse projects on desgnng stage nstead of tradtonal prototypng and testng approach. Abltes of analyss by desgn engneers durng the product development process strongly depend on the choce of tools that offer easy ntegraton wth CAD. One of ths knd of tools s the fnte element method. The orgns of the fnte element method reach back to the 1950s and 1960s, when t became very popular n engneerng use. Fast development of FEM can be seen n followng years together wth computer ndustry development. The accuracy and effcency of analyss carred out usng the FEM method manly depends on the qualty of dscrete model. We can dstngush two man methods to create dscrete models n practce: p- and h-method. Dscrete models usng h-elements approxmate the contnuous feld by a functon n the lnear form (or second order polynomals) nsde the element. However, n case of models usng p-elements, the contnuous feld s approxmated by hgher order polynomals. Addtonally, the process of adaptaton of the dscrete model bult usng p-elements mght be easly automated. P-methods have been worked out recently. However, only snce the md-1980s these methods have been used n commercal systems. In the present artcle examples of creatng dscrete models usng h- and p-elements are presented. Dfferences between dscrete models created usng h- and p-elements are llustrated. Addtonally, the paper presents examples of dscrete models made usng p-elements, whch were exposed to the process of mesh adaptaton. An example used n ths artcle s presented, thanks to the Pro/Mechanca Pro/Engneer Wldfre Idea of FEM The fnte element method s one of dscretzaton method of geometrcal contnuous arrangements,.e. dvson of contnuum nto a fnte number of subdoman. The dea of FEM method s based on modellng multple constructons through ther representaton wth the use of component elements of geometrcally smple shape. The man assumpton of the fnte element method s that a contnuous physcal doman s dvded (Fg. 1) nto a fnte Fg. 1. Meshng of contnuous model transformaton of fnte elements nto a set (mesh): a) geometrcal contnuous model, b) dscrete deal model, c) dscrete analytcal model Rys. 1. Dyskretyzacja modelu cągłego transformacja w zbór (satkę) elementów skończonych: a) model geometryczny cągły, b) model dyskretny dealny, c) model dyskretny oblczenowy

3 number of elements whch are joned n nodes. Ths results n creatng a dscrete model. It can be wrtten n a smple form as 227 n S = S n + S feld of model whch s meshed, S feld of element n dscrete. 1, (1) 3. Meshng usng h-method There are two man ways to create dscrete models n FEM method: usng h-elements or p-elements. In both h- and p-methods the geometrcal object under analyss s dvded nto a fnte number of fragments called elements. In solds mechancs, contnuous dsplacement feld s approxmated n the smplest case by the functons n lnear form (h-elements) and dsplacements n the nodes of elements. The element dsplacement felds, {u}, can be wrtten as { u } = Nd = [ N ]{ d} = 1 N shape functons of order 1, d vector of unknown nodal dsplacements, number of corner nodes. n (2) Fg. 2. Adaptaton usng h-elements: decreasng the szes of telements Rys. 2. Adaptacja typu h: zmnejszane wymarów elementów Usng the prncple of vrtual works, the approxmate dsplacement felds and the materal behavour relatons, the fnte element equaton can be wrtten as [ k ]{ d } = { f } (3)

4 228 k f stffness matrx, load vector. The contnuous dsplacement feld descrbed by approxmaton functons cause a dscretzaton error n the fnte element soluton. Ths error can be reduced by mesh adaptaton, whch can be done by decreasng the szes of the elements and ncreasng ther number. 4. Meshng usng p-method Meshng usng p-method means to use elements wth shape functons n a polynomal form and ncrease the order of ths polynomal durng calculaton. There are a few solutons of adaptaton usng p-elements. One of them can be done by addng hgher order shape functons nto the exstng shape functons present n the element. The element dsplacement felds n ths case can be wrtten as n { u} = N d + N d = [ N ]{ d } + [ N ]{ d } (4) = 1 s j = 1 h, j h, j h h N h d h s newly ntroduced shape functons, the vector of p dsplacements, the number of shape functons. In case of elements wth shape functon n the second order polynomal, stresses n the element descrbed by a lnear dependence (Fg. 3) and adaptaton usng p-elements can be done by ncreasng the order of polynomal wthout decreasng the szes of elements. Fg. 3. Adaptaton usng p-elements: ncreasng order of polynomal shape functon Rys. 3. Adaptacja typu p: zwększane stopna welomanu funkcj kształtu

Because of that, to mnmse the dscretzaton error and to get satsfactory results a mesh wth a large number of small elements s usually created.

5 5. Dscretzaton of geometrcal models 229 The accuracy of results analyss depends on the elements shape of the mesh. H-method of FEA uses elements wth the dsplacement feld descrbed by the frst or second order polynomals. Ths lmts the shapes of elements to smple geometrc prmtves (Fg. 2). Because of that, to mnmse the dscretzaton error and to get satsfactory results a mesh wth a large number of small elements s usually created. Geometry must be extensvely defeatured, geometry detals removed, whch are deemed unmportant for analyss, dealsed and cleaned up before t can be meshed. Fg. 4. Three-dmensonal h-elements Rys. 4. Trójwymarowe elementy typu h P-method of FEA uses elements of more complex shapes (Fg. 5) wth the dsplacement feld descrbed by hgher order polynomals (up to 9th order). Ths allows for larger elements that map precsely to geometry and correctly represent thn sold features. There s an advantage of automatc creatng elements of acceptable shapes of the model n meshng process. Adaptaton process of a dscrete model usng p-elements wth lttle nterference of the user s more effectve. Fg. 5. Three-dmensonal p-elements Rys. 5. Trójwymarowe elementy typu p

230 Examples of creatng dscrete models usng h- and p-elements. Fg. 6.

Porównane model dyskretnych wykonanych z użycem elementów typu h p Mesh adaptaton of a model usng p-elements can be carred out automatcally

7 there are presented dscrete models wth p-elements wth varous order polynomals created as a result of mesh adaptaton n Pro/Mechanca

6 230 Examples of creatng dscrete models usng h- and p-elements. Fg. 6. Comparson of dscrete models made usng h- and p- method Rys. 6. Porównane model dyskretnych wykonanych z użycem elementów typu h p Mesh adaptaton of a model usng p-elements can be carred out automatcally n selected elements of the model. In Fg. 7 there are presented dscrete models wth p-elements wth varous order polynomals created as a result of mesh adaptaton n Pro/Mechanca programme. 6. Conclusons The am of the paper was to compare dscretzaton methods of geometrcal models used n FEM systems. There were presented two basc methods of creatng dscrete models usng h- and p-elements. There are also llustrated dfferences n models usng both knds of elements together wth examples of adaptaton process of dscrete models usng p-elements, made n Pro/Mechanca programme.

, Metody komputerowe w lnowej mechance konstrukcj, Kraków 2003. [3] Ztka M., hp-fem for large-scale sngular 3D problems, Texas, May 2006. [4] Kurowsk P.M., Analyss Tools for Desgn Engneers, Socety of Automotve Engneers, 2001.

7 231 Fg. 7. Mesh adaptaton of models usng p-elements Rys. 7. Adaptacja satk model wykorzystujących elementy typu p References [1] Strang G., Fx G.J., An Analyss of the Fnte Element Method, Prentce Hall, Englewood Clffs, New York, [2] C c h o ń Cz., C e c o t W., K r o k J., P l u c ń s k P., Metody komputerowe w lnowej mechance konstrukcj, Kraków [3] Ztka M., hp-fem for large-scale sngular 3D problems, Texas, May [4] Kurowsk P.M., Analyss Tools for Desgn Engneers, Socety of Automotve Engneers, [5] Rbero Flho M., Pnho J.T., Slva J.P., Nobrega K.Z., Hernandez- -Fgueroa H.E., A FEM Mesh Generator for Large Sze Aspect Rato Problems wth Applcatons n Optoelectroncs, IEEE, [6] Y o s b a s h Z., H a r t m a n n S., H e s s e r e r U., D u s t e r A., R a n k E., S z a n t o M., Axsymmetrc pressure boundary loadng for fnte deformaton analyss usng p-fem, Scence Drect, September [7] Promwungkwa A., Data Structure and Error Estmaton for an Adaptve p-verson Fnte Element Method n 2-D and 3-D Solds, Vrgna, Aprl [8] Budzyń sk A., Krótk wstęp do zastosowana Metody Elementów Skończonych (MES) do numerycznych oblczeń nżynerskch, buletyn GM Vew 5/2006.

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