Introduction to Geometric Modelling - CAD and FEM for solid modeling and analysis

Transkriptio

1 Introduction to Geometric Modelling - CAD and FEM for solid modeling and analysis Kirsi Virrantaus Aalto University School of Engineering Department of Built Environment

2 Content of the lecture 1. Terms and concepts in geometric modeling 2. Approaches to geometric modeling 3. Models in engineering design 4. Data model transformations and exchange standards 2

3 1.Terms in Geometric Models (Mortenson, 2006) GMs are often complex models made of geometric primitives (shapes) Topology, algebraic geometry and Boolean algebra are used in definition of complex models Topology in model construction means the property of being invariant under transformations that stretch, bend, twist or compress a figure; topological properties like connectivity and adjacency are not metric Algebraic geometry is extension of analytic geometry including differential geometry; analytic geometry = coordinate geometry; differential geometry = using differential and integral calculus Computational geometry is on design and analysis of geometric algorithms 3

4 1. Keskeisiä termejä Geometriset mallit ovat usein kompleksisia malleja, jotka on konstruoitu geometrisistä primitiiveistä Kompleksisten mallien konstruoinnissa käytetään mm. topologiaa, algebrallista geometriaa and Boolen algebraa Topologia tarkoittaa tässä yhteydessä sitä, että mallin (topologinen) rakenne ei muutu vaikka sitä venytetään, taivutetaan, kierretään tai puristetaan; topologiset relaatiot, kuten yhdistävyys tai viereisyys (esim. verkossa) ei ole metrinen relaatio (kuten esim. etäisyys) Algebrallinen geometria on analyyttisen geometrian laajennos, joka sisältää differentiaaligeometriaa Laskennallinen geometria on geometristen algoritmien suunnittelua ja toteutusta 4

5 More common basic terms Model is constructed by using geometric primitives: points, lines, polygons, spheres, cylinders etc. The ways to connect the primitives in the models differ: mathematical functions (splines) mathematical operations (Boolean operations) topological relations (complex data structures) Models are presented as data structures from simple arrays to trees and more complicated structures including shared edges Topological relations between parts of the shapes play a role in data structures representing the model; topological relations can be defined in many ways; for example in geosciences adjacency and connectivity are the most popular topological relationships 5

6 Lisää perustermejä Mallit konstruoidaan käyttäen perusprimitiivejä: piste, viiva, polygoni, pallo, sylinteri Malleissa perusprimitiivit yhdistetään eri tavoin: Pisteet käyräksi splinillä; matemaattinen funktio Kappaleet yhteen Boolen operaatoilla, toteutus esim. puurakenteena Primitiivit yhteen topologisilla relaatioilla, kuten verkkorakenteessa, toteutus sopivalla tietorakenteella Käytetyt tietorakenteet voivat olla yksinkertaisia taulukoita ja puita tai monimutkaisempia topologisia rakenteita Geoinformaatiotieteen malleissa jatkuvuus ja viereisyys ovat suosituimpia topologisia relaatioita 6

7 2. 3d wireframes starting point for geometric modeling of solids 3d wireframes/rautalankamallit in 70 s Lines and edges representing edges of the solid Algorithm development for representations for example the hidden line presentation

8 Wireframe and hidden line presentations of an object

9 More developed approaches - Combining simple shapes to complex models Curves, surfaces and solids are created by mathematical means Various transformations applied to them Parametizations, sweepings and deformations applied to them More complex models generated by using simple obejcts Boolean models and topology-based boundary representations Surface (mesh) models and voxel models

10 Elementary methods for solid generations These elementary methods appear as parts in the more developed approaches Cell model The model is made of parts Sweep method The model is created by sweeping the space by a shape Patch method The model is made of patches, irregular or regular

11 Model made of parts Model created by sweep 11

12 Voxel model of sphere Geometric Modeling for Engineering Applications; Franz- Erich Wolter, Martin Reuter, Niklas Peinecke

13 Voxel models, spatial occupancy enumeration Voxel is a 3d pixel Voxels fill the entire space and objects can be defined by assiging each voxel by a Boolean value (1 or 0) Low resolution poor accuracy; high resolution high requirements of memory space Compression and indexing methods Oct tree structures In voxel model topology is implicite, the matrix structure defines the adjacent cells without referencing Easy computations by matrices Can be seen as a special case of CSG models, and as cell methods Conversions to boundary representations can be made

14 Basic methods behind the models Topology and topological data structures Parametrization Feature based modeling 14

15 Topology Topology is maybe the strongest development in the gemetric models Geometry is presented by: points, curves and surfaces When topology is considered we talk about: vertices, edges, faces; adjacency, connectivity We come to topological data structures (Jussi Nikander s lecture)

16 Parametric modeling Means just defining solids by using variables instead of constant values Parametric functions describe coordinates as functions including variables Parametric curves are defned by parametric functions Examples of parametric curves are splines and NURBS (Antti Niemi s lecture) Do not mix with procedural modeling!!

17 Feature based modeling Feature based modeling means that solid can be constructued by using more complex parts than elementary primitives Features can be stored in libraries

18 3. Models in engineering design solid models, CAD and FEM models in engineering design models are made about various components and parts of machines, equipments, vehicles, structures etc models are made by using CAD (computer assisted design) and CAM (computer assisted manufacturing) software models are typically solid models two main approaches for solid modeling are Constructive solid geometry (CSG) B-rep models 18

19 CSG and b-rep presentations of an object (Stroud & Nagy, 2011) CSG malli voidaan muuntaa b-rep muotoon ohjelmallisesti CSG model can be transformed into b-rep automatically

20 Constructive solid geometry, CSG Complex solids as compositions of simpler solids Boolean operators are used to create a kind of procedural model of a complex solid Each object is either one or a set of simple objects, the primitives, like ball, cylinder, box, cone Operations on the primitives should include: translation, rotation, union, intersection, difference Data structure of a GSG model is a binary tree CSG can be called as procedural modeling, because the solid is created by mathematical functions and operations, not by reading coordinate lists An example of CSG model is in the Mortenson s book page 372, Fig 11.46

21 Constructive Solid Geometry, CSG Kohteet kuvataan pienempien osien kokoonpanona Osat organisoidaan puurakenteeseen; binääripuu Osien yhdistely perustuu Boolen operaatioiden käyttöön ja, tai, ei Kohteet muodostuvat primitiiveistä Pallo, sylinteri, kuutio, kartio Operaatioihin tulee kuulua näiden primitiivien Kierto, siirto, unioni, leikkaus, erotus Funktioihin perustuvaa mallinnusta kutsutaan proseduraaliseksi mallinnukseksi kohde muodostetaan matemaattisilla funktioilla ja operaatioilla, ei lukemalla koordinaattilistoja

22 22

23 Procedural modeling Procedural modeling means techniques for creating models by applying (mathematical) rules L-systems, fractals etc. Used often for creating natural forms CSG model can be transformed into so-called boundary representation Algorithms used are called boundary evaluator

24 24

25 Proseduraalinen mallinnus Proseduraalisella mallinnuksella tarkoitetaan kohteen geometrian luomista automaattisesti, jonkin matemaattisen säännön avulla Laajasti katsoen CSG on proseduraalista mallinnusta (Boolen algebra), mutta varsinaisesti tarkoitetaan hieman toisenlaista tapaa Muotoja voidaan luoda esimerkiksi fraktaaleilla Fraktaalissa sama muoto iteroituu niin, että jos kuviosta otetaan pieni pala ja suurennetaan sitä, sama muoto jatkuu ja kuvio näyttää samanlaiselta Fraktaaleilla voidaan mallintaa esim. kasvillisuutta Proseduraalista mallinnusta esim. pelien kaupunkimaisemaan (ESRI CityEngine; Geodesign)

26 (Diplomityö, Viinikka, 2014)

27 Boundary representation, b-rep B-rep model is built by a collection of surfaces A solid is a union of faces (surfaces), bounded by edges (curves) which in turn are bounded by vertices (points) Each edge is adjoined by two and only two faces and vertices Set of rules make it sure that there are no open surfaces Boundaries can be curves, NURBS Data structures like DCEL and Winged edge Boundary representation of a solid has two parts: Topological information Geometric information

28 Brep-model The model is composed of geometric features and topological relaationships; face, edge, vertex arelinked to each others IFC schema that shows B-rep-model of an object IFC = Industry foundation classes; Open file format specification for BIM software Schema in which objects are linked by relationships Lähde:

29 Brep-malli Muodostuu geometrisista osista (geometric features) ja topologisista relaatioista; esim. face, edge ja vertex Esimerkki IFC skeemasta jossa esitetään kappaleen B-rep-malli Kaavioesitys, jossa kohteet yhdistetään relaatioilla (osa topologisia) Lähde:

30 Winged-edge ja DCEL tietorakenteet Brep-malli voidaan esittää tehokkaasti topologisella tietorakenteella, kuten esimerkiksi winged-edge tai DCEL Kummatkin perustuvat siihen, että viereisten face-kohteiden yhteinen raja (edge) jaetaan kahteen puolisivuun (half-edge) ja facet muodostetaan puolisivujen ketjusta Jokaiselle puolisivulle ilmaistaan edeltäjä ja seuraaja ja tällä tiedolla muodostetaan facet (sulkeutuvana puolisivujen ketjuna) Jokainen puolisivu tuntee kaksosparinsa ja sitä kautta viereisen facen Winged edge muodostuu yhdesta taulukosta DCEL koostuu kolmesta taulukosta Olennaista näissä rakenteissa on, että topologia on tallennettu eikä sitä tarvitse laskea; tehostaa algoritmin suoritusta These data structures are taught in Jussi Nikander s lecture

31 4. Data model transformations and exchange standards Most CAD software have their own main memory data structures which are not documented They import and expert data in standard formats, like STEP STEP Industrial automation systems and integration Product data representation and exchange. However the functionality and limitations of the software are based on the internal data modeling structure

32 Coversions from one model type to another in the previous presentation we have mentioned that for example CSG models can be converted to b-reps, solid models can also be converted to mesh models When the model is used for analysis (FEM,FEA) the a mesh model is used You will learn more in Jarkko Niiranen s lecture about FEM and FEA In Juho-Pekka Virtanen s lecture you learn about laser scanning that is the most modern methods for digitizing 3d objects and shapes This data is called as point cloud data and by using several algorithmic data processing methods solid objects can be reconized and transformed in for example CSG and Brep models

33 From geometric models to analysis and design process Geometric models can be used for geometric design They can also be used as tools for more comprehensive design and analysis Solid models can be transformed into mesh models and then used for FEM and FEA Geometric models can also be used as BIM models when they are connected with data bases and design processes

34 For analysis purposes CAD model is transformed to FEM model The CAD model is transformed into a mesh model Mesh model is then used for analysis purposes (Jarkko Niiranen s lecture) Cottrel, J., et al) 34

35 CAD MODEL FEM MODEL 35

36 FEM and FEA Typically (in solid mechanics), to calculate component displacements, strains, and stresses under internal and external loads, but also multi-physical analysis is possible Solid or plane model (geometry) is converted into a mesh structure: triangles/ tetrahedra, quadrangles/hexahedra Analysis is performed in this mesh structure producing piecewise polynomial approximations for field quantities Advanced software integrate CAD-modeling and FEA together so that analysis can be made during the design Isogeometric analysis: nonpolynomial approximations

37 Examples of FEM models Finite Element Model A combination of FEM and BEM models Boundary Element Model (Jani Romanoff, lectures 2013) 37

38 38 Balobanov, 2014)

39 39 (Balobanov, 2014)