The pipe junction challenge Tor Dokken SINTEF Oslo, Norway Pictures and examples by Vibeke Skytt, SINTEF 1

What is a pipe junction? A composition of cylindrical pipes meeting. For structural use the pipes are welded without cut-outs For use for transport of fluids or gas, cut-outs are made. Welding seams smooth the transition between pipes (fillet volumes) 2

How to represent the pipe-junction in the computer The representation approach depends on the application domain (the purpose of the computer program): Visualization of the model Animation of the model Design of the model Production of the model Analysis of the model, and type of analysis Structural, flow,… 3

Application domains and shape representations DomainQuality criteria:State-of-the-art: 3D visualizationVisual impressionTriangulations & texture mapping Animation moviesVisual impressionSubdivision surfaces Computer Aided Design (CAD) Face connectivity + Shape accuracy Boundary structures of elementary and NURBS surfaces Manufacturing & robotics Proper control of movements Curves as input to movement controllers. Finite Element Analysis (FEA) Volumetric connectivity Structures of 3-variate parametric polynomials, most often of degree 1 or 2. Isogeometric AnalysisVolumetric connectivity + Shape accuracy Block structures of 3-variate parametric NURBS, any degree. T-spline and LR-splines emerging 4

Example of pipe junction from the age of curved based CAD, Example from the mid 1970s of the geometry of cylinder junction from an offshore platform in the oil industry. Cylinders made from steel plates, one cylinder is flattend for flame cutting. Accurate geometry of cut-outs important. Flame cutters controlled by curve data. In current industry welding robots plays a central role Robots controlled by curve data Navigation of robots need an approximate surface based model for collision detection and navigation. 5

Representation of the pipe junction for design and analysis Target quality criteria for this presentation: Design stage: Face connectivity + Shape accuracy Analysis stage: Volumetric connectivity + Shape accuracy The ideal pipe junction can be composed of pieces of cylindrical tubes. During structural analysis loads are applied to the structures, the shape will be deflected becoming slightly sculptured NURBS * used in CAD for representing sculptured surfaces NURBS used in isogeometric analysis for representing sculptured volumes * NURBS - NonUniform Rational B-splines 6

Elementary surfaces play a central role in human made structures Elementary surfaces dominates design of modern human made industrial produced shapes: Plane Cylinder Sphere Cone Torus Surfaces of more sculptured type relates to Terrain Actual shape of produced parts (elementary shapes slightly deflected) Styled and designed products Shapes made by artists Vegetation is in general fractal 7

Some properties of elementary surfaces Elementary surfaces all have an exact rational parameterization The deflected elementary surface can be efficiently approximated by a NURBS-surface or by an algebraic surface of somewhat higher degree Elementary surfaces have low algebraic degree: Degree 1: Plane Degree 2: Cylinder, Sphere, Cone Degree 4: Torus Algorithms for handling elementary surfaces can both use the algebraic and rational parametric representation 8

Design representation - Volumetric CAD – Boundary structures (STEP ISO 10303) Representation of outer and inner hulls by surface patchwork Small gaps between surface allowed Edges of NURBS surfaces represented by 3 curves: A 3D curve One curve in the parameter domain of each NURBS surface Each of the 3 curves is an approximations of the exact edge curve 9

Challenge: A possible isogeometric volume structure 10

Why a volume structure? Parametric NURBS surfaces without trimming (curves removing parts of the domain) have 4 edge curves. Parametric NURBS volumes have 6 outer faces and is the mapping of an axis parallel box in the parameter domain. 11

Elementary shapes are not so simple as we used to think. Why? Isogeometric analysis allows in principle direct coupling of CAD and FEA. However, The models are respectively 2-variate and 3-variate, model restructuring necessary The elementary surface of CAD has to be given a suitable NURBS representation The CAD approach of 3 version of intersection curves cannot be allowed Isogeometric analysis demands accurate tri-variate parametric representations of the objects to be analyzed. No gaps allowed unless they reflect the actual geometry (e.g., a crack in the object). 12

Independent evolution of CAD and Finit Element Analysis (FEA) CAD (NURBS) and Finite Element Analysis evolved in different communities before electronic data exchange FEM developed to improve analysis in Engineering CAD developed to improve the design process Information exchange was drawing based, consequently the mathematical representation used posed no problems Manual modelling of the element grid Implementations used approaches that best exploited the limited computational resources and memory available. FEA was developed before the NURBS theory FEA evolution started in the 1940s and was given a rigorous mathematical foundation around 1970 (E.g,,1973: Strang and Fix's An Analysis of The Finite Element Method)StrangFix B-splines: 1972: DeBoor-Cox Calculation, 1980: Oslo Algorithm 13

Why are splines important to isogeometric analysis? Splines are polynomial, same as Finite Elements B-Splines are very stable numerically B-splines represent regular piecewise polynomial structure in a more compact way than Finite Elements NonUniform rational B-splines can represent elementary curves and surfaces exactly. (Circle, ellipse, cylinder, cone…) Efficient and stable methods exist for refining the piecewise polynomials represented by splines Knot insertion (Oslo Algorithm, 1980, Cohen, Lyche, Riesenfeld) B-spline has a rich set of refinement methods 14

Challenge 1: Topology How to split the object into proper 3-variate parametric NURBS 15

Steps in making the NURBS volume 1.Calculate the intersection of the cylinders. 2.Subdivide the cylinders into four sided regions by superimposing an edge and vertex structure  The only situation when C 1 continuity is simple is when a vertex has 4 vertices, and opposing edges across the vertex meet with proper C 1 continuity.  The regions should be made to simplify the making of the NURBS surfaces, and C 1 or higher continuity between surfaces. 16

In more detail Construct two pipes, 1 and 2 Intersect 1 and 2, selecting the boundary piece of 1 as intersection surface Trim 2 Trim the boundary of 1 Adapt 2 to the new boundary information to remove trimming Split 1 to remove boundary trimming Split 2 to meet the volumes originating from 1 corner-to-corner Update topology for each step Ensure continuity along boundary surfaces 17

In more detail 2. Parameter domain of boundary surface of volume 1 Ruled based approach Volume 1 touches the updated volume 2 along the white ring Splitting will be performed along the dotted lines The inner circle will get corner singularities Approximation is required as the geometry is not planar The topology of the split will be uniform in the thickness direction, i.e. the volume is split as the surface 14 blocks for volume 1 4 blocks for volume 2 18

Construction of pipe junction Two pipes represented as spline volumes We want to make a block structured isogeometric model Initial method: Boolean operations on volumes 19

Intersecting all boundary surfaces The boundary surfaces of one volume are not suited for the topology structure due to two surfaces along the seam The method is partly based on stable SISL intersections and partially on experimental or prototype GoTools code Tolerance issues: Accuracy versus data size Surface types: The method expects spline surfaces, but the boundary surfaces are SurfaceOnVolume 20

Splitting the initial models Pipe 1 Pipe 2 21

Modifying pipe 2 The outer part of the pipe is selected One boundary surface is fetched from pipe 1 This boundary surface must be approximated within the spline space of the initial volume Modification/construction of volume Adapt the volume to the new boundary surface. Volume smoothing is used Recreate the volume by linear loft between the new and the initial end surface Create volume interpolating all boundary surfaces (Coons approach) 22

The middle part Extra boundary surfaces extracted from the boundary surfaces of pipe 1 23

Pipe 1 This volume gets a hole by the Boolean operation Lets consider the outer cylinder surface Split the trimmed surface to get 4- sided surfaces that can be represented by spline surfaces 24

Pipe junctions gallery 25

Challenge 2: Geometry Representation by rational parametric Curves Surfaces Volumes 26

The intersection of two cylinders Let the first cylinders be represented implicitly: The centre c A unit vector d specifying the direction of the axis The radius The implicit description of the cylinder is then Let the second cylinder have radius 1, and have the z-axis as its axis, and let a quarter be described by the rational parameterization 27

Combining the cylinders Inserting the parametric represented cylinder in the implicit represented cylinder yields a polynomial of total degree 4, up to degree 4 in u and up to degree 2 in v This is an algebraic curve of total degree 4 in u and v. The general degree 4 algebraic curve do not have a rational parameterization, and this is the case for the cylinders when they are in general positions. 28

Approximation of shape cannot be avoided! Two components contribute to the need for approximation The intersection of two cylinders cannot in the general case be represented by a rational parameterization The block structuring might impose shape approximation to work properly Which approximation qualities are important? Approximation error Approximation ensured to be inside or outside of the real object Distribution of the error Degrees of the NURBS approximation Ensure that approximation lies in one of the cylinders The oscillatory behaviour of the approximation 29

Error when controlling tangent lengths cubic in Hermit interpolation Simplest power expansion of tangents Outside error Radius 1, opening angle 1 30

More examples Near near equioscillating Frist second and third derivate of error zero at midpoint 31

And even more examples 32 Error with zero integral q(p(t)) Error when square sum of Bezier coefficients is minimal

Summery of methods Examples from my doctorate thesis from 1997, that can be found at menu the GAIA project. (94, 95,… in the table refers to page numbers in the thesis) 33

Some shape approximation challenges Curve level: Controlling the quality of the approximation of the intersection curve of two cylinders: Cubic or higher degree polynomial approximation Rational approximation Surface level: Approximating the cylinder pieces resulting from segmentation of the trimmed cylinder in rectangular regions Volume level: Approximating the tube segments resulting from the segmentation 34

Example smoothing of NURBS volume parameterization 35 Video Courtesy: Kjell-Fredrik Pettersen, SINTEF

Challenge 3: Blends Often the transition between cylinders is made smooth by adding blends. Fillets can, e.g., represent grinded welds resulting from the manufacturing process. The simplest way of making the blend is by rolling a ball that touch both cylinders, and using the surface traced out as the outer surface of the blend volume. 36

Generating a rolling ball blend - 1 (Pictures by Rimas) 37

Generating a rolling ball blend - 2 (Pictures by Rimas) 38

SAGA already addresses some blend surfaces Heidi is addressing new approaches for making the fillet surface between a plane and cone together with Rimas. I hope a next step is that we can address the fillet volume between the fillet surface, the cone and the plane. A following challenge can the be to address the fillet volume between two cylinders and the fillet surface 39