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1
The pipe junction challenge
Tor Dokken
SINTEF
Oslo, Norway
Pictures and examples by Vibeke Skytt, SINTEF
2
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)
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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,…
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Application domains and shape representations Domain Quality criteria: State-of-the-art:
3D visualization Visual impression Triangulations & texture mapping
Animation movies Visual impression Subdivision 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 Analysis Volumetric connectivity + Shape accuracy
Block structures of 3-variate parametric NURBS, any degree. T-spline and LR-splines emerging
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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.
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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
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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
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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
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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
Topology Geometry
solid brep
shell
face
loop
edge, coedge
vertex
surface
curve
point
Is limited by
Limited by
Limited by
Defined by a number of
Defined by a number of
Shape given by
Shape given by
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Challenge: A possible isogeometric volume structure
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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.
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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).
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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)
B-splines: 1972: DeBoor-Cox Calculation, 1980: Oslo Algorithm
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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
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Challenge 1: Topology
How to split the object into proper 3-variate parametric NURBS
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Steps in making the NURBS volume1. 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 C1
continuity is simple is when a vertex has 4 vertices, and opposing edges across the vertex meet with proper C1 continuity.
The regions should be made to simplify the making of the NURBS surfaces, and C1 or higher continuity between surfaces.
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In more detail 1.
1 2
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
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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
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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
20
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
21
Splitting the initial models
Pipe 1 Pipe 2
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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)
23
The middle part
Extra boundary surfaces extracted from the boundary surfaces of pipe 1
24
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
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Pipe junctions gallery
26
Challenge 2: Geometry
Representation by rational parametric Curves Surfaces Volumes
27
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
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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.
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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
30
Error when controlling tangent lengths cubic in Hermit interpolation Simplest power
expansion of tangents
Outside error
Radius 1, opening angle 1
31
More examples
Near near equioscillating
Frist second and third derivate of error zero at midpoint
32
And even more examples
Error with zero integral q(p(t))
Error when square sum of Bezier coefficients is minimal
33
Summery of methods
Examples from my doctorate thesis from 1997, that can be found at http://www.sintef.no/IST_GAIA menu the GAIA project.
(94, 95,… in the table refers to page numbers in the thesis)
34
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
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Example smoothing of NURBS volume parameterization
Video Courtesy: Kjell-Fredrik Pettersen, SINTEF
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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.
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Generating a rolling ball blend - 1(Pictures by Rimas)
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Generating a rolling ball blend - 2(Pictures by Rimas)
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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