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The pipe junction challenge

Tor DokkenSINTEFOslo, Norway

Pictures and examples by Vibeke Skytt, SINTEF

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What is a pipe junction?

n A composition of cylindrical pipes meeting.

n For structural use the pipes are welded without cut-outs

n For use for transport of fluids or gas, cut-outs are made.

n 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):n Visualization of the modeln Animation of the modeln Design of the modeln Production of the modeln Analysis of the model, and type of analysis

n Structural, flow,…

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Application domains and shape representations n 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,

n Example from the mid 1970s of the geometry of cylinder junction from an offshore platform in the oil industry.

n Cylinders made from steel plates, one cylinder is flattend for flame cutting.

n Accurate geometry of cut-outs important.

n Flame cutters controlled by curve data.

n In current industry welding robots plays a central role

n Robots controlled by curve datan Navigation of robots need an

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Representation of the pipe junction for design and analysisn Target quality criteria for this presentation:

n Design stage: Face connectivity + Shape accuracyn Analysis stage: Volumetric connectivity + Shape

accuracyn The ideal pipe junction can be composed of pieces

of cylindrical tubes.n During structural analysis loads are applied to the

structures, the shape will be deflected becoming slightly sculptured

n NURBS* used in CAD for representing sculptured surfacesn 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 structuresn Elementary surfaces dominates design of modern human

made industrial produced shapes:n Planen Cylindern Spheren Conen Torus

n Surfaces of more sculptured type relates ton Terrainn Actual shape of produced parts (elementary shapes slightly

deflected)n Styled and designed productsn Shapes made by artists

n Vegetation is in general fractal

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Some properties of elementary surfaces n Elementary surfaces all have an exact rational

parameterizationn The deflected elementary surface can be efficiently

approximated by a NURBS-surface or by an algebraic surface of somewhat higher degree

n Elementary surfaces have low algebraic degree:n Degree 1: Planen Degree 2: Cylinder, Sphere, Conen Degree 4: Torus

n 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)

n Representation of outer and inner hulls by surface patchwork

n Small gaps between surface allowed

n Edges of NURBS surfaces represented by 3 curves:

n A 3D curven One curve in the

parameter domain of each NURBS surface

n Each of the 3 curves

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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

Challenge: A possible isogeometric volume structure

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Why a volume structure?n Parametric NURBS surfaces

without trimming (curves removing parts of the domain) have 4 edge curves.

n 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?n Isogeometric analysis allows in principle direct

coupling of CAD and FEA. However,n The models are respectively 2-variate and 3-variate,

model restructuring necessaryn The elementary surface of CAD has to be given a suitable

NURBS representationn The CAD approach of 3 version of intersection curves

cannot be allowedn Isogeometric analysis demands accurate tri-variate

parametric representations of the objects to be analyzed.

n 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)

n CAD (NURBS) and Finite Element Analysis evolved in different communities before electronic data exchange

n FEM developed to improve analysis in Engineeringn CAD developed to improve the design processn Information exchange was drawing based, consequently the

mathematical representation used posed no problemsn Manual modelling of the element grid

n Implementations used approaches that best exploited the limited computational resources and memory available.

n FEA was developed before the NURBS theory n 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)

n B-splines: 1972: DeBoor-Cox Calculation, 1980: Oslo Algorithm

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Why are splines important to isogeometric analysis?n Splines are polynomial, same as Finite Elementsn B-Splines are very stable numericallyn B-splines represent regular piecewise polynomial

structure in a more compact way than Finite Elements

n NonUniform rational B-splines can represent elementary curves and surfaces exactly. (Circle, ellipse, cylinder, cone…)

n Efficient and stable methods exist for refining the piecewise polynomials represented by splines

n Knot insertion (Oslo Algorithm, 1980, Cohen, Lyche, Riesenfeld)

n B-spline has a rich set of refinement methods1414

Challenge 1: Topology

n 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

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In more detail 1.1 2

n Construct two pipes, 1 and 2n Intersect 1 and 2, selecting

the boundary piece of 1 as intersection surface

n Trim 2n Trim the boundary of 1n Adapt 2 to the new boundary

information to remove trimming

n Split 1 to remove boundary trimming

n Split 2 to meet the volumes originating from 1 corner-to-corner

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In more detail 2.

Parameter domain of boundary surface of volume

n Ruled based approachn Volume 1 touches the

updated volume 2 along the white ring

n Splitting will be performed along the dotted lines

n The inner circle will get corner singularities

n Approximation is required as the geometry is not planar

n The topology of the split will be uniform in the thickness direction, i.e. the volume is split as the surface

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Construction of pipe junction

n Two pipes represented as spline volumesn We want to make a block structured isogeometric

modeln Initial method: Boolean operations on volumes

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Intersecting all boundary surfacesn The boundary surfaces of one

volume are not suited for the topology structure due to two surfaces along the seam

n The method is partly based on stable SISL intersections and partially on experimental or prototype GoTools code

n Tolerance issues: Accuracy versus data size

n Surface types: The method expects spline surfaces, but the boundary surfaces are SurfaceOnVolume

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Splitting the initial models

Pipe 1 Pipe 2

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Modifying pipe 2n The outer part of the pipe is

selectedn One boundary surface is

fetched from pipe 1n This boundary surface must be

approximated within the spline space of the initial volume

n Modification/construction of volume

n Adapt the volume to the new boundary surface. Volume smoothing is used

n Recreate the volume by linear loft between the new and the initial end surface

n Create volume interpolating all boundary surfaces (Coons

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The middle part

Extra boundary surfaces extracted from the boundary surfaces of pipe

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Pipe 1

n This volume gets a hole by the Boolean operation

n Lets consider the outer cylinder surface

n Split the trimmed surface to get 4-sided surfaces that can be represented by spline surfaces

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Pipe junctions gallery

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Challenge 2: Geometry

n Representation by rational parametricn Curvesn Surfacesn Volumes

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The intersection of two cylindersn Let the first cylinders be represented implicitly:

n The centre c n A unit vector d specifying the direction of the axisn The radius

The implicit description of the cylinder is then

n 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

n 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

n This is an algebraic curve of total degree 4 in u and v.

n 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!n Two components contribute to the need for

approximationn The intersection of two cylinders cannot in the general

case be represented by a rational parameterizationn The block structuring might impose shape approximation

to work properlyn Which approximation qualities are important?

n Approximation errorn Approximation ensured to be inside or outside of the real

objectn Distribution of the errorn Degrees of the NURBS approximationn Ensure that approximation lies in one of the cylindersn The oscillatory behaviour of the approximation

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Error when controlling tangent lengths cubic in Hermit n Simplest power

expansion of tangents

n Outside error

Radius 1, opening angle 1

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More examples

n Near near equioscillating

n Frist second and third derivate of error zero at midpoint

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And even more examples

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n Error with zero integral q(p(t))

n Error when square sum of Bezier coefficients is minimal

Summery of methods

n Examples from my doctorate thesis from 1997, that can be found at http://www.sintef.no/IST_GAIA menu the GAIA project.

n (94, 95,… in the table refers to page numbers in the thesis)

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Some shape approximation challengesn Curve level: Controlling the quality of the

approximation of the intersection curve of two cylinders:

n Cubic or higher degree polynomial approximationn Rational approximation

n Surface level: Approximating the cylinder pieces resulting from segmentation of the trimmed cylinder in rectangular regions

n Volume level: Approximating the tube segments resulting from the segmentation

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Example smoothing of NURBS volume parameterization

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Video Courtesy: Kjell-Fredrik Pettersen, SINTEF

Challenge 3: Blends

n Often the transition between cylinders is made smooth by adding blends.

n Fillets can, e.g., represent grinded welds resulting from the manufacturing process.

n 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

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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 surfacesn Heidi is addressing new approaches for making the

fillet surface between a plane and cone together with Rimas.

n I hope a next step is that we can address the fillet volume between the fillet surface, the cone and the plane.

n A following challenge can the be to address the fillet volume between two cylinders and the fillet surface

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