Abstract In this paper we present a software tool that con-verts any two-dimensional black and white image into aCAD/CAM file in IGES format. It can be used to inte-grate topology optimization into CAD/CAM manufacturingsystems, and that is indeed the main motivation of thework. The software is very versatile and allows to solvethe interoperability issue between numerical computationand CAD/CAM software packages. The main novelty ofthe paper is the IGES translator code lacking in most usednumerical computation software packages.
Topology optimization is a major tool for conceptual designof structures nowadays (see Bendsøe and Sigmund 2003;Christensen and Klarbring 2009, and the recent survey
J. M. Chacon (�)Institute of Applied Mathematics in Scienceand Engineering (IMACI), Universidad de Castilla - La Mancha,13071 Ciudad Real, Spaine-mail: [email protected]
J. C. Bellido · A. DonosoDepartamento de Matematicas, ETSII Universidadde Castilla - La Mancha, 13071 Ciudad Real, Spain
Sigmund and Maute 2013). Through it, we can gener-ate structures that optimize a certain cost (like compli-ance or weight, for instance), while fulfilling requiredconstraints (typically a volume constraint or local con-straints, for instance). The topology optimization theoryis already very developed, and there are available meth-ods and algorithms for many physical situations, andfurther, the amount of examples of industrial applica-tions of topology optimization is really overwhelming.In this paper, we address the integration of Topology Opti-mization with Computer Aided Design and Computer AidedManufacturing (CAD/CAM) systems.
Once the topology of an optimized structure is obtained,that is, a black and white image corresponding to a 0-1design, the following natural step is to manufacture the finalgeometry. This process is crucial and it must be done withhigh accuracy in order not to distort the expected responseof the designed structure.
At this point, CAD/CAM tools are very helpful becauseonce the geometry of an object has been designed by usingCAD program, the CAM system generates the G-code forthe manufacturing machine tool trajectories. These trajecto-ries are the inputs for a Computer Numerical Control (CNC)machine that finally manufactures the exact geometry ofthe optimized structure at the macroscale, or for a machineusing laser cutting techniques at the microscale.
The path between topology optimized designs and CAD/CAM is not trivial, since black and white bitmap imagesare not, in general, compatible with most CAD/CAM sys-tems, based on the representation of curves and surfacesin the standard Non-Uniform Rational B-Spline (NURBS)(Piegl and Tiller 1997; Farin 1999). Consequently, inter-faces between structure and void areas of the topologyoptimized designs have to be approximated, typically byobtaining, by different means, a polynomial representation
of those interfaces (Olhoff et al. 1991; Kumar and Gossard1996; Maute and Ramm 1997; Koguchi and Kikuchi 2006;Donoso et al. 2010).
A first attempt, beyond the interface approximation issuementioned above, to efficiently integrate topology opti-mization and CAD, and as far as the authors knowledgeis concerned the only one, was presented by Tang andChang (2001). In this work, the authors use cubic B-splinecurves and surfaces to approximate the boundary points ofthe structure layout, obtaining the information needed toapproach the boundaries. Then, through a CAD API (Appli-cation Programming Interface), these B-spline curves andsurfaces are brought into the CAD environment for planarand solid model construction SolidWorks�. This approachcan be extended to other CAD/CAM tools but it wouldimply to develop a new API, as the authors outlined intheir paper. A way to circumvent this problem would be togenerate a standard neutral file, like IGES (Initial Graph-ics Exchange Specification), more appropriate to exchangeinformation between CAD/CAM systems.
The aim of this paper is to present a graphical user inter-face software for incorporating into any CAD/CAM systemthe 0-1 layouts coming from a two-dimensional topologyoptimization problem. The input data is a jpg image andthe output is an IGES file (igs format) of the vectorizedimage that can be opened into any CAD/CAM program.This application has been created using the software pack-age Mathematica�9 (Wolfram 2013). This post-processingapplication is composed of three modules or steps (seeFig. 1):
1. Boundary detection: detect the interface border of ablack and white bitmap image.
2. B-spline fitting: compute a B-spline curve approxima-tion of each interface border.
3. IGES translator: generate the IGES file of the B-splinecurves.
This tool integrates all the functions needed to vectorizea jpg black and white image within a graphical interface.Moreover, it incorporates an IGES translator that is notimplemented into the most important and popular numericalcomputation packages, such as Matlab� (Attaway 2013)or Mathematica�. The IGES translator allows to trans-fer the geometry to any CAD/CAM system in order tomanufacture the optimized structure, solving the interoper-ability problems between numerical computation packagesand CAD/CAM systems. We believe our code will be use-ful both from a professional and an academic or educationalpoints of view. Although the software and its interface arevery versatile permitting the user many possibilities regard-ing approximation options, the techniques used for Modules1 and 2 are well-known and we do not claim any originality
on them. However we would like to emphasize that the con-verter to IGES format is fairly new as it is not available innumerical computations packages.
It is worth mentioning that we are not concerned in thispaper with shape optimization. This is an important part ofstructural optimization since after having obtained a concep-tual topological design, it is usually polished and optimizedthrough shape optimization techniques without changing thetopology (see the recent survey Sigmund and Maute 2013),and then brought into a CAD/CAM system. This last issueis what we address in this paper.
The paper is organized as follows: the next three sectionsare devoted to explain in detail each one of the three mod-ules mentioned above. Finally, some conclusions and futurework are pointed out in the last section.
2 Boundary detection
As we have mentioned above, the first step we have to dois to detect the interfaces or boundaries of the optimizeddesign. The input data for this module is a black and whitejpg image. Module 1 contains a boundary detection algo-rithm that provides the position of points on the borderbetween the black and white regions. The problem consistsin identifying the different regions of the bitonal image. Todo that, we consider the image as a graph where each pixelcorresponds to a vertex. This algorithm follows a three-stepprocedure (see Fig. 2):
1. Convert the black and white image into a 0-1 matrix.2. Compute the Laplacian of the image-matrix by finite
differences (Gonzalez and Woods 2007). As a resultof this, we obtain the matrix of the so-called bound-ary image that represents only the boundaries betweenblack and white regions.
3. Numerate each vertex of the pixel of the boundaryimage (columns upward). This allows us to compute theconnected components of the graph associated to theboundary image (Shapiro 1996). The output of Mod-ule 1 is the number of boundaries and the set of pixelsassociated to each boundary.
Figure 3 shows an example1 of boundary detection. First,the user inserts the file name of the jpg image and opens it.When the user clicks the “boundary detection button”, thesoftware provides the number of boundaries (five, in this
1The example chosen to illustrate our method along the paper corre-sponds to an optimized piezoelectric modal sensor/actuator followingthe ideas developed in Donoso and Bellido (2009). Notice that the lay-out is a pure 0-1 design. Black and white areas mean regions withopposite polarity, and that is the reason why it does not matter whetherthere are disconnected black areas or not.
Integration of topology optimized designs into CAD/CAM
Fig. 1 Graphical user interface
Module 1: Boundary detection
Import .jpg image ""
Open image Boundary detection
Zoom
Insert File.jpg form file directory=/Users
Module 2: B-Spline fitting
Input data Segments Precision
Fitting Yes Non
Boundary number 1
Zoom Small Medium Large
Polynomial degree 1 2 3 4 5 6
End conditions Yes Non
Smoothing leve 0 1 2 3 5
Parametrization Uniform Centripetal Chord-length
Number of segments
Module 3: IGES translator
Units inch mm
Scale 1
Tolerance 0.001
Filename ".igs"
1º: Generate IGES file 2º: Save IGES file
example) and the pixels positions of each boundary. For thesake of simplicity of use, the software does not show thenumerical values of the pixels position. Also, in order todisplay a description of each tool, tooltips appear when theuser hovers the pointer over a button. Notice that numer-ical computation software packages, such as Matlab�or Mathematica�, include interface detection functions,however these functions do not provide the number of
curves and the connected component to which each pixelbelongs to, which is information needed for the subsequentapproximation.
3 B-spline fitting
Once we have obtained a representation of each boundary,the software Module 2 computes a global B-spline curve
Fig. 2 Boundary detectionprocess: 0-1 design input dataand its matrix (left column);detected borders between the 0and 1 phases and itscorresponding position matrix(center column); three differentconnected components of theinterface or border and itsconnectivity matrix (rightcolumn)
J. M. Chacon et al.
Fig. 3 Module 1: Boundarydetection. Open a bitonal image(left). Detection of boundaries(right)
Open image
Module 1: Boundary detection
Import .jpg image "Fig2.jpg"
Open image Boundary detection
Zoom
Open image Boundary detection
Zoom
5 Boundaries
Click to import .jpg image Click to draw the boundaries
Module 1: Boundary detection
Import .jpg image "Fig2.jpg"
File name and extension .jpg in quotes
approximation in R2 (Piegl and Tiller 1997). A B-spline
curve is a piecewise polynomial curve represented in theBernstein basis. A B-spline curve of degree n, with controlpoints {d0, · · · , dL} (boldface stands for vector quantities)and a knot sequence {u0, · · · , uK}, with K = L+ n+ 1, isdefined by
b(u) =L∑
i=0
diNni (u), (1)
where the basis functions Nni (u) are defined recursively as
N0i (u) =
{1, if u ∈ [ui, ui+1],0 otherwise,
(2)
and
Nni (u) =
u− ui
ui+n − uiNn−1i (u)+ ui+n+1 − u
ui+n+1 − ui+1Nn−1i+1 (u).
(3)
Knots lead the union of two curve segments that composethe B-spline function (see an example for n = 3 in Fig. 4).The polygon formed by the control points di is called B-spline polygon or control polygon.
The Module 2 computes a B-spline fitting to the pointsp = {p0, · · · , pP } of the boundaries obtained in the Module
1. The points {p0, · · · , pP } are associated with the parame-ters w = {w0, · · · , wP }, which are chosen in a suitable wayexplained below (see (10)). As a result of this, we obtain aglobal approximation B-spline curve b(u) of degree n withknots {u0, · · · , uK} that is close to the data points p byminimizing the mean square error (Farin 2002),
P∑
j=0
‖ pj − b(wj ) ‖2 . (4)
Control points
Internal knots
d0
u=0u
u=1/4u=1/2 u=3/4
u=1
d1 d5 d6
d2 d4
d3
Fig. 4 B-spline curve of degree n = 3
Integration of topology optimized designs into CAD/CAM
This expression, rewritten for the B-spline functions definedabove, becomes
P∑
j=0
∣∣∣∣∣
∣∣∣∣∣pj −L∑
i=0
diNni (wj )
∣∣∣∣∣
∣∣∣∣∣
2
. (5)
Finally, to minimize this quantity is equivalent to solve thelinear system
L∑
i=0
di
P∑
j=0
Nni (wj )N
nk (wj )
=P∑
j=0
pjNnk (wj ), k = 0, . . . , L, (6)
where the coordinates of the control points di are theunknown parameters (Piegl and Tiller 1997; Farin 2002).
Module 2 has two interactive options to control the shapeof the approximation: “segments” and “precision”. In thesegments option the user introduces the number of poly-nomial curve segments by a slide bar (Fig. 5 top). Usingthe second option, the user introduces the precision of theapproximation, measured through the Hausdorff distance(Fig. 5b bottom).
The Hausdorff distance of two sets A and B is given by
dH (A,B) = max{d(A,B), d(B,A)}, (7)
where
d(A,B) = maxa∈A
(minb∈B
‖ a − b ‖). (8)
Both approximation methods, either controlling number ofsegments or approximation precision, have the followingoptions and inputs:
– Boundary number. The software treats each curve indi-vidually. In this menu the user can choose the boundaryto be approximated.
– Polynomial degree. The default option is that the soft-ware chooses cubic B-spline curves because in mostcases it seems to be a reasonable choice, and it istypically the most used by the CAD/CAM systems.However, the software allows the user to modify thisvalue. Figure 6 shows the differences between a linearand a cubic approximation.
– End conditions. The B-spline curve can interpolate boththe initial (p0) and end (pP ) points. This is achievedincluding end conditions, d0 = p0 and dL = pP , intothe system of linear equations (6). This option lets theuser adjust the shape of the approximations near theendpoints as it is shown in Fig. 7.
– Smoothing level. When we use a coarse mesh, the posi-tions of the points p of the boundaries can changeabruptly, and consequently, the resulting B-spline curveapproximation has wiggles. In order to improve the
Input data Segments Precision
Fitting Yes Non
Boundary number 1 2 3 4 5
Zoom Small Medium Large
Polynomial degree 1 2 3 4 5 6
End conditions Yes Non
Smoothing level 0 1 2 3 5
Parametrization Uniform Centripetal Chord-length
Fitting precision 0.65
15 Segments
Module 2: B-Spline fitting
Module 2: B-Spline fitting
Input data Segments Precision
Fitting Yes Non
Boundary number 1 2 3 4 5
Zoom Small Medium Large
Polynomial degree 1 2 3 4 5 6
End conditions Yes Non
Smoothing level 0 1 2 3 5
Parametrization Uniform Centripetal Chord-length
Number of segments
15
Control points
Internal knots
Choose the input data to fit the boundary
Choose the number of segments
Choose the fitting precision
Fig. 5 Module 2: B-spline fitting
shape of the approximation, smoothing equations areintroduced into the system of linear equations (6). Thisprocess is often called Laplacian smoothing, and thecorresponding smoothing equations are given by thefollowing expressions
⎧⎪⎨
⎪⎩
d0 − 2d1 + d2 = 0...
...
dL−2 − 2dL−1 + dL = 0
(9)
In order to adapt the curve to different boundaries,the effect of the smoothing conditions is weighted(Farin 2002). The user can choose among six levelsof smoothing. The effect of the smoothing options isshown in Fig. 8.
J. M. Chacon et al.
Fig. 6 Polynomial degreeoptions: cubic (left) and linear(right)
Polynomial degree 1 2 3 4 5 6
Cubic approximation
Polynomial degree 1 2 3 4 5 6
Linear approximation
1
– Parametrization. The values of the parameters associ-ated with the points p are defined as
wj −wj−1 = (‖ pj − pj−1 ‖)f∑P
i=0 (|pi + pi−1|)f, j = 1, . . . , P ,
(10)
where the initial value is w0 = 0 (Lee 1989). Choos-ing the parameter f we typically get three differentparametrizations: uniform for f = 0, centripetal forf = 0.5 and chord length for f = 1. Figure 9 showsthat, when the data are uniformly spaced, the threeparametrizations provide the same results (Lee 1989).However, there are exceptions in which one worksbetter than the others.
As a result of this, the outputs of Module 2 are the controlpoints and the knot vector of the polynomial B-spline curveapproximation of each boundary. Now, this representationcan be incorporated into any CAD/CAM program.
4 IGES translator
Once a B-spline representation of the optimized structureis generated, we need to incorporate it into a CAD/CAMsystem in order to manipulate the geometry and generatethe G-code for the manufacturing machine tool trajecto-ries. The simplest solution is to generate an eps vec-tor graphic file using a numerical computation packagesuch as Matlab� or Mathematica�. However, this is aquite rude way to make it, since in general, although
Fig. 7 Ends conditionsinfluence
End conditions Yes NonEnd conditions Yes Non
Contact at the endpoints Contact at the endpoints
Integration of topology optimized designs into CAD/CAM
eps admits B-spline curves of degree at most 3, in prac-tice, when one generates an eps file with Matlab� andMathematica� the approximation is reduced to continuouspiecewise linear, wasting most of the fine work made before.Further to that, the eps generated is not admissible for theprincipal CAD/CAM programs. Consequently, we need totranslate the B-spline curve into a neutral format that can beopened in any CAD/CAM system. To this purpose, Module3 generates an IGES file of the B-spline curves (see Fig. 10).
IGES file is a neutral text format that allows thedigital exchange of information between CAD/CAMsystems (IGES/PDES Organization 1996). IGES wasdesigned for the exchange of mechanical engineering data(Bloor and Owen 1991), and in general, it describesproduct design and manufacturing information that hasbeen created in a CAD/CAM system (Pasquill 1988).However, the IGES translator presented in this paper con-verts a B-spline curve generated in a numerical computationpackage, Mathematica� for instance, to any CAD/CAMsystem.
An IGES file is an ASCII (American Standard Code forInformation Interchange) code composed of 80-characterdivided into 3 columns. Text strings are represented inHollerith format defined by the number of characters in
the string, followed by the letter H and the text string (e.g.2HMM). An IGES files has 5 sections (Fig. 11):
– Start (S). Incorporate information for human readingonly.
– Global (G). Provide information about the software thathas created the IGES files, the scale, the units, theresolution of the geometric model and the IGES version5.3 (denoted by the number 11).
– Directory (D). Contains an entry for each geometricelement. In this case, we use the entity number 126corresponding with a rational B-spline curve.
– Parameter (P). Enclose the data of the geometric enti-ties. This section contains, among other data, the inputof Module 3 corresponding with the degree, number ofcontrol points, knots vector, weights and coordinates ofthe control points.
– Terminate (T). Contains a resume of the IGES file.
The user chooses the name of the file, the units andthe scale. Then, the software converts the B-splinecurves into an IGES file that can be read by anyCAD/CAM system. In particular, the IGES format (ver-sion 5.3) generated by this translation tool is supportedby any software based on the geometric modelling kernel
Click to generate the IGES file Click to save the IGES file
Parasolid� (Siemens 2013), i.e. NX� (Tickoo 2013a),Solid Edge� (Tickoo 2013b), SolidWorks� (Tran 2012)and MicroStation� (Conforti 2009). Therefore, compati-bility with most CAD/CAM systems is assured as geo-metric modelling kernel Parasolid is integrated into morethan 350 commercial software applications (Allen 2013).Further, we have ensured that the IGES translator is com-patible with widely used CAD programs that employother kernels, such as Rhinoceros� (Cheng 2013) and
AutoCAD� (Omura and Benton 2013). We would liketo emphasize that beyond the variability of implementa-tions of IGES format (Weissflog 1984), our IGES trans-lator gives an output compatible with most CAD/CAMsystem.
The point of the IGES translator is the organization ofinformation, in Fig. 12 is shown how this has been done inour software in a schematic way. Figure 11 shows an exam-ple of an IGES file with all the required information. Finally,
Fig. 13 shows how the IGES output file looks like openedin SolidWorks�.
5 Conclusions and future work
In this paper we present a software that covers the gapbetween topology optimization and manufacturing, solving
the interoperability problem between mathematical com-putation and fabrication software packages. Given a 2Dblack and white topology optimized design, our softwaredetects the boundaries of the structure, approximated themand generates an IGES file of the optimized structure.IGES output is acceptable for any CAD/CAM program. Our
Fig. 13 Imported curves intoSolidWorks�
J. M. Chacon et al.
Fig. 14 Manufacturing process in a CNC milling machine (top) andfinal manufactured structure (bottom)
software is very versatile as regards the selection of optionsin each step of the procedure. The main novelty of our codeis the IGES translator. We believe that our code will beuseful both for professional or educational activities.
In Fig. 14 we show respectively CNC milling machinemanufacturing the structure shown as example along thispaper and the final manufactured structure.
Despite the wide range of applications this software mayhave, our motivation to develop it was the manufacturingof piezoelectric microtransducers designed through topol-ogy optimization (Donoso and Bellido 2009; Sanchez-Rojaset al. 2010; Ruiz et al. 2013). These transducers are verysensitive to perturbations on the interfaces, not only on theground structure, but also on the electrode profile of thepiezoelectric material bound to the host structure.
We plan to develop an on-line applet of our softwarethat will be at the disposal of anyone for free use. We willbe working in the near future on the 3D version of thiscode, where to devise a proper approximation algorithm bypolynomial patches is one of the main issues.
Acknowledgments This work is supported by the Spanish Min-isterio Ciencia e Innovacion, under research grant DPI2012-32278,co-financed by the ERDF (European Regional Development Fund),(Jesus M. Chacon) and MTM2010-19739 (J.C. Bellido and A.Donoso). We would like to acknowledge interesting suggestions onthe subject of this paper due to our colleagues J. Sanchez-Reyes,J.L. Sanchez-Rojas, D. Ruiz and E. Aranda. Special thanks to E. GarcıaPlaza for his help in the manufacturing of the structure shown in
Fig. 14, and an anonymous referee for his valuable comments, sincethey have greatly helped us to improve the manuscript.
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