Min. error | Max. error | Mean error | RMS | |
A | $ 7.5102 \times 10^{-3} $ | $ 1.0764 $ | $ 2.9600 \times 10^{-1} $ | $ 3.5253 \times 10^{-1} $ |
B | $ 7.5103 \times 10^{-3} $ | $ 1.0764 $ | $ 2.9600 \times 10^{-1} $ | $ 3.5254 \times 10^{-1} $ |
A key step during industrial design is the passing of design information from computer aided design (CAD) to analysis tools (CAE) and vice versa. Here, one is faced with a severe incompatibility in geometry representation: While CAD is usually based on surface representations, analysis mostly relies on volumetric representations. The forward pass, i.e., converting CAD data to computational meshes, is well understood and established. However, the same does not hold for the inverse direction, i.e., CAD reconstruction of deformed geometries resulting from analysis. This is particularly important for industrial workflows in which the shape optimization of an initial product is outsourced. Such shape optimization is the focus of this work. The few reconstruction methods reported mainly rely on spline fitting, in particular on creating new splines similar to shape reconstruction from 3D imaging. In contrast, this paper studies a novel approach that reuses the CAD data given in the initial design. We show that this concept enables one to shape reconstruct mediocre deformations while preserving the initial notion of features defined during design. Furthermore, reusing the initial CAD representation reduces the shape reconstruction problem to a shape modification problem. We study this unique feature and show that it enables the reconstruction of CAD data from computational meshes by composing each spline in the initial CAD data with a single, global deformation spline. While post-processing is needed for use in current CAD software, most notably, this novel approach enables reconstructing complete CAD models even from defeatured computational meshes.
Citation: |
Figure 1. Shape optimization of the initial design (left) modifies the boundaries of existing geometric features (right). Modified from [21]
Figure 2. Topology optimization of the design space (left) determines the material distribution within the design space (right). Modified from [21]
Figure 3. Shape optimization in an industrial design workflow from top to bottom: For efficient computations, the CAD model is defeatured, and a computational mesh is obtained by tesselating this defeatured model. Shape-optimizing CAE yields a deformed computational mesh and this deformation needs to be imposed onto the CAD model
Figure 4. Topological connection of three single surfaces to a closed cylinder in CAD data. (A) shows the geometric representation of the single surfaces, (B) the intersected visualized parts, and (C) their topological connection as a surface representation. Modified from [19]
Figure 8. The mapping $ \boldsymbol{\mathcal{{T}}} $ is defined on the defeatured computational mesh without the stiffeners (A), and applied to the CAD model (B) resulting in updated CAD data. The approach yields watertight reconstructions even if defeatured models are used in CAE (compare missing stringers)
Figure 9. Initial geometry of plate with hole shown with different color for each entity in the geometry data. Results will show the identical coloring for identical entities. The dimensions are $ 200\ {\rm{mm}} \times 100\ {\rm{mm}} \times 1.5\ {\rm{mm}} $ and the hole diameter is $ 50\ {\rm{mm}} $
Table 1. The reconstruction error for the plate with a hole (in mm) of (A): the composed model, and (B): the composed and degree-reduced model (cf. Sec 4.3)
Min. error | Max. error | Mean error | RMS | |
A | $ 7.5102 \times 10^{-3} $ | $ 1.0764 $ | $ 2.9600 \times 10^{-1} $ | $ 3.5253 \times 10^{-1} $ |
B | $ 7.5103 \times 10^{-3} $ | $ 1.0764 $ | $ 2.9600 \times 10^{-1} $ | $ 3.5254 \times 10^{-1} $ |
Table 2.
Computational time (in s) for the plate with a hole. Each major processing step (functional composition, trivariate fit and low-degree approximation) is individually timed for each single spline
Process step | Cumulative | Minimal | Maximal |
Splinefit | $ 20.000 $ | – | – |
FC | $ 0.564 $ | 6.664 × 10^{−3} | 6.438 × 10^{−2} |
Low-degree approx. | 3.139 | 6.610 × 10^{−2} | 0.657 |
Overall | $ 23.703 $ | – | – |
Individual splinefit (reference) | 4.304 × 10^{4} | $ {20.010} $ | 1.746 × 10^{4} |
Table 3. The reconstruction error (in mm) for the pressure sensor of: (A) composed model, (B) composed and degree-reduced model (cf. Sec 4.3)
Min. error | Max. error | Mean error | RMS | |
A | $ 3.3194 \times 10^{-5} $ | $ 4.5816 \times 10^{-2} $ | $ 3.5244 \times 10^{-3} $ | $ 5.2402 \times 10^{-3} $ |
B | $ 3.3213 \times 10^{-5} $ | $ 4.5816 \times 10^{-2} $ | $ 3.5373 \times 10^{-3} $ | $ 5.2446 \times 10^{-3} $ |
Table 4. Computational time (in s) for the pressure sensor with our method at each individual step and overall. Similar to Tab. 2, timings are compared to an individual splinefit of each curve or surface spline in the geometry (Individual splinefit)
Process step | Cumulative | Minimal | Maximal |
Global splinefit | 5.785 × 10^{3} | – | – |
FC | 16.120 | 1.094 × 10^{−3} | 7.839 × 10^{−2} |
Low-degree approx. | 4.343 × 10^{4} | 6.340 | 6.341 × 10^{2} |
Overall | 4.923 × 10^{4} | – | – |
Individual splinefit (reference) | 4.192 × 10^{5} | 20.010 | 1.166 × 10^{4} |
[1] | NASA-IGES Translator and Viewer, 1995. |
[2] | An Advanced NURBS Fitting Procedure for Post-Processing of Grid-Based Shape Optimizations, American Institute of Aeronautics and Astronautics, 2011. |
[3] | Freecad, 2018. Software. |
[4] | Seamless integration of analysis and design: Automatic cad reconstruction of post-analysis geometries - data, 2021, Collection of reconstructed IGES files. doi: 10.18154/RWTH-2023-05453. |
[5] | G. Allaire, C. Dapogny and F. Jouve, Shape and topology optimization, in Handbook of Numerical Analysis, 22 (2021), 1-132. Elsevier. doi: 10.1016/bs.hna.2020.10.004. |
[6] | G. Beer, B. Marussig, J. Zechner, C. Dünser and T.-P. Fries, Boundary element analysis with trimmed NURBS and a generalized IGA approach, CoRR, abs/1406.3499, 2014. doi: 10.1016/j.cma.2016.03.035. |
[7] | M. P. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, Springer Science & Business Media, 2003. |
[8] | K.-U. Bletzinger, Shape optimization, in Encyclopedia of Computational Mechanics Second Edition, (2017), 1-42. John Wiley & Sons, Ltd. doi: 10.1002/9781119176817.ecm2109. |
[9] | M. Carraturo, P. Hennig, G. Alaimo, L. Heindel, F. Auricchio, M. Kästner and A. Reali, Additive manufacturing applications of phase-field-based topology optimization using adaptive isogeometric analysis, GAMM-Mitt., 44 (2021), e202100013. doi: 10.1002/gamm.202100013. |
[10] | J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, Ltd., Chichester, 2009. doi: 10.1002/9780470749081. |
[11] | J. D. Deaton and R. V. Grandhi, A survey of structural and multidisciplinary continuum topology optimization: Post 2000, Structural and Multidisciplinary Optimization, 49 (2014), 1-38. doi: 10.1007/s00158-013-0956-z. |
[12] | T. D. DeRose, R. N. Goldman, H. Hagen and S. Mann, Functional composition algorithms via blossoming, ACM Trans. Graph., 12 (1993), 113-135. doi: 10.1145/151280.151290. |
[13] | G. Elber, Free Form Surface Analysis using a Hybrid of Symbolic and Numeric Computation, PhD thesis, Dept. of Computer Science, University of Utah, 1992. |
[14] | G. Elber, Precise construction of micro-structures and porous geometry via functional composition, in Michael Floater, Tom Lyche, Marie-Laurence Mazure, Knut Mørken, and Larry L. Schumaker, editors, Mathematical Methods for Curves and Surfaces, 108-125, Cham, 2017. Springer International Publishing. doi: 10.1007/978-3-319-67885-6_6. |
[15] | G. Foucault, J.-C. Cuillière, V. François, J.-C. Léon and R. Maranzana, Adaptation of CAD model topology for finite element analysis, Computer-Aided Design, 40 (2008), 176-196. doi: 10.1016/j.cad.2007.10.009. |
[16] | C. González-Lluch, P. Company, M. Contero, J. D. Camba and R. Plumed, A survey on 3d cad model quality assurance and testing tools, Computer-Aided Design, 83 (2017), 64-79. |
[17] | M. Hojjat, E. Stavropoulou and K.-U. Bletzinger, The Vertex Morphing method for node-based shape optimization, Computer Methods in Applied Mechanics and Engineering, 268 (2014), 494-513. doi: 10.1016/j.cma.2013.10.015. |
[18] | S, G. Johnson, The NLopt Nonlinear-Optimization Package, 2011. |
[19] | P. R. Kennicott, IGES/PDES Organization and U. S. Product Data Association, Initial Graphics Exchange Specification, IGES 5.3, IGES/PDES Organization and U. S. Product Data Association, 1996. |
[20] | A. Koshakji, A. Quarteroni and G. Rozza, Free form deformation techniques applied to 3D shape optimization problems, Communications in Applied and Industrial Mathematics, 4 (2013), e452, 26 pp. doi: 10.1685/journal.caim.452. |
[21] | M. C. Lang, Simultaneous Structural and Material Optimization, PhD thesis, University of Leoben, 2021. |
[22] | C. Le, T. Bruns and D. Tortorelli, A gradient-based, parameter-free approach to shape optimization, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 985-996. doi: 10.1016/j.cma.2010.10.004. |
[23] | D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Mathematical Programming, 45 (1989), 503-528. doi: 10.1007/BF01589116. |
[24] | B. Louhichi, G. N. Abenhaim and A. S. Tahan, CAD/CAE integration: Updating the CAD model after a FEM analysis, The International Journal of Advanced Manufacturing Technology, 76 (2015), 391-400. doi: 10.1007/s00170-014-6248-y. |
[25] | B. Louhichi, N. Aifaoui, M. Hamdi and B. Abdelmajid, An optimization-based computational method for surface fitting to update the geometric information of an existing b-rep cad model, International Journal of Computer Applications in Technology, 9 (2009), 17-25. |
[26] | W. Ma and J. P. Kruth, NURBS curve and surface fitting for reverse engineering, The International Journal of Advanced Manufacturing Technology, 14 (1998), 918-927. doi: 10.1007/BF01179082. |
[27] | B. Marussig and T. J. R. Hughes, A review of trimming in isogeometric analysis: Challenges, data exchange and simulation aspects, Archives of Computational Methods in Engineering, 25 (2018), 1059-1127. doi: 10.1007/s11831-017-9220-9. |
[28] | J. Nocedal, Updating quasi-Newton matrices with limited storage, Mathematics of Computation, 35 (1980), 773-782. doi: 10.1090/S0025-5718-1980-0572855-7. |
[29] | L. Piegl and W. Tiller, The NURBS Book, Springer-Verlag, New York, NY, USA, second edition, 1996. |
[30] | D. F. Rogers, An Introduction to NURBS: With Historical Perspective, Morgan Kaufmann, 2001. |
[31] | T. W. Sederberg and S. R. Parry, Free-form deformation of solid geometric models, SIGGRAPH Comput. Graph., 20 (1986), 151-160. doi: 10.1145/15922.15903. |
[32] | S. C. Subedi, C. S. Verma and K. Suresh, A review of methods for the geometric post-processing of topology optimized models, Journal of Computing and Information Science in Engineering, 20 (2020), 060801. doi: 10.1115/1.4047429. |
[33] | N. Valizadeh, Y. Bazilevs, J. S. Chen and T. Rabczuk, A coupled IGA-meshfree discretization of arbitrary order of accuracy and without global geometry parameterization, Computer Methods in Applied Mechanics and Engineering, 293 (2015), 20-37. doi: 10.1016/j.cma.2015.04.002. |
[34] | B. van Sosin and G. Elber, Crossing knot lines in composition of freeform B-spline geometry, Comput. Aided Geom. Des., 62 (2018), 217-227. doi: 10.1016/j.cagd.2018.03.009. |
[35] | V. Weiss, L. Andor, G. Renner and T. Várady, Advanced surface fitting techniques, Computer Aided Geometric Design, 19 (2002), 19-42. doi: 10.1016/S0167-8396(01)00086-3. |
[36] | J. Zwar, G. Elber and S. Elgeti, Shape optimization for temperature regulation in extrusion dies using microstructures, Journal of Mechanical Design, 145 (2023), 012004. doi: 10.1115/1.4056075. |
Shape optimization of the initial design (left) modifies the boundaries of existing geometric features (right). Modified from [21]
Topology optimization of the design space (left) determines the material distribution within the design space (right). Modified from [21]
Shape optimization in an industrial design workflow from top to bottom: For efficient computations, the CAD model is defeatured, and a computational mesh is obtained by tesselating this defeatured model. Shape-optimizing CAE yields a deformed computational mesh and this deformation needs to be imposed onto the CAD model
Topological connection of three single surfaces to a closed cylinder in CAD data. (A) shows the geometric representation of the single surfaces, (B) the intersected visualized parts, and (C) their topological connection as a surface representation. Modified from [19]
We embed an initial point cloud (A) in a spline's parameter space (B). By adapting the control points, the trivariate mapping approximates the deformation between initial and shape-optimized point cloud (C)
Composition of a sphere and a deformation spline. The initial sphere template is bi-quadratic. The resulting composed sphere has degrees
The physical space of a disk (pink) is contained within the parametric space of another surface spline (green)
The mapping
Initial geometry of plate with hole shown with different color for each entity in the geometry data. Results will show the identical coloring for identical entities. The dimensions are
Functional composition applied to a simple geometry
The proposed method deforms an initial CAD model (A) to the shape of a deformed computational grid (B)
Resulting reconstructed shape with degree-reduction