{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:37:24Z","timestamp":1759970244481,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Alma Idea initiative from University of Bologna, Italy"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Additive manufacturing technologies have allowed the production of complex geometries that are typically obtained by applying topology optimization techniques. The outcome of the optimization process is a tessellated geometry, which has reduced aesthetic quality and unwanted spikes and cusps. Filters can be applied to improve the surface quality, but volume shrinking and geometry modification can be noticed. The design practice suggests manually re-designing the object in Computer-Aided Design (CAD) software, imitating the shape suggested by topology optimization. However, this operation is tedious and a lot of time is wasted. This paper proposes a methodology to automate the conversion from topology optimization output to a CAD-compatible design for industrial components. Topology optimization usually produces a dense triangle mesh with a high topological genus for those objects. We present a method to automatically generate a collection of spline (tensor-product) patches joined watertight and test the approach on real-wise industrial components. The methodology is based on the use of quadrilateral patches which are built on the external surface of the components. Based on the tests carried out, promising results have been obtained. It constitutes a first step towards the automatic generation of shapes that can readily be imported and edited in a CAD system.<\/jats:p>","DOI":"10.3390\/axioms14010072","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T12:32:52Z","timestamp":1737376372000},"page":"72","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["From Topological Optimization to Spline Layouts: An Approach for Industrial Real-Wise Parts"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0048-6034","authenticated-orcid":false,"given":"Carolina Vittoria","family":"Beccari","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7947-3932","authenticated-orcid":false,"given":"Alessandro","family":"Ceruti","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6174-9109","authenticated-orcid":false,"given":"Filip","family":"Chudy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"},{"name":"Institute of Computer Science, University of Wroc\u0142aw, ul. Joliot-Curie 15, 50-383 Wroc\u0142aw, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"e2401929","DOI":"10.1080\/17452759.2024.2401929","article-title":"Enhancing surface roughness of material extrusion additive manufacturing components via an innovative ironing process","volume":"19","author":"Montalti","year":"2024","journal-title":"Virtual Phys. Prototyp."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1814","DOI":"10.1108\/RPJ-09-2022-0320","article-title":"Efficient toolpath planning for collaborative material extrusion machines","volume":"29","author":"Bacciaglia","year":"2023","journal-title":"Rapid Prototyp. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1177\/0954406220917717","article-title":"Additive manufacturing and topology optimization: A design strategy for a steering column mounting bracket considering overhang constraints","volume":"235","author":"Mantovani","year":"2021","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Rosso, S., Uriati, F., Grigolato, L., Meneghello, R., Concheri, G., and Savio, G. (2021). An Optimization Workflow in Design for Additive Manufacturing. Appl. Sci., 11.","DOI":"10.3390\/app11062572"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5443","DOI":"10.1007\/s00170-024-14039-z","article-title":"Efficient part orientation algorithm for additive manufacturing in industrial applications","volume":"133","author":"Bacciaglia","year":"2024","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1089\/3dp.2020.0178","article-title":"Structural Analysis of Voxel-Based Lattices Using 1D Approach","volume":"9","author":"Bacciaglia","year":"2022","journal-title":"Print. Addit. Manuf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01650949","article-title":"Optimal shape design as a material distribution problem","volume":"1","year":"1989","journal-title":"Struct. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1016\/0045-7949(95)00235-9","article-title":"Evolutionary structural optimization for dynamic problems","volume":"58","author":"Xie","year":"1996","journal-title":"Comput. Struct."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1007\/s001580100140","article-title":"A simple checkerboard suppression algorithm for evolutionary structural optimization","volume":"22","author":"Li","year":"2001","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mortazavi, A. (2024). A novel binomial strategy for simultaneous topology and size optimization of truss structures. Eng. Optim., 1\u201335.","DOI":"10.1080\/0305215X.2024.2349100"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103819","DOI":"10.1016\/j.advengsoft.2024.103819","article-title":"A novel binomial-based fuzzy type-2 approach for topology and size optimization of skeletal structures","volume":"199","author":"Mortazavi","year":"2025","journal-title":"Adv. Eng. Softw."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Desbrun, M., Meyer, M., Schr\u00f6der, P., and Barr, A.H. (1999, January 8\u201313). Implicit fairing of irregular meshes using diffusion and curvature flow. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques-SIGGRAPH \u201999, Los Angeles, CA, USA.","DOI":"10.1145\/311535.311576"},{"key":"ref_13","first-page":"53","article-title":"Laplacian Mesh Processing","volume":"4","author":"Sorkine","year":"2005","journal-title":"Eurograph. (State Art Rep.)"},{"key":"ref_14","unstructured":"Belyaev, A., and Ohtake, Y. (2003, January 12\u201314). A comparison of mesh smoothing methods. Proceedings of the Israel-Korea Bi-National Conference on Geometric Modeling and Computer Graphics, Tel-Aviv, Israel."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2910","DOI":"10.1109\/TVCG.2018.2865363","article-title":"Mesh Denoising Guided by Patch Normal Co-Filtering via Kernel Low-Rank Recovery","volume":"25","author":"Wei","year":"2019","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"ref_16","unstructured":"Taubin, G. (1995, January 20\u201323). Curve and surface smoothing without shrinkage. Proceedings of the IEEE International Conference on Computer Vision-ICCV-95, Cambridge, MA, USA."},{"key":"ref_17","first-page":"6","article-title":"Extraction of the Quad Layout of a Triangle Mesh Guided by Its Curve Skeleton","volume":"35","author":"Usai","year":"2016","journal-title":"ACM Trans. Graph."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., and Stuetzle, W. (1995, January 6\u201311). Multiresolution analysis of arbitrary meshes. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques-SIGGRAPH \u201995, Los Angeles, CA, USA.","DOI":"10.1145\/218380.218440"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Eck, M., and Hoppe, H. (1996, January 4\u20139). Automatic reconstruction of B-spline surfaces of arbitrary topological type. Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques-SIGGRAPH \u201996, New York, NY, USA.","DOI":"10.1145\/237170.237271"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11003","DOI":"10.1115\/1.4051121","article-title":"A Local Approach for Computing Smooth B-Spline Surfaces for Arbitrary Quadrilateral Base Meshes","volume":"22","author":"Mosbach","year":"2022","journal-title":"J. Comput. Inf. Sci. Eng."},{"key":"ref_21","first-page":"120","article-title":"Towards the Integration of Topology Optimization into the CAD Process","volume":"11","author":"Francois","year":"2013","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.cag.2008.02.002","article-title":"A direction-oriented sharpness dependent filter for 3D polygon meshes","volume":"32","author":"Chen","year":"2008","journal-title":"Comput. Graph."},{"key":"ref_23","first-page":"107","article-title":"Automatic construction of structural CAD models from 3D topology optimization","volume":"15","author":"Nana","year":"2017","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"407","DOI":"10.3722\/cadaps.2009.407-418","article-title":"Converting Topology Optimization Results into Parametric CAD Models","volume":"6","author":"Larsen","year":"2009","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00366-006-0023-0","article-title":"A surface reconstruction algorithm for topology optimization","volume":"22","author":"Koguchi","year":"2006","journal-title":"Eng. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"113102","DOI":"10.1016\/j.cma.2020.113102","article-title":"Topologically robust CAD model generation for structural optimisation","volume":"369","author":"Yin","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_27","first-page":"197","article-title":"Surface reconstruction from FE mesh model","volume":"6","author":"Park","year":"2018","journal-title":"J. Comput. Des. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1080\/16864360.2014.902677","article-title":"Integration of CAD, FEA and Topology Optimization through a Unified Topological Model","volume":"11","author":"Francois","year":"2014","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.compstruc.2017.04.018","article-title":"Automatic reconstruction of beam structures from 3D topology optimization results","volume":"189","author":"Nana","year":"2017","journal-title":"Comput. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"103183","DOI":"10.1016\/j.cad.2021.103183","article-title":"Automated Lofting-Based Reconstruction of CAD Models from 3D Topology Optimization Results","volume":"145","author":"Amroune","year":"2022","journal-title":"Comput.-Aided Des."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3453","DOI":"10.1007\/s00158-021-03027-6","article-title":"Surface smoothing for topological optimized 3D models","volume":"64","author":"Bacciaglia","year":"2021","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1080\/02533839.2008.9671429","article-title":"Automated structural optimization system for integrated topology and shape optimization","volume":"31","author":"Lin","year":"2008","journal-title":"J. Chin. Inst. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s00158-008-0352-2","article-title":"Improved image interpreting and modeling technique for automated structural optimization system","volume":"40","author":"Chou","year":"2009","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/PL00013282","article-title":"Integration of topology and shape optimization for design of structural components","volume":"22","author":"Tang","year":"2001","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Piegl, L., and Tiller, W. (1996). The NURBS Book, Springer. [2nd ed.].","DOI":"10.1007\/978-3-642-59223-2"},{"key":"ref_36","unstructured":"Raymer, D.P. (2006). Aircraft Design: A Conceptual Approach, American Institute of Aeronautics and Astronautics. [4th ed.]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1002\/nme.2579","article-title":"Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities","volume":"79","author":"Geuzaine","year":"2009","journal-title":"Int. J. Numer. Methods Eng."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/72\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:32:00Z","timestamp":1759919520000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/72"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,20]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010072"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010072","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,20]]}}}