pith. sign in

arxiv: 1907.01370 · v2 · pith:UUHKHC5Inew · submitted 2019-06-26 · 💻 cs.CE · cs.SY· eess.SY· math.OC

Parametric shape optimization for combined additive-subtractive manufacturing

Christian Altenhofen , Marco Attene , Oliver Barrowclough , Michele Chiumenti , Marco Livesu , Federico Marini , Massimiliano Martinelli , Vibeke Skytt
show 1 more author
Lorenzo Tamellini
This is my paper

Pith reviewed 2026-05-25 15:16 UTC · model grok-4.3

classification 💻 cs.CE cs.SYeess.SYmath.OC
keywords additive manufacturingsubtractive machiningshape optimizationsparse-grid surrogatesinner structuresparametric optimizationdistortion reductionextra material thickness
0
0 comments X

The pith

Parametric shape optimization determines the minimal extra material thickness for 3D printed parts that undergo subtractive finishing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical methodology that applies parametric shape optimization to the thickness of extra material added during 3D printing. This choice lets manufacturers perform the least possible subtractive machining afterward while still meeting surface quality and dimensional requirements. The method pairs the thickness optimization with a new algorithm that generates inner structures to lower distortion and material use. Computation is kept practical by replacing the expensive objective and constraint functions from the manufacturing simulations with sparse-grid surrogates.

Core claim

The central claim is that a parametric shape optimization procedure, augmented by an inner-structure generation algorithm and accelerated through sparse-grid surrogate models, can systematically minimize the extra material volume required on additively manufactured parts so that subsequent subtractive operations become as small as possible while still satisfying finishing constraints.

What carries the argument

parametric shape optimization of extra material thickness, with sparse-grid surrogates standing in for the objective and constraint functions from manufacturing simulations

If this is right

  • The optimized thickness yields minimal machining operations while still meeting surface and accuracy requirements.
  • The inner-structure algorithm produces parts with reduced distortion and lower weight.
  • Substitution of objective and constraint functions by sparse-grid surrogates makes the constrained optimization computationally feasible.
  • Overall manufacturing costs drop through reduced printing time, material volume, and energy consumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The surrogate replacement could let designers include higher-fidelity physics models that would otherwise remain too slow for repeated evaluation.
  • The inner-structure generator might be combined with topology optimization routines that already exist for pure additive manufacturing.
  • The same parametric-thickness idea could be tested on other hybrid process chains that add material first and then remove it.

Load-bearing premise

The sparse-grid surrogates accurately reproduce the true objective and constraint functions of the underlying manufacturing simulations across the design space of interest.

What would settle it

Evaluating the full manufacturing simulations on the final optimized designs and checking whether those designs still satisfy all finishing constraints and remain near the predicted minimum machining volume.

Figures

Figures reproduced from arXiv: 1907.01370 by Christian Altenhofen, Federico Marini, Lorenzo Tamellini, Marco Attene, Marco Livesu, Massimiliano Martinelli, Michele Chiumenti, Oliver Barrowclough, Vibeke Skytt.

Figure 1
Figure 1. Figure 1: Gear model provided by STAM (left) and an approximation of a single tooth representing the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Minkowski sum of a spoon and a star (image courtesy of Peter Hachenberger and the CGAL project). [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Nominal geometry with different offset radii. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Nominal geometry and points cloud used to measure the distance of each surface of the warped geometry. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Software flowchart for the optimization methodology implemented. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: surrogate model for objective (left) and constraint (right) functions for w=4 when optimizing the offset radius. The plot of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: surrogate models for objective and constraint functions for w=4 when optimizing the offset and the minimum wall thickness. [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
read the original abstract

In the industrial practice, additive manufacturing processes are often followed by post-processing operations such as subtractive machining, milling, etc. to achieve the desired surface quality and dimensional accuracy. Hence, a given part must be 3D printed with extra material to enable such finishing phase. This combined additive/subtractive technique can be optimized to reduce manufacturing costs by saving printing time and reducing material and energy usage. In this work, a numerical methodology based on parametric shape optimization is proposed for optimizing the thickness of the extra material, allowing for minimal machining operations while ensuring the finishing requirements. Moreover, the proposed approach is complemented by a novel algorithm for generating inner structures leading to reduced distortion and improved weight reduction. The computational effort induced by classical constrained optimization methods is alleviated by replacing both the objective and constraint functions by their sparse-grid surrogates. Numerical results showcase the effectiveness of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a parametric shape optimization methodology to determine the thickness of extra material added during additive manufacturing to enable subsequent subtractive finishing operations, while minimizing machining effort and ensuring surface quality. It introduces a novel algorithm for generating inner structures to reduce distortion and achieve weight reduction, and replaces both objective and constraint functions with sparse-grid surrogates to lower computational cost. Numerical results are presented to demonstrate the approach's effectiveness.

Significance. If the sparse-grid surrogates prove accurate and the inner-structure algorithm delivers the claimed benefits, the work could provide a practical computational framework for optimizing hybrid additive-subtractive processes, potentially reducing material waste and production time in industrial settings. The use of sparse grids for surrogate modeling in this context is a positive technical choice for managing expensive simulations.

major comments (1)
  1. [Numerical results] Numerical results section: the paper replaces objective and constraint functions with sparse-grid surrogates but reports no a-posteriori verification by re-evaluating the true manufacturing simulation at the reported optimal designs. Without explicit error estimates or full-model checks at convergence, it is impossible to confirm that the optimized extra-material thicknesses remain feasible under the original constraints (e.g., surface-quality or distortion bounds). This verification is load-bearing for the central claim that the designs satisfy finishing requirements while minimizing operations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: Numerical results section: the paper replaces objective and constraint functions with sparse-grid surrogates but reports no a-posteriori verification by re-evaluating the true manufacturing simulation at the reported optimal designs. Without explicit error estimates or full-model checks at convergence, it is impossible to confirm that the optimized extra-material thicknesses remain feasible under the original constraints (e.g., surface-quality or distortion bounds). This verification is load-bearing for the central claim that the designs satisfy finishing requirements while minimizing operations.

    Authors: We agree that a-posteriori verification of the surrogate optima against the full manufacturing simulation is necessary to rigorously confirm constraint satisfaction. In the revised manuscript we will add explicit re-evaluations of the true model at the reported optimal designs, together with error estimates between surrogate and full-model values at those points, to demonstrate that the obtained extra-material thicknesses remain feasible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; forward numerical procedure with independent surrogates

full rationale

The paper presents a parametric shape optimization methodology that replaces objective and constraint functions with sparse-grid surrogates to reduce computational cost. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain. The derivation chain consists of standard optimization techniques applied to manufacturing simulations, with the surrogate construction described as an approximation tool rather than a tautological re-expression of the target result. The central claims remain externally falsifiable via full-model re-evaluation, satisfying the criteria for a self-contained numerical method.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on domain assumptions about the fidelity of the manufacturing simulations and the accuracy of the chosen surrogates; no free parameters or invented entities are explicitly named in the abstract.

free parameters (2)
  • extra-material thickness variables
    Optimization variables whose values are determined by the procedure; their number and bounds are not specified.
  • sparse-grid level and refinement parameters
    Parameters controlling the surrogate construction that must be chosen to balance accuracy and cost.
axioms (1)
  • domain assumption The underlying additive and subtractive process simulations can be treated as black-box functions whose values are adequately approximated by sparse-grid surrogates over the relevant design space.
    The surrogate replacement step is justified only if this modeling assumption holds.

pith-pipeline@v0.9.0 · 5713 in / 1279 out tokens · 31651 ms · 2026-05-25T15:16:19.683366+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Christian Altenhofen (a,b)

    work page

  2. [2]

    Oliver Barrowclough (d)

    work page

  3. [3]

    Michele Chiumenti (e)

    work page

  4. [4]

    Massimiliano Martinelli (f)

    work page

  5. [5]

    nominal geometry

    Lorenzo Tamellini (corresponding author, tamellini@imati.cnr.it) (f) Affiliations: a. Fraunhofer Institute for Computer Graphics Research IGD, Interactive Engineering Technologies, Darmstadt, Germany b. Technische Universität Darmstadt, Interactive Graphics Systems Group, Darmstadt, Germany c. Consiglio Nazionale Delle Ricerche, Istituto di Matematica App...

    work page 2020

  6. [6]

    The grid resolution (size of the cavities)

    work page

  7. [7]

    wall thickness

    The minimum wall thickness between the cavities and the outer surface of the object. Both parameters are used to define the parameter space in which the optimization is performed. Table 2 shows some examples of inner structures created varying wall thickness from 0.0 mm to 0.5 mm. It can be seen that the number of repetitions, as well as the amount of mat...

    work page 2008

  8. [8]

    Nocedal and S

    J. Nocedal and S. Wright, Numerical Optimization, New York: Springer-Verlag New York, 2006

    work page 2006

  9. [9]

    Structural optimization using sensitivity analysis and a level-set method,

    G. Allaire, F. Jouve and A. Toader, "Structural optimization using sensitivity analysis and a level-set method," Journal of computational physics, 2004. Parametric shape optimization for combined additive-subtractive manufacturing P a g e | 18

    work page 2004

  10. [10]

    Stress-based shape and topology optimization with the level set method,

    E. Picelli, S. Townsend, C. Brampton, J. Norato and A. Kim, "Stress-based shape and topology optimization with the level set method," Computer Methods in Applied Mechanics and Engineering, vol. 329, pp. 1- 23, 2018

    work page 2018

  11. [11]

    A level set method for structural topology optimization,

    Y. M. Wang, X. Wang and D. Guo, "A level set method for structural topology optimization," Computer Methods in Applied Mechanics and Engineering, vol. 192, pp. 227-246, 2003

    work page 2003

  12. [12]

    A level set method for shape and topology optimization of coated structures,

    Y. Wang and Z. Kang, "A level set method for shape and topology optimization of coated structures," Computer Methods in Applied Mechanics and Engineering, vol. 329, pp. 553-574, 2018

    work page 2018

  13. [13]

    Shape and topology optimization considering anisotropic features induced by additive manufacturing processes,

    C. Dapogny, R. Estevez, A. Faure and G. Michailidis, "Shape and topology optimization considering anisotropic features induced by additive manufacturing processes," Computer Methods in Applied Mechanics and Engineering, vol. 344, pp. 626-665, 2019

    work page 2019

  14. [14]

    Predicting Shrinkage and Warpage in Injection Molding: Towards Automatized Mold Design,

    F. Zwicke, M. Behr and S. Elgeti, "Predicting Shrinkage and Warpage in Injection Molding: Towards Automatized Mold Design," in AIP Conference Proceedings, 2017

    work page 2017

  15. [15]

    Structural optimization under overhang constraints imposed by additive manufacturing technologies,

    G. Allaire, C. Dapogny, R. Estevez, A. Faure and G. Michailidis, "Structural optimization under overhang constraints imposed by additive manufacturing technologies," Journal of Computational Physics, vol. 351, pp. 295-328, 2017

    work page 2017

  16. [16]

    A level set based method for fixing overhangs in 3D printing,

    S. Cacace, E. Cristiani and L. Rocchi, "A level set based method for fixing overhangs in 3D printing," Applied Mathematical Modelling, vol. 44, pp. 446-455, 2017

    work page 2017

  17. [17]

    Finite element modeling of multi-pass welding and shaped metal deposition processes,

    M. Chiumenti, M. Cervera, A. Salmi, C. Agelet de Saracibar, N. Dialami and K. Matsui, "Finite element modeling of multi-pass welding and shaped metal deposition processes," Computer Methods in Applied Mechanics and Engineering, vol. 199, pp. 2343-2359, 2010

    work page 2010

  18. [18]

    Numerical Modelling and Experimental Validation in Selective Laser Melting,

    M. Chiumenti, E. Neiva, E. Salsi, M. Cervera, S. Baida, J. Moya, Z. Chen, C. Lee and C. Davies, "Numerical Modelling and Experimental Validation in Selective Laser Melting," Additive Manufacturing, vol. 18, pp. 171-185, 2017

    work page 2017

  19. [19]

    A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing,

    E. Neiva, S. Badia, A. F. Martin and M. Chiumenti, "A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing," International Journal for Numerical Methods in Engineering, p. In press, 2019

    work page 2019

  20. [20]

    Sparse Grids,

    H. J. G. M. Bungartz, "Sparse Grids," Acta Numerica, vol. 13, pp. 147-269, 2004

    work page 2004

  21. [21]

    Dimension-Adaptive Tensor-Product Quadrature,

    T. Gerstner and M. Griebel, "Dimension-Adaptive Tensor-Product Quadrature," Computing, vol. 73, no. 1, pp. 65-87, 2003

    work page 2003

  22. [22]

    Stochastic Spectral Galerkin and Collocation Methods for PDEs with random coefficients: A numerical comparison,

    J. Back, F. Nobile, L. Tamellini and R. Tempone, "Stochastic Spectral Galerkin and Collocation Methods for PDEs with random coefficients: A numerical comparison," in Spectral and High Order Methods for Partial Differential Equations, Lecture notes in Computational Science and Engineering, vol. 76, Springer, 2011

    work page 2011

  23. [23]

    A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code,

    M. McKay, R. Beckman and W. Conover, "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code," Technometrics, vol. 21, no. 2, pp. 239-245, 1979. Parametric shape optimization for combined additive-subtractive manufacturing P a g e | 19

    work page 1979

  24. [24]

    Approximation of high-dimensional parametric PDEs,

    A. Cohen and R. DeVore, "Approximation of high-dimensional parametric PDEs," Acta Numerica, vol. 24, 2015

    work page 2015

  25. [25]

    Buhmann, Radial basis functions : theory and implementations, Cambridge University Press, 2003

    M. Buhmann, Radial basis functions : theory and implementations, Cambridge University Press, 2003

    work page 2003

  26. [26]

    Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations,

    F. Ballarin, A. Manzoni, A. Quarteroni and G. Rozza, "Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations," International Journal for Numerical Methods in Engineering, vol. 102, no. 5, pp. 1136-1161, 2014

    work page 2014

  27. [27]

    Deep Learning: An Introduction for Applied Mathematicians,

    C. Higham and D. Higham, "Deep Learning: An Introduction for Applied Mathematicians," arXiv, 2018

    work page 2018

  28. [28]

    Multiscale topology optimization using neural network surrogate models,

    D. A. White, W. J. Arrighi, J. Kudo and S. E. Watts, "Multiscale topology optimization using neural network surrogate models," Computer Methods in Applied Mechanics and Engineering, vol. 346, pp. 1118-1135, 2018

    work page 2018

  29. [29]

    Hierarchical Gradient-Based Optimization with B-Splines on Sparse Grids,

    J. Valentin and D. Pflüger, "Hierarchical Gradient-Based Optimization with B-Splines on Sparse Grids," in Sparse Grids and Applications - Stuttgart 2014, 2016

    work page 2014

  30. [30]

    Recursively generated B-spline surfaces on arbitrary topological meshes,

    E. Catmull and J. Clark, "Recursively generated B-spline surfaces on arbitrary topological meshes," Computer-Aided Design, vol. 10, no. 6, pp. 350-355, 1978

    work page 1978

  31. [31]

    Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values,

    J. Stam, "Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values," in SIGGRAPH '98. Proceedings of the 25th annual conference on Computer graphics and interactive techniques, 1998

    work page 1998

  32. [32]

    CGAL, The Computational Geometry Algorithms Library,

    A. T. M. Fabri, "CGAL, The Computational Geometry Algorithms Library," in 10ème Colloque National en Calcul des Structures, 2011

    work page 2011

  33. [33]

    Boolean Operations on 3D Selective Nef Complexes: Optimized Implementation and Experiments,

    P. K. L. Hachenberger, "Boolean Operations on 3D Selective Nef Complexes: Optimized Implementation and Experiments," in Proc. of 2005 ACM Symposium on Solid and Physical Modeling. SPM. , Boston, MA, 2005

    work page 2005

  34. [34]

    A generic software design for Delaunay refinement meshing,

    R. Y. M. Laurent, "A generic software design for Delaunay refinement meshing," Comput. Geom. Theory Appl., , vol. 38, pp. 100-110, 2007

    work page 2007

  35. [35]

    Numerical simulation and experimental calibration of Additive Manufacturing by blown powder technology. Part I: thermal analysis,

    M. Chiumenti, X. Lin, M. Cervera, W. Lei, Y. Zheng and W. Huang, "Numerical simulation and experimental calibration of Additive Manufacturing by blown powder technology. Part I: thermal analysis," Rapid Prototyping Journal, vol. 23, no. 2, pp. 448-463, 2017

    work page 2017

  36. [36]

    Finite element analysis and experimental validation of the thermomechanical behavior in laser solid forming of Ti-6Al-4V,

    X. Lu, X. Lin, M. Chiumenti, M. Cervera, J. Li, L. Ma, L. Wei, Y. Hu and W. Huang, "Finite element analysis and experimental validation of the thermomechanical behavior in laser solid forming of Ti-6Al-4V," Additive Manufacturing, vol. 21, pp. 30-40, 2018

    work page 2018

  37. [37]

    Residual stresses and distortion of rectangular an S-shaped Ti-6Al-4V Laser Solid Forming parts: modelling and experimental calibration,

    X. Lu, X. Lin, M. Chiumenti, M. Cervera, Y. Hu, X. Ji, H. Yang and W. Huang, "Residual stresses and distortion of rectangular an S-shaped Ti-6Al-4V Laser Solid Forming parts: modelling and experimental calibration," Additive Manufacturing, vol. 26, pp. 166-179, 2019

    work page 2019

  38. [38]

    In situ measurements and thermo-mechanical simulation of Ti–6Al–4V laser solid forming processes, I, 153-154, 119-130,

    X. Lu, X. Lin, M. Chiumenti, M. Cervera, Y. Hu, X. Ji, L. Ma and W. Huang, "In situ measurements and thermo-mechanical simulation of Ti–6Al–4V laser solid forming processes, I, 153-154, 119-130," International Journal of Mechanical Sciences, Vols. 153-154, pp. 119-130, 2019. Parametric shape optimization for combined additive-subtractive manufacturing P a...

    work page 2019

  39. [39]

    Numerical modelling of heat transfer and experimental validation in Powder-Bed Fusion with the Virtual Domain Approximation,

    E. Neiva, M. Chiumenti, M. Cervera, E. Salsi, G. Piscopo, S. Badia, A. Martin, Z. Chen, C. Lee and C. Davies, "Numerical modelling of heat transfer and experimental validation in Powder-Bed Fusion with the Virtual Domain Approximation," Computational Mechanics, p. Submitted, 2019

    work page 2019

  40. [40]

    Empirical methodology to determine inherent strains in additive manufacturing,

    I. Setien, M. Chiumenti, S. van der Veen, M. San Sebastian, F. Garciandía and A. Echeverría, "Empirical methodology to determine inherent strains in additive manufacturing," Computers and Mathematics with Applications, p. In press, 2018

    work page 2018

  41. [41]

    COMET: Coupled Mechanical and Thermal Analysis. Data Input Manual, Version 5.0. Technical Report IT-308.,

    M. Cervera, C. Agelet de Saracibar and M. Chiumenti, "COMET: Coupled Mechanical and Thermal Analysis. Data Input Manual, Version 5.0. Technical Report IT-308.," CIMNE, Barcelona, 2002

    work page 2002

  42. [42]

    GiD: the Personal Pre and Post-Processor,

    CIMNE, Technical University of Catalonia., "GiD: the Personal Pre and Post-Processor," 2002. [Online]. Available: http://gid.cimne.upc.edu

    work page 2002

  43. [43]

    A method for registration of 3-D shapes.,

    P. J. Besl and N. D. McKay, "A method for registration of 3-D shapes.," IEEE Transactions on pattern analysis and machine intelligence, vol. 14, no. 2, pp. 239-256, February 1992

    work page 1992

  44. [44]

    CGAL 4.11.1 - 3D Mesh Editing,

    P. Alliez, C. Jamin, L. Rnieau, S. Tayebv, J. Tournois and M. Yvinec, "CGAL 4.11.1 - 3D Mesh Editing," [Online]. Available: https://doc.cgal.org/latest/Mesh_3/index.html

    work page

  45. [45]

    Brezinski and M

    C. Brezinski and M. Redivo Zaglia, Extrapolation methods. Theory and Practice, vol. 73, Amsterdam: North-Holland, 1991

    work page 1991

  46. [46]

    Build-to-Last: Strength to Weight 3D Printed Objects,

    L. Lu, A. Sharf, H. Zhao, Y. Wei, Q. Fan, X. Chen, Y. Savoye, C. Tu, D. Cohen-Or and B. Chen, "Build-to-Last: Strength to Weight 3D Printed Objects," in ACM Transactions on Graphics (TOG), 2014

    work page 2014

  47. [47]

    Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs,

    F. Nobile, L. Tamellini and R. Tempone, "Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs," Numerische Mathematik, vol. 134, no. 2, pp. 343-388, 2016

    work page 2016

  48. [48]

    Pflüger, Spatially Adaptive Sparse Grids for Higher-Dimensional Problems, Munchen: Verlag Dr

    D. Pflüger, Spatially Adaptive Sparse Grids for Higher-Dimensional Problems, Munchen: Verlag Dr. Hut, 2010

    work page 2010