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Multi-material structural optimization for additive manufacturing based on a phase field approach

Luise Blank, Maximilian Urmann

TL;DR

This work addresses phase-field based, multi-material topology optimization for additive manufacturing with the goal of suppressing gravity-induced deformations during layer-by-layer construction. It develops a height-aware optimization framework that combines mean compliance, a construction-penalty, and a diffuse-interface perimeter term, solved via a VMPT method with PDAS subproblem Solvers; layer-by-layer and height-continuous formulations are both analyzed and implemented. The authors prove well-posedness and differentiability, establish height-independent Korn-type estimates, and demonstrate convergence and efficiency of the nested VMPT approach across 2D and 3D problems, including multiphase and removable-support scenarios. The numerical results show that iteration counts are robust to mesh refinement and layer count, and that nested strategies and problem-structure-aware metrics yield substantial speedups while enabling detailed parameter studies and practical designs for complex AM tasks. The framework provides a path toward manufacturable, optimized multi-material components with controlled overhangs, reducing post-processing and improving constructability in diverse AM contexts.

Abstract

A topology optimization problem in a phase field setting is considered to obtain rigid structures, which are resilient to external forces and constructable with additive manufacturing. Hence, large deformations of overhangs due to gravity shall be avoided during construction. The deformations depend on the stage of the construction and are modelled by linear elasticity equations on growing domains with height-dependent stress tensors and forces. Herewith, possible hardening effects can be included. Analytical results concerning the existence of minimizers and the differentiability of the reduced cost functional are presented in case of a finite number of construction layers. By proving Korn's inequality with a constant independent of the height, it is shown that the cost functional, formulated continuously in height, is well-defined. The problem is numerically solved using a projected gradient type method in function space, for which applicability is shown. Second-order information can be included by adapting the underlying inner product in every iteration. Additional adjustments enhancing the solver's performance, such as a nested procedure and subsystem solver specifcations, are stated. Numerical evidence is provided that for all discretization level and also for any number of construction layers, the iteration numbers stay roughly constant. The benefits of the nested procedure as well as of the inclusion of second order information are illustrated. Furthermore, the choice of weights for the penalization of overhangs is discussed. For various problem settings, results are presented for one or two materials and void in two as well as in three dimensions.

Multi-material structural optimization for additive manufacturing based on a phase field approach

TL;DR

This work addresses phase-field based, multi-material topology optimization for additive manufacturing with the goal of suppressing gravity-induced deformations during layer-by-layer construction. It develops a height-aware optimization framework that combines mean compliance, a construction-penalty, and a diffuse-interface perimeter term, solved via a VMPT method with PDAS subproblem Solvers; layer-by-layer and height-continuous formulations are both analyzed and implemented. The authors prove well-posedness and differentiability, establish height-independent Korn-type estimates, and demonstrate convergence and efficiency of the nested VMPT approach across 2D and 3D problems, including multiphase and removable-support scenarios. The numerical results show that iteration counts are robust to mesh refinement and layer count, and that nested strategies and problem-structure-aware metrics yield substantial speedups while enabling detailed parameter studies and practical designs for complex AM tasks. The framework provides a path toward manufacturable, optimized multi-material components with controlled overhangs, reducing post-processing and improving constructability in diverse AM contexts.

Abstract

A topology optimization problem in a phase field setting is considered to obtain rigid structures, which are resilient to external forces and constructable with additive manufacturing. Hence, large deformations of overhangs due to gravity shall be avoided during construction. The deformations depend on the stage of the construction and are modelled by linear elasticity equations on growing domains with height-dependent stress tensors and forces. Herewith, possible hardening effects can be included. Analytical results concerning the existence of minimizers and the differentiability of the reduced cost functional are presented in case of a finite number of construction layers. By proving Korn's inequality with a constant independent of the height, it is shown that the cost functional, formulated continuously in height, is well-defined. The problem is numerically solved using a projected gradient type method in function space, for which applicability is shown. Second-order information can be included by adapting the underlying inner product in every iteration. Additional adjustments enhancing the solver's performance, such as a nested procedure and subsystem solver specifcations, are stated. Numerical evidence is provided that for all discretization level and also for any number of construction layers, the iteration numbers stay roughly constant. The benefits of the nested procedure as well as of the inclusion of second order information are illustrated. Furthermore, the choice of weights for the penalization of overhangs is discussed. For various problem settings, results are presented for one or two materials and void in two as well as in three dimensions.

Paper Structure

This paper contains 19 sections, 7 theorems, 32 equations, 13 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The optimization problem
  3. Analytical results for the optimization problem
  4. Numerical method for the optimization problem
  5. Numerical Experiments
  6. Numerical analysis of the optimization solver
  7. Iteration numbers for increasing mesh sizes and number of layers M
  8. Benefits of the nested ansatz and of variable metrics
  9. Impact of the choice of beta1, beta2, Fc, Cc and GammaB on the solutions
  10. Impact of the parameters beta1, beta2
  11. Comparison of three different choices of Fc
  12. Hardening of the structure
  13. Impact of different building plate locations on the final structure
  14. Numerical examples with three phases and in 3D.
  15. Example in 2D with three phases (multiphase setting)
  16. ...and 4 more sections

Key Result

Theorem 1

The control-to-state operators $S:H^1(\Omega)^N \cap L^\infty(\Omega)^N \rightarrow H^1_D(\Omega)^d$, where $S(\varphi):=u$ is the solution of eq:WeakFormLinElast, and $S^c(h,.):H^1(\Omega)^N \cap L^\infty(\Omega)^N \rightarrow H^1_B(\Omega_{h})^d$, where $S^c(h,\varphi):=u^c(h,.)$ is the solutio hold with constants $c$ and $c^c$ independent of $\varphi$.

Figures (13)

  • Figure 1: Error reductions in the scalar case
  • Figure 2: Test example with low and high perimeter
  • Figure 3: Test example with increasing $\beta_1$
  • Figure 4: Overhangs violating the 45° angle condition marked black
  • Figure 5: Level sets of intermediate structures (black) and their corresponding displaced version (green) for different $\Omega_{k\delta}$
  • ...and 8 more figures

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1: Choice of the bilinear forms $a_k$.
  • Remark 2: Choice of the scaling $\rho_k$ of $j'(\varphi^{(k)})$.
  • Theorem 5
  • Remark 3: Spatial discretization
  • ...and 6 more