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Topology optimization of multimaterial aircraft pylons using generalized shape function approach

Swagatam Islam Sarkar, Prabhat Kumar

arXiv:2609.27685v1cs.CE

TL;DR

Aircraft pylons must meet structural and resource demands, while most pylon topology-optimization studies use a single material and conventional multimaterial methods can require more design variables. This paper applies gSF with density filtering and projection to optimize pylon compliance, reporting near-discrete material layouts and 3D designs extruded from a symmetric 2D midplane.

  • Problem

    Pylons must transfer engine and aerodynamic loads within structural and mass constraints, while most pylon topology-optimization studies assume one material and conventional multimaterial methods can require growing design-variable sets.

    Source: §1 · pp. 1–2

  • Method

    The gSF formulation uses natural-coordinate variables per element with density filtering and projection to optimize pylon compliance under volume constraints.

    Source: §1 · p. 2, §2 · p. 3, §5 · p. 9

  • Results

    The optimized designs approach crisp 0–1 material states, retain active volume constraints, and show smooth objective convergence.

    Source: §5 · p. 9, §4 · p. 9

  • Takeaways & Limitations

    The study demonstrates multimaterial pylon designs with up to 14 materials by optimizing a symmetric 2D midplane and extruding it into 3D.

    Source: §5 · p. 9

    The 3D topology is obtained by extrusion rather than 3D optimization, relying on geometric and loading symmetry along the z-direction.

    Source: §4 · p. 8

Abstract

from arXiv · show

As the primary structural component connecting the engine to the wing or fuselage, an aircraft pylon requires optimized structural efficiency; this paper provides topology optimization of multimaterial pylons using the generalized shape function (gSF) approach. The gSF method uses $n$ natural-coordinate design variables per element to provide optimized designs up to $2^n$ distinct material phases while promoting close to discrete material layouts in conjunction with the density and formulated Heaviside projection filters. Pylon structural compliance is minimized subject to volume constraints. Exploiting the geometric features of a typical pylon structure, multimaterial evolution is performed on a corresponding 2D design domain representing the midplane, with up to 14 candidate materials. The optimized two-dimensional layout is then extruded to achieve the corresponding three-dimensional optimized pylon structure. The Method of Moving Asymptotes is employed to achieve the final design variables. The resulting convergence histories exhibit smooth and stable objective minimization. The results highlight the capability of the multimaterial topology optimization framework to effectively optimized aircraft pylons with multiple candidate materials, without requiring a considerable expansion of the design variable set.

1 Introduction

Aircraft pylons must transfer engine and aerodynamic loads while meeting stiffness, strength, and resource requirements. The paper motivates multimaterial topology optimization with a gSF approach that supports many material phases using few design variables.

1 Introduction

  • Pylons transfer engine and aerodynamic loads to the airframe while facing stiffness, strength, fatigue, and mass constraints.These combined load and resource demands make pylon design a topology-optimization challenge.Source: §1 · p. 1
  • Most pylon topology-optimization studies assume one homogeneous material, despite growing use of materials with differing stiffness, density, and fatigue behavior.The paper identifies multimaterial design as a potential route to further mass savings.Source: §1 · p. 2
  • Conventional multimaterial methods often increase design variables and require parameter tuning to suppress intermediate-density interfaces as candidate materials increase.Some approaches reduce variable count but still require substantial tuning for many candidate materials.Source: §1 · p. 2
  • The gSF method maps n natural-coordinate variables to material fractions, supporting 2^n phases and near-discrete layouts with filtering and projection.The paper applies this formulation to multimaterial aircraft-pylon topology optimization.Source: §1 · p. 2

2 Multimaterial model with the gSF approach

The gSF formulation maps element-level natural coordinates to material densities, then filters and projects them to produce near-discrete material layouts.

2 Multimaterial model with the gSF approach

  • For an n-dimensional material element, gSF calculates each material's density from the element's natural-coordinate design variables.The n variables correspond to coordinates of the material-simplex vertices represented by a centered unit-side hypercube.Source: §2 · p. 3
  • Density filtering controls minimum feature size and prevents common topology-optimization instabilities; filtering design variables is equivalent to filtering material densities.The formulation computes filtered element design variables using neighboring elements within a filter radius.Source: §2 · p. 3
  • Because filtered material densities can remain nonbinary, a modified projection-based filter is used to obtain crisp, discrete solutions.Its projection parameters control design discreteness and smoothness, and projected density is the element's physical density for that material.Source: §2 · p. 3
  • Mnd measures material-layout non-discreteness; values approaching zero indicate nearly crisp distributions across the design domain.The measure is defined using physical densities across candidate materials and finite elements.Source: §2 · p. 4

3 Optimization Formulation

The formulation interpolates element stiffness from the physical densities of candidate materials and minimizes structural compliance subject to material-volume constraints. The Method of Moving Asymptotes updates the design variables.

3 Optimization Formulation

  • Element Young’s modulus is interpolated from the physical densities of its materials, with a penalization factor of 3 and a small void modulus to avoid stiffness-matrix singularities.The modified SIMP interpolation uses each material’s Young’s modulus and physical density.Source: §3 · p. 4
  • The optimization minimizes structural compliance while enforcing an allowable volume constraint for each material.The formulation defines compliance, element volume, and material-volume terms in the objective and constraints.Source: §3 · p. 4
  • The Method of Moving Asymptotes updates the element design variables using a gradient-based algorithm.Source: §3 · p. 5

4 Results and Discussions

Across the optimized cases, material distributions approach pure-material states, and compliance nearly converges before the 400-iteration limit. The study optimizes a two-dimensional pylon midplane and extrudes the result to represent the three-dimensional structure.

4 Results and Discussions

  • A 30%-wide portion of the top edge is fixed, and a distributed force acts along the bottom edge; the considered load is fuselage weight.Source: §4 · p. 5
  • The optimized pylon study presents designs up to 14 materials, using 2D shape functions for 2–3 materials, 3D for 4–7, and 4D for 8–14; its 2D domain uses 150 × 60 finite elements.The maximum allowable volume fraction is 0.2, 0.08, and 0.04 per material with 2D, 3D, and 4D shape functions, respectively.Source: §4 · p. 5
  • Candidate material moduli are scaled relative to the stiffest material, whose normalized Young’s modulus is 1; void modulus is 10^-9 and Poisson’s ratio is 0.3.The normalized moduli are assigned proportionally from stiffest to softest across the material sets.Source: §4 · pp. 7–8
  • Geometric and static-loading symmetry along z supports extruding the converged 2D design into a 3D representation, reducing computational cost.The 3D topologies are shown alongside corresponding 2D counterparts in Tables 1 and 2.Source: Table 1 · p. 6, Table 2 · p. 7, §4 · p. 8
  • Mnd values close to zero indicate near-binary material distributions, while compliance nearly converges before 400 iterations; optimization continues to activate volume constraints.The reported results describe crisp designs approaching pure-material states.Source: §4 · p. 9

5 Conclusions

The optimization minimizes pylon compliance while respecting prescribed material-volume limits, producing crisp material layouts and extruded 3D designs using the gSF approach.

5 Conclusions

  • The gSF approach optimizes the pylon mid-plane with up to 14 materials using n design variables per element, then extrudes the layout into a 3D design.The paper reports using two-, three-, and four-dimensional material elements.Source: §5 · p. 9
  • The optimized designs approach near 0–1 material states, satisfy prescribed volume constraints, and show smooth convergence of objective values.The final layouts approach pure physical phases rather than intermediate mixtures.Source: §5 · p. 9
  • Each material is placed where it contributes most to stiffness as the optimization minimizes pylon compliance under prescribed loading and material-volume limits.At the final stage, each material's volume constraint remains active.Source: §5 · p. 9
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