PaperlayerView exact revision on arXiv

Source-linked AI summary

Multi-Material Topology Optimization for Targeted Structural Dynamic Vibration Attenuation

Bir Bahadur Chaudhary, Bidhan Pandey, Bin Ouyang, Shiguang Deng

arXiv:2610.31996v1cs.CE

TL;DR

Resonance-induced vibration can degrade structural performance and service life, motivating designs that place natural frequencies relative to excitation frequencies. This paper proposes multi-material topology optimization with new interpolation and frequency-control formulations; numerical examples demonstrate distinct dynamic responses across those formulations.

  • Problem

    Resonance-induced excessive vibration can degrade performance, reliability, and service life, motivating structural designs that shift natural frequencies away from excitation frequencies.

    Source: §1 · p. 1

  • Method

    The paper combines multivariate weight-based stiffness and mass interpolation with three eigenfrequency objectives and a rational-polynomial frequency-band constraint.

    Source: §1 · p. 2, §3 · p. 3, §4 · p. 5

  • Results

    Numerical examples produced distinct structural dynamic responses across the three formulations; adjacent-frequency-gap optimization yielded 847 rad/s for the three-material design and 800 rad/s for the four-material design.

    Source: §7.1.2 · p. 14, §8 · p. 20

  • Takeaways & Limitations

    The formulations provide different design routes: raise the lowest natural frequency, widen an adjacent-frequency gap, or exclude natural frequencies from a prescribed interval.

    Source: §4 · p. 5, §8 · p. 20

    The model is limited to linear elasticity, small deformations, and undamped free vibration, with external forces and damping neglected.

    Source: §2 · p. 2

Abstract

from arXiv · show

Resonance-induced excessive vibrations can degrade performance, reduce reliability, and shorten the service life of engineering structures. To reduce the risk of resonance, structural natural frequencies are generally engineered to be shifted away from excitation frequencies. To achieve these frequency shifts, topology optimization provides a systematic approach to tailor structural stiffness and mass as an effective inverse design approach. Multi-material topology optimization exploits the high-dimensional design space of spatially varying material constituents by simultaneously optimizing material selection and spatial distribution over the design domain, offering great flexibility in controlling natural frequencies. This study proposes a multi-material topology optimization that attenuates excessive structural dynamic vibrations in free-vibration scenarios via a novel multivariate weight-based interpolation method. Additionally, to address non-smoothness due to mode switching amid optimization, we employ a bound formulation with an auxiliary scalar design variable to replace the max-min eigenvalue objective with constraints on individual eigenvalues. Furthermore, we study and compare three optimization formulations for multi-material design to attenuate vibrations, including maximizing fundamental eigenfrequency, maximizing the gap between adjacent eigenfrequencies, and excluding frequencies from targeted frequency bands, which are smoothly approximated by a proposed rational polynomial formulation. Numerical examples are presented to demonstrate the effectiveness and applicability of our proposed method.

1. Introduction

The introduction motivates resonance attenuation by placing structural natural frequencies relative to excitation frequencies and identifies multi-material topology optimization as a flexible design approach. It presents a multivariate interpolation scheme and three formulations for controlling the spectrum.

1. Introduction

  • Resonance can produce large-amplitude oscillations, while closely spaced natural frequencies or repeated eigenvalues can involve multiple modes and complex dynamic behavior.The response depends on the relationship between excitation frequencies, natural frequencies, and mode shapes.Source: §1 · p. 1
  • Passive damping or isolation can add mass and work over limited frequency ranges, while active control requires sensors, actuators, and energy supplies.An alternative is to keep natural frequencies outside the operating range; the introduction identifies fundamental-eigenfrequency maximization as a common approach.Source: §1 · p. 1
  • Topology optimization tailors dynamic characteristics by distributing materials and voids, and prior work extended fundamental-frequency optimization to multiple-eigenfrequency spectral tailoring.The introduction also notes work on repeated-eigenvalue sensitivity and artificial localized modes in low-density regions.Source: §1 · p. 1
  • Multi-material methods include interpolation schemes and phase-based formulations, while prior frequency-band constraints used Heaviside functions to keep natural frequencies out of prescribed bands.The introduction identifies mode switching and determining relevant natural frequencies in a prescribed interval as challenges for existing objectives.Source: §1 · p. 2
  • Prior multi-material vibration designs used fundamental-frequency maximization or adjacent-frequency separation, while another approach combined penalized stiffness interpolation with linear mass interpolation to address pseudo modes and mode switching.Existing objectives also include avoiding resonance in prescribed operating ranges.Source: §1 · p. 2
  • The study proposes multivariate weight-based stiffness and mass interpolation and compares formulations for fundamental-frequency maximization, adjacent-frequency-gap maximization, and targeted frequency-band generation.It also uses a rational-polynomial frequency-band constraint to address non-differentiability.Source: §1 · p. 2

2. Mathematical model of structural vibration

The model treats structural response as undamped free vibration under linear elasticity and small deformations. It derives natural frequencies and mode shapes from a generalized eigenvalue problem, then filters and projects design variables to support stable, discrete material layouts.

2. Mathematical model of structural vibration

  • The finite-element model assumes linear elasticity, small deformations, and undamped free vibration, neglecting external forces and damping effects.The resulting generalized eigenvalue problem yields natural frequencies and corresponding mode shapes.Source: §2 · p. 2

2.1. Governing equation for structural vibration

  • A separable harmonic solution reduces the equation of motion to a generalized eigenvalue problem, whose eigenpairs represent natural frequencies and mode shapes.The eigenvectors are normalized with respect to the mass matrix and form a linearly independent basis.Source: §2.1 · p. 2

2.2. Design variable filtering and projection

  • Density filtering and smoothed Heaviside projection are applied to design variables to prevent checkerboards, reduce mesh dependence, and promote distinct material distributions.Projection steepness is increased gradually through a continuation scheme.Source: §2.2 · p. 3

2.2.1. Density filter

  • The density filter weights neighboring elements within a prescribed radius, with closer centroids contributing more to each filtered density.The filter is applied to suppress checkerboard patterns and mesh dependence.Source: §2.2.1 · p. 3

2.2.2. Heaviside projection

  • The smoothed Heaviside projection reduces intermediate filtered densities toward discrete values; its threshold and sharpness govern the mapping.Because large sharpness makes optimization highly nonlinear and can cause divergence, sharpness is increased gradually; projection sensitivities are transferred by the chain rule.Source: §2.2.2 · p. 3

3. Multivariate weight-based multi-material interpolation

The section develops a hierarchical multivariate interpolation for assigning material contributions, uses those properties to assemble the structural vibration model, and smooths frequency-band exclusion for optimization.

3. Multivariate weight-based multi-material interpolation

  • The multivariate interpolation represents n_MAT candidate materials with n_MAT−1 design fields per element, with later fields selecting among materials after earlier selections.In the three-material example, the first field selects the first material versus the remaining two, and the second distinguishes the latter pair.Source: §3 · p. 3, Figure 1 · p. 3

3.1. Weights for multi-material interpolation scheme

  • Normalized material weights form complementary fractions that bound interpolated element properties by the candidate materials’ properties.The weights are obtained from filtered and projected fields; the first, intermediate, and last materials have separately defined unnormalized weights.Source: §3.1 · pp. 3–4

3.2. Finite element formulation for structural vibration

  • Effective material properties scale the element stiffness and mass matrices, which are assembled into global matrices for the generalized eigenvalue analysis.The formulation assumes plane stress and a shared Poisson’s ratio across candidate materials; unit-property element matrices are reused across iterations.Source: §3.2 · p. 4

3.3. Frequency band constraint by rational polynomial function

  • The rational polynomial function replaces the discontinuous Heaviside measure with a smooth, differentiable frequency-band measure whose value is 1/2 at either band edge.Its transition sharpness is controlled by β_FBC; the 1/2 threshold separates frequencies inside and outside the prohibited band.Source: §3.3 · p. 5
  • A p-norm aggregation combines mode-wise contributions into one differentiable constraint per band, approximating their maximum from below and approaching it as p increases.Increasing β_FBC steepens the transition and reduces smoothing conservatism; increasing p better approximates the maximum but increases constraint nonlinearity.Source: §3.3 · p. 5

4. Topology optimization for vibration attenuation

The section presents three multi-material vibration-attenuation formulations: raising the fundamental frequency, widening an adjacent-mode gap, and excluding modes from a prescribed frequency band.

4.1. Maximization of fundamental frequency

  • The fundamental-frequency formulation maximizes the lowest eigenfrequency using an auxiliary bound variable and smooth constraints on individual eigenvalues.This bound formulation avoids the nonsmooth min operator and mode-switching discontinuities by keeping the lowest modes in separate constraints.Source: §4.1 · p. 6

4.2. Maximizing adjacent eigenfrequency gap

  • The adjacent-gap formulation maximizes the separation between modes n_b and n_b+1 using two auxiliary bounds on the lower and upper eigenfrequency groups.The lower group is constrained below λ1, while the upper group is constrained above λ2; maximizing λ2 − λ1 widens the gap.Source: §4.2 · p. 6

4.3. Maximizing targeted frequency gap

  • The targeted-gap formulation raises the lower spectrum while constraining every monitored eigenfrequency to remain outside a prescribed frequency interval.It combines a bound-variable frequency-maximization objective with a frequency-band constraint for the targeted interval.Source: §4.3 · pp. 6–7

5. Design sensitivity derivations

The section derives sensitivities for the material interpolation and for three eigenfrequency-based formulations, including a treatment of repeated eigenvalues that improves convergence stability.

5.1. Design sensitivity of material candidates

  • Material-weight sensitivities follow the chain rule, with distinct derivative cases for the first, intermediate, and last material candidates and normalization of the resulting gradients.The derivative cases depend on whether a design variable appears in the product defining an unnormalized weight.Source: §5.1 · p. 7, §5.1.2 · p. 7, §5.1.3 · p. 7

5.2. Sensitivity for fundamental frequency maximization

  • Fundamental-frequency maximization uses an auxiliary scalar alongside density variables, with sensitivities for the objective, eigenfrequency constraints, and material-volume constraints.Sensitivity propagation applies the chain rule through design-variable projection and density filtering.Source: §5.2 · p. 7, §5.2.2 · pp. 7–8, §5.2.3 · p. 8

5.3. Sensitivity for adjacent frequency gap maximization

  • For adjacent-frequency gap maximization, the objective separates two auxiliary bounds and is independent of density variables; its sensitivities push the bounds apart.The formulation also derives sensitivities for lower and upper eigenfrequency bounds and material-volume constraints.Source: §5.3.1 · p. 8, §5.3.2 · p. 8, §5.3.3 · p. 9

5.4. Sensitivity for targeted frequency gap

  • The targeted-frequency formulation excludes eigenfrequencies from prescribed bands through a rational-polynomial constraint, with sensitivities propagated through the design and auxiliary variables.The frequency-band constraint sensitivity is expanded using the eigenfrequency sensitivity and the chain rule.Source: §5.4.3 · pp. 9–10

5.5. Sensitivity for repeated eigenvalues

  • Repeated-eigenvalue sensitivities use directional derivatives from the associated eigenvector subspace, suppressing oscillation and mode switching to improve convergence stability.For a cluster, elemental stiffness and mass operators are projected onto the repeated-mode subspace; when multiplicity is one, the method reduces to the classical simple-eigenvalue sensitivity.Source: §5.5 · p. 10

6. Numerical implementations

The implementations test three-, four-, five-, and seven-material numerical sets, specify optimization parameters for the examples, and use MMA in an iterative optimization procedure.

6.1. Material properties

  • Candidate-material tests use three-, four-, five-, and seven-material sets, with Young’s modulus and density varied and Poisson’s ratio fixed at 0.3.The extreme materials share an E/ρ ratio of 10 MPa m3kg−1; the listed properties are numerical test cases, not real engineering materials.Source: §6.1 · pp. 10–11

6.2. Experiment parameter setup

  • Across examples, the setup specifies penalization continuation, six eigenmodes, material-volume fractions, and termination after the objective stops changing at maximum continuation.Stiffest-material volume fraction is 0.3 for fundamental-frequency and adjacent-gap maximization; all material fractions are identical for targeted-band gap generation.Source: §6.2 · pp. 11–12
  • Filter radii and projection parameters vary by example and formulation, while the frequency-band constraint uses a steepness parameter increased to 32.The projection threshold is 0.5, and the p-norm parameter is 40.Source: §6.2 · pp. 11–12

6.3. Optimization algorithm

  • The MMA algorithm iterates over design updates, evaluates objectives and constraints with sensitivities, applies continuation updates, and returns the optimized design variables.The algorithm takes candidate-material data, mesh data, and the number of eigenmodes as inputs.Source: §6.3 · p. 12

7. Numerical examples

Numerical examples apply the three vibration-attenuation formulations to a pinned beam, a square plate, and a C-clamp. They demonstrate fundamental-frequency maximization, adjacent-eigenfrequency gap maximization, and exclusion of natural frequencies from targeted bands.

7. Numerical examples

  • The validations use a pinned beam, a corner-fixed square plate, and a C-clamp with fixed right arms.The beam is 8 m × 1 m; the plate is 1 m × 1 m.Source: §7 · p. 12, §7.1 · p. 12, §7.2 · p. 16

7.1.1. Maximum fundamental frequency

  • For the five-material beam, the proposed interpolation reaches 167 rad/s versus 166 rad/s with ordered SIMP; seven candidates reach 166 rad/s, showing more materials need not improve the maximum.The five-material designs have similar layered distributions, and the stiffest, densest material tends to occupy high modal-strain-energy regions.Source: §7.1.1 · pp. 13–14, Figure 4 · p. 13, Figure 5 · p. 13

7.1.2. Gap maximization between adjacent eigenfrequencies (a)

  • For the beam’s third–fourth gap, three- and four-material designs reach 847 and 800 rad/s; in the latter, mode 3 is axial and modes 5–6 localize near supports.The gap widens as the third eigenfrequency moves down and the fourth moves up; layout depends on the difference between their modal-energy distributions.Source: §7.1.2 · p. 14, Figure 6 · p. 14, Figure 7 · p. 14

7.1.3. Creating a gap between targeted frequency range

  • Beam designs exclude [500, 1000] rad/s or both [300, 500] and [700, 1000] rad/s; the two-band spectrum stabilizes after a continuation-driven frequency drop.The single-band gap slightly exceeds its prescribed interval, which the authors attribute to p-norm aggregation error.Source: §7.1.3.1 · p. 15, Table 2 · p. 15, §7.1.3.2 · p. 16

7.2. Square plate design

  • On the square plate, fundamental-maximizing material concentrates near supports; third–fourth gap optimization reaches 8,499 rad/s by lowering the first three frequencies.For gap maximization, candidate materials are placed around the domain center.Source: §7.2.1 · p. 16, §7.2.2 · p. 17, Figure 13 · p. 17

7.2.3. Creating the gap between targeted frequency range

  • For the square plate, excluding [2000, 10000] rad/s produces a clear gap and concentric material layers, from the stiffest material at the center to the least dense at the boundary.The resulting first six eigenfrequencies are reported in Table 3.Source: §7.2.3 · p. 17, Table 3 · p. 17, Figure 14 · p. 18

7.3. C-clamp design

  • The five-material C-clamp’s third–fourth gap reaches 16285 rad/s, opening predominantly through reduction of the third eigenfrequency; the optimized layout is symmetric about the horizontal midplane.The C-clamp fundamental-frequency optimization and its optimized design are shown in Figure 16.Source: §7.3.2 · p. 18, §7.3.1 · p. 18, Figure 17 · p. 19

7.3.3. Creating gaps between targeted frequency ranges

8. Conclusions

The study presents a multi-material topology optimization framework for structural vibration attenuation and compares three formulations. Numerical experiments show distinct dynamic responses, while future work proposes extending the framework to other design settings and coupled systems.

8. Conclusions

  • The framework uses multivariate weight-based interpolation and an auxiliary scalar variable in a bound formulation to replace max-min objectives with individual eigenvalue constraints.Source: §8 · p. 20
  • Maximizing the fundamental eigenfrequency raises the lowest natural frequency and delays the onset of the first structural resonance.Source: §8 · p. 20
  • Maximizing the gap between adjacent eigenmodes separates targeted neighboring eigenfrequencies and enlarges their interval.Source: §8 · p. 20
  • Frequency-band constraints shift natural frequencies outside a prescribed operating interval, using a rational polynomial for smooth approximation and sensitivity evaluation.Source: §8 · p. 20
  • Future work may extend the framework to programmable attenuation, multi-material metamaterials and periodic structures, and multiphysics systems.The proposed multiphysics scope includes coupled mechanical, thermal, acoustic, or electromechanical systems.Source: §8 · p. 20

Replication of results

The authors state that the reported results should be reproducible from the methodologies described in the publication.

Replication of results

  • The reported results should be reproducible through implementation of the methodologies described in the publication.Source: Replication of results · p. 20

Data availability

The paper states that all data involved are included in the paper.

Data availability

Conflict of interest

The authors report no competing financial or non-financial interests.

Conflict of interest

Loading 2610.31996v1…