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Robust topology optimization with non-Gaussian material fields using polygonal finite elements
Nilton Cuellar, Anderson Pereira, Ivan F. M. Menezes, Americo Cunha
TL;DR
Robust topology optimization needs to represent spatially varying material uncertainty without violating physical admissibility or making repeated analyses impractical. The paper combines polygonal finite elements, translated non-Gaussian random fields, and non-intrusive polynomial chaos; non-Gaussian fields reallocate 6–12% of structural volume and alter load paths, while the surrogate reproduces intrusive results within a few percent. The study is limited to two-dimensional benchmarks and does not yet include manufacturing-oriented constraints or model-form uncertainty.
Problem
Robust topology optimization requires a framework that combines physically admissible spatially varying material uncertainty with polygonal finite-element optimization and consistent transfer between discretizations.
Method
The framework couples polygonal finite elements, translation-based non-Gaussian random fields for Young’s modulus, and non-intrusive polynomial-chaos surrogates.
Results
6–12% of solid volume is reallocated under Uniform, Gamma, or Beta modulus fields, with altered load paths and reduced compliance scatter.
Takeaways & Limitations
The combined approach provides a practical, computationally affordable route to topology-optimized designs under realistic material uncertainty.
Takeaways & Limitations
The study is limited to two-dimensional benchmarks and does not yet address large-scale three-dimensional problems, manufacturing-oriented constraints, or model-form uncertainty.
Abstract
from arXiv · showhide
We present a computational framework for robust topology optimization that integrates polygonal finite-element discretizations, spatially correlated non-Gaussian material modeling, and non-intrusive polynomial-chaos surrogates. Spatial uncertainty in Young's modulus is represented as a homogeneous non-Gaussian random field obtained via a memoryless transformation of a truncated Karhunen-Loève expansion, ensuring physical admissibility through positivity of stiffness while preserving the prescribed autocovariance. Polygonal finite elements provide a stable discretization for density-based optimization on unstructured meshes and mitigate checkerboard artefacts and mesh bias, while the sparse polynomial-chaos expansion enables efficient estimation of low-order statistical moments required by the robust objective at a fraction of the cost of intrusive or Monte Carlo approaches. Numerical studies on a cantilever and a curved beam show that introducing non-Gaussian material variability leads to systematic load-path redistribution and a reallocation of 6-12% of the structural volume, together with a reduction in compliance scatter. The non-intrusive surrogate reproduces intrusive reference results within 3% using an order of magnitude fewer full finite-element analyses. These results demonstrate that the proposed framework offers a physically consistent and computationally efficient route to topology-optimized designs that remain reliable under realistic material uncertainty.
1 Introduction
The paper addresses robust topology optimization under spatially varying material uncertainty by combining physically consistent non-Gaussian fields, polygonal finite elements, and non-intrusive polynomial chaos. This integration targets numerical stability, physical fidelity, and computational efficiency.
- Topology optimization must account for material and loading variability because deterministic designs can perform poorly in service.
- Gaussian modeling can produce physically invalid negative Young’s modulus values and cannot represent heavy or bounded material-property tails.
- Polygonal finite elements mitigate checkerboard artefacts and mesh bias in density-based optimization on unstructured meshes.
- The proposed framework couples translation-based non-Gaussian Young’s-modulus fields with polygonal meshes and non-intrusive polynomial-chaos uncertainty propagation.
- The combined framework is characterized by numerical stability, physical fidelity, and computational efficiency.
2 Random fields modeling
The paper represents spatial uncertainty through truncated Karhunen–Loève expansions and memoryless transformations of Gaussian random fields. These constructions support homogeneous non-Gaussian fields with prescribed marginal behavior while retaining computationally manageable spatial structure.
- A random field is a spatially indexed random vector, while Gaussian fields are characterized by their mean and autocovariance functions.
- Non-Gaussian fields require more characterization than first and second moments because those moments do not uniquely determine the field.
- The Karhunen–Loève expansion discretizes a Gaussian random field using covariance eigenvalues, eigenfunctions, and a finite set of random variables.
- Truncation accuracy improves monotonically in mean square as the number of expansion terms increases.
- N = 8 terms accurately approximate an exponential autocovariance function even with a reasonably coarse discretization.
- A memoryless nonlinear translation maps a homogeneous Gaussian field to a homogeneous non-Gaussian field using a prescribed marginal cumulative distribution function.
3 Polynomial chaos surrogate
Polynomial chaos represents random functions with orthogonal polynomial bases and computes coefficients using non-intrusive numerical integration. Sparse grids and Monte Carlo methods are selected according to stochastic dimension to avoid prohibitive tensor-product costs.
- Polynomial chaos expands a function of independent random variables as a weighted sum of multidimensional orthogonal polynomials.
- Practical PCE truncates the random-variable dimension and polynomial order, producing a finite number of polynomial terms.
- PCE coefficients can be computed non-intrusively using Monte Carlo, tensor-product, or sparse-grid quadrature.
- Tensor-product quadrature becomes computationally prohibitive in higher stochastic dimensions because collocation points grow rapidly.
- The paper uses sparse grids for low stochastic dimension and Monte Carlo regression when the problem is high-dimensional.
4 Topology optimization framework
The framework combines density-based topology optimization with stochastic compliance modeling for spatially varying, non-Gaussian material properties. Polygonal finite elements, field-to-mesh interpolation, and a non-intrusive polynomial-chaos surrogate support robust optimization using compliance statistics.
- Deterministic topology optimization: The topology optimization minimizes structural volume subject to an upper compliance limit under static, linear-elastic behavior.The finite-element formulation uses global displacement, loading, and stiffness quantities to solve the governing linear system.
- Deterministic topology optimization: SIMP parameterizes intermediate element densities by penalizing their stiffness while retaining a small compliant material in void regions.Emin prevents ill-posed void regions, while p controls the penalization of intermediate densities.
- Robust topology optimization: The robust formulation treats compliance as stochastic and uses its mean and standard deviation in a weighted objective or constraint framework.The volume is independent of stiffness uncertainty, whereas compliance depends on the material field and random variables.
- Robust topology optimization: A non-intrusive polynomial-chaos surrogate efficiently estimates compliance statistics from surrogate coefficients during robust optimization.Orthonormality enables direct estimators for the mean and standard deviation, while the surrogate propagates uncertain random variables and fields.
- Material uncertainty model: Young’s modulus is modeled as a spatially varying non-Gaussian field generated by a memoryless transformation of a Gaussian field with truncated Karhunen–Loève representation.The correlation structure is described through an autocovariance kernel and its eigenfunctions; uniform, generalized beta, and gamma marginals are used.
- Material uncertainty model: The random-field and finite-element meshes have distinct roles, with centroid-based interpolation transferring Young’s modulus values onto polygonal elements.The structured quadrilateral mesh represents the field, while the unstructured polygonal mesh assembles stiffness, compliance, and sensitivities; polygonal elements help suppress checkerboards and mesh dependency.
5 Results and discussion
The numerical studies compare deterministic and robust topology optimization for cantilever and curved-beam settings under Gaussian and non-Gaussian material uncertainty. Results show that non-Gaussian statistics alter load paths and volume allocation, while polygonal discretization and non-intrusive surrogates support stable, accurate computations.
- Cantilever beam: The cantilever study compares deterministic and robust topologies under Gaussian and non-Gaussian random fields for uncertain Young’s modulus.The robust designs are evaluated against deterministic results and prior literature.
- Cantilever beam: 0.233 is the minimum deterministic volume obtained with 11560 polygonal elements without ad-hoc density filters.The polygonal shape functions produce mesh-stable layouts without checkerboard patterns.
- Cantilever beam: ≤3% agreement in both µC and σC confirms that the non-intrusive polynomial-chaos surrogate reaches intrusive-level accuracy.The comparison uses the log-normal Gaussian-field benchmark in Table 1.
- Cantilever beam: Up to 0.494 versus 0.249 in allocated volume occurs when higher material scatter or shorter correlation length increases robustness requirements.The non-Gaussian cantilever designs add material to secure compliance targets.
- Discussion: Non-Gaussian material statistics primarily induce global load-path redistribution and systematic volume reallocation rather than consistent refinement of structural features.Differences between Gaussian and non-Gaussian designs reflect skewness and boundedness in addition to variance.
- Curved beam: Gamma-field variability is more visible because its nonlinear marginal CDF amplifies fluctuations, while polygonal discretization preserves artefact-free contours without post-processing filters.Uniform and Beta fields appear visually similar for the reported parameter choices.
- Curved beam: 32% lower mean compliance, 7.20 versus 10.56, and 72% lower standard deviation are obtained for the Beta field than the Uniform field at k = 0.The comparison demonstrates sensitivity of robust designs to the selected marginal distribution.
- Discussion: The density-based formulation permits intermediate-density regions rather than enforcing strictly binary layouts.Boundary-based and level-set extensions remain a future direction when strict phase separation is required.
6 Conclusions
The framework combines polygonal finite elements, physically consistent non-Gaussian material variability, and non-intrusive polynomial-chaos surrogates for robust topology optimization. It produces computationally affordable designs whose layouts respond to non-Gaussian uncertainty while remaining stable under discretization changes.
- The framework integrates polygonal finite elements, non-Gaussian material modeling, and non-intrusive polynomial chaos within one topology-optimization loop.The polygonal mesh mitigates checkerboard artefacts and mesh bias, while the random-field model enforces positive Young’s modulus and the sparse surrogate estimates compliance moments efficiently.
- 6–12% of the solid volume is reallocated under non-Gaussian material models, with altered load paths and reduced compliance scatter.These changes arise primarily from load-path redistribution and volume reallocation rather than systematic feature refinement.
- Within a few percent, the surrogate reproduces intrusive reference solutions while leaving the deterministic solver unmodified.The robust loop requires wall-clock effort comparable to a handful of deterministic re-analyses.
- The study is limited to two-dimensional benchmarks and does not yet include manufacturing-oriented constraints or model-form uncertainty.Future work includes large-scale three-dimensional problems, overhang and print-path constraints, and stochastic or Bayesian model-form treatments.
- The combined framework provides a practical, computationally affordable route to topology-optimized designs that remain reliable under realistic material uncertainty.
Funding
The research received financial support from Tecgraf/PUC-Rio, CNPq, CAPES, and FAPERJ.
- The research received support from Tecgraf/PUC-Rio and CNPq grants 317319/2021-3 and 305476/2022-0.
- CAPES supported the research under Finance Code 001, alongside FAPERJ grants 211.037/2019 and 204.477/2024.