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arxiv: 1908.07157 · v2 · submitted 2019-08-20 · ⚛️ physics.app-ph · physics.comp-ph

An efficient evolutionary structural optimization method for multi-resolution designs

Hongxin Wang , Jie Liu , Guilin Wen This is my paper

Pith reviewed 2026-05-24 16:11 UTC · model grok-4.3

classification ⚛️ physics.app-ph physics.comp-ph
keywords topology optimizationBESOXFEMmulti-resolutionlarge-scale optimizationevolutionary structural optimizationadjoint sensitivity analysis
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The pith

A modified BESO algorithm paired with XFEM solves topology optimization problems with millions of design variables by calculating material properties on fine sub-regions while solving equilibrium only on coarse elements.

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

The paper develops an algorithm that combines modified bi-directional evolutionary structural optimization with the extended finite element method to address large-scale topology optimization. Enriched nodes partition each finite element into uniform sub-regions such as sub-triangles or sub-tetrahedrons, where material grids and shape functions are defined for accuracy. The equilibrium equation remains at the coarse finite element level to maintain efficiency, and all standard and enriched nodes act as design variables under a modified material interpolation model. An enrichment function models the transition between solid and void material, with sensitivities obtained via the adjoint method to support the BESO process. Numerical examples with millions of variables confirm that the approach produces optimized designs.

Core claim

The proposed method efficiently solves topology optimization problems involving millions of design variables using a modified BESO combined with XFEM. Within XFEM, a set of enriched nodes are defined to divide the finite element into several uniform sub-regions. The material grid and shape functions are defined on each sub-region to improve the computational accuracy, whereas the equilibrium equation is established on the level of coarse finite elements to increase the computational efficiency. All standard FE nodes and the enriched nodes serve as design variables under a modified material interpolation model, with an enrichment function characterizing the discontinuity between solid andvoid

What carries the argument

XFEM sub-region partitioning via enriched nodes, which computes material properties and interpolation on fine sub-elements while restricting equilibrium solution and global solve to the original coarse mesh, integrated into modified BESO with adjoint sensitivities.

If this is right

  • The algorithm can optimize structures containing millions of design variables without solving the full fine-mesh equilibrium at every iteration.
  • Both two- and three-dimensional problems are handled through the same sub-region construction using sub-triangles and sub-tetrahedrons.
  • Adjoint-based sensitivity analysis supplies the gradients required for the evolutionary BESO update at the scale of the enriched design variables.
  • Typical benchmark problems demonstrate convergence to designs comparable to those obtained by conventional methods but at reduced computational cost.

Where Pith is reading between the lines

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

  • The same coarse-mesh equilibrium plus fine-sub-region material model could be inserted into other gradient-based topology optimizers besides BESO.
  • Engineering fields that already use XFEM for crack or interface problems could reuse the enrichment function to add topology optimization at little extra implementation cost.
  • Parallel solvers for the coarse equilibrium step would further increase the maximum number of design variables that remain tractable.

Load-bearing premise

The material properties and design sensitivities computed from the sub-region interpolation and enrichment function remain accurate enough for the optimization to converge to correct designs even though equilibrium is solved only on the coarse mesh.

What would settle it

A side-by-side run on an identical problem where the final topology from the proposed coarse-equilibrium method differs in load-bearing members or compliance value from a reference optimization performed entirely on a uniformly fine mesh.

read the original abstract

To solve large-scale or high-resolution topology optimization problem, a novel algorithm is developed based on modified bi-directional evolutionary structure optimization (BESO) and extended finite element method (XFEM). Within XFEM, a set of enriched nodes are defined to divide the finite element into several uniform sub-regions, i.e. sub-triangles and sub-tetrahedrons. The material grid and shape functions are defined on each sub-region to improve the computational accuracy, whereas the equilibrium equation is established on the level of coarse finite elements to increase the computational efficiency. We set all the standard FE nodes and the enriched nodes as the design variables, and a modified material interpolation model is introduced to calculate the material properties for sub-regions. An enrichment function originating from modeling voids scheme is adopted to character the discontinuity between solid material to void material. To efficiently use the gradient-based algorithm, BESO, sensitivity analysis is performed with the aid of adjoint method. Typical numerical examples, involving millions of design variables, are carried to verify the effectiveness of the proposed method.

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

2 major / 2 minor

Summary. The paper claims to develop a novel algorithm based on modified bi-directional evolutionary structural optimization (BESO) combined with the extended finite element method (XFEM) to solve large-scale topology optimization problems involving millions of design variables. Enriched nodes divide coarse finite elements into uniform sub-regions (sub-triangles/tetrahedra) where material grids and shape functions are defined; equilibrium is solved only at the coarse-element level while a modified material interpolation model and an enrichment function (from a voids-modeling scheme) characterize solid-void discontinuities. All standard and enriched nodes serve as design variables, adjoint sensitivities are computed for the gradient-based BESO updates, and effectiveness is asserted via typical numerical examples.

Significance. If the accuracy of the coarse-mesh adjoint sensitivities holds, the approach could enable practical multi-resolution topology optimization at scales previously limited by computational cost, by trading fine-mesh equilibrium solves for sub-region enrichment while retaining BESO's evolutionary update rule. The method extends established BESO/XFEM frameworks with concrete modifications for efficiency, and successful validation would directly address scalability in evolutionary structural optimization.

major comments (2)
  1. [Abstract] Abstract: the claim that 'typical numerical examples' verify effectiveness supplies no quantitative results, convergence histories, compliance values, or error metrics. This leaves the central assumption—that material properties and adjoint sensitivities derived from the enrichment function plus modified interpolation on sub-regions remain accurate when equilibrium is solved only on the coarse mesh—unverified against high-fidelity fine-mesh references.
  2. [XFEM formulation and adjoint sensitivity analysis] The description of the XFEM enrichment and sensitivity analysis: the assumption that the enrichment function and modified material interpolation model on sub-triangles/tetrahedra produce sufficiently accurate material properties and sensitivities (when global equilibrium is solved only on coarse elements) is load-bearing for the claim that BESO updates converge to correct designs. No mesh-convergence studies, sensitivity error norms, or benchmark comparisons to standard fine-mesh BESO are reported, so O(1) errors in the adjoint derivatives cannot be ruled out.
minor comments (2)
  1. The parameters of the modified material interpolation model and the coefficients of the enrichment function are introduced as free quantities but are not explicitly listed or subjected to sensitivity analysis; their values should be stated for reproducibility.
  2. [Introduction] Standard references to the foundational BESO algorithm and XFEM enrichment schemes should be added in the introduction to clarify the precise modifications introduced here.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting the need for stronger quantitative verification of the method's accuracy. We address the major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'typical numerical examples' verify effectiveness supplies no quantitative results, convergence histories, compliance values, or error metrics. This leaves the central assumption—that material properties and adjoint sensitivities derived from the enrichment function plus modified interpolation on sub-regions remain accurate when equilibrium is solved only on the coarse mesh—unverified against high-fidelity fine-mesh references.

    Authors: We agree that the abstract and results section would be strengthened by explicit quantitative metrics. The manuscript's examples demonstrate handling of millions of design variables, but direct compliance values, convergence histories, and error metrics versus fine-mesh references are not reported. In revision we will add these, including comparisons to high-fidelity solutions where feasible to verify the coarse-mesh adjoint sensitivities. revision: yes

  2. Referee: [XFEM formulation and adjoint sensitivity analysis] The description of the XFEM enrichment and sensitivity analysis: the assumption that the enrichment function and modified material interpolation model on sub-triangles/tetrahedra produce sufficiently accurate material properties and sensitivities (when global equilibrium is solved only on coarse elements) is load-bearing for the claim that BESO updates converge to correct designs. No mesh-convergence studies, sensitivity error norms, or benchmark comparisons to standard fine-mesh BESO are reported, so O(1) errors in the adjoint derivatives cannot be ruled out.

    Authors: The current validation relies on the obtained topologies in the numerical examples. We acknowledge the absence of explicit mesh-convergence studies, sensitivity error norms, or fine-mesh BESO benchmarks. To address the concern about possible O(1) errors in adjoint derivatives, the revised manuscript will include such studies and direct comparisons to quantify accuracy of the sub-region enrichment and modified interpolation. revision: yes

Circularity Check

0 steps flagged

No circularity; standard extension of BESO/XFEM with independent verification

full rationale

The derivation chain consists of adapting established BESO evolutionary updates and XFEM enrichment to a coarse-mesh equilibrium plus sub-region interpolation. Adjoint sensitivities are computed from the standard variational form; no parameter is fitted to the target performance metric and then re-used as a 'prediction.' No self-citations are invoked as uniqueness theorems or load-bearing premises. Numerical examples serve as external checks rather than tautological confirmation. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that coarse-mesh equilibrium combined with sub-region material properties yields usable optimization sensitivities; the modified interpolation model and enrichment function are introduced without independent evidence of their accuracy beyond the method description itself.

free parameters (2)
  • parameters of the modified material interpolation model
    The abstract states a modified interpolation model is introduced but does not specify whether its coefficients are chosen by hand or fitted.
  • enrichment function coefficients
    An enrichment function is adopted from a voids modeling scheme; any tunable parameters are not detailed.
axioms (1)
  • domain assumption Equilibrium equations solved on coarse elements with sub-region material properties produce accurate adjoint sensitivities for the BESO update.
    This assumption underpins the sensitivity analysis step described in the abstract.

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    To make the design space evolve into a better one, one may increase the number of design variables

    presented the parallel topology optimization to deal with large-scale structural eigenvalue-related design problems. In a recent work, Liu et al. [6] presented a fully parallel parameterized level set method to realize large-scale or high-resolution structural topology optimization design. In their work, the whole optimization process is parallelized, con...

    work page

  2. [2]

    Discretize the design domain using the finite mesh and define the loading and boundary condition

    work page

  3. [3]

    Calculate the stiffness matrix and the volume for each sub-region

    Define a set of enriched nodes and divide the element into several sub-regions. Calculate the stiffness matrix and the volume for each sub-region

    work page

  4. [4]

    Assemble the global stiffness matrix as section 4.2 and perform FE analysis to obtain the nodal sensitivity numbers

    work page

  5. [5]

    (21) and save the resulting sensitivity number for next iteration

    Average the sensitivity number with its history information as Eq. (21) and save the resulting sensitivity number for next iteration. 1 2 k k i i i    (21)

    work page

  6. [6]

    Once the objective volume is reached, the volume 1kV  will be kept constant for the remaining iteration

    Determine the target volume for the next iteration as the regulation 1 (1 )k kV V ER   , where ER is called the evolutionary volume ratio. Once the objective volume is reached, the volume 1kV  will be kept constant for the remaining iteration

    work page

  7. [7]

    Define the threshold of sensitivity numbers for removing and adding nodes th del and th add

    Update the nodal density. Define the threshold of sensitivity numbers for removing and adding nodes th del and th add . For solid nodes, it’s nodal density will be switched to 0 if th i del  . On the contrary, the nodal density will be switched to 1 if th i add  for void nodes. Project the gray sub-regions into solid or void material by Heaviside function

    work page

  8. [8]

    (22) is satisfied

    Go back to step 3 until the objective volume is reached and the convergence criterion Eq. (22) is satisfied. 1 1 1 1 1 ( ) M k i k M i i M k i i C C change C                (22) where  is the teeny allowable convergence criterion, and M is the selected iteration number that are expected with stable compliance. 5 Filter scheme As we know, t...

    work page 2001

  9. [9]

    Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 1988;71(2):197-224

    Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 1988;71(2):197-224

    work page 1988

  10. [10]

    W., & Stucker, B

    Gibson, I., Rosen, D. W., & Stucker, B. Additive manufacturing technologies (Vol. 17). New York: Springer, 2004

    work page 2004

  11. [11]

    On the (non-)optimality of Michell structures

    Sigmund O, Aage N, Andreassen E. On the (non-)optimality of Michell structures. Structural and Multidisciplinary Optimization 2016;54:361–73

    work page 2016

  12. [12]

    Large-scale topology optimization in 3D using parallel computing

    Borrvall T, Petersson J. Large-scale topology optimization in 3D using parallel computing. Computer Methods in Applied Mechanics and Engineering 2001;190(46-47):6201-6229

    work page 2001

  13. [13]

    Parallelized structural topology optimization for eigenvalue problems

    Kim TS, Kim JE, Kim YY. Parallelized structural topology optimization for eigenvalue problems. International Journal of Solids and Structures 2004;41(9–10):2623–2641

    work page 2004

  14. [14]

    Fully parallel level set method for large-scale structural topology optimization

    Liu H, Tian Y, Zong H, Ma Q, Wang M, Zhang L. Fully parallel level set method for large-scale structural topology optimization. Computers and Structures 2019;221:13–27

    work page 2019

  15. [15]

    Megapixel topology optimization on a graphics processing unit

    Wadbro E, Berggren M. Megapixel topology optimization on a graphics processing unit. SIAM Review 2009;51(4):707–21

    work page 2009

  16. [16]

    Efficient topology optimization using GPU computing with multilevel granularity

    Martínez-Frutos J, Martínez-Castejón PJ, Herrero-Pérez D. Efficient topology optimization using GPU computing with multilevel granularity. Advances in Engineering Software 2017;106:47–62

    work page 2017

  17. [17]

    Evolutionary and GPU computing for topology optimization of structures

    Ram L, Sharma D. Evolutionary and GPU computing for topology optimization of structures. Swarm and Evolutionary Computation 2017;35:1-13

    work page 2017

  18. [18]

    A novel Multi-Grid assisted reanalysis for re-meshed finite element models

    Huang G, Wang H, Li G. A novel Multi-Grid assisted reanalysis for re-meshed finite element models. Computer Methods in Applied Mechanics and Engineering 2017;313:817-833

    work page 2017

  19. [19]

    Reanalysis method for second derivatives of static displacement

    Zuo, W., Fang, J., & Feng, Z. Reanalysis method for second derivatives of static displacement. International Journal of Computational Methods. DOI: 10.1142/S0219876219500567

    work page doi:10.1142/s0219876219500567

  20. [20]

    Approximate reanalysis in topology optimization

    Amir O, Bendsøe MP, Sigmund O. Approximate reanalysis in topology optimization. International Journal for Numerical Methods in Engineering 2009;78(12):1474-1491

    work page 2009

  21. [21]

    A novel minimum weight formulation of topology optimization implemented with reanalysis approach

    Long K, Gu C, Wang X, Liu J, Du Y, Chen Z, Saeed N. A novel minimum weight formulation of topology optimization implemented with reanalysis approach. International Journal for Numerical Methods in Engineering DOI: 10.1002/nme.6148

    work page doi:10.1002/nme.6148

  22. [22]

    Structural modal reanalysis for topological modifications of finite element systems

    Huang C, Chen S, Liu Z. Structural modal reanalysis for topological modifications of finite element systems. Engineering Structures 2000;22(4):304-310

    work page 2000

  23. [23]

    Robust topology optimization considering load uncertainty based on a semi-analytical method

    Zheng, Y., Gao, L., Xiao, M., Li, H., & Luo, Z. Robust topology optimization considering load uncertainty based on a semi-analytical method. The International Journal of Advanced Manufacturing Technology 2018; 94(9-12): 3537-3551

    work page 2018

  24. [24]

    Design and experimental validation of self-supporting topologies for additive manufacturing

    Fu Y, Rolfe B, Chiu L, Wang Y, Huang X, Ghabraie K. Design and experimental validation of self-supporting topologies for additive manufacturing. Virtual and Physical Prototyping 2019;14(4), 382-394

    work page 2019

  25. [25]

    An efficient evolutionary structural optimization method with smooth edges based on the game of building blocks

    Wang H, Liu J, Wen G. An efficient evolutionary structural optimization method with smooth edges based on the game of building blocks. Engineering Optimization DOI: 10.1080/0305215X.2018.1562550

    work page doi:10.1080/0305215x.2018.1562550 2018

  26. [26]

    The partition of unity finite element method: basic theory and applications

    Melenk JM, BabusKa I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996;139(1-4): 289-314

    work page 1996

  27. [27]

    Fronts propagating with curvature dependent speed: algorithms based on Hamilton-jacobi formulations

    Osher S. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-jacobi formulations. Journal of Computational Physics 1988;79

    work page 1988

  28. [28]

    Xiao SP, Parimi C

    Belytschko T. Xiao SP, Parimi C. Topology optimization with implicit functions and regularization. International Journal for Numerical Methods in Engineering 2003;57(8):1177-1196

    work page 2003

  29. [29]

    A study on XFEM in continuum structural optimization using a level set model

    Wei P, Wang M, Xing X. A study on XFEM in continuum structural optimization using a level set model. Computer-Aided Design 2010;42(8):708-719

    work page 2010

  30. [30]

    Evolutionary topology optimization using the extended finite element method and isolines

    Abdi M, Wildman R, Ashcroft I. Evolutionary topology optimization using the extended finite element method and isolines. Engineering Optimization 2014;46(5):628-647

    work page 2014

  31. [31]

    Design space optimization using a numerical design continuation method

    Kim IY, Kwak BM. Design space optimization using a numerical design continuation method. International Journal for Numerical Methods in Engineering 2002;53(8):1979-2002

    work page 2002

  32. [32]

    M., & Steven, G

    Xie, Y. M., & Steven, G. P. A simple evolutionary procedure for structural optimization. Computers & Structures 1993; 49(5): 885-896

    work page 1993

  33. [33]

    M., Steven, G

    Querin, O. M., Steven, G. P., & Xie, Y. M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Engineering Computations 1998; 15(8): 1031-1048

    work page 1998

  34. [34]

    Huang, X., & Xie, Y. M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design 2007; 43(14): 1039-1049

    work page 2007

  35. [35]

    Liu, J., Wen, G., & Xie, Y. M. Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model. Structural and Multidisciplinary Optimization 2016; 53(1): 81-100

    work page 2016

  36. [36]

    Modeling holes and inclusions by level sets in the extended finite-element method

    Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extended finite-element method. Computer Methods in Applied Mechanics and Engineering 2001;190:6183-200

    work page 2001

  37. [37]

    A finite element method for crack growth without remeshing

    Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999;46:131-50

    work page 1999

  38. [38]

    3D corrected XFEM approach and extension to finite deformation theory

    Loehnert S, Mueller-Hoeppe DS, Wriggers P. 3D corrected XFEM approach and extension to finite deformation theory. International Journal for Numerical Methods in Engineering 2011; 86(4-5):431-452

    work page 2011

  39. [39]

    Multi-material topology optimization considering interface behavior via XFEM and level set method

    Liu P, Luo Y, Kang Z. Multi-material topology optimization considering interface behavior via XFEM and level set method. Computer Methods in Applied Mechanics and Engineering 2016;245-246:75-89

    work page 2016

  40. [40]

    Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima

    Sigmund O, Petersson J. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural and Multidisciplinary Optimization 1998;16(1):68-75

    work page 1998

  41. [41]

    Doing topology optimization explicitly and geometrically—a new moving morphable components based framework

    Guo, X., Zhang, W., & Zhong, W. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. Journal of Applied Mechanics 2014; 81(8): 081009

    work page 2014

  42. [42]

    Explicit three dimensional topology optimization via Moving Morphable Void (MMV) approach

    Zhang, W., Chen, J., Zhu, X., Zhou, J., Xue, D., Lei, X., & Guo, X. Explicit three dimensional topology optimization via Moving Morphable Void (MMV) approach. Computer Methods in Applied Mechanics and Engineering 2017; 322: 590-614

    work page 2017

  43. [43]

    N., & Hu, P

    Wang, X., Long, K., Hoang, V. N., & Hu, P. An explicit optimization model for integrated layout design of planar multi-component systems using moving morphable bars. Computer Methods in Applied Mechanics and Engineering 2018; 342: 46-70

    work page 2018