Structural topology optimization (TO) aims to find the optimal material distribution within a given design domain such that an objective function is minimized and constraints are satisfied [1]. Compared with size optimization and shape optimization, TO offers a greatly enlarged design space and consequently better design solutions. It is, however, the most challenging, as it deals with the creation of new geometries from scratch.

Most existing topology optimization (TO) methods implicitly represent structures using a fixed grid of pixels or voxels and employ the widespread finite element method (FEM) to evaluate objective functions and constraints. Despite their broad applicability, these grid-based, FEM-reliant approaches face certain limitations. For instance, the low-order continuity between neighboring finite elements [1]—illustrated in Fig. 1a—can significantly reduce computational accuracy, especially for stress-related metrics. This makes such methods less suitable for stress-constrained TO design. To address this, some studies have proposed using adaptive mesh refinement to improve the smoothness of structure boundaries and enhance stress calculations [2–4]. The main idea is to refine the mesh in regions where higher accuracy is needed and coarsen it elsewhere to improve computational efficiency. However, even with a refined mesh, the design is still represented as a pixel-based image with non-smooth boundaries, as shown in Fig. 1b. Beyond the issue of stress calculation, optimized designs often require additional post-processing to be compatible with computer-aided design (CAD) software for manufacturing, which can further compromise design quality.

To overcome the limitations of traditional FEM, Hughes and his co-workers [5,6] introduced the isogeometric analysis (IGA) framework, which unifies CAD and computer-aided engineering (CAE) by employing the same spline basis functions for both geometry representation and numerical analysis. This seamless integration eliminates the need for mesh generation and offers higher continuity and numerical accuracy compared to conventional FEM. All these advantages make IGA particularly advantageous for TO and form a new paradigm: isogeometric topology optimization (ITO). In particular, for stress-constrained design problems, the higher-order continuity of spline basis functions significantly enhances the accuracy of both structural representation and stress evaluation.

Significant efforts have been made to extend classical TO methods by incorporating IGA [7–9]. Representative examples include its integration with level-set [10], evolution method [11], and moving morphable component [12] approaches. Moreover, IGA has been combined with the boundary element method to tackle optimization problems in acoustics [13–15]. The density-based TO methods can be integrated with IGA by assigning design variables to the control points of the Non-uniform Rational B-splines (NURBS) basis function [16–19]. Stress-related ITO problems, typically believed to be more challenging due to factors such as the non-linear behavior of stress and the localized nature of stress constraints [20], have also been studied in recent years. For example, Kazemi et al. [21] proposed an ITO method for minimizing structural weight under local stress constraints using NURBS-based IGA, with control points’ density as design variables. The optimization process utilizes direct sensitivity analysis combined with the Method of Moving Asymptotes (MMA) for iterative updates [22]. To handle stress singularities, the ε relaxation method is employed. However, the optimized structures exhibit insufficient smoothness, likely due to the limited number of NURBS patches. Additionally, the reliance on direct sensitivity analysis (without using adjoint method) reduces efficiency for large-scale problems. Jahangiry and Tavakkoli [23] developed a level-set-based ITO method. The design variables are the control points of the NURBS-based level set function, which are iteratively updated by solving the Hamilton-Jacobi equation using the forward Euler scheme. Numerical examples, including weight minimization under local stress constraints and the simultaneous minimization of weight and strain energy, demonstrate the method’s capability to achieve reasonable results with smooth boundaries. Building on Gao’s work [17], Liu et al. [24] extended the SIMP-based IGA to tackle global stress-constrained TO problems. Density values at NURBS control points are used to construct the density distribution function, with a filter applied to ensure smoothness. The optimization problem is solved using MMA, while a global p-norm stress aggregation function constrains the stress during volume minimization. Numerical examples confirm that this approach can deliver smooth geometry representations and reliable stress analysis. To improve the optimization efficiency of Liu’s SIMP-based IGA work, Zhai [25] proposed a hybrid optimizer combining the alternating direction method of multipliers (ADMM) with MMA. Different from minimizing weight, Villalba et al. [26] focused on stress-constrained compliant mechanisms, using quadratic B-splines for material layout and structural analysis. By incorporating an overweight constraint to manage stress, their method achieves a 40%–60% reduction in computational time compared to classical FEM. In the monograph of Gao et al. [27], they studied the volume-constrained stress minimization problems in the ITO framework. Two global stress aggregation formulations (p-norm and Kreisselmeier-Steinhauser) are utilized to measure stress, which effectively eliminates boundary stress concentration issues. Vo et al. [28] integrated a gradient-free proportional topology optimization scheme into the isogeometric analysis framework, where NURBS basis functions simultaneously represent geometry, displacement, and material density. The approach defines densities at integration points in proportion to compliance, derives their relation to control points, and constructs a global NURBS-based density description.

Despite the significant development of ITO, the efficiency challenge remains and in fact becomes even more critical in large-scale stress-constrained TO designs. Compared with compliance design, stress-constrained TO is significantly more complex, often requiring many more iterations and longer convergence times. To address this, we extend a previously developed multi-grid, single-mesh online learning framework [29]—originally designed for compliance-based TO—to the ITO setting. This extension leverages the ITO framework’s inherent capacity for smooth geometric representation, enabling more accurate stress calculations.

The use of artificial neural network (ANN)-based surrogate models to accelerate topology optimization (TO) is an active field of research [30–34]. While various methods have been proposed, they often require large, expensive-to-generate training datasets. For applications involving only a few large-scale design problems, generating a large training dataset certainly is not efficient and sometimes even infeasible. For such cases, online learning presents another promising approach [35]. Unlike offline methods that rely on pre-collected datasets, online learning collects training data from designs and analysis results obtained during previous TO iterations. This eliminates the need for extensive upfront data generation and allows the neural networks to adapt specifically to the problem at hand, making online learning particularly well-suited for large-scale TO problems. To ensure accuracy, ANN models are continuously updated throughout the TO process as new training data is available. This iterative refinement ensures that the surrogate models remain closely aligned with the evolving design space, improving both computational efficiency and solution quality.

Online learning framework for TO was first proposed in [35] for compliance design. The method employs a localized patch-based prediction strategy to greatly enhance the scalability of the method and training data diversity. However, it neglects the structural information outside each local region when predicting patch responses. This may lead to an undetermined mapping and suboptimal prediction accuracy. Additionally, the coarse-fine two-grid mapping technique employed introduces additional computational overhead associated with generating coarse grid solutions. To overcome these limitations, our group recently developed a multigrid, single-mesh online learning TO method [29], which has demonstrated a better performance. In the present work, we further extend this advanced framework to stress-constrained design problems and validate its efficacy through a series of illustrative examples.

This section outlines the TO formulation of stress-constrained design problems based on IGA. The NURBS-based density distribution function is used to formulate the ITO approach, with IGA applied to compute structural responses. The Method of Moving Asymptotes (MMA) is employed to update design variables.

Nodal densities ρi,j associated with control points are defined as the design variables, which are evolved during optimization. A Shepard function [17] is utilized to filter the nodal densities, ensuring overall smoothness. The smoothed nodal density 𝒢i,j is expressed as: (12)𝒢i,j=∑l=1𝒩∑h=1ℳψ(ρl,h)ρi,j,where 𝒩 and ℳ are the numbers of control points within the local support area of the current control point. The Shepard function ψ(ρl,h) is defined as: (13)ψ(ρl,h)=w(ρl,h)∑l∗=1𝒩∑h∗=1ℳw(ρl∗,h∗),where w(ρl,h) is the weight function for the (l,h) nodal density, constructed using compactly supported radial basis functions [37] with C4 continuity: (14)w(ρl,h)=max(0,(1−r))6(35r2+18r+3),where r=d/dm. d is the Euclidean distance between the current (l,h) control point and another (i,j) control point. dm is the radius of the local support domain.

After filtering, a projection function [38] is applied to reduce intermediate density values: (15)𝓂(𝒢i,j)=tanh⁡(βη)+tanh⁡(β(𝒢i,j−η))tanh⁡(βη)+tanh⁡(β(1−η)),where η is the threshold (default: 0.5) and β is the shape control parameter.

Based on the projection function, m˙(𝒢i,j), the sensitivity of the projection function, is given by: (16)m˙(𝒢i,j)=∂𝓂(𝒢i,j)∂𝒢i,j=βsech(β(𝒢i,j−η))2tanh⁡(βη)+tanh⁡(β(1−η)).

In isogeometric analysis, the knot vector discretizes the design domain into IGA elements. For the stress-constrained topology optimization problem studied in this work, a constant density distribution within each IGA element is employed for problem formulation and sensitivity calculations. The density at the centroid of an IGA element is chosen as the elemental relative density, ρe, which is calculated as: (17)ρe=∑i=1n∑j=1mRi,jp,q(ξec,ηec)𝓂(𝒢i,j),where (ξec,ηec) are the parametric coordinates of the eth element centroid. Based on the IGA discretization, the volume minimization problem with a maximum von Mises stress constraint is formulated as: (18){infρ:J(ρ)=∑e=1Neρeves.t.:G(𝒳)=max(σavm)−σlim≤0 for a=1,2,…,Na,where J represents the structural volume. ve is the IGA element volume, and Ne is the number of IGA elements. σavm is the von Mises stress and σlim is the prescribed stress limit. Na is the total number of evaluation points. Typically, 2 × 2 Gauss points are employed for quadratic IGA elements. The elemental constitutive matrix is derived using a power-law relationship, De=ρeγD0, which is used to obtain the stiffness matrix.

The von Mises stress at a stress evaluation point a is expressed as: (19)σavm=(σaTVσa)0.5,where σa=ρesD0Baue is the stress vector. The parameter s is set to a recommended value of 0.5 [39]. Ba is the strain-displacemnet matrix. ue is the elemental displacement vector. V is the constant matrix defined as: (20)V=[1−0.50−0.510003].

To alleviate the difficulty in the optimization caused by local stress constraint, a p-norm global stress measure is employed to approximate the local maximum stress: (21)G(𝒳)=c⋅σpn−σlim≤0,where σpn=(∑a=1Na(σavm)p)−p. Na is the total number of evaluation points. The scaling coefficient c is updated at each iteration as cn=max(σa,nvm)/σnpn to minimize the gap between the p-norm stress σnpn and the maximum stress, where subscript n refers to the iteration number.

Based on the adjoint method detailed in the next subsection, the sensitivity of the objective and the constraint are obtained and summarized as: (22){∂J∂ρi,j=∑e∈Ii,jRi,jp,q⁡(ξec,ηec)⁢m˙⁡(Gi,j)⁢ψ⁡(ρi,j)⁢ve∂G∂ρi,j=c⁡∑a∈Ji,j(σp⁢n)1−p⋅(σav⁢m)p−2⁢VT⁢σa⁢s⁢ρes−1⁢Ri,jp,q⁡(ξec,ηec)⁢m˙⁡(Gi,j)⁢ψ⁡(ρi,j)⁢D0⁢Ba⁢ue−c⁡(λT⁢∂K∂ρi,j⁢u),where Ii,j is the set of elements connected to the control point Pi,j. Ji,j is the set of stress evaluation points associated with Pi,j. The adjoint vector λ is calculated by solving the following equation: (23)Kλ=∑a=1Na(σpn)1−p⋅(σavm)p−2(ρesD0Ba)TVTσa.

The term ∂K/∂ρi,j in Eq. (22) is composed of the derivative of element stiffness matrix w.r.t. the design variable: (24)∂Ke∂ρi,j=∫Ωe.BTγρeγ−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0BdΩe.

After obtaining the sensitivity, the MMA optimizer [22] is used to update the control density ρi,j. To improve the numerical stability of the topology optimization process, several schemes are employed. The coefficient c is averaged over the last 4 iterations, which means at the nth iteration, it is calculated as cn=0.25(cn+cn−1+cn−2+cn−3). The design variable ρ is also updated based on the average value: ρn=0.5(ρn+ρn−1). In the power law relationship to obtain the elemental constitutive matrix, γ starts at 0.8 and increases by 0.2 per iteration until reaching a maximum of 3.

Sensitivity Analysis

Based on the IGA discretization, the volume minimization problem with a maximum von Mises stress constraint is formulated in Eq. (18). The sensitivity of the objective function J and constraint G to the design variable ρi,j are detailed in this subsection.

First, the derivative of objective J w.r.t. the ρi,j can be expressed as: (25)∂J∂ρi,j=∑e=1Ne∂ρe∂ρi,jve.

Since the density at the centroid of an IGA element is chosen as the elemental relative density, ρe, which is given by: (26)ρe=∑i=1n∑j=1mRi,jp,q(ξec,ηec)m(𝒢i,j),where (ξec,ηec) is the parametric coordinate of the eth element centroid. Substituting Eqs. (26) into (25), we derive: (27)∂J∂ρi,j=∑e∈Ii,jRi,jp,q(ξec,ηec)∂m(𝒢i,j)∂ρi,jve,where Ii,j is the set of elements connected to the control point Pi,j. Incorporating the formulation of projection function in Eq. (15) and filter function in Eq. (13), the (27) becomes: (28)∂J∂ρi,j=∑e∈Ii,jRi,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)ve,where m˙(𝒢i,j), the sensitivity of the projection function, is given by Eq. (16).

Next, the derivative of objective G w.r.t. the ρi,j is derived. Since we use p-norm stress σ~pn=c⋅σpn to approximate the max(σavm), we have: (29)∂G∂ρi,j=c∑a=1Na∂σpn∂σavm(∂σavm∂σa)T∂σa∂ρi,j.

The intermediate terms are as follows: (30)∂σpn∂σavm=(σpn)1−p⋅(σavm)p−1. (31)∂σavm∂σa=1σavmσaTV.

Since σa=ρesD0Baue, the last term ∂σa/∂ρi,j in Eq. (29) becomes: (32)∂σa∂ρi,j=∂ρes∂ρi,jD0Baue+ρesD0Ba∂u∂ρi,j=sρes−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0Baue+ρesD0Ba∂u∂ρi,j.

Substituting Eqs. (32) into (29) yields: (33)∂G∂ρi,j=c∑a=1Na∂σpn∂σavm(∂σavm∂σa)T(sρes−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0Baue+ρesD0Ba∂u∂ρi,j).

Since external load is a constant, ∂f/∂ρi,j=0. By differentiating Ku=f, we have: (34)∂K∂ρi,ju+K∂u∂ρi,j=0.

Multiplying Eq. (34) by λT and substituting into Eq. (33), we obtain: (35)∂G∂ρi,j=c∑a=1Na∂σpn∂σavm(∂σavm∂σa)T(sρes−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0Baue+ρesD0Ba∂u∂ρi,j)−λT∂K∂ρi,ju−λTK∂u∂ρi,j.

Collecting all terms related to ∂u/∂ρi,j and let it equal to 0, we derive the adjoint equation: (36)Kλ=∑a=1Na∂σpn∂σavm(ρesD0Ba)T∂σavm∂σa=∑a=1Na(σpn)1−p⋅(σavm)p−2(ρesD0Ba)TVTσa.

Solving Eq. (36) and substituting λ back to Eq. (35) gives the final expression: (37)∂G∂ρi,j=c∑a∈Ji,j∂σpn∂σavm(∂σavm∂σa)Tsρes−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0Baue−cλT∂K∂ρi,ju=c∑a∈Ji,j(σpn)1−p⋅(σavm)p−2VTσasρes−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0Baue−cλT∂K∂ρi,ju,where Ji,j is the set of stress evaluation points associated with Pi,j. The term ∂K/∂ρi,j in Eq. (37) is composed of the derivative of element stiffness matrix w.r.t. the design variable: (38)∂Ke∂ρi,j=∫Ωe.BTγρeγ−1Ri,jp,q(ξec,ηec)m˙(𝒢i,j)ψ(ρi,j)D0BdΩe.

The most time-consuming step in the IGA process is the calculation of the sensitivities ∂G/∂ρi,j, which depend on state variables. To speedup this calculation, the proposed online learning framework based on FNO model is employed to perform this sensitivity calculation. In this framework, design variables are the control densities, and an FNO surrogate model is constructed to predict sensitivity fields in selected iterations, replacing the conventional IGA calculations. The training data is collected dynamically using the design results obtained from previous TO iterations, and the FNO model is continuously updated using the newly generated data during the TO process.

The flowchart of the online learning process is shown in Fig. 3. During the initialization phase, the traditional TO process based on isogeometric analysis (TO-IGA) is conducted for n0 iterations. It is followed by another n1 TO-IGA iterations. The designs and analysis results generated during these n1 iterations are saved and used to train the initial FNO model. This initial FNO model is then integrated with TO process (TO-FNO) and optimization is carried out for n2 iterations. If the design is unsatisfactory, another n1 TO-IGA iterations are executed, and results generated from this process are collected. The optimized design at the last iteration is then evaluated. If it is still unsatisfactory, the FNO model is re-trained using the newly collected data, and the TO-FNO process continues for another n2 iterations. This alternative process between TO-IGA and TO-FNO continues until the design meets the specified objectives and constraints.

Two L-bracket design problems with different load positions, with their problem setups illustrated in Fig. 7, are used as the benchmark examples to validate the effectiveness of the online learning framework. The optimization parameters are set as follows. The filter radius is 4, which means the adjacent 4 control densities are used for smoothing. The default norm value for the p-norm is 8. In this work, a uniformly distributed load t is applied. To compute the load assigned to the control points, numerical integration is performed using five Gaussian quadrature points [17]. For example, in Fig. 7a, the load position in parametric space is ξ=[0.95,1.0],η=1.0. Load applied on control points (i,j) is calculated as: (41)Fi,j=∫0.951.0Ri,j(ξ,η)t(ξ,η)‖dS(ξ,η)dξ‖dξ≈∑k=1nRi,j(ξk)t(ξk)‖dS(ξk)dξ‖wkwhere t is the load per unit length, ξk and wk are the kth Gauss points location and weight within the interval [0.95, 1.0]. Each design problem considered a total load of 2.0 N, and the stress constraint limit is set at 25 Pa.

We also consider another design domain of quarter annulus, illustrated as Fig. 8. A total load of 2.0 N, and the stress constraint limit is set at 40 Pa.

All numerical experiments are conducted on a Windows 10 Pro operating system with an i7-11700 CPU, 16 GB RAM, and an NVIDIA 3060 GPU with 12 GB RAM. The implementation was done in MATLAB 2020b, with the online learning component executed using Python 3.8 and PyTorch 1.10.2 framework. In the FNO model, the number of modes kept after the Fourier transform is 12. The channel width is 64 after the lifting layer.

This work presents an online learning framework to improve efficiency of isogeometric topology optimization, where the FNO surrogate is employed to predict sensitivity fields. A multi-grid decomposition strategy is employed, enabling the augmentation of training data and embedding global information across different levels in the input. This strategy enhances framework scalability and FNO prediction accuracy. The results of numerical experiments demonstrate the effectiveness of the proposed framework across two stress constrained design problems. By incorporating FNO as surrogate models, the framework significantly reduces the number of IGA evaluations and sensitivity calculations, leading to substantial computational savings while maintaining results comparable to ground-truth IGA solutions in large-scale examples. Reducing the usage ratio of IGA updates can further improve efficiency; however, this also reduces the amount of available data, which may negatively impact the accuracy of FNO predictions. Therefore, there’s a tradeoff between efficiency and accuracy. Overlapping strategy can help increase the training dataset size, which can enhance the FNO’s prediction accuracy.

There remain multiple promising directions for future research. For example, future efforts could focus on dynamically adjusting the FNO usage ratio during inference, possibly guided by design change indicators. Additionally, while the framework is validated on medium-scale problems, its scalability to giga-scale and 3D stress-constrained designs remain undemonstrated. Future work will include such large-scale 3D stress-constrained design problems and extensions to other application problems, such as acoustic wave analysis.