This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures. These structures, commonly encountered in engineering applications, often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables. As a result, sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges. Over the past several decades, topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds. In comparison to the large-scale equation solution, sensitivity analysis, graphics post-processing, etc., the progress of the sequential approximation functions and their corresponding optimizers make sluggish progress. Researchers, particularly novices, pay special attention to their difficulties with a particular problem. Thus, this paper provides an overview of sequential approximation functions, related literature on topology optimization methods, and their applications. Starting from optimality criteria and sequential linear programming, the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables. In addition, recent advancements have led to the emergence of approaches such as Augmented Lagrange, sequential approximate integer, and non-gradient approximation are also introduced. By highlighting real-world applications and case studies, the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area. Furthermore, to provide a comprehensive overview, this paper offers several novel developments that aim to illuminate potential directions for future research.

In a seminal work published in 1988, Bendsøe et al. introduced the homogenization theory into structural design and proposed the so-called homogenization methodology for topology optimization of continuum structures [

A multitude of design variable update schemes are now available in the topology optimization community [

Let us consider the general topology optimization problem with an objective function g_{0} total number of J constraint functions _{j}_{i}

Assuming linear elasticity for continua, the equilibrium equation can be stated as

By writing the displacements as functions of the design variables via solving equilibrium equations, we acquire the nested formulation of the optimization problem as follows:

In the following section, we will introduce the most popular optimization solvers in topology optimization, such as optimality criteria (OC), sequential linear programming (SLP), convex linearization (CONLIN), method of moving asymptotes (MMA), sequential quadratic programming (SQP), and others. Topology optimization of continuum structure process explained through a flowchart as shown in

Step 1: Define the design area for optimization

Step 2: Determine the design model-related parameters for topology optimization

Step 3: Implement the interpolation scheme for SIMP (Solid isotropic material with penalization) to predict material distribution within space.

Step 4: Perform sequential approximation (OC, SLP, CONLIN, SQP, MMA) for continuum structure optimization.

Step 5: Determine whether the convergence condition is satisfied, if satisfied then stop the iteration otherwise return to Step 4.

The variable density method, especially with the solid isotropic material with penalization (SIMP) interpolation, is undoubtedly one of the most renowned approaches due to its simple concept. The penalization scheme assumes a relationship between the elemental Young’s modulus _{i}_{0}

The term _{i}

In the density-filter approach [_{min} has a similar function to the parameter _{0}/10^{9}_{i}

In the presented OC algorithm, the Lagrange multiplier associated with a constraint is typically obtained via bisection search. Kumar and Suresh proposed a direct manner for the Lagrange multiplier, which exhibits several benefits including, fewer iterations, robust convergence, and insensitivity to the given material and load [

The intuition-based OC scheme permits efficient solutions of the computationally demanding problems in a relatively low number of iterations, especially for the compliance minimization issue with a single volume restriction. As all gradients have negative indicators, removing material will always increase compliance. However, conservative variable update by adjusting parameters allows the OC scheme to be applied to the flexible mechanism design and material design issues with material usage limits [

For optimization problem ^{(k)}

_{0}, _{j}

At present, solving topology optimization problems with SLP is uncommon. It is not surprising because the accuracy of this approximation is inferior to those that will follow in this paper. In the composite optimization problem, manufacturing constraints take the form of linear constraints, which makes the SLP algorithm more efficient at finding a solution [

Even for seasoned researchers, sequential structural approximation using reciprocal variables can be somewhat confounding. This is demonstrated by introducing an intervening variable

The partial derivative of _{i}

When we choose the equations _{i}_{i}_{i}_{i}

The

Fleury defined the approximation of ^{(k)}^{L}^{R}

In

The COLIN is also known as a conservative approximation [^{+}^{−}

Thus, the sub-problem using CONLIN approximation can be reformulated:

Based on inequality

The CONLIN was furthered by multiple researchers. For example, Zhang et al. proposed a modified CONLIN approximation, which strengthens the convexity of the problem by introducing a convex factor [^{TM}, even as a milestone of topology optimization in this code [^{TM}.

_{1 }= 1_{2 }= 6

By differentiating _{1}_{2}^{C}^{L}^{R}^{C}^{L}^{R}

Despite the fact that COLIN has demonstrated its efficacy for a variety of structural optimization problems, it occasionally converges slowly due to excessively conservative approximations. In contrast, it does not converge at all, indicating that it is insufficiently conservative. To stabilize the optimization process, Svanberg developed a variant of COLIN by constructing artificial asymptotes [_{i}_{i}

In MMA, the approximating function at ^{(k)}

Thus, the approximate structural optimization problem using MMA can be rewritten:

For the SIMP method, the MMA algorithm is widely regarded as one of the most dependable and efficient optimizers [

^{4}.

We can see that as the upper asymptote approaches infinity from

In addition, the MMA algorithm is also used as the optimizer in various topology optimization methods, such as the stiffness spreading method [^{TM} is the basis of the MMA algorithm. ^{TM}.

The preceding approximation function is characterized by the first order approximation and makes use of current data. On the basis of previous optimization iterations, it is anticipated that more precise approximations over a broader range can be attained.

Typically, Fadel et al. proposed a two-point approximation with the intervening variables [

According to

In _{i}

It is not surprising that several two-point or three-point approximation functions were proposed to enhance approximate accuracy and expand the approximate range, the majority of which were numerically tested by mathematical problems and truss optimization problems [

When the second-order term is appended in the Taylor expansion of the objective function in SLP, the following SQP-based approximation occurs:

One way to construct the SQP is the utilization of Newton’s method to find the stationary point of the Lagrangian function:
_{j}_{j}^{(k)}

The complete Hessian matrix

The computation and storage of Hessian matrix _{i}

In certain practical situations, the second-order sensitivity information is undesirable. To overcome these difficulties, Grovenwold et al. proposed an incomplete series expansion (ISE) in which the approximate Hessian matrix

TopSQP is an efficient second-order SQP algorithm developed by Rojas-Labanda and Stople for structural topology optimization [

Rojas-Labanda and Stople conducted a comprehensive benchmark of topology optimization problems in conjunction with various optimizers, such as OC, MMA, and SQP. They concluded that the second-order information aids in obtaining accurate results and that SQP outperforms all other solutions for classical benchmark solvers [

Recently, Zhang et al. [

As long as the

Generally, the SQP with approximate Hessian has garnered the interest of numerous academicians in structural optimization. No matter how the approximate methodology differs, one common pursuit is to obtain faster convergence at the lower calculation cost of sensitivity information.

Different from the aforementioned SQP family algorithms constructed second-order information based on mathematical programming, Sui et al. proposed a novel formulation, also known as the independent continuous mapping (ICM) method in 1996 [_{i}_{E}_{i}_{v}_{i}

A typical formulation for the function in

The design variables _{i}_{E}_{i}

When

The elemental volume can be rewritten as follows:

According to _{i}

In contrast to prevalent density methods, the ICM method focuses on minimizing the total volume or weight while maintaining constraints on various structural responses. Taking the compliance constraint as an example, the topology optimization formulated can be mathematically stated as follows:

The volume function and compliance function can therefore be expressed by first-order and second-order Taylor expansion series. The original topology optimization problem can be converted as a quadratic program with second-order information, which provides another distinguishing feature over the widely used first-order method. From

Since the ICM approach was proposed, its applications have undergone tremendous developments with the efforts of their groups, also propelling the industry forward at the breakneck speed [

The ICM method aroused the attention of other scholars. Long et al. extended this method into the framework of meshless analysis [

In recent years, the AL method has emerged as a viable approach to topology optimization, especially for extensive constraints. The AL method addresses constraints directly by appending them to the objective function as a penalty term with variable parameters. AL is not a novel concept in the field of structural optimization. For instance, the parameterized level set method has been successfully implemented to enforce a sole volume constraint [

Utilizing three phase projections including eroded, intermediate, and dilated, da Silva et al. employed the AL function for stress-constrained topology optimization problems while accounting for manufacturing uncertainties [

BESO is the predominant discrete variable-based topology optimization approach. More recently, Sivapuram et al. treated topology optimization as a discrete variables-based optimized problem [^{TM} for distribution [

Liang et al. suggested a sequential approximate integer programming with a trust region framework to restrict the range of discrete design variables by linearizing the non-linear trust region constraint [

The majority of the existing topology optimization method is solved by the gradient-based algorithm, which is due in large part to the efficient sensitivity analysis approach. Sigmund gave a comprehensive analysis of the non-gradient topology optimization from multiple aspects including global solution, discrete designs, simple implementation, and efficiency, particularly for the SIMP method [

The two-point or three-point approximation belongs to the mid-range approximation. Since topology optimization requires repeated iterations until convergence, it is a natural choice to construct approximation functions using multi-point information to expand its approximate range. The approximation of this type can also be regarded as the connection of many local approximations, such as response surface and kriging model. Wang et al. presented a Hermite interpolation function using multi-point data generated during the iterative process of optimization [

Luo et al. described structural topologies using the material-field series expansion, with the series expansion coefficients serving as the design variables [

Recent years have witnessed rapid progress in artificial intelligence and neural networks. Some researchers have focused on topology optimization using these techniques, in an effort to accelerate the optimization iterations or enhance graphics post-processing. AI technology is used to establish the implicit connection between structural response and design variables. Woldseth et al. performed a comprehensive analysis of the combination of artificial neural networks and topology optimization [

Authors are aware that the number of applicable optimizers is relatively limited, particularly for a wide range of multiphysics topology optimization with nontrivial and multiple constraints. For decades, the MMA and its globally convergent variant have been regarded as the most reliable optimizers. The authors conclude that inadequate research has been conducted on the use of contemporary mathematical programming techniques to solve large-scale, complex topology optimization problems. The AL method, ILP, and optimization algorithm based on non-gradient approximation require further development.

Sequential approximation, a crucial technique in topology optimization, has attracted a great deal of interest since the beginning of structural optimization. After decades of research advancements in topology optimization, the community has settled on a handful of sequential approximations. MMA and its global convergent version become dominant among them. This allows researchers to concentrate on other essential technologies. This paper provides a comprehensive overview of sequential approximation, its related topology optimization methods, and its applications. The initial section provides a concise introduction to the optimality criteria and sequential linear programming. The subsequent section introduces the intervening variables in order to explore various forms of sequential approximation, including COLIN, MMA, two-point or three-point approximation, and SQP. This paper presents the latest improvements in the field, including AL function, sequential approximate integer programming, and non-gradient approximation, aiming to aid researchers effectively choosing the most suitable approximate form for their studies. It is anticipated that a forthcoming proposal will present a notable advancement in the field of topology optimization, specifically in relation to sequential approximation.

_{0}

Objective function

_{j}

Constraint functions

The design variable vector containing the component _{i}

The global stiffness matrix

External load vector

Nodal displacement vector

Surrogate function of g_{0}(

Surrogate function of g_{j}

_{i}

The

_{0}

Young’s modulus of solid material

_{min}

The minimum stiffness with typical value of _{0}/10^{9}

Lagrangian multiplier

Approximate terms of Hessian matrix

Euclidean norm

Replacement of Hessian matrix

Penalization factor

Total volume in the structural design domain

Static compliance

Upper limit of the static compliance

_{E}

Interpolation for young’s modulus

_{v}

Interpolation for volume fraction

_{i}

The ith independent topological variable

Maximum and minimum design variable

Intervening variable

_{i}, l

_{i}

Lower and upper bound for moving asymptotes

_{i}, β

_{i}, γ

Parameters in MMA algorithm

Hessian matrix

Hessian matrix in Lagrangian function

We thank Professor Yunkang Sui (Beijing University of Technology) partly for his pioneering research on topology optimization solved by SQP since 1996, and partly for his personality on the first author’s scientific career.

This work was financially supported by the National Key R&D Program (2022YFB4201302), Guang Dong Basic and Applied Basic Research Foundation (2022A1515240057), and the Huaneng Technology Funds (HNKJ20-H88).

The authors confirm their contribution to the paper as follows: study concept, writing, and interpretation of results: Kai Long, Ayesha Saeed; data collection: Jinhua Zhang, Yara Diaeldin, and Feiyu Lu; analysis and design: Tao Tao, Yuhua Li; draft manuscript preparation: Pengwen Sun, Jinshun Yan. All authors reviewed the results and approved the final version of the manuscript.

Data will be provided on request.

The authors declare that they have no conflicts of interest to report regarding the present study.