Topology optimization of constrained layer damping plates with frequency- and temperature-dependent viscoelastic core via parametric level set method

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

Vibration analysis of simply supported beams with enhanced self-sensing active constrained layer damping treatments

Traditional methods based on an element-wise parameterization of the material distribution applied to the topology optimization of fluidic systems often suffer from slow convergence of the optimization process, as well as robustness issues at increased Reynolds numbers. The local influence of the design variables in the traditional approaches is seen as a possible cause for the slow convergence. Non-smooth material distributions are suspected to trigger premature onset of instationary flows which cannot be treated by steady-state flow models. In the present work, we study whether the convergence and the versatility of topology optimization methods for fluidic systems can be improved by employing a parametric level-set description. In general, level-set methods allow controlling the smoothness of boundaries, yield a non-local influence of design variables, and decouple the material description from the flow field discretization. The parametric level-set method used in this study utilizes a material distribution approach to represent flow boundaries, resulting in a non-trivial mapping between design variables and local material properties. Using a hydrodynamic lattice Boltzmann method, we study the performance of our level-set approach in comparison to a traditional material distribution approach. By numerical examples, the parametric level-set approach is validated through comparison with traditional material distribution based methods. While the parametric level-set approach leads to similar optimal designs, the present study reveals no general improvements of the convergence of the optimization process and of the robustness of the nonlinear flow analyses when compared to the traditional material distribution approach. Instead, our numerical experiment suggests that a continuation method operating on the volume constraint is needed to achieve optimal designs at higher Reynolds numbers.

This paper focuses on two kinds of bi-objective topology optimization problems with uniform-stress constraints: compliance-volume minimization and local frequency response–volume minimization problems. An adaptive volume constraint (AVC) algorithm based on an improved bisection method is proposed. Using this algorithm, the bi-objective uniform-stress-constrained topology optimization problem is transformed into a single-objective topology optimization problem and a volume-decision problem. The parametric level set method based on the compactly supported radial basis functions is employed to solve the single-objective problem, in which a self-organized acceleration scheme based on shape derivative and topological sensitivity is proposed to adaptively adjust the derivative of the objective function and the step length during the optimization. To solve the volume-decision problem, an improved bisection method is proposed. Numerical examples are tested to illustrate the feasibility and effectiveness of the self-organized acceleration scheme and the AVC algorithm based on the improved bisection method. An extended application to the bi-objective stress-constrained topology optimization of a structure with stress concentration is also presented.

Topology optimization has developed rapidly in the past three decades; as a creative and efficient optimization technique, it has been applied in engineering fields of aerospace and mechanical. However, there are a few attempts in bridge form design. In this paper, the parametric level set method is utilized to solve the form finding of arch bridges. The optimization model for minimizing the structural compliance under the volume constraint is built. Three numerical examples of form finding of arch bridges are studied. Results show that the optimal structures which have well‐distributed stress and smooth force transmission are almost identical with the actual forms of arch bridges. The optimal forms can be treated as alternatives in the preliminary design stage, and topology optimization has a bright prospect in form finding of arch bridges.

Non-probabilistic Reliability-based Topology Optimization (NRBTO) Scheme for Continuum Structures Based on the parameterized Level-Set method and Interval Mathematics

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

Topology optimization has been used to maximize the damping performance of constrained damping layer structures with viscoelastic core [1-3]. Studies show that using nanotubes as reinforcements in polymer material might improve the damping and stiffness properties [4,5]. Experimental investigations using nanotube reinforcements in constrained damping layers showed a significant improvement in damping properties of the structure [6]. Optimizing the topology of structures involving nanotube reinforcements can further enhance the damping properties by making optimal use of the nanotubes as reinforcements. In preliminary studies performed with volume fraction of nanotubes as a design variable in the optimization [7,8] substantial improvements in the damping properties were observed. In this work, the topology of a cantilever beam structure involving nanotube reinforced polymer layers is optimized with the objective of maximizing the system loss factor for the first resonance frequency of the base beam. The volume fractions and orientations of the nanotubes in each finite element and the material fraction of the nanotube reinforced polymer (NRP) and the elastic material are the design variables in the optimization process. ABAQUS is used for the finite element modeling and an optimization code NLPQL which uses a sequential quadratic programming algorithm is used for the optimization.

ABSTRACTThis paper extends the energy‐based homogenization method (EBHM) to evaluate the effective elastic properties of two‐dimensional arc unit cells, where the periodic boundary conditions are constructed based on the polar strain–displacement relationships, and the displacement field is solved by the polar finite element method. Combining the extended EBHM with the parametric level set method (PLSM) and the augmented Lagrangian multiplier method (ALM), it is able to design the topological configurations of arc unit cells for seeking superior effective elastic properties. In addition, the level set function is first filtered by compactly supported radial basis functions (CSRBFs) in order to eliminate small holes in the configuration. Numerical examples are performed to demonstrate the advantages of the proposed method. The periodic boundary condition performs well when the circumferential angle of the arc unit cell is in the range of [0.0002, 0.2]. Results indicate that the effective elastic properties of optimized arc unit cells override those of geometrically transformed optimal square unit cells. This superiority mainly benefits from the fact that unisymmetrical configurations are allowed in arc unit cells but not in square unit cells. Results also imply that the capability of nucleation of new holes is retained even though small holes are eliminated by filtering. It contributes to the trade‐off between performance and manufacturability.

Optimal arrangement of damping materials is an important problem in vibration control of thin walled structure treated with constrained layer damping, which directly affects vibration energy dissipation and global energy flow distribution. In this thesis, an evolutionary topology optimization method is used to establish the optimization model for constrained layer damping processing panel, with the optimization goal of maximizing the modal loss factor and limiting the usage of damping material. Taking into account the sensitivity of modal shapes, based on an improved modal superposition method, the sensitivity of the objective function with respect to the design variables is obtained. The level set function constructed from the node sensitivity determines the implicit smooth structure topology of the restricted damping layer. Numerical examples suggest that the algorithm is feasible, correct and valid. Studies show that this method not only allows the objective function to converge stably to the optimal value, but also obtains smooth and clear boundaries.

A new fuzzy strategy for size and topology optimization of truss structures

An efficient aerodynamic topology optimization method was constructed based on the parameterized level set method (PLSM) and large neighborhood search algorithm. The traditional aerodynamic optimization design process is to optimize the shape of a basic shape, and the aerodynamic configuration is determined. The determination of the basic shape relies on the experience of human designers, making it difficult to obtain innovative aerodynamic configurations. To this end, the PLSM is used to obtain aerodynamic configurations of supersonic Busemann biplane with different topological structures. In response to the characteristics of large sample size and low computational efficiency in aerodynamic topology optimization design, a large neighborhood search algorithm is adopted as the optimization algorithm. Aerodynamic topology optimization design was carried out to achieve good drag reduction effect for the Busemann biplane. Compared with shape optimization design, aerodynamic topology design can generate airfoils with different topological structures, greatly improving the design freedom and making it possible to find different aerodynamic configurations in a wide design space.

Conventional Passive Constrained Layer Damping (PCLD) treatments with viscoelastic cores are provided with built-in sensing and actuation capabilities to actively control and enhance their vibration damping characteristics. The design parameters and control gains of the resulting Active Constrained Layer Damping (ACLD) treatments are optimally selected, in this paper, for fully-treated beams using rational design procedures. The optimal thickness and shear modulus of the passive visco-elastic core are determined first to maximize the modal damping ratios and minimize the total weight of the damping treatment. The control gains of the ACLD are then selected using optimal control theory to minimize a weighted sum of the vibrational and control energies. The theoretical performance of beams treated with the optimally selected ACLD treatment is determined at different excitation frequencies and operating temperatures. Comparisons are made with the performance of beams treated with optimal PCLD treatments and untreated beams which are controlled only by conventional Active Controllers (AC). The results obtained emphasize the potential of the optimally designed ACLD as an effective means for providing broad-band attenuation capabilities over wide range or operating temperatures as compared to PCLD treatments.

This paper proposes an effective method for the design of mechanical metamaterials by combining the parametric level set method (PLSM) with the energy-based homogenisation method (EBHM), where the topologies and shapes of microstructures periodically distributed in the material with negative Poisson's ratios are optimised simultaneously. Firstly, the effective properties of the materials are calculated by the EBHM under the topologies of microstructures, where the stress and strain theorems work as the basic theoretical framework rather than the asymptotic theory. Secondly, under the definition of the objective function to seek for the negative Poisson's ratio, the PLSM is used to evolve the topological configurations of microstructures until the bulk material is to be featured with a negative Poisson's ratio. Thirdly, the discrete wavelet transform (DWT) is utilised to deal with the full matrix of the globally supported radial basis function (GSRBF) interpolation. The proposed method can efficiently design the microstructure with the negative Poisson's ratio on the premise of ensuring the calculation accuracy. Several numerical examples are provided to illustrate the effectiveness of the proposed method for design of microstructures considering the orthotropy and isotropy. [Submitted 31 August 2019; Accepted 26 November 2019]