Topology Optimization of Arc Unit Cells by Extending Energy‐Based Homogenization and Parametric Level Set Methods

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

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.

Frequency- and temperature-dependent properties of viscoelastic materials have a significant effect on vibration characteristics of constrained layer damping (CLD) structures. In this article, parametrized level set method (PLSM) is extended to solve the topology optimization problem of CLD plates, in which the viscoelastic material with frequency- and temperature-dependent characteristics is employed as damping core. The single/weighted modal loss factors, which are solved by an iterative method, are defined as objective functions, and topology optimization problem is formulated based on the finite element model for CLD plates which are described by combining parametric level set (PLS) functions. The sensitivities of objective functions with respect to expansion coefficients in PLS are calculated by the use of adjoint vector method. As the optimized results with complex constant shear modulus for viscoelastic core indicates, the proposed method is effective for topology optimization of CLD structures and insensitive to initial design. Numerical examples with the consideration of frequency- and temperature-dependent characteristics for “soft” and “hard” viscoelastic core are implemented. Through comparing the optimal results with different models for viscoelastic core, the influences of frequency- and temperature-dependent characteristics for viscoelastic core on natural frequencies, modal loss factors and optimized layouts are analyzed. Similarity indices are defined for distinguishing the difference among optimized layouts and initial design induced by modal strain energy contour. The optimal design dependence on layer thicknesses is further discussed with additional mass of CLD materials unchanged.

Herein new lattice unit cells with buckling load 261–308% higher than the classical octet unit cell were reported. Lattice structures have been widely used in sandwich structures as lightweight core. While stretching dominated and bending dominated cells such as octahedron, tetrahedron and octet have been designed for lightweight structures, it is plausible that other cells exist which might perform better than the existing counterparts. Machine learning technique was used to discover new optimal unit cells. An 8-node cube containing a maximum of 27 elements, which extended into an eightfold unit cell, was taken as representative volume element (RVE). Numerous possible unit cells within the RVE were generated using permutations and combinations through MATLAB coding. Uniaxial compression tests using ANSYS were performed to form a dataset, which was used to train machine learning algorithms and form predictive model. The model was then used to further optimize the unit cells. A total of 20 optimal symmetric unit cells were predicted which showed 51–57% higher capacity than octet cell. Particularly, if the solid rods were replaced by porous biomimetic rods, an additional 130–160% increase in buckling resistance was achieved. Sandwich structures made of these 3D printed optimal symmetric unit cells showed 13–35% higher flexural strength than octet cell cored counterpart. This study opens up new opportunities to design high-performance sandwich structures.

Stress-driven nonlocal homogenization method for cellular structures

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]

A homogenization method for the vibration analysis of periodic sandwich panels is presented. Periodic boundary conditions are applied to a representative volume element (RVE) of the structures. The complex responses of the RVE under eight sinusoidal excitations are calculated and the complex stiffness matrix of the sandwich panels is obtained by the direct-solution steady-state dynamic analysis (DSDA) using the finite element method (FEM) in the frequency domain. Based on equivalent single-layer (ESL) theory and finite element-modal strain energy (FE-MSE) method, the natural frequencies, the mode shapes and the loss factors of the structures are obtained by shell models. The effectiveness of the homogenization method is validated by using pyramidal lattice truss sandwich panels made of carbon fiber reinforced plastics (CFRP). The results obtained by the homogenization method are compared with those obtained by experiments and solid models. The effects of the number of cells and damping layers are discussed.

Stiffness design of a multilayer arbitrary BCC lattice structure with face sheets

The characterization of the flexural performances of short glass fiber reinforced polyether‐ether‐ketone (SGFR‐PEEK) composites fabricated by injection molding is a crucial and challenging task due to the process‐induced fiber orientation at different locations, resulting layered shell‐core‐shell microstructures and exhibiting large variations in mechanical performances. This article proposes a method for predicting the bending performance of SGFR‐PEEK and indicates the influence of fiber orientation on its bending behavior. This article reports a new scheme that combines multiscale homogenization with periodic microstructures and presents an experimental investigation of the flexural properties of SGFR‐PEEK composites. The distributions of fiber orientation through the thickness extracted from the micro‐computed tomography (μCT) were analyzed and reproduced with a layered skin‐shell‐core structure along the thickness at meso‐scale finite element model within representative volume elements (RVEs) of the macroscopic composites. Then, the effective properties of the RVEs were predicted using the homogenization method with the periodic boundary condition. The elastic modulus of skin‐shell‐core layers in the direction of flow are 4260.51, 4541.27, 4628.52, and 4630.84 MPa, respectively. The classical laminate theory (CLT) and the second homogenization implementation were then utilized in conjunction with the obtained mechanical properties of RVEs to predict the effective macrostructural behavior. The flexural modulus of the composite material was determined to be 5448.7 MPa through a three‐point bending test. The results obtained through lamination plate reinforcement theory and finite element simulation were 5887.4 and 5785.9 MPa, respectively, showing good agreement with the experimental findings. It is shown that the proposed multiscale scheme yields satisfactory agreement with experimental measurements. In addition, the effect of layered skin‐shell‐core microstructures on the flexural behavior was discussed.Highlights RVEs of layered shell‐core‐shell microstructures were generated with specified fiber orientations. The homogenization scheme with the periodic boundary condition was adopted to accurately predict each layer mechanical properties. The proposed multiscale scheme was compared with experimental results and yielded satisfactory predictions. The effect of layered skin‐shell‐core microstructures on the flexural behavior was thoroughly studied by the multiscale analytical approach.

The homogenization method has been known as a powerful tool to investigate the macroscopic material properties of composite materials, whose microstructures are periodic. This methodology can also be considered to be an advanced version of unit cell approach. However, the assumption of periodic structure becomes somewhat suspicious after the material has undergone a large deformation, because each unit cell also deforms finitely. In the context of homogenization method presented by Guedes and Kikuchi, the periodicity condition has to be satisfied rigorously. In this paper, a formulation which can rigorously satisfy the condition of periodicity even after the microstructure has undergone a finite deformation, is presented. It is accomplished by referring the initial undeformed configuration of microstructure, when we define the periodicity condition. Some simple numerical results of two-dimensional finite strain elasto-plastic problem are also presented and discussed. The outcome of the present research may find some application in analyzing Metal Matrix Composite Materials.

Numerical evaluation of the thermomechanical effective properties of a functionally graded material using the homogenization method

Optimization of elastic properties and weaving patterns of woven composites

The homogenization method is examined in conjunction with the method of representative volume elements (RVE). When the digital image-based modeling technique enables us to model the complex geometry of microstructures, the boundaries of unit cells fail to have the geometric periodicity Nonetheles, the effective moduli are reasonably estimated by the homogenition with periodic boundary conditions. This fact is justified theoretically and simulated along with the notion of homogenization convergence. In adition to the convegence of effective moduli with respect to the size of RVEs, the microscopic mechanical behaviors are examined in comparison with macroscopic ones. Thus, the multiscale nature of the analysis modeling by the asymptotic homogenization is characterized by carrying out a series of linear and nonliinear numerical analyses.

Periodic boundary condition is commonly used in the FE homogenization method for predicting the thermo-mechanical properties of composites. However, its existing numerical algorithms are generally inefficient. Linear displacement boundary condition is much easier and more efficient to implement for the Representative Volume Elements (RVEs) with conformal and nonconformal surface meshes. Aiming at replacing the periodic boundary condition by the linear displacement boundary condition to easily and efficiently predict the thermo-mechanical properties of composites, this paper compares the implementation efficiency and prediction accuracy of the periodic and linear displacement boundary conditions. Regarding the prediction of the elastic properties of composites, the balance of micro-energy and macro-energy of the RVEs imposed the periodic and linear displacement boundary conditions is checked, and the results demonstrate that these two boundary conditions guarantee the Hill lemma (i.e. the balance of micro-energy and macro-energy of the RVEs). Moreover, comparing with the experimental test and the multi-step micro-mechanical method, the effectiveness of the periodic and linear displacement boundary conditions to predict the elastic properties of composites is validated.

Crashworthiness unit cell design investigation for energy absorption analysis