Multi-scale design of composite material structures for maximizing fundamental natural frequency

Topological optimization is an innovative method to realize the lightweight design. This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method. In the method, SIMP is used to update the chromosomes, which can accelerate convergence. The filtering scheme in the SIMP method can filter unconnected elements to ensure the connectivity of the structure. We studied the influence of varying the filtering radius on the optimized structure. Simultaneously, in the SIMP-GA method, each element is regarded as a gene, which controls the population number to a certain extent, reduces the amount of calculation, and improves the calculation efficiency. The calculation of some typical examples proves that the SIMP-GA method can obtain a better solution than the gradient-based method. Compared with the conventional genetic algorithm and GA-BESO (Bi-directional Evolutionary Structural Optimization) method, the calculation efficiency of the proposed method is higher and similar results are obtained. The innovative topology optimization method could be an effective way for structural lightweight design.

In this work, a topology optimization (TO) based framework for functional grading of triply periodic minimal surfaces (TPMS) based lattice structures is developed, implemented and demonstrated. Material interpolation laws of the gyroid, G-prime and Schwarz-D surfaces are derived by numerical homogenization for transversely isotropic elasticity and are represented as convex combinations of solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP) models. These convex combinations are implemented in the TO-based compliance problem with new upper and lower bounds on the density variables representing the volume fraction limits of the lattices. The lower bound on the density variables is treated by introducing a sigmoid filter in the optimization loop forcing densities below the lower boundary towards zero. The optimal density solution is represented by Shepard interpolations or radial basis function networks, which, in turn, are utilized for the thickness grading of the TPMS-based lattices. In addition, the global boundary of the lattice structure is identified by support vector machines. Finally, a standard triangle language (STL) file is generated from the implicit surfaces by using marching cubes, which is utilized for further studies by nonlinear finite element analysis and to set up 3D printing of the optimal component quickly. The framework is demonstrated for the established L-shaped benchmark and the well-known General Electric engine bracket.

Research on multi-stage topology optimization method based on latent diffusion model

Conventional topology optimization is concerned with the structures modeled by classical theory of mechanics. Since it does not consider the effects of the microstructures of materials, the classical theory can not reveal the size effect due to material’s heterogeneity. Couple-stress theory, which takes account of the microscopic properties of the material, is capable of describing the size effect in deformations. The purpose of this paper is to investigate the formulation for topology optimization of couple-stress material structures. The artificial material density of each element is chosen as design variable. Based on the basic idea of SIMP (Solid Isotropic Material with Penalization) method, the effective material stiffness matrix of couple-stress material is related to the artificial density by power law with penalty. The structural analysis is implemented by finite element method for couple-stress materials, and a 4-noded quadrilateral couple-stress element is formulated in which C1 continuity requirement is relaxed. Some typical problems are solved and the optimal results based on the couple-stress theory are compared with the conventional ones. It is found that the optimal topologies of couple-stress continuum show remarkable size effect.

This work proposes an improved topology optimization model for optimizing continuum structures with self-weight loading conditions. A modified Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, the penalty factor of the SIMP model is increased to maintain the activeness of the prescribed volume constraint and to drive the design domain to binary distribution. The optimization objectives include minimizing the total strain energy of the design domain and minimizing the total displacement of the fixed domain. The shape optimization procedure is used to furtherly enhance structural performance. The whole optimization procedure is implemented with a two-dimensional model under the loading conditions of self-weight and external force. The classic MBB beam and self-weight arch are utilized to verify the proposed method, and conceptual layout designs of the steel structure bridge are conducted. It is proved that the proposed model is effective for topology optimization of continuum structures including self-weight. And it is found that the optimal structural topology is affected by the ratio of the external force to self-weight.

Design of component structure in assemblies for simultaneously regulating contact pressure distribution and natural frequencies

Topology Optimization (TO) redistributes the material within a design space to optimize certain objective functions under given constraints. Currently available TO methods do not consider the designer's preferences about the final material layout in the optimized design. Contrarily, often an improved design similar to a reference design is required because of the economic, manufacturing or assembly restrictions. In this article, the proposed heuristic similarity control methods like the Energy Scaling Method (ESM), weak passive material method, and an intuitive method of modified design domain are compared with the formal mathematical method of Optimality Criteria (OC)-based Solid Isotropic Material with Penalization (SIMP) with a similarity constraint. Initially, the methods are coupled with Hybrid Cellular Automata (HCA) and OC-based SIMP for the TO of a cantilever beam under a static point load. The ESM is found to be the most effective and further tested for similarity-based TO with HCA in crash loading scenarios. The results show that ESM can be used to control the TO process effectively.

Concurrent multi-material and multi-scale design optimization of fiber-reinforced composite material and structures for minimum structural compliance

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0---1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton---Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.

This paper discusses a geometrical optimization method for thermoelectric generators. Our method provides the optimized geometry in accord with various arbitrary conditions such as types of materials, volume of materials, temperatures of installation position, and shape of installation position. It has the potential to contribute to improving thermoelectric devices in supporting design stages. By establishing the coupled equations of state for the thermoelectric problem, an analytical model subject to these equations is introduced that mimics the closed circuit composed of the thermoelectric materials, electrodes and a resistor. The total electric power lost through the resistor is formulated as an objective function for optimization. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization (SIMP) method used in topology optimizations. Simple relations are formulated between the density function of SIMP method and physical properties of thermoelectric material. Sensitivity analysis for the objective function is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes, which has numerous benefits in various optimization problems by virtue of its combination with topology optimization. Finally, the numerical examples are provided to demonstrate the validity of our method.

This paper describes the influence of fibre orientation, delamination size and location on the natural frequencies of the single mid-plane delaminated unidirectional flax fibre reinforced epoxy (FFRE) composite plates under different boundary conditions numerically by using ANSYS MAPDL. The delaminated composite plate was modelled as two separate volumes divided at the midplane of the plate, with the nodes on the intact surfaces merged. In contrast, the nodes on the delaminated region remained separate. The results show that the CCCC delaminated composite plate has the highest fundamental natural frequency (219.996 Hz). Furthermore, for increasing fibre orientation from 0° to 45°, the fundamental natural frequencies decreased by 10.12% for the CCCC condition and increased by 6.01% for the SSSS condition. The fundamental natural frequency for the cantilever condition decreased 60.91% when fibre orientation increased to 90°. Moreover, the CCCC condition significantly reduces the fundamental natural frequency (up to 38.95%) with increasing delamination size. For CCCC and SSSS conditions, the centre delamination possesses the highest fundamental natural frequency (219.996 Hz and 116.525 Hz, respectively). The highest fundamental natural frequency for cantilever conditions is 33.081 Hz, with delamination located at the middle of the width and near the free end.

Fundamental natural frequencies of rotating functionally graded cylindrical shells have been calculated through the improved wave propagation approach using three different volume fraction laws. The governing shell equations are obtained from Love’s shell approximations using improved rotating terms and the new equations are obtained in standard eigenvalue problem with wave propagation approach and volume fraction laws. The effects of circumferential wave number, rotating speed, length-to-radius, and thickness-to-radius ratios have been computed with various combinations of axial wave numbers and volume fraction law exponent on the fundamental natural frequencies of nonrotating and rotating functionally graded cylindrical shells using wave propagation approach and volume fraction laws with simply supported edge. In this work, variation of material properties of functionally graded materials is controlled by three volume fraction laws. This process creates a variation in the results of shell frequency. MATLAB programming has been used to determine shell frequencies for traveling mode (backward and forward) rotating motions. New estimations show that the rotating forward and backward simply supported fundamental natural frequencies increases with an increase in circumferential wave number, for Type I and Type II of functionally graded cylindrical shells. The presented results of backward and forward simply supported fundamental natural frequencies corresponding to Law I are higher than Laws II and III for Type I and reverse effects are found for Type II, depending on rotating speed. Our investigations show that the decreasing and increasing behaviors are noted for rotating simply supported fundamental natural frequencies with increasing length-to-radius and thickness-to-radius ratios, respectively. It is found that the fundamental frequencies of the forward waves decrease with the increase in the rotating speed, and the fundamental frequencies of the backward waves increase with the increase in the rotating speed. This investigation has been made with three different volume fraction laws of polynomial (Law I), exponential (Law II), and trigonometric (Law III). The presented numerical results of nonrotating isotropic and rotating functionally graded simply supported are in fair agreement with parts of other earlier numerical results.

Seismic behavior of a structure is directly related to its dynamic characteristics, which include natural frequency, damping ratio, and mode shape. This study focuses on the long-term monitoring of dynamic characteristics of six selected target structures. The covariance-driven stochastic subspace identification (SSI) approach is used to estimate the fundamental natural frequency and damping ratio of target buildings based on long-term motion records in order to examine the temporal variation of dynamic properties. The fundamental natural frequency and damping ratio variations over time are first discussed. It is found that the fundamental natural frequency of some structures reduces dramatically after the 2011 Tohoku Earthquake, accompanied by a rise in damping ratio. Then, regression analysis is used to assess the relationship between dynamic characteristics and ground motion parameters (Peak ground acceleration (PGA), magnitude, focal depth, and epicentral distance) and structural response (root mean square acceleration, maximum response amplitude). It is discovered that the identified natural frequency has no clear correlation with the focal depth, a slight negative correlation with the epicentral distance, and a strong negative correlation with the magnitude and PGA. The root mean square acceleration and the maximum response amplitude are negatively correlated to the target buildings’ natural frequencies. Finally, the influence of environmental factors on dynamic properties is investigated.

Graphene oxide (GO) has been recognized as one of the most potential nanomaterial for reinforcement in cementitious composites due to its extraordinary properties. In the present investigation, the static and dynamic mechanical characteristics of GO and fly ash based concrete (FA-GO-Concrete) have been studied experimentally. The static characteristics have been determined by conducting a compressive strength test and dynamic properties such as fundamental natural frequencies and damping ratios of FA-GO-Concrete beams in free-free conditions were measured using the impact hammer technique. In this paper, the influence of GO addition at 0.15% and variation of fly ash content at 0, 10, 20 and 30% as a replacement of cement was studied. The experimental findings showed that the addition of GO and replacement of fly ash increased the compressive strength and fundamental natural frequencies, whereas reduced the damping ratio of FA-GO-concrete compared to the control concrete. The damping ratios of FA-GO-Concrete increased with fly ash content increment, while the fundamental natural frequencies were reduced with fly ash content increment with respect to the GO-Concrete. The static and dynamic elasticity modulus of FA-GO-Concrete were also determined. It has been found that the desired dynamic properties can be achieved by adding GO and partially replacing the cement with fly ash.KeywordsGraphene oxideFly ashStatic mechanicalDynamic mechanical

The composite structures in the aerospace industry for in recent decades are widely applied however, at the beginning of the 21st century composites are growing rapidly. The largest companies in the aerospace industry are increasing the volume of composites application of in the structures, and nowadays the volume of composites reaches 50%. The different elements of aircraft and even highly loaded structures such as spars, ribs, skin, etc., are currently made from composites. First of all, this is due to the possibility of a significant reduction in the weight of the structure, as well as a decreasing in production costs. The advanced technologies in the engineering software allows to solute different complex problems. One of the main direct of research in the composites is optimization of composite structure due to improving the relative strength and relative stiffness of the composite structure, and improving the efficiency of manufacturing processes. There are a lot of methods of optimizations but currently the topological optimization is the most conceptual and forward-looking method. The main goal of the article is to analyze and estimate the approach for designing wing rib with symmetric laminated plates with the different fiber orientation based on the topology optimization. The following tasks were solved for this: firstly, a topological optimization model was determined. This model was based on maximum stiffness with a specified volume constraint is established. The next step was optimization by the solid isotropic material with penalization (SIMP) model and sensitivity filtering technique; as a result of optimization the topological structures of wing rib with different fibre orientations were obtained. The topological structure and stiffness of the wing rib depend on the fibre orientation. Finally, the corresponding morphing analysis of wing rib with laminated plates is implemented by adopting ANSYS, which verified the anti-deforming capability of topology structure and illustrated the feasibility for designing the wing rib. The result shows that the maximum deformation of optimized structure is 1.57mm, whereas the maximum deformation of the un-optimized structure is 2.02 mm. Under the condition of the same material removals, the optimized structure can decrease by more than 20% deformations.