Optimal Material Distribution Research Articles

Variable-density topology optimization and bending test of quasi-honeycomb sandwich structures

The topology optimization is considered to enhance the strength, stiffness, and specific energy absorption of a quasi-honeycomb sandwich structure since it can achieve the optimal distribution of materials. However, the existing material interpolation models do not make the topology structure have good clarity and stability. In the present study, a novel variable-density topology optimization based on an improved interpolation model is developed for the design of a quasi-honeycomb core. The combined cells are circularly spreading outward from the geometric center of the design domain, and the wall thickness function of the equivalent cell (i.e. a simplified model of the combined cell) is established on the basis of relative density. The SR material interpolation model is modified by adding the minimum elastic modulus term, fitting the weighting and penalty factors, and adopting the volume constraint of Bi-directional Evolutionary Structural Optimization (BESO). The optimization is conducted to minimize the compliance of the quasi-honeycomb core, in which the relative densities of equivalent cells are chosen as the design variables due to the mapping between topology elements and equivalent cells. The three-dimensional (3D) geometric models of original and optimized quasi-honeycomb cores, upper face sheets, and lower face sheets are built using SolidWorks and printed on a 3D printer. The deflection, strain, bending stiffness, and energy absorption under different material retention rates, restraint positions, and face sheet thicknesses are investigated by three-point bending tests. It is found that the improved SR material interpolation model is efficient for suppressing the grid dependence and checkerboard, and the mechanical properties of optimized quasi-honeycomb sandwich structures are significantly enhanced in comparison with original structures, confirming that the developed method provides important guidance for designing and optimizing the quasi-honeycomb sandwich structure.

Two-scale concurrent optimization of composites with elliptical inclusions under microstress constraints within the FE2 framework

In order to avoid the adverse effects of structural damage caused by stress concentration, the optimization problem considering stress constraints is very important in structural design. This paper aims at introducing microstress constraints within a multilevel finite element (FE2) analysis framework to perform the two-scale concurrent optimization and reduce the stress concentration while exhibiting better overall stiffness properties for composites with elliptical inclusions. In order to do that, at the macroscale, relying on the solid isotropic material with penalization (SIMP), a density-based optimization algorithm is raised for a maximum stiffness design and free material distribution optimization under a volume constraint, these displacement solutions and material distributions at the macroscopic structural scale provide kinematic constraints for microscale optimization. At the microscale, the parameterized composite microstructure (inclusion orientation and aspect ratio) is optimized under the principal strain direction constraints of the macroscale element’s Gauss integration point. The optimized microstructure in turn updates the macroscopic effective stress directly from the volume average of the microscopic stress field. The clustering strategy is used in order to improve the computation efficiency of the optimization. The novelty of the work lies in the use of microstress constraints to solve the concurrent optimization problems of two-phase composites within the FE2 framework. The effectiveness of the proposed concurrent optimization approach is validated through three numerical examples.

Lightweight Optimization Design of Structures with Multiple Cellular Materials

Cellular materials have been widely applied to a lightweight design of structures. The mechanical properties of those materials depend on their microstructures at the microlevel/mesolevel, and the optimizaiton design of lightweight structures using multiple cellular materials is still challenging. This paper develops a topology optimization algorithm for a lightweight design of structures constructed by multiple cellular materials with specified microstructures. The mechanical properties of cellular materials are homogenized according to their microstructures and then integrated into topology optimization. The topology optimization problem is defined by minimizing structural compliance subject to a specified mass constraint. In order to identify the distribution of multiple cellular materials within the design domain, the multiple design variables are introduced based on the volume fractions of multiple cellular materials within each element. Meanwhile, the homogenized mechanical properties are linearly interpolated, and multiple floating projection constraints are imposed on the relaxed design variables to push them toward 0 or 1. Numerical examples demonstrate the successful implementation of the proposed algorithm by the optimal distribution and selection of multiple cellular materials.

Topology optimization of nonlinear flexoelectric structures

Recent advances in nanotechnology have allowed for the manufacturing of nanostructures and nanodevices with optimized topologies that outperforms their traditional counterparts based on simple geometry in terms of efficiency and function. In this work, a novel nonlinear topology optimization procedure is developed to design optimal layouts of flexoelectric structures undergoing large displacement. The optimal material distribution is determined by optimizing the energy conversion efficiency. Two material properties such as the elasticity coefficient and the permittivity coefficient are interpolated through the solid isotropic material with a penalization approach via an energy interpolation scheme to overcome the numerical instability. Obtained results have revealed that accounting for the geometric nonlinearity leads to a different final optimal topology with a better energy converting efficiency as compared to the linear model. We also found the significant influences of size effects on the optimized structure emphasizing the importance of nonlocal elastic behavior in characterizing and designing micro-structures and flexoelectricity.

Topological optimization of a symmetrical adhesive joint. Island model of genetic algorithm

Modern additive technologies make it possible to create structures of variable thickness and of any shape. Thus, designers face problems of optimal design of a new type, and these are problems of topological optimization. Such problems are to determine the optimal form of the structure or the optimal distribution of material over the structure. As a rule, the criterion of optimality is the mass of the structure. However, the structure must retain its bearing capacity under a certain load. The symmetric two-shear adhesive joint of the main plate with two overlays of the same shape on both sides is the object of study in this article. The main goal of this study was to determine the optimal form of overlays with variable thicknesses under certain restrictions. The main restriction is the strength of the structure. Furthermore, additional restrictions are imposed on the minimum and maximum thickness of the overlay. Therefore, the solution to the problem is presented in the form of a set of the following tasks: building a mathematical model of the adhesive joint, building a numerical solution to the primal problem using the finite difference method, and building a genetic optimization algorithm. In the presented article, to improve the convergence of the genetic algorithm is proposed to use an island model that consists of several populations. The main feature of the proposed model of the genetic algorithm lies in the fact that on one of the "islands" mutations occur more frequently and with higher dispersion than on the other two "islands". On the one hand, this decision ensures a high rate of evolutionary selection, and on the other hand, the stability of the results is achieved. Several modeling problems are solved in this article. The main results of this research include the following: nonlinear dependence of the overlay length on the applied load was determined; restrictions on the minimum thickness of the overlay, which cause the appearance of a certain “plateau” at the edge of the overlay, the thickness of which is equal to the minimum allowable were defined.

Computational morphology design of duplex structure considering interface debonding

A finite volume interface is an interface region with interface strength, and is most often generated during the fabrication (3D printing) of a duplex structure. However, it is often neglected in morphology design due to numerical complexity and computational difficulties. In addition, a sharp and perfect-bonding interface is usually assumed in literatures. However, such assumptions bring failure risks, thus limiting the industrial applicability of the morphology designs of duplex structures. This study aims to identify the optimal morphology design though a computational design method, considering the finite volume interface and debonding of a duplex structure. This method is based on topology optimization, which utilizes a level-set function for optimizing material distribution in the design space. To introduce finite volume interfaces in morphology design, a simple interface debonding model is integrated into implicit finite element analysis, based on the finite strain theory. Moreover, a distance function is employed to describe the interface region in addition to a level-set function for topology optimization. Further, a topological derivative based on an adjoint variable method is formulated for a debonding interface state in a nonlinear finite element analysis, which is incorporated in topology optimization to obtain the optimal duplex structures. The numerical demonstrations verified the applicability of the proposed approach. The zigzag interface was proven to be one of the features of the optimal duplex structures, considering interface debonding. The results also indicated the optimal duplex structures considering interface debonding to be non-symmetric, in which the interfaces are primarily enriched in compression areas to ensure structural integrity.

Discussing a draft of an example working program of basic general education in Russian language

The article analyzes the Draft of an exemplary work program for basic general education in the Russian language for grades 5–9. The purpose of the article is to determine the compliance of the Project with the tasks of the modern school and the Federal State Educational Standard, determine the continuity in the content of education, identify controversial issues and formulate proposals for improving the Project. Methodology: theoretical (linguo-methodological) analysis of the document from the standpoint of traditional and modern linguistics and methodology, practice of teaching the Russian language in grades 5-9. Positively evaluating the fact of creating the Draft Exemplary Work Program, the author dwells in detail on the identified theoretical linguistic inaccuracies, primarily of a classification nature, as well as methodological shortcomings. Comments were made both of a general plan, relating to the entire program, and private, concerning individual topics, sections, and the distribution of material by class. There are cases of violation of continuity and prospects in the content of teaching the Russian language and insufficient specification of a number of didactic units and skills. The overload of the 5th grade program is argued in detail and recommendations are made for its correction by studying part of the linguistic information at the propaedeutic level. At the end of the article, it is concluded that it is necessary to improve the Draft Exemplary Work Program, proposals are presented aimed at correcting it, in particular, eliminating the identified theoretical errors, cases of discontinuity, more optimal distribution of linguistic material by year of study.

Automatic Design of PM Motor Using Monte Carlo Tree Search in Conjunction With Topology Optimization

A novel automatic design method for permanent magnet (PM) motors using a Monte Carlo tree search is presented. The optimal motor structures are determined through a tree search, in which the motors with different numbers of poles, current phase angles, PM configurations, and numbers of PMs are simultaneously considered. At the leaf nodes, parameter and topology optimizations are performed to obtain the optimal material shape and distribution. The proposed method was applied to the optimization of a 24-slot motor. It was shown to be effective in finding the optimal motor structure and geometry to maximize the average torque while considering iron loss. The proposed method can be applied not only to the design of PM motors but also to many types of electric apparatus and other systems.

Crack initiation and growth simulations of functionally graded dental implant subjected to cyclic axial loads

AbstractTwo essential characteristics in designing dental implants are having high fracture resistance and being compatible with jawbone. One of the best implant choices to achieve this aim is titanium‐hydroxyapatite functionally graded (FG) material. Also, fracture of implants under cyclic loads is one of the most common causes of destruction of dental implants. However, the crack initiation location and growth path as well as the optimum material distribution for the FG implants are questions that have not been answered yet. The aim of this study was introducing a new and powerful numerical approach to predict the crack initiation behavior and progressive damage in the FG dental implants, subjected to cyclic axial loadings. The approach is developed by employing extended finite element method (XFEM) in ABAQUS software, Visual Studio, and FORTRAN Compiler. Also, the Python code was modified for crack growth modeling in fatigue fracture analysis based on Paris law. The result showed that the micro‐cracks below the outer surface of FG implant are created, due to the application of cyclic load, which is invisible. This research shows that FG implants with power factor less than 1 show much more resistance to crack growth under pre‐tightening and cyclic loadings.

Infill Microstructures for Additive Manufacturing

Additive Manufacturing (AM) is a well-known and rapidly advancing method, especially in the manufacturing of high-strength and lightweight microstructures. Utilizing AM, it is possible to fabricate any structure as complicated as it is. For an efficient and cost-effective printing, a critical parameter is the infill, which can be characterized from an easy 2D shape to high complexity. At the same time, Topology Optimization (TO) is an appropriate method to create high-strength and mass optimized microstructure lattices. In the current study, TO starts from a solid cubic volume of 15 × 15 mm, and different boundary conditions of two new cellular microstructures designed with 0.4 and 0.1 relative density are applied, respectively. The adopted TO method was Solid Isotropic Material with Penalization (SIMP), which predicts an optimal material distribution within a given design domain. TO methods do not check other characteristics of the structure, such as anisotropy. To evaluate and characterize the optimized microstructure, a general purpose homogenization method is utilized to calculate the Zener ratio and the elastic modulus. Using Fused Filament Fabrication (FFF), which is a material extrusion 3D printing method, lattice structure samples are fabricated and then tested in compression and tensile strength tests. The comparative results from the homogenization study showed that both microstructures have anisotropic behavior and an accepted response in the stress test similar to the homogenized material. The experimental results show that the mechanical behavior of the lattice structure changes significantly when the cell mapping angle differs.

Concurrent topology and fiber orientation optimization of CFRP structures in space-borne optical remote sensor

Lightweight design is one of the most important concepts in the design of space-borne optical remote sensors. Carbon fiber reinforced polymer (CFRP), as one material with high ratio of modulus to density, is widely employed to realize the lightweight design. To maximize the stiffness and strength of CFRP with prescribed mass, fiber orientation and topology optimization are accomplished and optimum design result is acquired in this paper. First, the optimization algorithm of fiber orientation is put forward to find the optimal fiber direction and verified by theoretical analysis and simulations, illustrating that [90°/+ 45°/− 45°/0°] fiber orientation and symmetrical laminas are indispensable. Furthermore, the topology optimization algorithm including the density interpolation method and sensitivity analysis method is brought about to find the optimal material distribution. The algorithm is validated to be convergent and can provide a conceptual model which offers a reference for the critical design. Finally, acquired in the mechanical test is the performance of the optical remote sensor whose first three characteristic frequencies are respectively 66.95 Hz, 70.37 Hz, and 98.84 Hz. The mechanical amplification factor is 5.29, which meet the performance requirement of the stiffness and strength for the optical remote sensor.

Topology optimization of porous solid structures for heat transfer and flow channels in reactors with fluid-solid reaction coupling

In this paper, the optimization models for the heat transfer and flow channels with porous structures are developed. To enhance the heat transfer capacity of the thermal system, the topological optimization of the fin structure in the radial direction of the heat transfer unit is proposed and the influence of the initial conditions on the optimization of fins is studied. At the same time, the transient process of the heat transfer unit with topology optimization is investigated to verify the feasibility of the optimization model. The results show that the fins designed by topology optimization have coral-likestructure, and the width of the fin decreases as it moves outward from the center. For the optimization of flow channels, the optimum distribution of porous material with theoptimal reaction rate in laminar flow reactor is analyzed. The optimal structures for flow channels in porous media is obatined theoretically and simulatednumerically for obtaining the related transport characteristics. In addition, for the topologically optimized reactor, as the initial reaction parameters change, the fluid channels in the porous domain show different structures withtopology optimization.

Allying topology and shape optimization through machine learning algorithms

Structural optimization is part of the mechanical engineering field and, in most cases, tries to minimize the overall weight of a given design domain, subjected to functionality constraints given in terms of stresses or displacements. The most relevant techniques are topology and shape optimization. Topology optimization provides the optimal material distribution layout into a given, static, design domain. On the other hand, shape optimization provides the optimal combination of the parameters that define the required parametrization of the domain's boundary. Both techniques have strengths and weaknesses, thus a hybrid optimization approach that combines the former techniques will define a more general structural optimization framework that will take advantage of their synergistic combination. The difficulty arises when communicating both techniques for which, in this paper, we propose a machine learning-based methodology. • We proposed a methodology that allies topology and parametrized shape optimization algorithms, obtaining an hybrid optimization algorithm. • We studied the strengths and weaknesses of topology and shape optimization algorithms, which justified its combination to obtain a synergetic effect. • The use of manifold learning techniques, such as the Locally Linear Embedding (LLE), to extract the principal geometrical modes from a set of material distributions that topology optimization algorithm provides. • In order to properly use the LLE technique, the quasi-boolean material distribution provided by the topology optimization algorithm must be transformed into a richer level-set with the distance information. • The geometrical modes, extracted with the LLE algorithm, may be used to generate brand new geometries, as they define a parametric geometrical model. • The parametric space of geometrical modes may be considered by a shape optimization algorithm as its design space.

Multi-Material Topology Optimization of Flexure Hinges Using Element Stacking Method.

Traditional flexure hinges are designed by using a single material, and their performance is inadequate, compared to the ideal hinge. This paper presents a topology-optimization design method for multi-material flexure hinges based on the element stacking method. A topology optimization model for multi-material flexure hinges is constructed to find the optimal distribution of various materials, where the objective function is to maximize the compliance in the rotational direction, whilst minimizing the compliance in the axial direction. In order to ensure the rotation precision of the hinge, the position constraint of the rotation center is proposed. The gradient information of objective and constraint functions is derived by the adjoint method, and the method of moving asymptotes (MMA) is used to update the design variable. Several numerical examples are performed to verify the effectiveness of the proposed method, and the results show that the multi-material flexure hinge has a higher rotation ratio than the single-material flexure hinge.

Optimized design and additive manufacture of double-sided metal mirror with self-supporting lattice structure

Lattice structure has excellent mechanical properties to develop lightweight devices, and additive manufacture makes it possible to construct complex lattice without extra cost. Especially, with the development of structural optimization technology, design of gradient lattice structure with optimized material distribution has become a research hotspot. In this paper, a double-sided metal mirror infilled with topology optimization (TO) based self-supporting lattice structure and fabricated with selective laser melting (SLM) was proposed. Considering the manufacturing constraints of SLM technology, commonly used body-centered cubic (BCC) lattice was modified and coordinate transformed to construct the self-supporting infill of the mirror. To further improve the mechanical properties of the structure, density-based TO method was introduced to optimize the diameters of all struts in the infill lattice. Finally, an SLM 3D printer was utilized to manufacture the mirror, and both stiffness and anti-vibration performance of the original and two different lattice infilled metal mirrors were evaluated through FEA. Compression tests were also conducted to verify the compressive property of all mirrors. Even though with 20% less material cost, both computational and experimental verifications have shown that the optimized lattice infilled mirror has much better mechanical properties compared with the original design.

Mechanical Performance of Lightweight-Designed Honeycomb Structures Fabricated Using Multijet Fusion Additive Manufacturing Technology.

Cellular structures including three-dimensional lattices and two-dimensional honeycombs have significant benefits in achieving optimal mechanical performance with light weighting. Recently developed design techniques integrated with additive manufacturing (AM) technologies have enhanced the possibility of fabricating intricate geometries such as honeycomb structures. Generally, failure initiates from the sharp edges in honeycomb structures, which leads to a reduction in stiffness and energy absorption performance. By material quantity, these hinges account for a large amount of material in cells. Therefore, redesigning of honeycomb structures is needed, which can improve aforementioned characteristics. However, this increases the design complexity of honeycombs, such that novel manufacturing techniques such as AM has to be employed. This research attempts to investigate the optimal material distribution of three different topologies of honeycomb structures (hexagonal, triangular, and square) with nine different design configurations. To achieve this, higher amount of material was distributed at nodes in the form of fillets while keeping overall weight of the structure constant. Furthermore, these design configurations were analyzed in terms of stiffness, energy absorption, and the failure behavior by performing finite element analysis and experimental tests on the samples manufactured using Multijet fusion AM technology. It was found that adding material to the edges can improve the mechanical properties of honeycombs such as stiffness and energy absorption efficiency. Furthermore, the failure mechanism is changed due to redistribution of material in the structure. The design configurations without fillets suffer from brittle failure at the start of the plastic deformation, whereas the configurations with increased material proportion at the nodes have larger plastic deformation zones, which improves the energy absorption efficiency.

A genetic algorithm of optimal design of beam at restricted sagging

A genetic algorithm for solving the problem of optimal beam material distribution along length at a given restriction on maximum sagging value is suggested. A review of literature sources is conducted and it was shown that existing solutions cover partial cases only in which the position of the point with maximum sagging was defined previously. In the paper presented I-section beam with constant proportions is considered, i.e., beam width, caps, and web thickness are proportional to beam height in the current cross-section. A statically determined beam is being considered. The load applied to a beam can be arbitrary, including cases of non-symmetrical loads and differently oriented ones. The position of point(s) at which beam sagging is maximum are unknown at the beginning of optimization and are found in the process solution. The problem is solved in the linear definition. Beam mass was assumed to be an optimization criterion. The method of finite differences is used for beam sagging finding, i.e., for the solution of the differential equation of the bending beam with a variable cross-section. Discretization allows transforming the problem of design into the problem of beam height determination at a system of reference points. At this stage, found values of beam height must satisfy restrictions on reference point displacements. The suggested technique allows controlling beam displacement quite flexibly because restrictions on point displacement are considered separately and do not depend on each other. The suggested objective function is the linear superposition of beam mass and the possible penalty in case of beam maximum sagging over exceeding predefined values. The application of a genetic algorithm allows getting sets of beam thicknesses those, which guaranty reaching the minimum of the objective function. The model problem is solved. It is shown that the suggested algorithm allows effectively solves problems of optimal design of beams with restrictions on the maximum sagging value. The suggested approach can be developed for strength restrictions, statically undetermined structures, etc.

Topology design and equivalent mechanical properties of a three-dimensional star-shaped auxetic structure

Topology design is the most important method of obtaining optimal material distribution that is directly related to the mechanical properties of structural materials. Aimed at improving the accuracy and efficiency of topology design, this paper explores an innovative topological technique that integrates a sigmoid-rational approximation of material properties approach, with a modified optimality criteria algorithm. Furthermore, a three-dimensional (3D) star-shaped auxetic structure, based on the sigmoid-rational approximation of material properties approach and a modified optimality criteria technique is proposed. The unit cell structural parameters are extracted, and the mechanical properties of the 3D star-shaped structure, including the relative density, equivalent elastic modulus, and equivalent Poisson's ratio, are investigated using mechanical derivation, numerical, and experimental methods. The subsequent results demonstrate that the sigmoid-rational approximation of material properties approach and a modified optimality criteria technique provides a feasible novel idea for multicellular structure development and that the advantageous 3D star-shaped auxetic structure has great potential within the field of structural safety.

Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials.

One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylindrical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies.

Design of the Battery Thermal Management System With Phase Change Material Coupled Cold Plates

The battery thermal management system with phase change material coupled cold plate was investigated using numerical simulation method. The results show that the temperature and temperature difference of the battery pack can be reduced by increasing the flow rates of the cold plate in the system, but the power consumption of the cold plate is significantly increased, leading to poor efficiency of the system. In order to improve the efficiency of the coupled thermal management system, an adjusting strategy was proposed to optimize the thickness distribution of phase change materials with the system volume fixed. The optimized results of typical case show that the optimal phase change material distribution can be obtained by only five adjusting steps. Compared with the original system, the maximum temperature of the battery pack is reduced by 1.1 K and the temperature difference is reduced by 29% after the optimization. When achieving the same temperature difference in battery pack, the power consumption of the optimized system is reduced by 64% compared with that before optimization.