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As an important class of natural and engineered materials, periodic microstructural composites have drawn substantial attention from the material research community for their excellent flexibility in tailoring various desirable physical behaviors. To develop periodic cellular composites for multifunctional applications, this paper presents a unified design framework for combining stiffness and a range of physical properties governed by quasi-harmonic partial differential equations. A multiphase microstructural configuration is sought within a periodic base-cell design domain using topology optimization. To deal with conflicting properties, e.g. conductivity/permeability versus bulk modulus, the optimum is sought in a Pareto sense. Illustrative examples demonstrate the capability of the presented procedure for the design of multiphysical composites and tissue scaffolds.

The heavy duty diesel engine must have a large output for maintaining excellent mobility. In this study, a three-dimensional finite element model of a heavy-duty diesel engine was developed to conduct the stress analysis by using property of CGI. The compacted graphite iron (CGI) is a material currently under study for the engine demanded for high torque, durability, stiffness, and fatigue. The FE model of the heavy duty diesel engine section consisting of four half cylinders was selected. The heavy duty diesel engine section includes a cylinder block, a cylinder head, a gasket, a liner, a bearing cap, bearing and bolts. The loading conditions of engine are pre-fit load, assembly load, and gas load. A structural analysis on the result was performed in order to optimize on the cylinder block of the diesel engine.

The imperative for lightweighting technologies, paramount in mission-critical cyber-physical systems (CPSs) including aerospace, automotive and allied sectors, hinges upon optimizing energy efficiency and curbing product weight. Honeycomb structures, celebrated for their exceptional strength-to-weight ratio, have indisputably guided the pursuit of lightweight design. This paper expounds upon the versatility of honeycomb structures by scrutinizing their in-plane mechanical attributes. Leveraging finite element simulations and polynomial fitting, we enhance the prevailing equivalent elastic modulus model for uniform honeycomb structures, expanding its domain to encompass a broader spectrum of relative density values. Deliberations ensue concerning the model’s constraints and its inapplicability to nonuniform honeycomb structures. The investigation introduces nodes as pivotal influencers in the mechanical comportment of nonuniform honeycomb structures, delineating the nexus between the equivalent elastic modulus and node dimensions through a fusion of finite element simulations and mechanical experimentation. Furthermore, this research delves into the tenets and constructs of density-based variable density methodologies within the ambit of topology optimization, with an overarching goal of maximizing stiffness. We furnish a holistic design protocol for optimizing honeycomb structures, underscored by a pragmatic instantiation of the density-based variable density approach. Scrutinizing the geometric interplay between honeycomb structures and design spaces, we posit an innovative paradigm employing concentric circles to approximate cellular envelopes, streamlining numerical cartography and the conversion of optimization outputs into variable density honeycomb configurations. Evaluation of the in-plane mechanical attributes of variable density honeycomb structures reveals that TPU material augments the resilience of both uniform and variable density honeycomb structures, whereas topology optimization amplifies specific stiffness and resilience modulus in variable density honeycomb structures relative to their uniform counterparts. This study sheds light on the complexities of honeycomb structures, providing valuable insights for their optimization in lightweight applications.

Topology optimization for fluid flow aims at finding the location of a porous medium minimizing a cost functional under constraints given by the Navier–Stokes equations. The location of the porous media is usually taken into account by adding a penalization term $\alpha u$, where $\alpha$ is a kinematic viscosity divided by a permeability and $u$ is the velocity of the fluid. The fluid part is obtained when $\alpha =0$ while the porous (solid) part is defined for large enough $\alpha$ since this formally yields $u=0$. The main drawback of this method is that only solid that does not let the fluid to enter, that is perfect solid, can be considered. In this paper, we propose to use the porosity of the media as optimization parameter hence to minimize some cost function by finding the location of a porous media. The latter is taken into account through a singular perturbation of the Navier–Stokes equations for which we prove that its weak-limit corresponds to an interface fluid-porous medium problem modeled by the Navier–Stokes–Darcy equations. This model is then used as constraint for a topology optimization problem. We give necessary condition for such problem to have at least an optimal solution and derive first order necessary optimality condition. This paper ends with some numerical simulations, for Stokes flow, to show the interest of this approach.

We introduce a model and several constraints for shape and topology optimization of structures, built by additive manufacturing techniques. The goal of these constraints is to take into account the thermal residual stresses or the thermal deformations, generated by processes like Selective Laser Melting, right from the beginning of the structural design optimization. In other words, the structure is optimized concurrently for its final use and for its behavior during the layer-by-layer production process. It is well known that metallic additive manufacturing generates very high temperatures and heat fluxes, which in turn yield thermal deformations that may prevent the coating of a new powder layer, or thermal residual stresses that may hinder the mechanical properties of the final design. Our proposed constraints are targeted to avoid these undesired effects. Shape derivatives are computed by an adjoint method and are incorporated into a level set numerical optimization algorithm. Several 2D and 3D numerical examples demonstrate the interest and effectiveness of our approach.

A directional coupler in a two-dimensional photonic crystal slab waveguide was optimized to operate in wideband by applying topology optimized bends at input/output ports. We fabricated the sample based on the optimized design and demonstrated that the directional coupler performs properly in the transmittance and the bandwidth comparable to the straight waveguide. We experimentally confirmed the excellent extinction ratio (12 ~ 15dB) of bar and cross port.

This paper presents a topology optimization method for dynamic problems with an improved bi-directional evolutionary structural optimization (BESO) technique. The sensitivity derivation for the frequency optimization problem in the case of multiple eigenvalues and for the stiffness–frequency optimization problem is proposed. Algorithms for a filter scheme, sensitivity history-averaging, and sensitivity global-ranking are used in the present method. Techniques for adaptively removing alternative elements and eliminating singular and single-hinged elements are proposed. Solution-convergence and localized modes are discussed through numerical examples. Results show that the improved BESO method is capable of solving the frequency optimization problem and the multi-objective optimization problem for stiffness and frequency effectively.

This paper addresses the design problem of piezoelectric actuators for multimodal active vibration control. The design process is carried out by a topology optimization procedure which aims at maximizing a control performance index written in terms of the controllability Gramian, which is a measure that describes the ability of the actuator to move the structure from an initial condition to a desired final state in a finite time interval. The main work contribution is that independent sets of design variables are associated with each modal controllability index, then the multi-objective problem can be split into independent single-objective problems. Thus, no weighting factors are required to be tuned to give each vibration mode a suitable relevance in the optimization problem. A material interpolation scheme based on the Solid Isotropic Material with Penalization (SIMP) and the Piezoelectric Material with Penalization (PEMAP) models is employed to consider the different sets of design variables and the sensitivity analysis is carried out analytically. Numerical examples are presented by considering the design and vibration control for a cantilever beam and a beam fixed at both ends to show the efficacy of the proposed formulation. The control performance of the optimized actuators is analyzed using a Linear-Quadratic Regulator (LQR) simulation.

In this paper, we study vibro-acoustic behavior of auxetic sandwich panels subjected to different excitations and boundary conditions. The core of this panel has the auxetic feature (with negative Poisson’s ratio or NPR) with anti-tetrachiral honeycomb structure. Mechanical behavior of the core is formulated using theoretical relations presented for this kind of auxetic. Using the Finite Element Method, the modal analysis and spectral analysis of the structure are accomplished. Different random colored noises are applied as the system excitation. First, a parametric study is performed; and some interesting results are observed from investigating the effects of geometric parameters, boundary conditions, and noise color on the vibro-acoustic behavior of the structure. These parameters affect the natural frequencies, level of radiated sound, and mass of the structure. An optimization algorithm is applied to the geometrical parameters in order to simultaneously reduce the level of radiated sound and preserve the amount of total mass. By the use of the Genetic Algorithm (GA), we could achieve a remarkable noise attenuation gain. It is shown that the GA choses different optimized parameters for the structure according to the location of the load and frequency content of the load spectrum.

This paper aims at to improve the vibration behavior of the train floor panel by the use of a cellular auxetic layer. A field measurement is performed to obtain the vibrational frequency content of the body floor moving on the tangent track. Using acceleration sensors, the vibrational response is measured on the bogie (as the input excitation) and on the floor panel (as the observation response). Finite Element modeling for the floor panel is accomplished and measurement data are used for both the input excitation and the verification of the numerical results. The floor panel is a sandwich panel containing multiple layers. In this study, the conventional wooden layer of the panel is substituted with a cellular auxetic one with a re-entrant hexagonal pattern. Then, an optimization problem is defined while the topological parameters of the auxetic layer are the design variables and the dynamic performance of the panel is the objective function. The parameter of power–mass–ratio (PMR) is defined taking the effects of both weight and dynamic response amplitude into calculation. It is found that the PMR is reduced to almost 0.6 by replacing the wooden layer with an auxetic one, and after topological optimization, it is reduced to 0.35.

Objective: To reduce the shielding effect caused by the large elastic modulus of metal fusion devices after osteotomy for the treatment of scoliosis. Methods: A personalized fusion device was designed using reverse engineering techniques, three-dimensional modeling, topology optimization, and finite element analysis. A finite element model of the lumbar spine after orthopedic surgery was established from an actual case. A fusion device was implanted into the lumbar spine before and after the optimized design for simulation calculations, respectively. The similarities and differences in the mechanical properties of the different fusion devices and vertebrae were compared. Based on the topological optimization of the mechanical properties of the fusion system, a lightweight fusion device was designed and the stress-shielding effect on the lumbar vertebrae was improved. Results: The topologically optimized fusion device described in the current case was 60% lighter while maintaining the strength of the fusion. After optimization, the average strain of the fusion was increased by up to 200% and the stiffness was significantly decreased. The average equivalent stress of cancellous bone was increased by up to 9.55% for the PEEK fusion device. In contrast, the topologically optimized fusion can increase the average equivalent stress of cortical bone by 5.63% and reduce the stress-masking effect of vertebrae. Conclusion: Topologically optimized titanium alloy fusion devices can significantly reduce the mass of the implant as well as reduce fusion stiffness. Additionally, a topology-optimized titanium alloy fusion device can effectively improve the stress-shielding effect of cortical bone.

We establish topology optimization model in terms of Independent Continuum Map method (ICM), so as to avoid the difficulties caused by multiple objective functions of compliance, owing to referring to weight as objective function. Using the distorted-strain-energy criterion, we transform stress constraints on all elements into structure strain-energy constraints in global sense. Then, the problem of topological optimum of continuum structure subjected to global strain-energy constraints is formulated and solved. The process of optimization is conducted through three basic steps which include the computation of the minimum strain energy of structure corresponding to the maximum strain-energy under the load case due to prescribing weight constraint, the determination of the allowable strain-energy of structure for every load case by using a formula from our numerical tests, as well as the establishment and solution of optimization model with the weight function due to all allowable strain energies. A strategy that is available to cope with complicated load ill-posedness in terms of different complementary approaches one by one is presented in the present work. Several numerical examples demonstrate that the topology path of transferring forces can be obtained more readily by global strain energy constraints rather than local stress constraints, and the problem of load ill-posedness can be tackled very well by the weighting method with regard to structural strain energy as weighting coefficient.

The modal topology optimization method of continuum structure based on element-free Galerkin (EFG) method is presented by integrating solid isotropic material with penalization (SIMP) method with the optimality criteria method, and the penalty method is used to impose essential boundary conditions. The density of Gauss point and nodal density are selected as the design variables respectively, and the maximum of the first-order natural frequency is specified as the objective function. The sensitivity analysis algorithm is derived by using direct differential method. The examples are finished by selecting the two types of design variables respectively. The results obtained show that the checkerboard phenomenon does not appear when nodal density is selected as the design variable, and also verify that topology optimization method presented is feasible.

Various formulations of smoothed particle hydrodynamics (SPH) have been presented by scientists to overcome inherent numerical difficulties including instabilities and inconsistencies. Low approximation accuracy could cause a result of particle inconsistency in SPH and other meshfree methods. In this study, centroid Voronoi tessellation (CVT) topology optimization is used for rearrangement of particles so that the inconsistency due to irregular particle arrangement can be corrected. Using CVT topology optimization method, the SPH particles, which are generated randomly inside a predetermined domain, are moved to the centroids, i.e., the center of mass of the corresponding Voronoi cells based on Lloyd’s algorithm. The volume associated with each particle is determined by its Voronoi cell. On the other hand, it has been shown that particle methods with stress point integration are more stable than the ones using nodal integration. Conventional SPH approximations only use SPH particles, and it results in the so-called tensile instability. In this paper, a new approach of using stress points is introduced to assist SPH approximations and stabilize the SPH methods.

This paper presents an algorithm for structural topology optimization involving linear buckling. In this algorithm, finite element analysis (FEA) is conducted only in a domain with solid and gray elements, eliminating the contribution of low density elements; and the response function is constructed in the full design domain accounting the contribution of removed low density elements. The errors induced by removing void elements in FEA on eigenvalue and eigenvectors are analyzed. By introducing a dynamic low bound of the first eigenvalue and a load-path coefficient, the algorithm allows converged, nondisjointed and accurate solutions for topology optimization of structures involving buckling. Numerical results are presented for plate and column-beam structures against linear buckling to illustrate the efficiency and effectiveness of the present algorithm. Buckling experiments of the plates manufactured from the obtained topologies further verify the present algorithm.

This paper presents a simple yet efficient method for the topology optimization of continuum structures considering interval uncertainties in loading directions. Interval mathematics is employed to equivalently transform the uncertain topology optimization problem into a deterministic one with multiple load cases. An efficient soft-kill bi-directional evolutionary structural optimization (BESO) method is proposed to solve the problem, which only requires two finite element analyses per iteration for each external load with directional uncertainty regardless of the number of the multiple load cases. The presented algorithm leads to significant computational savings when compared with Monte Carlo-based optimization (MCBO) algorithms. A series of numerical examples including symmetric and nonsymmetric loading variations demonstrate the considerable improvement of computational efficiency of the proposed approach as well as the significance of including uncertainties in topology optimization when to design a structure. Optimums obtained from the proposed algorithm are verified by MCBO method.

This work performs a topology optimization of the interior structure of engine blades in compressors with any given geometry of the desired outer-surface shape that may be determined by CFD and aerodynamic design software for the desired performance for thermal and fluid flows. A lofted compressor airfoil surface from the aerodynamic design was used to create a three-dimensional (3D) solid in SolidWorks. This was converted to an .IGS file that would be imported into HyperMesh® for the meshing and submitted to OptiStruct® for optimization. An optimization process is designed to produce an optimal interior structure, considering both pressure on the outer surface and centrifugal forces produced by rotational movements. The optimized blade becomes hollow in an optimal pattern with minimum materials needed for the pressure loading on outer skin and the distributed centrifugal forces. The final design was compared to the initial design using finite element method (FEM) to confirm that the mass, stress, strain, and displacement were reduced. The mass was reduced by 59.8% and the stresses reduced by a factor of 3.66! These results were validated by conducting a mesh independence study. 3D printers were used to produce the optimized blades in both plastic and metal.

We combine isogeometric analysis (IGA), level set (LS) and pointwise density-mapping techniques for design and topology optimization of piezoelectric/flexoelectric materials. We use B-spline elements to discretize the fourth-order partial differential equations of flexoelectricity, which require at least $C^1$ continuous approximations. We adopt the multiphase vector LS model which easily copes with various numbers of material phases and multiple constraints. In case studies, we first confirm the accuracy of the IGA model and then provide numerical examples for both pure and composite flexoelectric structures. The results demonstrate the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization and particularly by multi-material topology optimization for flexoelectric composites.

Modal analysis is widely used to investigate the dynamic characteristics of large and complex structures. For finite element models, iterative solvers are needed to precisely calculate eigenpairs or frequency and vibration mode. However, in cases such as large-scale analysis or reanalysis studies, or optimization design of a huge structure, computational cost may become too time consuming. This paper focuses on the quick structural modal analysis based on the reanalysis technique for large complex structures. Based on the stiffness and mass matrix of the analytical structures, a high precision and efficiency eigensolution is generated by the proposed modal analysis method (the Pseudo Random Independent and Coupling Inverse Iteration (PRICII) method), which combines the pseudo random number initialization, ICII (Independent and Coupling Inverse Iteration) strategy with the double Rayleigh–Ritz analysis. By comparing with the Subspace iteration method, Lanczos method, etc. the large-scale numerical examples show that the actual computational savings of the proposed method are usually higher than 75% with sufficient precision. Also, its applications in topology optimization are greatly effective.

Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson’s ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.