This work presents benchmark examples related to the modelling of sound absorbing porous media with rigid frame based on the periodic geometry of their microstructures. To this end, rigorous ...mathematical derivations are recalled to provide all necessary equations, useful relations, and formulae for the so-called direct multi-scale computations, as well as for the hybrid multi-scale calculations based on the numerically determined transport parameters of porous materials. The results of such direct and hybrid multi-scale calculations are not only cross verified, but also confirmed by direct numerical simulations based on the linearised Navier-Stokes-Fourier equations. In addition, relevant theoretical and numerical issues are discussed, and some practical hints are given.
Advancements in additive manufacturing have led to the widespread adoption of topology and shape optimization in the design process across various industrial applications. Topology optimization has ...emerged as a promising approach for creating ultra-lightweight structures with exceptional functionality. By designing porous structures and addressing the layout of pores, concurrent multiscale topology optimization schemes offer a means to achieve such outcomes. This paper focuses on the development of a multiscale topology optimization tool and presents a numerical investigation of the initial design domain's impact on maximizing heat conductivity through concurrent multiscale topology optimization. The study highlights the significant influence of the initial design domain in the microscale on the final microscale design. Moreover, modifying the initial micro design domain has a direct impact on the macro design domain. The findings underscore the importance of considering the initial design domain when aiming to maximize heat conductivity through concurrent multiscale topology optimization. To facilitate the replication of the results presented in this paper, comprehensive details regarding parameter settings and implementation aspects are provided.
In this study, we present a shape optimization approach for designing the shapes of periodic microstructures using the homogenization method and the H1 gradient method. The compliance of a ...macrostructure is minimized under the constraint conditions of the total area of the microstructures distributed in the macrostructure, the elastic equation of the macrostructure and the homogenization equation of the unit cells. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function involving the state and adjoint variables for both the macro- and micro-structures is theoretically derived. Clear and smooth boundary shapes of the unit cells can be determined with the H1 gradient method. The proposed method is applied to multiscale structures, in which the numbers of domains with the microstructures are varied and the optimized shapes of the unit cells and the compliances obtained are compared. The numerical results confirm the effectiveness of the proposed method for creating the optimal shapes of microstructures distributed in macrostructures
► This review puts emphasis on their aspects of mechanics and computations. ► Auxetic materials are technologically and theoretically important and fascinating. ► Up until now, the research has been ...dominated by periodic/ordered microstructures. ► The future is with disordered microstructures using homogenization method.
This paper summarizes research work related to materials with zero, or negative Poisson’s ratio, materials which are also referred to as auxetic materials. This review puts an emphasis on computations and aspects of their mechanics. It also considers diverse examples: from large structural, to biomedical applications. It is concluded that auxetic materials are technologically and theoretically important. While the development of the research has been dominated by periodic/ordered microstructures, the author predicts that future research will be in the direction of disordered microstructures utilizing the homogenization method.
A topology optimization method is presented for the design of periodic microstructured materials with prescribed homogenized nonlinear constitutive properties over finite strain ranges. Building upon ...an existing computational homogenization method for periodic materials undergoing finite strain, generalized sensitivity equations are derived for the gradient-based topology optimization of periodic unit cells that account for the nonlinear homogenized response of the unit cell. The mechanical model assumes linear elastic isotropic materials, geometric nonlinearity at finite strain, and a quasi-static response. The optimization problem is solved by a nonlinear programming method and the sensitivities computed via the adjoint method. Several topology optimization examples are presented to evaluate the performance of the developed method. Furthermore, two-dimensional structures identified using this optimization method are additively manufactured and their uniaxial tensile strain response compared with the numerically predicted behavior. The optimization approach herein enables the design and development of lattice-like materials with prescribed nonlinear effective properties, for use in myriad potential applications, ranging from stress wave and vibration mitigation to soft robotics.
This paper is dedicated to the conceptual design of periodically microstructured metafilters/waveguides, made of thermoelastic material, described by a generalized thermoelastic theory with two ...relaxation times. In the general case of a three-dimensional material, the governing equations are first transformed into the s-space and the wave propagation is analysed through the multidimensional Floquet–Bloch transform. Special focus is then given to the case of a layered material. By exploiting the transfer matrix method, the spectral properties are analysed, focusing on the propagation of Bloch waves and the interaction between mechanical and thermal variables. Furthermore, two novel non-local continualization methods are proposed, addressing high-frequency approximations across different wave regimes and identifying non-local continua in general thermoelasticity contexts. The paper introduces innovative methods for spectral characterization, including a representation in the space of invariants and comparisons between continualization and homogenization techniques. Additionally, a novel method based on Padé approximants accelerates spectrum convergence.
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•Conceptual design of periodically microstructured thermoelastic waveguides.•Multifield continualization scheme in non-standard thermoelasticity.•Bloch wave propagation in periodic thermo-elastic Cauchy materials.•The Green–Linsdsay model accounting for two relaxation times is leveraged.•Illustrative examples, validating the accuracy of the approach, are presented.
•A novel shape optimization method for reducing thermal stress of porous structures.•Introduced homogenization method considering heat conduction and thermal elasticity.•Derived the sensitivity ...function theoretically for a multiscale shape design problem.•Achieved smooth pore design using the H1 gradient method, a non-parametric method.•Total volume can be properly distributed into the plural microstructures.
In this paper, we propose a shape optimization method to minimize the maximum thermal stress induced in the microstructures of a multiscale structure by heat conduction and thermal expansion. The weak- coupling problem is solved by applying the temperature distribution, obtained by solving the heat conduction problem, to the thermoelastic problem to find the maximum thermal stress caused by thermal expansion. The homogenization method is used to bridge the macrostructure and the porous microstructures, in which the elastic tensor, the tensor of the coefficients of thermal expansion, the thermal conductivity tensor and the thermal transfer coefficient are homogenized. The local thermal stress in the porous structure is minimized by shape optimization. The difficulty posed by non-differentiability of the local maximum stress is avoided by introducing a Kreisselmeier-Steinhauser function. It is assumed that the macrostructure consists of multiple subregions, in which the homogenized coefficients can be independently defined. This problem is formulated as a distributed-parameter optimization problem subject to a volume constraint, including all the microstructures. The shape gradient function for this design problem is derived for each subregion using the Lagrange multiplier method, the material derivative method and the adjoint method. The H1 gradient method is used to determine the optimal shape of the porous unit cell while reducing the objective function and maintaining smooth design boundaries. The effectiveness of the proposed method for minimizing the microstructural thermal stress of porous structures is confirmed by the numerical examples presented.
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Due to a lack of analytical solution to the issue of dynamics of asymmetric sandwich plates with a periodic microstructure, the only available method of modelling of such structures is FEM. However, ...such approach is usually a time-consuming process, which additionally requires a lot of computing resources due to a highly refined mesh. In this paper the analytical model of the mentioned structure, based on the tolerance averaging technique, is presented and discussed. The obtained model can be formulated with a system of partial differential equations with constant coefficients, which is relatively simple to solve. As a result, the proposed method is convenient for engineers and accurate. Moreover, unlike the asymptotic homogenisation method, it can be used to investigate the influence of microstructure on the overall behaviour of the plate. Eventually, in the calculation example a comparative simulations were conducted to investigate the influence of certain set of assumptions on the obtained results of free vibration frequencies and to verify the effectiveness and superiority of proposed calculation method over the FEM.
In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are ...addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.
Summary
An efficient second‐order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the ...proposed approach are (i) an asymptotic higher‐order nonlinear homogenization that does not require higher‐order continuity of the coarse‐scale solution and (ii) an efficient model reduction scheme for solving higher‐order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher‐order coarse‐scale derivatives by postprocessing from the zeroth‐order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth‐order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.
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