Optimization of Bodies with Locally Periodic Microstructure (original) (raw)
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Optimization of bodies with locally periodic microstructure by varying the periodicity pattern
Networks and Heterogeneous Media, 2014
This paper describes a numerical study of the optimization of elastic bodies featuring a locally periodic microscopic pattern. Our approach makes the link between the microscopic level and the macroscopic one. It adds a new component (variation of the periodicity pattern) to previously published works by the same authors. Two-dimensional linearly elastic bodies are considered; the same techniques can be applied to threedimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. The macroscopic domain is divided in (rectangular) finite elements and in each of them the microstructure is supposed to be periodic; the periodic pattern is allowed to vary from element to element. Each periodic microstructure is discretized using a finite element mesh on the periodicity cell, by identifying the opposite sides of the cell in order to handle the periodicity conditions in the cellular problem. Shape, topology and periodicity optimization are used at the microscopic level, following an alternate directions algorithm. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.
Shape optimization of periodic structures
Computational Mechanics, 2003
This paper describes a numerical approach to the optimization of effective properties of periodic perforations in an infinite body, in the frameworks of heat conduction and of linear elasticity. We implement a special finite element mesh in order to deal with the periodic nature of the problem. We compute the gradient of the functional to be minimized. We describe the process of mesh deformation and mesh regeneration. We give several numerical examples, some of them having practical relevance.
International Journal for Numerical Methods in Engineering, 2002
The problem of determining highly localized buckling modes in perfectly periodic cellular microstructures of inÿnite extent is addressed. A double scale asymptotic technique is applied to the linearized stability problem for a periodic structure built from linearly elastic microstructures. The obtained stability condition for the microscale level is then used to establish a comparative analysis between di erent material distributions in the base cell subjected to the same strain ÿeld at the macroscale level. The idea is illustrated by some two-dimensional ÿnite element examples and used to design materials with optimal elastic properties that are less prone to localized instability in the form of local buckling modes at the scale of the microstructure.
2008
The present paper deals with the implementation of an optimization algorithm for periodic problems which alternates shape and topology optimization (the theoretic al background about shape and topological derivatives was developed in Part I [7]). The proposed numerical code lays on a special implementation of the periodicity conditions based on differential geometry concepts: periodic functio s are viewed as functions defined on a torus. Moreover the notion of periodicity is extended and cases where the periodicity cell is a general parallelogram are admissible. Thi s approach can be adapted to other frameworks (fluids, cable design, etc.). The numerical method was tested for the design of periodic microstructures. Several examples of opti mal microstructures are given for bulk modulus maximization, maximization of rigidity for shear response, maximiz ation of rigidity in a prescribed direction, minimization of the Poisson coefficient.
Topology optimization for microstructural design under stress constraints
Structural and Multidisciplinary Optimization, 2018
This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits un
Multiscale Topology Optimization of Structures and Periodic Cellular Materials
Volume 3A: 39th Design Automation Conference, 2013
Topology optimization allows designers to obtaining lightweight structures considering the binary distribution of a solid material. Further material savings and increased performance may be achieved if the material and the structure topologies are concurrently optimized. The use of homogenization methods promotes the introduction of material-scale parameters in the problem's formulation. While some research has been focused on material parameters and periodic topology optimization, this work deals with non-periodic material topologies. Since no preconceived material and structure geometries are considered, the multiscale approach is capable of driving the design to innovative and potentially better configurations at both length scales. The proposed methodology is applied to minimum compliance problems and compliant mechanism synthesis. The multiscale results are compared with the traditional structural-level designs in the context of Pareto solutions, demonstrating benefits of ultra-lightweight configurations. 2.
Acta Mechanica Sinica, 2007
An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases.
Topology optimization of three-dimensional structures using optimum microstructures
Structural Optimization, 1998
In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic compliance or minimum total elastic energy at equilibrium. The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem. Several examples of optimum topology designs of threedimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual twodimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.
Journal of Mechanics of Materials and Structures, 2017
Additive manufacturing has enabled the fabrication of lightweight materials with intricate cellular architectures. These materials are interesting due to their properties which can be optimized upon the choice of the parent material and the topology of the architecture, making them appropriate for a wide range of applications including lightweight aerospace structures, energy absorption, thermal management, metamaterials, and bioscaffolds. In this paper we present the simplest initial computational framework for the analysis, design, and topology optimization of low-mass metallic systems with architected cellular microstructures. A very efficient elastic-plastic homogenization of a repetitive Representative Volume Element (RVE) of the microlattice is proposed. Each member of the cellular microstructure undergoing large elastic-plastic deformations is modeled using only one nonlinear three-dimensional (3D) beam element with 6 degrees of freedom (DOF) at each of the 2 nodes of the beam. The nonlinear coupling of axial, torsional, and bidirectional-bending deformations is considered for each 3D spatial beam element. The plastic hinge method, with arbitrary locations of the hinges along the beam, is utilized to study the effect of plasticity. We derive an explicit expression for the tangent stiffness matrix of each member of the cellular microstructure using a mixed variational principle in the updated Lagrangian corotational reference frame. To solve the incremental tangent stiffness equations, a newly proposed Newton homotopy method is employed. In contrast to the Newton's method and the Newton-Raphson iteration method, which require the inversion of the Jacobian matrix, our homotopy methods avoid inverting it. We have developed a code called CELLS/LIDS (CELLular Structures/Large Inelastic DeformationS), which provides the capabilities to study the variation of the mechanical properties of the low-mass metallic cellular structures by changing their topology. Thus, due to the efficiency of this method we can employ it for topology optimization design and for impact/energy absorption analyses.