scholarly journals Stress minimization for lattice structures. Part I: Micro-structure design

2021 ◽  
Vol 379 (2201) ◽  
pp. 20200109 ◽  
Author(s):  
A. Ferrer ◽  
P. Geoffroy-Donders ◽  
G. Allaire
Keyword(s):  
Orientation Angle ◽  
Homogenization Method ◽  
Structure Design ◽  
Smoothing Parameter ◽  
Periodicity Cell ◽  
Lattice Structures ◽  
Compliance Minimization ◽  
Macroscopic Stress ◽  
Special Cases ◽  
Stress Minimization

Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

Two-Scale Topology Optimization With Parameterized Cellular Structures

2021 ◽  
Author(s):  
Sina Rastegarzadeh ◽  
Jun Wang ◽  
Jida Huang
Keyword(s):  
Topology Optimization ◽  
High Resolution ◽  
Structural Optimization ◽  
Optimization Problem ◽  
Material Design ◽  
Homogenization Method ◽  
Elasticity Tensor ◽  
Structure Design ◽  
Cellular Structures ◽  
Mapping Technique

Abstract Advances in additive manufacturing enable the fabrication of complex structures with intricate geometric details. It also escalates the potential for high-resolution structure design. However, the increasingly finer design brings computational challenges for structural optimization approaches such as topology optimization (TO) since the number of variables to optimize increases with the resolutions. To address this issue, two-scale TO paves an avenue for high-resolution structural design. The design domain is first discretized to a coarse scale, and the material property distribution is optimized, then using micro-structures to fill each property field. In this paper, instead of finding optimal properties of two scales separately, we reformulate the two-scale TO problem and optimize the design variables concurrently in both scales. By introducing parameterized periodic cellular structures, the minimal surface level-parameter is defined as the material design parameter and is implemented directly in the optimization problem. A numerical homogenization method is employed to calculate the elasticity tensor of the cellular materials. The stiffness matrices of the cellular structures derived as a function of the level parameters, using the homogenization results. An additional constraint on the level parameter is introduced in the structural optimization framework to enhance adjacent cellulars interfaces’ compatibility. Based on the parameterized micro-structure, the optimization problem is solved concurrently with an iterative solver. The reliability of the proposed approach has been validated with different engineering design cases. Numerical results show a noticeable increase in structure stiffness using the level parameter directly in the optimization problem than the state-of-art mapping technique.

Lattice Structure Design for Additive Manufacturing: Unit Cell Topology Optimization

2019 ◽  
Author(s):  
Bradley Hanks ◽  
Mary Frecker
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Lattice Structure ◽  
Optimization Method ◽  
Structure Design ◽  
Unit Cell ◽  
Lattice Structures ◽  
Design For Additive Manufacturing ◽  
Structure Topology

Abstract Additive manufacturing is a developing technology that enhances design freedom at multiple length scales, from the macroscale, or bulk geometry, to the mesoscale, such as lattice structures, and even down to tailored microstructure. At the mesoscale, lattice structures are often used to replace solid sections of material and are typically patterned after generic topologies. The mechanical properties and performance of generic unit cell topologies are being explored by many researchers but there is a lack of development of custom lattice structures, optimized for their application, with considerations for design for additive manufacturing. This work proposes a ground structure topology optimization method for systematic unit cell optimization. Two case studies are presented to demonstrate the approach. Case Study 1 results in a range of unit cell designs that transition from maximum thermal conductivity to minimization of compliance. Case Study 2 shows the opportunity for constitutive matching of the bulk lattice properties to a target constitutive matrix. Future work will include validation of unit cell modeling, testing of optimized solutions, and further development of the approach through expansion to 3D and refinement of objective, penalty, and constraint functions.

Mesoscale Lattice Structure Design and Simulation with the Support of a Property Database

2021 ◽  
pp. 247-273
Author(s):  
Guoying Dong ◽  
Yunlong Tang ◽  
Yaoyao Fiona Zhao
Keyword(s):  
Mechanical Performance ◽  
Lattice Structure ◽  
Cellular Materials ◽  
Structure Design ◽  
Lattice Structures ◽  
Strut Thickness ◽  
Design And Optimization ◽  
The Third ◽  
Unit Cells ◽  
The Difference

The lattice structure is a type of cellular materials [1] that has truss-like structures with interconnected struts and nodes in a three-dimensional (3D) space. Compared to other cellular materials such as random foams and honeycombs, the lattice structures exhibit better mechanical performance [2]. Some examples of lattice structures are shown in Figure 8.1. The first one is a randomized lattice structure. Due to the disordered lattice cells, the properties of this type of lattice structures are stochastic and difficult to control. But it can be used as implants in orthopedic surgeries. The second and the third are lattice structures with periodic unit cells. The difference is that the strut thickness of the second one is uniform, which is called homogeneous lattice structures. However, the third one has non-uniform strut thickness for specific loading conditions, which is called heterogeneous lattice structures. By properly adjusting the material in vital parts of the lattice structure, the heterogeneous periodic lattice structure can have a better mechanical performance than the homogeneous one with the same weight. Plenty of design and optimization methods [3-5] have been proposed for lattice structures to pursue better performance in different engineering applications. For example, the lattice structure is applied to achieve lightweight [3, 4], energy absorption [6], and thermal management [7]. Due to the complexity of the geometry, the fabrication of lattice structures had been the most critical issue. However, with the development of Additive Manufacturing (AM) processes, the difficulty in the fabrication was largely relieved.

Additive Manufacturing-Oriented Design of Graded Lattice Structures Through Explicit Topology Optimization

2017 ◽  
Vol 84 (8) ◽  
Author(s):  
Chang Liu ◽  
Zongliang Du ◽  
Weisheng Zhang ◽  
Yichao Zhu ◽  
Xu Guo
Keyword(s):  
Topology Optimization ◽  
Structure Design ◽  
Lattice Structures ◽  
New Approach ◽  
Optimization Framework ◽  
Concurrent Optimization ◽  
Design Variables ◽  
Graded Material ◽  
Graded Lattice ◽  
Moving Morphable Components

In the present work, a new approach for designing graded lattice structures is developed under the moving morphable components/voids (MMC/MMV) topology optimization framework. The essential idea is to make a coordinate perturbation to the topology description functions (TDF) that are employed for the description of component/void geometries in the design domain. Then, the optimal graded structure design can be obtained by optimizing the coefficients in the perturbed basis functions. Our numerical examples show that the proposed approach enables a concurrent optimization of both the primitive cell and the graded material distribution in a straightforward and computationally effective way. Moreover, the proposed approach also shows its potential in finding the optimal configuration of complex graded lattice structures with a very small number of design variables employed under various loading conditions and coordinate systems.

scholarly journals A homogenization method for thermomechanical continua using extensive physical quantities

2012 ◽  
Vol 468 (2142) ◽  
pp. 1696-1715 ◽  
Author(s):  
Kranthi K. Mandadapu ◽  
Arkaprabha Sengupta ◽  
Panayiotis Papadopoulos
Keyword(s):  
Heat Flux ◽  
Continuum Mechanics ◽  
Equations Of Motion ◽  
Homogenization Method ◽  
Deformation Condition ◽  
Balance Laws ◽  
Macroscopic Stress ◽  
Stress Deformation ◽  
The One ◽  
Physical Quantities

This article proposes a continuum thermomechanical homogenization method inspired by the Irving–Kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics. This method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities. The resulting equation for macroscopic stress affords a rational comparison with the widely used Hill–Mandel stress-deformation condition, while the one for heat flux reduces, under certain assumptions, to a Hill–Mandel-like condition involving heat flux and the gradient of temperature.

scholarly journals Homogenization of fully nonlinear rod lattice structures: on the size of the RVE and micro structural instabilities

2021 ◽  
Author(s):  
Ludwig Herrnböck ◽  
Paul Steinmann
Keyword(s):  
Large Deformations ◽  
Mechanical Response ◽  
Three Dimensional ◽  
Homogenization Method ◽  
Volume Element ◽  
Lattice Structures ◽  
Micro Scale ◽  
Macro Scale ◽  
Unit Cells ◽  
Structural Instabilities

AbstractThis work investigates the possibility of applying two-scale computational homogenization to rod lattice structures emerging, for instance, from additive manufacturing. The influence of the number of unit cells within the representative volume element (RVE), thus, the RVE’s size on the homogenized mechanical response is studied for occurring microscopic structural instabilities. Therein, the macro-scale, described in terms of three-dimensional continuum mechanics, is coupled to the micro-scale described by geometrically exact rods, enabling arbitrary large deformations and rotations. A special feature of the presented framework is that the rods building the lattice structures are not restricted to deform purely elastically but may deform inelastically. The mechanical response of lattice structures is investigated by applying the developed homogenization method to an exemplary lattice. Under special loads the structure reaches an instable state and may buckle. The appearance of instabilities depends on the geometric properties of the lattice’s underlying rods and the RVE’s size.

scholarly journals Geometric and semantic quality assessments of building features in OpenStreetMap for some areas of Istanbul

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Melih Basaraner
Keyword(s):  
Spatial Data ◽  
Large Scale ◽  
Data Entry ◽  
Orientation Angle ◽  
Type Definition ◽  
Free Data ◽  
Special Cases ◽  
Geometric Quality ◽  
The One ◽  
Shape Complexity

AbstractNowadays volunteered geographic information (VGI) and collaborative mapping projects such as OpenStreetMap (OSM) have gained popularity as they not only offer free data but also allow crowdsourced contributions. Spatial data entry in this manner creates quality concerns for further use of the VGI data. In this regard, this article focuses on the assessments of geometric and semantic quality of the OSM building features (BFs) against a large-scale topographic (TOPO) data belonging to some areas of Istanbul. The comparison is carried out based on the one-to-one matched BFs according to a geometric matching ratio. In geometric terms, various parameters of position (i.e. X, Y), size (i.e. area, perimeter and granularity), shape (i.e. convexity, circularity, elongation, equivalent rectangular index, rectangularity and roughness index), and orientation (i.e. orientation angle) elements are computed and compared. In semantic terms, BF type coherences are evaluated. According to the findings of geometric quality, the average positional difference was less than three meters. In addition, the perimeter values tended to decrease while area and granularity values tended to increase in OSM data against TOPO data. Those showed that the level of the detail of the OSM BFs was lower than TOPO BFs in general. This was also confirmed by the decreasing tendency of shape complexity according to the parameters of shape element. Orientation angle differences was often low except for some special cases. It was found that the scale of the OSM dataset, even though not homogenous, approximately corresponded to the lower limit of medium scale maps (i.e. 1:10,000) or a slightly smaller scale. According to the findings of semantic quality, in case of the presence of specific type definition, the coherence was rather high between OSM and TOPO BFs while the most OSM BFs did not have a specific type attribute. This study showed that the matching process needed some improvements while the followed approach was largely successful in the evaluation of the matched buildings from geometric and semantic aspects.

Synthesis Methods for Lightweight Lattice Structures

2009 ◽  
Author(s):  
Gregory C. Graf ◽  
Jane Chu ◽  
Sarah Engelbrecht ◽  
David W. Rosen
Keyword(s):  
Lattice Structure ◽  
Problem Formulation ◽  
Structure Design ◽  
Synthesis Methods ◽  
Lattice Structures ◽  
Element Analysis ◽  
High Stiffness ◽  
Complex Design ◽  
Solution Methods ◽  
Lm Algorithm

Lattice structures are a type of cellular material (e.g., honeycombs, foams, trusses) that can achieve very high stiffness-to-weight and strength-to-weight ratios. When loaded, elements in lattice structures typically stretch and compress, rather than bend, enabling their advantageous performance characteristics. We desire efficient algorithms for searching the large, complex design spaces associated with lattice structure design. In this paper, we present a problem formulation for lattice structure design, using cross-section sizes as variables, and three proposed solution methods. The methods are Particle Swarm Optimization (PSO), Levenburg-Marquardt (LM) algorithm based on a least-squares minimization formulation, and a new non-iterative size matching and scaling method based on results of a finite-element analysis, which are evaluated for their capabilities in achieving light weight and high stiffness. Two-dimensional and 3-D examples are used to test the solution methods.

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