Topological sensitivity analysis in heterogeneous anisotropic elasticity problem. Theoretical and computational aspects

Giusti, Sebastián M.; Ferrer, A.; Oliver, J.

doi:10.1016/j.cma.2016.08.004

Cited by 38 publications

(35 citation statements)

References 34 publications

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“…As shown in the work of Novotny and Sokolowski, in the case of the compliance, the topological derivative can be written as follows:

scriptT false(\overset{x}{true} false) = normalσ false(\overset{x}{true} false) : double-struckP : \nabla^{s} u false(\overset{x}{true} false)

with

double-struckP

being the fourth‐order polarization tensor and ∇ s u the symmetric gradient of the displacements. For isotropic materials, plane stress, and circular inclusions, (see the work of Giusti et al) the polarization tensor adopts the form

P = - \frac{1}{2} \frac{1}{β γ + {normalτ}_{1}} [(1 + β) ({normalτ}_{1} - γ) I + \frac{1}{2} (α - β) \frac{γ (γ - 2 {normalτ}_{3}) + {normalτ}_{1} {normalτ}_{2}}{α γ + {normalτ}_{2}} (I \otimes I)],

where I and

double-struckI

denote the second‐ and fourth‐order identity tensors, respectively, and

α = \frac{1 + ν}{1 - ν}, β = \frac{3 - ν}{1 + ν}, γ = \frac{E^{⋆}}{E}, {normalτ}_{1} = \frac{1 + {normalν}^{⋆}}{1 + ν}, {normalτ}_{2} = \frac{1 - {normalν}^{⋆}}{1 - ν}, and {normalτ}_{3} = {normalν}^{⋆}

…”

Section: Topological Derivative For Multiscale Topology Optimizationmentioning

confidence: 99%

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Two‐scale topology optimization in computational material design: An integrated approach

Ferrer

Cante

Hernández

et al. 2018

Numerical Meth Engineering

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SummaryIn this work, a new strategy for solving multiscale topology optimization problems is presented. An alternate direction algorithm and a precomputed offline microstructure database (Computational Vademecum) are used to efficiently solve the problem. In addition, the influence of considering manufacturable constraints is examined. Then, the strategy is extended to solve the coupled problem of designing both the macroscopic and microscopic topologies. Full details of the algorithms and numerical examples to validate the methodology are provided.

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“…As shown in the work of Novotny and Sokolowski, in the case of the compliance, the topological derivative can be written as follows:

scriptT false(\overset{x}{true} false) = normalσ false(\overset{x}{true} false) : double-struckP : \nabla^{s} u false(\overset{x}{true} false)

with

double-struckP

P = - \frac{1}{2} \frac{1}{β γ + {normalτ}_{1}} [(1 + β) ({normalτ}_{1} - γ) I + \frac{1}{2} (α - β) \frac{γ (γ - 2 {normalτ}_{3}) + {normalτ}_{1} {normalτ}_{2}}{α γ + {normalτ}_{2}} (I \otimes I)],

where I and

double-struckI

denote the second‐ and fourth‐order identity tensors, respectively, and

α = \frac{1 + ν}{1 - ν}, β = \frac{3 - ν}{1 + ν}, γ = \frac{E^{⋆}}{E}, {normalτ}_{1} = \frac{1 + {normalν}^{⋆}}{1 + ν}, {normalτ}_{2} = \frac{1 - {normalν}^{⋆}}{1 - ν}, and {normalτ}_{3} = {normalν}^{⋆}

…”

Section: Topological Derivative For Multiscale Topology Optimizationmentioning

confidence: 99%

“…In this work, we set α 0 =10 −3 . For the derivation of the polarization tensor

double-struckP

for the case of anisotropic materials, the readers are referred to the work of Giusti et al…”

Section: Topological Derivative For Multiscale Topology Optimizationmentioning

confidence: 99%

Two‐scale topology optimization in computational material design: An integrated approach

Ferrer

Cante

Hernández

et al. 2018

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…The parameter E 0 is a reference Young's modulus, typically the modulus of the stiff phase of the designed composite.Therefore, after the graded material has been defined in the complete domain through the FMO methodology, the heuristic condition removes the subdomains, where the demanded material resource is low. A similar result can also be obtained using a more formal mathematical technique, for example that based on the topology optimization algorithm described in the work of Giusti et al Note that the topologies of Ω red and Ω may be different. A second FMO problem is solved in the domain Ω red by imposing the additional constraint

false(\overset{C}{true} - δ bold-italic1 false) \in S^{+} .

The scalar δ >0 is a small parameter ensuring that all the elasticity tensor eigenvalues are non‐null. This constraint has been proposed by Schury as a manufacture restriction.Even when constraint generates suboptimal solutions, it facilitates the microstructure design because it fixes lower bounds to the material properties.…”

Section: Overview Of the Two‐scale–based Approachmentioning

confidence: 58%

“…Therefore, after the graded material has been defined in the complete domain through the FMO methodology, the heuristic condition removes the subdomains, where the demanded material resource is low. A similar result can also be obtained using a more formal mathematical technique, for example that based on the topology optimization algorithm described in the work of Giusti et al Note that the topologies of Ω red and Ω may be different.…”

Section: Overview Of the Two‐scale–based Approachmentioning

confidence: 58%

Material design of elastic structures using Voronoi cells

Podestá

Méndez²,

Toro

et al. 2018

Numerical Meth Engineering

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Summary New tools for the design of metamaterials with periodic microarchitectures are presented. Initially, a two‐scale material design approach is adopted. At the structure scale, the material effective properties and their spatial distribution are obtained through a Free Material Optimization technique. At the microstructure scale, the material microarchitecture is designed by appealing to a Topology Optimization Problem (TOP). The TOP is based on the topological derivative and the level set function. The new proposed tools are used to facilitate the search of the optimal microarchitecture configuration. They consist of the following: (i) a procedure to choose an adequate shape of the unit cell domain where the TOP is formulated and shapes of Voronoi cells associated with Bravais lattices are adopted and (ii) a procedure to choose an initial material distribution within the Voronoi cell being utilized as the initial configuration for the iterative TOP.

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“…χ R 2 \Ω ε z z z Carpio (Carpio and Rapún, 2008) (Lopes et al, 2015, Giusti et al, 2016) Relationship between the optimal configuration with two materials and Young's modulus. Table 1 Values of the objective functional when various values of E 2 were set for each configuration shown in Fig.…”

Section: ·1mentioning

confidence: 99%

Topology optimization for multi-material structures based on the level set method

Kishimoto

Noguchi

Sato

et al. 2017

Transactions of the JSME (In Japanese)

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Topology optimization is the most flexible type of structural optimization method. This method has been applied in a variety of physics problems dealing with a multitude of design problems. In a given design problem, however, the optimization problem often has conflicting evaluative functions, such as the need for high rigidity in combination with minimal weight. The difficulty of simultaneously achieving high performance for two or more functions may be further compounded because current topology optimization methods typically only deal with a single material. On the other hand, when multiple kinds of materials having various properties can be selected for use, the range of a designer's choices is increased and an appropriate solution that greatly improves product functions may be achieved. Thus, this paper presents a new topology optimization method for multi-materials that obtains high-performance configurations. We apply the Multi-Material Level Set (MM-LS) topology description model in the topology optimization method, which uses a total of n level set functions to represent n materials, plus the void phase. The advantage of the MM-LS model is that clear optimal configurations are obtained and the design sensitivity for multi-material structures can be easily calculated. The level set functions that are design variables are updated using the topological derivatives, which also function as design sensitivities, and we derive the topological derivatives for multiple materials. Through several numerical examples, we demonstrate the validity of the proposed method.

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Topological sensitivity analysis in heterogeneous anisotropic elasticity problem. Theoretical and computational aspects

Cited by 38 publications

References 34 publications

Two‐scale topology optimization in computational material design: An integrated approach

Two‐scale topology optimization in computational material design: An integrated approach

Material design of elastic structures using Voronoi cells

Topology optimization for multi-material structures based on the level set method

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