scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Phillip Baumann ◽  
Kevin Sturm

PurposeThe goal of this paper is to give a comprehensive and short review on how to compute the first- and second-order topological derivatives and potentially higher-order topological derivatives for partial differential equation (PDE) constrained shape functionals.Design/methodology/approachThe authors employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint-based methods to compute higher-order topological derivatives. To illustrate the methodology proposed in this paper, the authors then apply the methods to a linear elasticity model.FindingsThe authors compute the first- and second-order topological derivatives of the linear elasticity model for various shape functionals in dimension two and three using Amstutz' method, the averaged adjoint method and Delfour's method.Originality/valueIn contrast to other contributions regarding this subject, the authors not only compute the first- and second-order topological derivatives, but additionally give some insight on various methods and compare their applicability and efficiency with respect to the underlying problem formulation.


1986 ◽  
Vol 16 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Donald E. Carlson ◽  
Anne Hoger

2015 ◽  
Vol 9 (1) ◽  
pp. 367-384 ◽  
Author(s):  
A. Diez ◽  
O. Eisen

Abstract. A preferred orientation of the anisotropic ice crystals influences the viscosity of the ice bulk and the dynamic behaviour of glaciers and ice sheets. Knowledge about the distribution of crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. However, the developed anisotropic fabric influences not only the flow behaviour of ice but also the propagation of seismic waves. Two effects are important: (i) sudden changes in COF lead to englacial reflections, and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and, thus, recorded travel times. A framework is presented here to connect COF data from ice cores with the elasticity tensor to determine seismic velocities and reflection coefficients for cone and girdle fabrics. We connect the microscopic anisotropy of the crystals with the macroscopic anisotropy of the ice mass, observable with seismic methods. Elasticity tensors for different fabrics are calculated and used to investigate the influence of the anisotropic ice fabric on seismic velocities and reflection coefficients, englacially as well as for the ice–bed contact. Hence, it is possible to remotely determine the bulk ice anisotropy.


Optimization ◽  
2013 ◽  
Vol 64 (2) ◽  
pp. 389-407 ◽  
Author(s):  
L. Minchenko ◽  
A. Tarakanov

2010 ◽  
Vol 09 (01) ◽  
pp. 219-231 ◽  
Author(s):  
XIAODONG LIU ◽  
YONGQING QIU ◽  
SHILING SUN ◽  
CHUNGUANG LIU ◽  
ZHONGMIN SU

DFT B3LYP method was employed to calculate the second-order nonlinear optical (NLO) responses of the derivatives of disubstituted seven-vertex cobaltacarborane metallocenyl. The results show that cobaltacarborane metallocenyl plays a pushing/pulling role and a bridge role to transfer electron in these molecules. The five-membered ring of cyclopentadiene is more beneficial to increase second-order NLO response than the five-membered ring composed of two C atoms and three B atoms in cobaltacarborane. Moreover, the second-order NLO response is more powerful when one substituent containing electron donor group and one substituent containing electron acceptor group are located at meta position. Accordingly, among the nine models, model c2 is the optimum model with largest value of βtot. The calculation results also show that cobaltacarborane metallocenyl and ferrocene parts play the same roles to increase second-order NLO response. Thus, cobaltacarborane metallocenyl can be a promising second-order NLO material.


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