Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction

In this work, we present an efficiently computational approach for designing material micro-structures by means of topology optimization. The central idea relies on using the isogeometric analysis integrated with the parameterized level set function for numerical homogenization, sensitivity calculation and optimization of the effective elastic properties. Design variables, which are level set values associated with control points, are updated from the optimizer and represent the geometry of the unit cell. We further improve the computational efficiency in each iteration by employing reduced order modeling when solving linear systems of the equilibrium equations. We construct a reduced basis by reusing computed solutions from previous optimization steps, and a much smaller linear system of equations is solved on the reduced basis. Two- and three-dimensional numerical results show the effectiveness of the topology optimization algorithm coupled with the reduced basis approach in designing metamaterials

Inverse design of metamaterials via topology optimization

Metamaterials are artificial composites with micro-structures that are systematically designed such that the macroscopic behavior can accommodate particular functionalities or exhibit extraordinary properties, which are not commonly found in natural materials. Topology and geometry of micro-structures play an important role in characterizing the properties of the metamaterials. Inverse design of metamaterials via topology optimization methods offer new topological features and helps in achieving novel physical mechanism or high-performance functionalities. Inverse design is an iterative process, that involves numerical analysis and requires much computational resources. This dissertation proposes the methodology for designing metamaterials using topology optimization with level set functions and model order reduction methods. Level set method enables design with smooth boundaries, while the computational effort required in solving large linear system of equations is eliminated with reduced basis approximations. An example of the inverse design method from the dissertation is to find a unit cell structure that results in macroscopic properties with intended elastic modulus for instance, with negative Poisson's ratio. The other example is to enhance hydrophone performance in 1-3 piezoelectric composites. Numerical examples demonstrate that the methodology is computationally efficient and robust for designing metamaterials. Taking advantage of inverse design as a powerful tool in designing metamaterials, it is adopted in this dissertation for the waveguides design. The second part of this thesis aims to design phononic crystals that offer robust transport of mechanical waves on the interfaces. The propagating wave modes in plate-like structures are topologically protected edge states and are analogous to quantum valley hall effect and quantum spin hall effect in the electronic systems. The computational inverse design methodology adopted is through topology optimization using genetic algorithm to find optimized unit cell geometries resulting from objective functions based on band structures and wave modes. The optimized phononic crystals support wave propagation against backscattering inspite of the presence of defects.Metamaterialien sind künstliche Verbundwerkstoffe mit Mikrostrukturen, die systematisch so gestaltet sind, dass das makroskopische Verhalten besondere Funktionen oder außergewöhnliche Eigenschaften aufweist, die in natürlichen Materialien nicht üblich sind. Topologie und Geometrie der Mikrostrukturen spielen eine wichtige Rolle bei der Charakterisierung der Eigenschaften der Metamaterialien. Das inverse Design von Metamaterialien mittels Topologie-Optimierungsmethoden bietet neue topologische Eigenschaften und hilft bei der Erreichung neuartiger physikalischer Mechanismen oder Hochleistungsfunktionen. Inverses Design ist ein iterativer Prozess, der numerische Analysen beinhaltet und viel Rechenleistung erfordert. In dieser Dissertation wird eine Methodik für den Entwurf von Metamaterialien unter Verwendung von Topologieoptimierung mit Level-Set-Funktionen und Methoden zur Reduzierung der Modellordnung vorgeschlagen. Die Level-Set-Methode ermöglicht ein Design mit glatten Grenzen, während der Rechenaufwand, der für die Lösung großer linearer Gleichungssysteme erforderlich ist, durch reduzierte Basisapproximationen entfällt. Ein Beispiel für die inverse Entwurfsmethode aus der Dissertation ist die Suche nach einer Einheitszellenstruktur, die zu makroskopischen Eigenschaften mit beabsichtigtem Elastizitätsmodul führt, beispielsweise mit negativer Poissonzahl. Ein anderes Beispiel ist die Verbesserung der Leistung von Hydrophonen in 1-3 piezoelektrischen Verbundwerkstoffen. Numerische Beispiele zeigen, dass die Methodik für die Entwicklung von Metamaterialien rechnerisch effizient und robust ist. Die Vorteile des inversen Designs als leistungsfähiges Werkzeug bei der Entwicklung von Metamaterialien werden in dieser Dissertation für die Entwicklung von Wellenleitern genutzt. Der zweite Teil dieser Arbeit zielt darauf ab, phononische Kristalle zu entwerfen, die einen robusten Transport von mechanischen Wellen an den Grenzflächen ermöglichen. Die sich ausbreitenden Wellenmoden in plattenförmigen Strukturen sind topologisch geschützte Randzustände und entsprechen dem Quanten-Tal-Hall-Effekt und dem Quanten-Spin-Hall-Effekt in elektronischen Systemen. Die angewandte rechnerische inverse Entwurfsmethodik besteht in der Topologieoptimierung mit Hilfe eines genetischen Algorithmus, um optimierte Einheitszellengeometrien zu finden, die sich aus Zielfunktionen auf der Grundlage von Bandstrukturen und Wellenmoden ergeben. Die optimierten phononischen Kristalle unterstützen die Wellenausbreitung trotz des Vorhandenseins von Defekten gegen Rückstreuung

Auxetic material in biomedical applications: a systematic review

This study reviews and analyzes the different auxetic materials that have been developed in recent years. The search for research articles was carried out through one of the largest databases such as ScienceDirect, where 845 articles were collected, of which several filters were carried out to have a base of 386 articles. There are a variety of materials depending on their structure, composition, and industrial application, highlighting biomedical applications from tissue engineering, cell proliferation, skeletal muscle regeneration, transportation, bio-prosthesis to biomaterial. The present paper provides an overview of auxetic materials and its applications, providing a guide for designers and manufacturers of devices and accessories in any industry

기하학적으로 정밀한 비선형 구조물의 아이소-지오메트릭 형상 설계 민감도 해석

학위논문 (박사)-- 서울대학교 대학원 : 공과대학 조선해양공학과, 2019. 2. 조선호.In this thesis, a continuum-based analytical adjoint configuration design sensitivity analysis (DSA) method is developed for gradient-based optimal design of curved built-up structures undergoing finite deformations. First, we investigate basic invariance property of linearized strain measures of a planar Timoshenko beam model which is combined with the selective reduced integration and B-bar projection method to alleviate shear and membrane locking. For a nonlinear structural analysis, geometrically exact beam and shell structural models are basically employed. A planar Kirchhoff beam problem is solved using the rotation-free discretization capability of isogeometric analysis (IGA) due to higher order continuity of NURBS basis function whose superior per-DOF(degree-of-freedom) accuracy over the conventional finite element analysis using Hermite basis function is verified. Various inter-patch continuity conditions including rotation continuity are enforced using Lagrage multiplier and penalty methods. This formulation is combined with a phenomenological constitutive model of shape memory polymer (SMP), and shape programming and recovery processes of SMP structures are simulated. Furthermore, for shear-deformable structures, a multiplicative update of finite rotations by an exponential map of a skew-symmetric matrix is employed. A procedure of explicit parameterization of local orthonormal frames in a spatial curve is presented using the smallest rotation method within the IGA framework. In the configuration DSA, the material derivative is applied to a variational equation, and an orientation design variation of curved structure is identified as a change of embedded local orthonormal frames. In a shell model, we use a regularized variational equation with a drilling rotational DOF. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. A design optimization method for a constrained structure in a curved domain is also developed, which focuses on a lattice structure design on a specified surface. We define a lattice structure and its design variables on a rectangular plane, and utilize a concept of free-form deformation and a global curve interpolation to obtain an analytical expression for the control net of the structure on curved surface. The material derivative of the analytical expression eventually leads to precise design velocity field. Using this method, the number of design variables is reduced and design parameterization becomes more straightforward. In demonstrative examples, we verify the developed analytical adjoint DSA method in beam and shell structural problems undergoing finite deformations with various kinematic and force boundary conditions. The method is also applied to practical optimal design problems of curved built-up structures. For example, we extremize auxeticity of lattice structures, and experimentally verify nearly constant negative Poisson's ratio during large tensile and compressive deformations by using the 3-D printing and optical deformation measurement technologies. Also, we architect phononic band gap structures having significantly large band gap for mitigating noise in low audible frequency ranges.본 연구에서는 대변형을 고려한 휘어진 조립 구조물의 연속체 기반 해석적 애조인 형상 설계 민감도 해석 기법을 개발하였다. 평면 Timoshenko 빔의 선형화된 변형률의 invariance 특성을 고찰하였고 invariant 정식화를 선택적 축소적분(selective reduced integration) 기법 및 B-bar projection 기법과 결합하여 shear 및 membrane 잠김 현상을 해소하였다. 비선형 구조 모델로서 기하학적으로 정밀한 빔 및 쉘 모델을 활용하였다. 평면 Kirchhoff 빔 모델을 NURBS 기저함수의 고차 연속성에 따른 아이소-지오메트릭 해석 기반 rotation-free 이산화를 활용하여 다루었으며, 기존의 Hermite 기저함수 기반의 유한요소법에 비해 자유도당 해의 정확도가 높음을 검증하였다. 라그랑지 승수법 및 벌칙 기법을 도입하여 회전의 연속성을 포함한 다양한 다중패치간 연속 조건을 고려하였다. 이러한 기법을 현상학적 (phenomenological) 형상기억폴리머 (SMP) 재료 구성방정식과 결합하여 형상의 프로그래밍과 회복 과정을 시뮬레이션하였다. 전단변형을 겪는 (shear-deformable) 구조 모델에 대하여 대회전의 갱신을 교대 행렬의 exponential map에 의한 곱의 형태로 수행하였다. 공간상의 곡선 모델에서 최소회전 (smallest rotation) 기법을 통해 국소 정규직교좌표계의 명시적 매개화를 수행하였다. 형상 설계 민감도 해석을 위하여 전미분을 변분 방정식에 적용하였으며 휘어진 구조물의 배향 설계 변화는 국소 정규직교좌표계의 회전에 의하여 기술된다. 최종 변형 형상에서 직교 변환 행렬의 전미분을 계산함으로써 대회전 문제에서 추가적인 반복 계산없이 변형 해석에서의 접선강성행렬에 의해 해석적 설계 민감도를 계산할 수 있다. 쉘 구조물의 경우 면내 회전 자유도 및 안정화된 변분 방정식을 활용하여 보강재(stiffener)의 모델링을 용이하게 하였다. 또한 본 연구에서는 휘어진 영역에 구속되어있는 구조물에 대한 설계 속도장 계산 및 최적 설계기법을 제안하며 특히 곡면에 구속된 빔 구조물의 설계를 집중적으로 다룬다. 자유형상변형(Free-form deformation)기법과 전역 곡선 보간기법을 활용하여 직사각 평면에서 형상 및 설계 변수를 정의하고 곡면상의 곡선 형상을 나타내는 조정점 위치를 해석적으로 표현할 수 있으며 이의 전미분을 통해 정확한 설계속도장을 계산한다. 이를 통해 설계 변수의 개수를 줄일 수 있고 설계의 매개화가 간편해진다. 개발된 방법론은 다양한 하중 및 운동학적 경계조건을 갖는 빔과 쉘의 대변형 문제를 통해 검증되며 여러가지 휘어진 조립 구조물의 최적 설계에 적용된다. 대표적으로, 전단 강성 및 충격 흡수 특성과 같은 기계적 물성치의 개선을 위해 활용되는 오그제틱 (auxetic) 특성이 극대화된 격자 구조를 설계하며 인장 및 압축 대변형 모두에서 일정한 음의 포아송비를 나타냄을 3차원 프린팅과 광학적 변형 측정 기술을 이용하여 실험적으로 검증한다. 또한 우리는 소음의 저감을 위해 활용되는 가청 저주파수 영역대에서의 밴드갭이 극대화된 격자 구조를 제시한다.Abstract 1. Introduction 2. Isogeometric analysis of geometrically exact nonlinear structures 3. Isogeometric confinguration DSA of geometrically exact nonlinear structures 4. Numerical examples 5. Conclusions and future works A. Supplements to the geometrically exact Kirchhoff beam model B. Supplements to the geometrically exact shear-deformable beam model C. Supplements to the geometrically exact shear-deformable shell model D. Supplements to the invariant formulations E. Supplements to the geometric constraints in design optimization F. Supplements to the design of auxetic structures 초록Docto

Deformation patterns in a second-gradient lattice annular plate composed of "Spira mirabilis" fibers

In this paper, we aim to explore the mechanical potentialities of a material made of an orthogonal net of fibers arranged in logarithmic spirals. Therefore, an annular plate described with a second-gradient model is envisaged to evaluate the behavior of such material in a nonlinear elastic regime when large displacements and deformations occur. Several mechanical tests are performed numerically under the finite element method approximation obtained directly with a weak formulation based on the elastic energy that it is assumed to be predictive for this kind of network system of fibers. Plots reporting the mechanical characteristics in all the considered tests are provided to illustrate the overall mechanical behavior of the evaluated system