Isogeometric topology optimization for auxetic metamaterials and structures

University of Technology Sydney. Faculty of Engineering and Information Technology.It is known that topology optimization is located at the conceptual design phase, which can effectively determine the numbers, connectivity and existence of holes in the structural design domain and evolve design elements to improve the concerned performance. General speaking, topology optimization works as an important tool to seek for the optimal material distribution, which has been identified as one of the most promising sub-field of structural optimization due to its superior features occurring in the conceptual design stage without prior knowledge of the design domain. In the current work, the main intention is to propose a novel numerical method for the topology optimization with more effectiveness and efficiency for the single-material structures and structures with multiple materials. Meanwhile, the proposed topology optimization method is also applied to implement the rational design of auxetic metamaterials and auxetic composites. In Chapter 1, we provide a brief description for the main intention of the current work. In Chapter 2, the comprehensive review about the developments of topology optimization, isogeometric topology optimization and the rational design of auxetic materials is provided. In Chapter 3, a more effective and efficient topology optimization method using isogeometric analysis is proposed for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF mainly involves two steps: (1) the smoothness of nodal densities is improved by the Shepard function; (2) the higher-order NURBS basis functions are combined with the smoothed nodal densities to construct the DDF with the continuity. A topology optimization formulation to minimize the structural mean compliance is developed using the DDF and isogeometric analysis (IGA) is applied to solve structural responses. An integration of the geometry parametrization and numerical analysis offer several benefits for the optimization. The Chapter 4 intends to develop a Multi-material Isogeometric Topology Optimization (M-ITO) method. Firstly, a new Multi-material Interpolation model is established with the use of NURBS (Non-uniform Rational B-splines), termed by the “N-MMI” model, which mainly involves three components: (1) Multiple Fields of Design Variables (DVFs); (2) Multiple Fields of Topology Variables (TVFs); (3) Multi-material interpolation. Two different M-ITO formulations are developed using the N-MMI model to address the problems with multiple volume constraints and the total mass constraint, respectively. The decoupled expression and serial evolving of the DVFs and TVFs can effectively eliminate numerical difficulties in the multi-material problems. In Chapter 5, the proposed ITO method is applied for the systematic design of both 2D and 3D auxetic metamaterials. An energy-based homogenization method (EBHM) to evaluate the macroscopic effective properties is numerically implemented by IGA, with the imposing of periodic boundary formulation on material microstructure. An ITO formulation for 2D and 3D auxetic metamaterials is developed using the DDF, where the objective function is defined as a combination of the homogenized elastic tensor. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. In Chapter 6, the proposed M-ITO method is applied for the systematic design of both 2D and 3D auxetic composites. The homogenization, that evaluates macroscopic effective properties of auxetic composites, is numerically implemented by IGA, with the imposing of the periodic boundary formulation on composite microstructures. The developed N-MMI model is applied to describe the multi-material topology and evaluate the multi-material properties. A topology optimization formulation for the design of both two-dimensional (2D) and three-dimensional (3D) auxetic composites is developed. Finite element simulations of auxetic composites are discussed using the ANSYS to show different deformation mechanisms. Finally, conclusions and prospects are given in Chapter 7

Topology Optimization of Structures with High Spatial Definition Considering Minimum Weight and Stress Constraints

Programa Oficial de Doutoramento en Enxeñaría Civil . 5011V01[Abstract] The first formulation of Topology Optimization was proposed in 1988. Since then, many contributions have been presented with the purpose of improving its efficiency and extending its applicability. In this thesis, a topology optimization algorithm that allows to obtain the structure of minimum weight that is able to support different loads is developed. For this purpose, the requirement that stresses have to be lower than a maximum value has been considered in its development. Although the structural topology optimization problem with stress constraints have been previously formulated with several different approaches, a Damage Constraint approach is developed in this thesis to incorporate them in a different way. The main objective of this modification is to reduce the CPU time required in the solution of the topology optimization problem. This reduction will allow to solve problems with a higher number of design variables what enables the attainment of solutions with high spatial definition. Moreover, two different approaches are used to define the material distribution in the domain: uniform density per element formulation and material density distribution by means of isogeometric interpolation. In the first approach the Finite Element Method (FEM) is used to solve the structural analysis and the relative density in each element of the mesh is chosen as design variable, while the second one uses the Isogeometric Analysis (IGA) for solving the structural analysis and the values of the relative density at a certain number of control points are used as design variables. On the other hand, the optimization is addressed by using Sequential Linear Programming, that requires a first order sensitivity analysis. All the sensitivities are obtained through analytic derivatives by using both, direct differentiation and the adjoint variable method. Finally, some application examples are solved by means of both methods (FEM and IGA) in the two-dimensional and three-dimensional space.[Resumen] La primera formulación de la Optimización Topológica fue propuesta en 1988. Desde entonces muchas aportaciones se han presentado para mejorar su eficiencia y extender su aplicabilidad. En esta tesis se desarrolla un algoritmo de optimización topológica que permita obtener la estructura de mínimo peso que sea capaz de soportar diferentes cargas. Para este propósito se ha considerado en su desarrollo la condición de que las tensiones sean inferiores a un cierto valor máximo. Aunque el problema de optimización topológica estructural con restricciones de tensión se formuló previamente con diferentes enfoques, en esta tesis se desarrolla un enfoque que considera una restricción de daño para incorporarlas de una forma diferente. El principal objetivo de esta modificación es reducir el tiempo de computación requerido en la solución del problema de optimización topológica. Esta reducción permitir ´a resolver problemas con un mayor número de variables de diseño lo que a su vez permite la obtención de soluciones con alta definición espacial. Para definir la distribución de material en el dominio se usan dos formulaciones diferentes: formulación de densidad uniforme por elemento y distribución de material por medio de una interpolación isogeométrica. El primer planteamiento usa el Método de los Elementos Finitos (MEF) para resolver el análisis estructural y toma como variable de diseño el valor de la densidad relativa en cada elemento de la malla, mientras que el segundo requiere del uso del Análisis Isogeométrico (IGA) para resolver el análisis estructural y los valores de la densidad relativa en un cierto número de puntos de control son las variables de diseño. El problema de optimización se resuelve con las técnicas de Programación Lineal Secuencial requiriendo ´únicamente el análisis de sensibilidad de primer orden. Todas las derivadas se calculan por derivación analítica haciendo uso de las técnicas de derivación directa y del método de la variable adjunta. Finalmente, se resuelven algunos ejemplos de aplicación con ambos métodos (MEF e IGA) en el espacio bidimensional y tridimensional.[Resumo] A primeira formulación da Optimización Topolóxica foi proposta en 1988. Desde entón moitas achegas se presentaron para mellorar a súa eficiencia e estender a súa aplicabilidade. Nesta tese desenvólvese un algoritmo de optimización topolóxica que permita obter a estrutura de mínimo peso que sexa capaz de soportar diferentes cargas. Para este propósito considerouse no seu desenvolvemento a condición de que as tensións sexan inferiores a un certo valor máximo. Aínda que o problema de optimización topolóxica estrutural con restricións de tensi´on formulouse previamente con diferentes enfoques, nesta tese desenvólvese un enfoque que considera unha restrición de dano para incorporalas dunha forma diferente. O principal obxectivo desta modificación é reducir o tempo de computación requirido na solución do problema de optimizaci´on topol´oxica. Esta reduci´on permitir´a resolver problemas cun maior número de variables de dese˜no o que ´a s´ua vez permite a obtención de solucións con alta definición espacial. Para definir a distribución de material no dominio úsanse dúas formulacións diferentes: formulación de densidade uniforme por elemento e distribución de material por medio dunha interpolación isoxeométrica. A primeira formulación usa o Método dos Elementos Finitos (MEF) para resolver a análise estrutural e toma coma variable de deseño o valor da densidade relativa en cada elemento da malla, mentres que o segundo require do uso da Análise Isoxeométrica (IGA) para resolver a análise estrutural e os valores da densidade relativa nun certo número de puntos de control son as variables de deseño. O problema de optimización resólvese coas técnicas de Programación Lineal Secuencial requirindo unicamente a análise de sensibilidade de primeira orde. Todas as derivadas calcúlanse por derivación analítica facendo uso das técnicas de derivación directa e do método da variable adxunta. Finalmente, resólvense algúns exemplos de aplicación con ámbolos métodos (MEF e IGA) no espazo bidimensional e tridimensionalMinisterio de Economía y Competitividad; DPI2015-68341-RMinisterio de Economía y Competitividad; RTI2018-093366-B-I00Xunta de Galicia; GRC2014/039Xunta de Galicia; GRC2018/4

Robust topological designs for extreme metamaterial micro-structures

We demonstrate that the consideration of material uncertainty can dramatically impact the optimal topological micro-structural configuration of mechanical metamaterials. The robust optimization problem is formulated in such a way that it facilitates the emergence of extreme mechanical properties of metamaterials. The algorithm is based on the bi-directional evolutionary topology optimization and energy-based homogenization approach. To simulate additive manufacturing uncertainty, combinations of spatial variation of the elastic modulus and/or, parametric variation of the Poisson’s ratio at the unit cell level are considered. Computationally parallel Monte Carlo simulations are performed to quantify the effect of input material uncertainty to the mechanical properties of interest. Results are shown for four configurations of extreme mechanical properties: (1) maximum bulk modulus (2) maximum shear modulus (3) minimum negative Poisson’s ratio (auxetic metamaterial) and (4) maximum equivalent elastic modulus. The study illustrates the importance of considering uncertainty for topology optimization of metamaterials with extreme mechanical performance. The results reveal that robust design leads to improvement in terms of (1) optimal mean performance (2) least sensitive design, and (3) elastic properties of the metamaterials compared to the corresponding deterministic design. Many interesting topological patterns have been obtained for guiding the extreme material robust design

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

Optimal Design of Functionally Graded Parts

Several additive manufacturing processes are capable of fabricating three-dimensional parts with complex distribution of material composition to achieve desired local properties and functions. This unique advantage could be exploited by developing and implementing methodologies capable of optimizing the distribution of material composition for one-, two-, and three-dimensional parts. This paper is the first effort to review the research works on developing these methods. The underlying components (i.e., building blocks) in all of these methods include the homogenization approach, material representation technique, finite element analysis approach, and the choice of optimization algorithm. The overall performance of each method mainly depends on these components and how they work together. For instance, if a simple one-dimensional analytical equation is used to represent the material composition distribution, the finite element analysis and optimization would be straightforward, but it does not have the versatility of a method which uses an advanced representation technique. In this paper, evolution of these methods is followed; noteworthy homogenization approaches, representation techniques, finite element analysis approaches, and optimization algorithms used/developed in these studies are described; and most powerful design methods are identified, explained, and compared against each other. Also, manufacturing techniques, capable of producing functionally graded materials with complex material distribution, are reviewed; and future research directions are discussed

Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization

An important feature that drives the auxetic behaviour of the star-shaped auxetic structures is the hinge-functional connection at the vertex connections. This feature poses a great challenge for manufacturing and may lead to significant stress concentrations. To overcome these problems, we introduced smoothed petal-shaped auxetic structures, where the hinges are replaced by smoothed connections. To accommodate the curved features of the petal-shaped auxetics, a parametrisation modelling scheme using multiple NURBS patches is proposed. Next, an integrated shape design frame work using isogeometric analysis is adopted to improve the structural performance. To ensure a minimum thickness for each member, a geometry sizing constraint is imposed via piece-wise bounding polynomials. This geometry sizing constraint, in the context of isogeometric shape optimization, is particularly interesting due to the non-interpolatory nature of NURBS basis. The effective Poisson ratio is used directly as the objective function, and an adjoint sensitivity analysis is carried out. The optimized designs – smoothed petal auxetic structures – are shown to achieve low negative Poisson’s ratios, while the difficulties of manufacturing the hinges are avoided. For the case with six petals, an in-plane isotropy is achieved.Singapore MOE Tier 2 Grant R30200013911

A Review on Topology Optimization Strategies for Additively Manufactured Continuous Fiber-Reinforced Composite Structures

Topology Optimization (TO) recently gained importance due to the development of Ad- ditive Manufacturing (AM) processes that produce components with good mechanical properties. Among all additive manufacturing technologies, continuous fiber fused filament fabrication (CF4) can fabricate high-performance composites compared to those manufactured with conventional technolo- gies. In addition, AM provides the excellent advantage of a high degree of reconfigurability, which is in high demand to support the immediate short-term manufacturing chain in medical, transportation, and other industrial applications. CF4 enables the fabrication of continuous fiber-reinforced compos- ite (FRC) materials structures. Moreover, it allows us to integrate topology optimization strategies to design realizable CFRC structures for a given performance. Various TO strategies for attaining lightweight and high-performance designs have been proposed in the literature, exploiting AM’s design freedom. Therefore, this paper attempts to address works related to strategies employed to obtain optimal FRC structures. This paper intends to review and compare existing methods, analyze their similarities and dissimilarities, and discuss challenges and future trends in this field