A new computational approach to topology optimization in solid mechanics problems

Tesi en modalitat compendi de publicacionsThis doctoral thesis addresses topology optimization problems at a single scale. Based on this purpose, a new topology optimization approach is developed in order to improve existing and widespread techniques in the research community on the topic. The proposed technique presents several characteristics that overcome some of the well-known difficulties in topological optimization while maintaining a considerable degree of simplicity. In the first place, the formulation of the topological optimization technique is presented, as well as its algorithm. The method is based on 4 fundamental features: (1) the use of a 1-0 characteristic function, as well as the precise identification of the material boundaries from a discrimination function (0-level-set function), (2) the definition of a topological derivative consistent with the ersatz method (used in the state problem), as an approximation to the exact topological derivative, (3) the inclusion of a Laplacian regularization with minimum size control of the different components, and (4) the formulation of an analytical optimality condition aiming at the optimal topology solution. The approach is applied to different topology optimization problems, well-reported in the literature and used as numerical benchmarks (in structural and thermal problems), to examine their performance. In these fields, stiffness and conductivity maximization problems are considered for validation, respectively. In addition, different topological optimization problems of major engineering interest are tackled, including the design of compliant mechanisms within the structural field and thermal cloaking devices within the thermal field. Finally, a comparison of the formulation with other existing topology optimization techniques is performed, including (1) SIMP, (2) ESO/BESO, and (3) Level-set with Hamilton-Jacobi as the updating equation. The analysis of the results provides a comparison in terms of the quality of the topology of each method, the computational cost of the optimal solutions, as well as the simplicity of implementation. The resulting study reveals the potential of the developed methodology in these specific comparison terms. In an attempt to bring the method closer to other researchers and to promote its use, an educational version of the method (written in MATLAB) has been published in an online repository, together with documentation, facilitating its dissemination and subsequent use in other applications of interest.El objetivo de esta tesis doctoral es abordar el problema de optimización topológica a una única escala. En base a este propósito, se desarrolla una nueva técnica de optimización capaz de competir con técnicas ya existentes y extendidas entre la comunidad investigadora sobre el tema. Esta técnica presenta características que superan algunas de las dificultades bien conocidas en optimización topológica manteniendo un buen grado de simplicidad. En primer lugar, se presenta la formulación de la técnica de optimización topológica, así como su algoritmia. El método se fundamenta en 4 aspectos básicos: (1) la utilización de una función característica 1-0, así como la definición precisa de las fronteras materiales a partir de una función de discriminación (isonivel 0 de la función level-set), (2) la definición de una derivada topológica coherente con el método ersatz (utilizado en la ecuación de estado), como aproximación a la derivada topológica exacta, (3) la inclusión de una regularización Laplaciana con control de tamaño mínimo de los diferentes componentes, y (4) la definición de una condición de optimalidad analítica para la determinación de la solución óptima de la topología. La metodología se aplica a diferentes problemas de optimización topológica bien detallados en la literatura y utilizados como ensayos numéricos para examinar su respuesta frente a problemas estructurales y térmicos. En estos campos, se incluyen problemas de maximización de la rigidez y de la conductividad, respectivamente. Además, se resuelven diferentes problemas de optimización topológica con gran interés ingenieril en los campos estructurales con el diseño de mecanismos y térmicos con el diseño de dispositivos de camuflaje térmicos. Finalmente, se realiza una comparación de la formulación con otras técnicas ya existentes, por ejemplo: (1) SIMP, (2) ESO/BESO, y (3) Level-set con Hamilton-Jacobi como ecuación de evolución. El análisis de los resultados permite comparar la calidad de la topología de cada método, el coste computacional de las soluciones óptimas, así como la simplicidad de implementación, demostrando el potencial de la metodología desarrollada principalmente en estos términos de comparación. Con la finalidad de acercar el método a otros investigadores y de promover su utilización, se ha publicado una versión educativa del mismo (en MATLAB) en un repositorio online, junto a documentación, permitiendo así la divulgación del mismo y la posible utilización en otras aplicaciones de interés.Postprint (published version

Study for the numerical resolution of conservation equations of mass, momentum and energy to be applied to solar thermal collectors

The aim of this project is to create a software in C++ language that allows to solve numerically the Navier-Stokes equations for cases with simple geometries and presupposing certain hypotheses, as well as the study and comprehension of the equations that govern the fluid dynamics. Furthermore, the procedure applied to cases of conduction and radiation is also introduced. The main purpose is to simulate the fluid-dynamic and thermal behaviour of already well-known cases as a way to verify the results. Principally, The studied problems are: "Smith-Hutton problem”, “Driven Cavity problem” as well as “Differentially heated cavity problem”. Finally, an introduction is done about the phenomenology that characterizes the turbulence as well as the theoretical and practical implementation of 1D cases.

Variational approach to relaxed topological optimization: closed form solutions for structural problems in a sequential pseudo-time framework

The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a non-smoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution. In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy. In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closed-form solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the workPeer ReviewedPostprint (author's final draft

Topology optimization using the unsmooth variational topology optimization (UNVARTOP) method: an educational implementation in MATLAB

The final publication is available at Springer via http://dx.doi.org/10.1007/s00158-020-02722-0This paper presents an efficient and comprehensive MATLAB code to solve two-dimensional structural topology optimization problems, including minimum mean compliance, compliant mechanism synthesis, and multi-load compliance problems. The unsmooth variational topology optimization (UNVARTOP) method, developed by Oliver et al. (Comput Methods Appl Mech Eng 355:779–819, 2019), is used in the topology optimization code, based on the finite element method (FEM), to compute the sensitivity and update the topology. The paper also includes instructions to improve the bisection algorithm, modify the computation of the Lagrangian multiplier by using an Augmented Lagrangian to impose the constraint, implement heat conduction problems, and extend the code to three-dimensional topology optimization problems. The code, intended for students and newcomers in topology optimization, is included as an appendix (Appendix A) and it can be downloaded from https://github.com/DanielYago/UNVARTOP together with supplementary material.This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Proof of Concept Grant agreement n 874481) through the project \Computational design and prototyping of acoustic metamaterials for target ambient noise reduction" (METACOUSTIC). The authors also acknowledge nancial support from the Spanish Ministry of Economy and Competitiveness, through the research grant DPI2017-85521-P for the project \Computational design of Acoustic and Mechanical Metamaterials" (METAMAT) and through the \Severo Ochoa Programme for Centres of Excellence in R&D" (CEX2018-000797-S). D. Yago acknowledges the support received from the Spanish Ministry of Education through the FPU program for PhD grants.Peer ReviewedPostprint (author's final draft

Computational design of locally resonant acoustic metamaterials

The so-called Locally Resonant Acoustic Metamaterials (LRAM) are considered for the design of specifically engineered devices capable of stopping waves from propagating in certain frequency regions (bandgaps), this making them applicable for acoustic insulation purposes. This fact has inspired the design of a new kind of lightweight acoustic insulation panels with the ability to attenuate noise sources in the low frequency range (below 5000 Hz) without requiring thick pieces of very dense materials. A design procedure based on different computational mechanics tools, namely, (1) a multiscale homogenization framework, (2) model order reduction strategies and (3) topological optimization procedures, is proposed. It aims at attenuating sound waves through the panel for a target set of resonance frequencies as well as maximizing the associated bandgaps. The resulting design’s performance is later studied by introducing viscoelastic properties in the coating phase, in order to both analyse their effects on the overall design and account for more realistic behaviour. The study displays the emerging field of Computational Material Design (CMD) as a computational mechanics area with enormous potential for the design of metamaterial-based industrial acoustic parts.Peer ReviewedPostprint (author's final draft

Study of topological optimization techniques applied to the design of aeronautical structures.

Study of topological optimization techniques applied to the design of aeronautical structures. This thesis deals with the study of di˙erent topology optimization techniques in which the aim of the optimization procedure is to maximize the sti˙ness of a typical structural component subject to a given reduction of volume. Specifically, three di˙erent topology optimization techniques are analysed and com-pared. The techniques that are treated in this dissertation are: (1) SIMP approach,(2) Topological derivative methodology, and (3) variational approach, which has been developed by the COMP-DES-MAT research group. These three techniques are compared in terms of computational cost, mass distribution and compliance function. The comparison is carried out in a set of benchmarking tests. The proposed optimization methodology described in this thesis was deeply analysed with a set of numerical experiments. In order to simulate those examples, an in-house code within MATLAB framework was implemented, in which two- and three-dimensional problems could be executed using triangular, quadrilateral, tetrahedral or hexahedral elements. Moreover, a set of loads and boundary conditions could be applied, such as concentrated, distributed or volumetric loads. Nevertheless, the code is limited to linear elements. Finally, the potential of the variational method is analysed optimizing the internal structure of a rib. The aerodynamics loads have been obtained from the open source program Xfoil. The optimization of the structure is carried out in a 2D wing profile by the use of quadrilateral elements

Study of topological optimization techniques applied to the design of aeronautical structures.

Study of topological optimization techniques applied to the design of aeronautical structures. This thesis deals with the study of di˙erent topology optimization techniques in which the aim of the optimization procedure is to maximize the sti˙ness of a typical structural component subject to a given reduction of volume. Specifically, three di˙erent topology optimization techniques are analysed and com-pared. The techniques that are treated in this dissertation are: (1) SIMP approach,(2) Topological derivative methodology, and (3) variational approach, which has been developed by the COMP-DES-MAT research group. These three techniques are compared in terms of computational cost, mass distribution and compliance function. The comparison is carried out in a set of benchmarking tests. The proposed optimization methodology described in this thesis was deeply analysed with a set of numerical experiments. In order to simulate those examples, an in-house code within MATLAB framework was implemented, in which two- and three-dimensional problems could be executed using triangular, quadrilateral, tetrahedral or hexahedral elements. Moreover, a set of loads and boundary conditions could be applied, such as concentrated, distributed or volumetric loads. Nevertheless, the code is limited to linear elements. Finally, the potential of the variational method is analysed optimizing the internal structure of a rib. The aerodynamics loads have been obtained from the open source program Xfoil. The optimization of the structure is carried out in a 2D wing profile by the use of quadrilateral elements

Study for the numerical resolution of conservation equations of mass, momentum and energy to be applied to solar thermal collectors

The aim of this project is to create a software in C++ language that allows to solve numerically the Navier-Stokes equations for cases with simple geometries and presupposing certain hypotheses, as well as the study and comprehension of the equations that govern the fluid dynamics. Furthermore, the procedure applied to cases of conduction and radiation is also introduced. The main purpose is to simulate the fluid-dynamic and thermal behaviour of already well-known cases as a way to verify the results. Principally, The studied problems are: "Smith-Hutton problem”, “Driven Cavity problem” as well as “Differentially heated cavity problem”. Finally, an introduction is done about the phenomenology that characterizes the turbulence as well as the theoretical and practical implementation of 1D cases.

Study for the numerical resolution of conservation equations of mass, momentum and energy to be applied to solar thermal collectors

Este proyecto tiene como objetivo la creación de un software en lenguaje C++ que permita resolver las ecuaciones de Navier-Stokes para casos con geometrías sencillas y presuponiendo ciertas hipótesis, así como el estudio y comprensión de las ecuaciones que rigen la dinámica de fluidos. Aunque también se introduce la metodología aplicada a casos de conducción y radiación. La finalidad principal es llegar a simular el comportamiento fluido-dinámico y térmico de casos ya conocidos y así verificar los resultados. Principalmente se estudiarán los problemas: “Smith-Hutton problem”, “Driven Cavity problem” así como “Differentially heated cavity problem”. Finalmente se procede a introducir la fenomenología que caracteriza la turbulencia así como la implementación teórica y práctica de casos 1D.The aim of this project is to create a software in C++ language that allows to solve numerically the Navier-Stokes equations for cases with simple geometries and presupposing certain hypotheses, as well as the study and comprehension of the equations that govern the fluid dynamics. Furthermore, the procedure applied to cases of conduction and radiation is also introduced. The main purpose is to simulate the fluid-dynamic and thermal behaviour of already well-known cases as a way to verify the results. Principally, The studied problems are: "Smith-Hutton problem”, “Driven Cavity problem” as well as “Differentially heated cavity problem”. Finally, an introduction is done about the phenomenology that characterizes the turbulence as well as the theoretical and practical implementation of 1D cases.

Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-020-01850-0This paper extends the nonsmooth Relaxed Variational Approach (RVA) to topology optimization, proposed by the authors in a preceding work, to the solution of thermal optimization problems. First, the RVA topology optimization method is brie y discussed and, then, it is applied to a set of representative problems in which the thermal compliance, the deviation of the heat flux from a given field and the average temperature are minimized. For each optimization problem, the relaxed topological derivative (RTD) and the corresponding adjoint equations are presented. This set of expressions are then discretized in the context of the finite element method (FEM) and used in the optimization algorithm to update the characteristic function. Finally, some representative (3D) thermal topology optimization examples are presented to asses the performance of the proposed method and the Relaxed Variational Approach solutions are compared with the ones obtained with the level set method in terms of the cost function, the topology design and the computational cost.This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Proof of Concept Grant agreement n 874481) through the project \Computational design and prototyping of acoustic metamaterials for target ambient noise reduction"" (META-COUSTIC). The authors also acknowledge nancial support from the Spanish Ministry of Economy and Competitiveness, through the research grant DPI2017-85521-P for the project \Computational design of Acoustic and Mechanical Metamaterials"" (METAMAT) and through the \Severo Ochoa Programme for Centres of Excellence in R&D"" (CEX2018-000797-S). D. Yago acknowledges the support received from the Spanish Ministry of Education through the FPU program for PhD grants.Peer Reviewe