Vademecum-based approach to multi-scale topological material design

The work deals on computational design of structural materials by resorting to computational homogenization and topological optimization techniques. The goal is then to minimize the structural (macro-scale) compliance by appropriately designing the material distribution (microstructure) at a lower scale (micro-scale), which, in turn, rules the mechanical properties of the material. The specific features of the proposed approach are: (1) The cost function to be optimized (structural stiffness) is defined at the macro-scale, whereas the design variables defining the micro-structural topology lie on the low scale. Therefore a coupled, two-scale (macro/micro), optimization problem is solved unlike the classical, single-scale, topological optimization problems. (2) To overcome the exorbitant computational cost stemming from the multiplicative character of the aforementioned multiscale approach, a specific strategy, based on the consultation of a discrete material catalog of micro-scale optimized topologies (Computational Vademecum) is used. The Computational Vademecum is computed in an offline process, which is performed only once for every constitutive-material, and it can be subsequently consulted as many times as desired in the online design process. This results into a large diminution of the resulting computational costs, which make affordable the proposed methodology for multiscale computational material design. Some representative examples assess the performance of the considered approach.Peer ReviewedPostprint (published version

Vademecum-based approach to multi-scale topological material design

The work deals on computational design of structural materials by resorting to computational homogenization and topological optimization techniques. The goal is then to minimize the structural (macro-scale) compliance by appropriately designing the material distribution (microstructure) at a lower scale (micro-scale), which, in turn, rules the mechanical properties of the material. The specific features of the proposed approach are: (1) The cost function to be optimized (structural stiffness) is defined at the macro-scale, whereas the design variables defining the micro-structural topology lie on the low scale. Therefore a coupled, two-scale (macro/micro), optimization problem is solved unlike the classical, single-scale, topological optimization problems. (2) To overcome the exorbitant computational cost stemming from the multiplicative character of the aforementioned multiscale approach, a specific strategy, based on the consultation of a discrete material catalog of micro-scale optimized topologies (Computational Vademecum) is used. The Computational Vademecum is computed in an offline process, which is performed only once for every constitutive-material, and it can be subsequently consulted as many times as desired in the online design process. This results into a large diminution of the resulting computational costs, which make affordable the proposed methodology for multiscale computational material design. Some representative examples assess the performance of the considered approac

Two-scale topology optimization in computational material design: an integrated approach

The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya–Babuška–Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations

Two-scale topology optimization in computational material design: an integrated approach

In 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.Peer ReviewedPostprint (published version

Multi-scale topological design of structural materials : an integrated approach

The present dissertation aims at addressing multiscale topology optimization problems. For this purpose, the concept of topology derivative in conjunction with the computational homogenization method is considered. In this study, the topological derivative algorithm, which is clearly non standard in topology optimization, and the optimality conditions are first introduced in order to a provide a better insight. Then, a precise treatment of the interface elements is proposed to reduce the numerical instabilities and the time-consuming computations that appear when using the topological derivative algorithm. The resulting strategy is examined and compared with current methodologies collected in the literature by means of some numerical tests of different nature. Then, a closed formula of the anisotropic topological derivative is obtained by solving analytically the exterior elastic problem. To this aim, complex variable theory and symbolic computation is considered. The resulting expression is validated through some numerical tests. In addition, different anisotropic topology optimization problems are solved to show the macroscopic topological implications of considering anisotropic materials. Finally, the two-scale topology optimization problem is tackled. As a first approach, an structural stiffness increase is achieved by considering the microscopic topologies as design variables of the problem. An alternate direction algorithm is proposed to address the high non-linearities of the problem. In addition, to mitigate the unaffordable time-consuming computations, a reduction technique is presented by means of pre-computing the optimal microscopic topologies in a computational material catalogue. As an extension of the first approach, besides designing the microscopic topologies, the macroscopic topology is also considered as a design variable, leading to even more optimal solutions. In addition, the proposed algorithms are modified in order to obtain manufacturable optimal designs. Two-scale topology optimization examples exhibit the potential of the proposed methodologyAquest treball té com a objectiu abordar els problemes d'optimització de topologia de múltiples escales. Amb aquesta finalitat, es consideren els conceptes de derivada topologia junt amb el mètode d'homogeneïtzació computacional. En aquest estudi, es presenta primer les condicions d'optimalitat i l'algorisme d'optimització utilitzat quan es considera la derivada topològica. A continuació, es proposa un tractament més precís dels elements de la interfície per reduir les inestabilitats numèriques i els elevats càlculs computacionals que apareixen quan s'utilitza l'algorisme de la derivada topològica. L'estratègia resultant s'examina i es compara amb les metodologies actuals, que es poden trobar sovint recollides a la literatura, mitjançant algunes proves numèriques. A més, s'obté una fórmula tancada de la derivada topològica anisotròpica quan es resol analíticament el problema exterior d'elasticitat. Per aconseguir-ho, es considera la teoria de variable complexa i la computació simbòlica. L'expressió resultant es valida a través d'algunes proves numèriques. A més, es resolen diferents problemes d'optimització topològica anisotròpica per mostrar les implicacions de la topològica macroscòpica en considerar materials anisòtrops. Finalment, s'aborda els problemes d'optimització topològica de dues escales. Com a primera estratègia, es considera les topologies microestructurals com a variables de disseny del problema per obtenir un augment de la rigidesa de l'estructura. Es proposa un algoritme de direcció alternada per fer front a les altes no linealitats del problema. A més, per mitigar els elevats càlculs computacionals, es presenta una tècnica de reducció per mitjà d'un precalcul de les topologies microestructural òptimes, que posteriorment són recollides en un catàleg de material. Com a una extensió de la primera estratègia, a més del disseny de les topologies microestructurals, la topologia macroscòpica també es considera com una variable de disseny, obtenint així solucions encara més òptimes. A més, els algoritmes proposats es modifiquen per tal d'obtenir dissenys que poden ser posteriorment fabricats. Alguns exemples numèrics d'optimització topològica de dues escales mostren el potencial de la metodologia proposada.Postprint (published version

Real-time simulation of surgery by Proper Generalized Decomposition techniques

La simulación quirúrgica por ordenador en tiempo real se ha convertido en una alternativa muy atractiva a los simuladores quirúrgicos tradicionales. Entre otras ventajas, los simuladores por ordenador consiguen ahorros importantes de tiempo y de costes de mantenimiento, y permiten que los estudiantes practiquen sus habilidades quirúrgicas en un entorno seguro tantas veces como sea necesario. Sin embargo, a pesar de las capacidades de los ordenadores actuales, la cirugía computacional sigue siendo un campo de investigación exigente. Uno de sus mayores retos es la alta velocidad a la que se tienen que resolver complejos problemas de mecánica de medios continuos para que los interfaces hápticos puedan proporcionar un sentido del tacto realista (en general, se necesitan velocidades de respuesta de 500-1000 Hz).Esta tesis presenta algunos métodos numéricos novedosos para la simulación interactiva de dos procedimientos quirúrgicos habituales: el corte y el rasgado (o desgarro) de tejidos blandos. El marco común de los métodos presentados es el uso de la Descomposición Propia Generalizada (PGD en inglés) para la generación de vademécums computacionales, esto es, metasoluciones generales de problemas paramétricos de altas dimensiones que se pueden evaluar a velocidades de respuesta compatibles con entornos hápticos.En el caso del corte, los vademécums computacionales se utilizan de forma conjunta con técnicas basadas en XFEM, mientras que la carga de cálculo se distribuye entre una etapa off-line (previa a la ejecución interactiva) y otra on-line (en tiempo de ejecución). Durante la fase off-line, para el órgano en cuestión se precalculan tanto un vademécum computacional para cualquier posición de una carga, como los desplazamientos producidos por un conjunto de cortes. Así, durante la etapa on-line, los resultados precalculados se combinan de la forma más adecuada para obtener en tiempo real la respuesta a las acciones dirigidas por el usuario. En cuanto al rasgado, a partir de una ecuación paramétrica basada en mecánica del daño continuo, se obtiene un vademécum computacional. La complejidad del modelo se reduce mediante técnicas de Descomposición Ortogonal Propia (POD en inglés), y el vademécum se incorpora a una formulación incremental explícita que se puede interpretar como una especie de integrador temporal.A modo de ejemplo, el método para el corte se aplica a la simulación de un procedimiento quirúrgico refractivo de la córnea conocido como queratotomía radial, mientras que el método para el rasgado se centra en la simulación de la colecistectomía laparoscópica (la extirpación de la vesícula biliar mediante laparoscopia). En ambos casos, los métodos implementados ofrecen excelentes resultados en términos de velocidades de respuesta y producen simulaciones muy realistas desde los puntos de vista visual y háptico.The real-time computer-based simulation of surgery has proven to be an appealing alternative to traditional surgical simulators. Amongst other advantages, computer-based simulators provide considerable savings on time and maintenance costs, and allow trainees to practice their surgical skills in a safe environment as often as necessary. However, in spite of the current computer capabilities, computational surgery continues to be a challenging field of research. One of its major issues is the high speed at which complex problems in continuum mechanics have to be solved so that haptic interfaces can render a realistic sense of touch (generally, feedback rates of 500–1 000 Hz are required). This thesis introduces some novel numerical methods for the interactive simulation of two usual surgical procedures: cutting and tearing of soft tissues. The common framework of the presented methods is the use of the Proper Generalised Decomposition (PGD) for the generation of computational vademecums, i. e. general meta-solutions of parametric high-dimensional problems that can be evaluated at feedback rates compatible with haptic environments. In the case of cutting, computational vademecums are used jointly with XFEM-based techniques, and the computing workload is distributed into an off-line and an on-line stage. During the off-line stage, both a computational vademecum for any position of a load and the displacements produced by a set of cuts are pre-computed for the organ under consideration. Thus, during the on-line stage, the pre-computed results are properly combined together to obtain in real-time the response to the actions driven by the user. Concerning tearing, a computational vademecum is obtained from a parametric equation based on continuum damage mechanics. The complexity of the model is reduced by Proper Orthogonal Decomposition (POD) techniques, and the vademecum is incorporated into an explicit incremental formulation that can be viewed as a sort of time integrator. By way of example, the cutting method is applied to the simulation of a corneal refractive surgical procedure known as radial keratotomy, whereas the tearing method focuses on the simulation of laparoscopic cholecystectomy (i. e. the removal of the gallbladder). In both cases, the implemented methods offer excellent performances in terms of feedback rates, and produce.<br /

Two-level Continuous Topology Optimization in Structural Mechanics

In the current industry, the development of optimized mechanical components able to satisfy the customer requirements evolves quickly. Therefore, companies are asked for efficient solutions to improve their products in terms of stiffness and strength. In this sense, Topology Optimization has been extensively used to determine the best topology of structural components from the mechanical point of view. Its main objective is to distribute a given amount of material into a predefined domain to reach the maximum overall stiffness of the component. Besides, high-resolution solutions are essential to define the final distribution of material. Standard Topological Optimization tools are able to propose an optimal topology for the whole component, but when small topological details are required (i.e. trabecular-type structures) the computational cost is prohibitive. In order to mitigate this issue, the present work proposes a two-level topology optimization method to solve high-resolution problems by using density-based methods. The proposed methodology includes three steps: The first one subdivides the whole component in cells and generates a coarse optimized low-definition material distribution assigning one different density to each cell. The second one uses an equilibrating technique that provides tractions continuity between adjacent cells, thus ensuring the material inter-cell continuity after the cells optimization process. Finally, each cell is optimized at fine scale taking as input data the densities and the equilibrated tractions obtained from the macro problem. The main goal of this work is to efficiently solve high-resolution topology optimization problems using density-based methods, which would be unaffordable with standard computing facilities and the current methodologies.Comment: 23 page

Virtual, Digital and Hybrid Twins: A New Paradigm in Data-Based Engineering and Engineered Data

Engineering is evolving in the same way than society is doing. Nowadays, data is acquiring a prominence never imagined. In the past, in the domain of materials, processes and structures, testing machines allowed extract data that served in turn to calibrate state-of-the-art models. Some calibration procedures were even integrated within these testing machines. Thus, once the model had been calibrated, computer simulation takes place. However, data can offer much more than a simple state-of-the-art model calibration, and not only from its simple statistical analysis, but from the modeling and simulation viewpoints. This gives rise to the the family of so-called twins: the virtual, the digital and the hybrid twins. Moreover, as discussed in the present paper, not only data serve to enrich physically-based models. These could allow us to perform a tremendous leap forward, by replacing big-data-based habits by the incipient smart-data paradigm

Computational material design for acoustic cloaking

A topology optimization technique based on the topological derivative and the level set function is utilized to design/synthesize the micro-structure of a pentamode material for an acoustic cloaking device. The technique provides a micro-structure consisting of a honeycomb lattice composed of needle-like and joint members. The resulting metamaterial shows a highly anisotropic elastic response with effective properties displaying a ratio between bulk and shear moduli of almost 3 orders of magnitude. Furthermore, in accordance with previous works in the literature, it can be asserted that this kind of micro-structure can be realistically fabricated. The adoption of a topology optimization technique as a tool for the inverse design of metamaterials with applications to acoustic cloaking problems is one contribution of this paper. However, the most important achievement refers to the analysis and discussion revealing the key role of the external shape of the prescribed domain where the optimization problem is posed. The efficiency of the designed micro-structure is measured by comparing the scattering wave fields generated by acoustic plane waves impinging on bare and cloaked bodies.Peer ReviewedPostprint (published version

Multiscale computational homogenization: review and proposal of a new enhanced-first-order method

This is a copy of the author 's final draft version of an article published in the Archives of computational methods in engineering. The final publication is available at Springer via http://dx.doi.org/10.1007/s11831-016-9205-0The continuous increase of computational capacity has encouraged the extensive use of multiscale techniques to simulate the material behaviour on several fields of knowledge. In solid mechanics, the multiscale approaches which consider the macro-scale deformation gradient to obtain the homogenized material behaviour from the micro-scale are called first-order computational homogenization. Following this idea, the second-order FE2 methods incorporate high-order gradients to improve the simulation accuracy. However, to capture the full advantages of these high-order framework the classical boundary value problem (BVP) at the macro-scale must be upgraded to high-order level, which complicates their numerical solution. With the purpose of obtaining the best of both methods i.e. first-order and second-order, in this work an enhanced-first-order computational homogenization is presented. The proposed approach preserves a classical BVP at the macro-scale level but taking into account the high-order gradient of the macro-scale in the micro-scale solution. The developed numerical examples show how the proposed method obtains the expected stress distribution at the micro-scale for states of structural bending loads. Nevertheless, the macro-scale results achieved are the same than the ones obtained with a first-order framework because both approaches share the same macro-scale BVP.Peer ReviewedPostprint (author's final draft