Efficient fast Fourier transform-based solvers for computing the thermomechanical behavior of applied materials

The mechanical behavior of many applied materials arises from their microstructure. Thus, to aid the design, development and industrialization of new materials, robust computational homogenization methods are indispensable. The present thesis is devoted to investigating and developing FFT-based micromechanics solvers for efficiently computing the (thermo)mechanical response of nonlinear composite materials with complex microstructures

Free and forced propagation of Bloch waves in viscoelastic beam lattices

Beam lattice materials can be characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. Within this framework, governing the free and forced wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical interest with applications in aerospace, chemical, naval, biomedical engineering. The first part of the Thesis addresses the free propagation of Bloch waves in non-dissipative microstructured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the bidimensional infinite material domain, according to the classic canons of linear solid or structural mechanics. Adopting the centrosymmetric tetrachiral cell as prototypical periodic topology, the frequency dispersion spectrum is obtained by applying the Floquet-Bloch theory. The dispersion spectrum resulting from a synthetic Lagrangian beam lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continua). Asymptotic perturbation-based approximations and numerical spectral solutions are compared and cross-validated. Adopting the low-frequency band gaps of the dispersion spectrum as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The microstructural design or tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material. Alternatively, band gaps in the material spectrum can be opened at target center frequencies by using metamaterials with inertial resonators. Based on these motivations, in the second part of the Thesis, a general dynamic formulation is presented for determining the dispersion properties of viscoelastic metamaterials, equipped with local dissipative resonators. The linear mechanism of local resonance is realized by tuning periodic auxiliary masses, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped Bloch waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent coefficients are obtained by applying the bilateral Laplace transform. The corresponding complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed. Particularly, the spectra corresponding to Taylor series approximations of the equation coefficients are investigated. The standard dynamic equations with linear viscous damping are recovered at the first order approximation. Increasing approximation orders determine non-negligible spectral effects, including the occurrence of pure damping spectral branches. Finally, the forced response to harmonic single frequency external forces in the frequency and the time domains is investigated. The response in the time domain is obtained by applying the inverse bilateral Laplace transform. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint

Higher-order multi-scale deep Ritz method for multi-scale problems of authentic composite materials

The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials with highly oscillatory and discontinuous coefficients. In this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method are designed to high-accuracy and mesh-free simulation for macroscopic homogenized equation without multi-scale property and microscopic lower-order and higher-order cell problems with highly discontinuous coefficients. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, extensive numerical experiments are presented to show the computational accuracy of the proposed HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep learning framework that enables the effective simulation and analysis of multi-scale problems of authentic composite materials

Efficient fast Fourier transform-based solvers for computing the thermomechanical behavior of applied materials

The mechanical behavior of many applied materials arises from their microstructure. Thus, to aid the design, development and industrialization of new materials, robust computational homogenization methods are indispensable. The present thesis is devoted to investigating and developing FFT-based micromechanics solvers for efficiently computing the (thermo)mechanical response of nonlinear composite materials with complex microstructures

Dynamic problems for metamaterials: Review of existing models and ideas for further research

Metamaterials are materials especially engineered to have a peculiar physical behaviour, to be exploited for some well-specified technological application. In this context we focus on the conception of general micro-structured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom, which may be electric or, more generally, related to some micro-motion. An interesting class of problems in this context regards the design of wave-guides aimed to control wave propagation. The description of the state of the art is followed by some hints addressed to describe some possible research developments and in particular to design optimal design techniques for bone reconstruction or systems which may block wave propagation in some frequency ranges, in both linear and non-linear fields. (C) 2014 Elsevier Ltd. All rights reserved

A review of nonlinear FFT-based computational homogenization methods

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform

Development of a finite volume method for elastic materials and fluid-solid coupled applications

This thesis presents the development of a parallel finite volume numerical method to analyse thermoelastic and hyperelastic materials and applied problems with mutual interaction between a fluid and a structure. The solid problem follows a cell-centred finite volume formulation for three-dimensional unstructured grids under the same framework that is frequently devoted to computational fluid dynamics. Second-order accurate schemes are used to discretise both in time and space. A direct implicit time integration promotes numerical stability when facing vibration and quasi-static scenarios. The geometrical non-linearities, encountered with the large displacements of both Saint Venant-Kirchhoff and neo-Hookean models, are tackled by means of an updated Lagrangian approach. Verification of the method is conducted with canonical cases which involve: static equilibrium, thermal stress, vibration, structural damping, large deformations, nearly incompressible materials and high memory usage. Significant savings in computation time are achieved owing to the acceleration strategies implemented within the system resolution, namely a segregated algorithm with Aitken relaxation and a block-coupled system arrangement. The similarities between the block-coupled method and the displacement-based finite element method, with regards to the matrix form of the resulting equations, allow for including Rayleigh viscous damping within a finite volume solver. The program for structures is to be coupled with the in-house fluid numerical models in order to produce a unified fluid-structure interaction platform, where an arbitrary Lagrangian-Eulerian approach is used to solve the flow in a conforming grid. As a first step, the method for incompressible Newtonian fluids is adapted to deal with structure-coupled problems. To do so, the Lagrangian-Eulerian version of the Navier-Stokes equations is presented, and automatic moving mesh techniques are developed. These techniques are designed to mitigate the mesh quality deterioration and to satisfy the space conservation law. Besides, a semi-implicit coupling algorithm, which only implicitly couples the fluid pressure term to the structure, is implemented. As a result, numerical stability for strongly coupled phenomena at a reduced computational cost is obtained. These new tools are tested on an applied case, consisting of the turbulent flow through self-actuated flexible valves. Finally, a pioneering coupled numerical model for the thermal and structural analysis of packed-bed thermocline storage tanks is developed. This thermal accumulation system for concentrated solar power plants has attracted the attention of the industry due to the economic advantage compared to the usual two-tank system. Dynamic coupling among the thermoelastic equations for the tank shell and the numerical models for all other relevant elements of the system is considered. After validating the model with experimental results, the commercial viability of the thermocline concept, regarding energetic effectiveness and structural reliability, is evaluated under real operating conditions of the power plants.Esta tesis presenta el desarrollo de un método numérico paralelo basado en volúmenes finitos para analizar materiales termoelásticos e hiperelásticos y problemas con una interacción mutua entre un fluido y una estructura. El problema del sólido sigue una formulación de volúmenes finitos centrada en las celdas para mallas no-estructuradas tridimensionales, bajo el mismo marco que se suele emplear en la dinámica de fluidos computacional. Se utilizan esquemas de segundo orden de precisión para discretizar el tiempo y el espacio. Una integración temporal directa implícita asegura estabilidad numérica al afrontar escenarios casi-estáticos o de vibración. Las no linealidades, que aparecen con los amplios desplazamientos de los modelos de Saint Venant-Kirchhoff y de neo-Hookean, son abordadas con un enfoque Lagrangiano actualizado. La verificación del método se realiza a través de casos canónicos que involucran: equilibrio estático, tensiones térmicas, vibración, amortiguación estructural, grandes deformaciones, materiales casi incompresibles y altos requerimientos de memoria. Se registra un ahorro significativo en el tiempo de cálculo gracias a las estrategias de aceleración implementadas dentro de la resolución del sistema, principalmente un algoritmo segregado con relajación Aitken y una disposición acoplada en bloques del sistema. Las similitudes entre este método acoplado en bloques y el método de los elementos finitos basados en el desplazamiento, con respecto a la forma matricial de las ecuaciones resultantes, permiten incluir la amortiguación viscosa tipo Rayleigh dentro de un solucionador de volúmenes finitos. El programa para estructuras se acoplará con los modelos numéricos internos para fluidos con el objetivo de generar una plataforma unificada de interacción fluido-estructura, donde se usa un enfoque arbitrario Lagrangiano-Euleriano sobre una malla conforme para resolver el fluido. Como primer paso, el método para flujos incompresibles Newtonianos se adapta para lidiar con problemas acoplados a una estructura. Para ello, se presenta la versión Lagrangiana-Euleriana de las ecuaciones de Navier-Stokes y se desarrollan técnicas automáticas de movimiento de malla. El diseño de estas técnicas se centra en mitigar el deterioro de la calidad de la malla y satisfacer la ley de conservación del espacio. Además, se implementa un algoritmo de acoplamiento semi-implícito, que sólo acopla implícitamente el término fluido de presión a la estructura. Como resultado, se obtiene estabilidad numérica para fenómenos fuertemente acoplados a un coste computacional reducido. Estas nuevas herramientas se prueban en un caso aplicado, que consiste el flujo turbulento a través de válvulas flexibles autoactivadas. Finalmente, se desarrolla un modelo numérico acoplado pionero para analizar estructuralmente y térmicamente los tanques termoclina de almacenamiento térmico. Este sistema de acumulación para centrales termosolares ha atraído la atención de la industria debido al ahorro económico comparado con el sistema de doble tanque habitual. Se tiene en cuenta el acoplamiento dinámico entre las ecuaciones gobernantes de la pared del tanque y las de todos los elementos relevantes del sistema. Tras validar el modelo con datos experimentales, se evalúa la viabilidad comercial de estos tanques, en cuanto a rendimiento energético y fiabilidad estructural, bajo condiciones reales de operación de las centrales.Postprint (published version

Free vibration analysis of FGM plates by using layer wise displacement model

In this paper, the free vibration analysis of simply supported functionally graded material (FGM) plate is analyzed. The displacement model based on Generalized Laminate Plate Theory (GLPT) assumes layer wise (LW) linear variation of in–plane displacements, constant transverse displacement, linear strain–displacement relations and linear material properties. The effective material properties of FGMs are assumed to be given by the Voigt’s rule of mixture (ROM). The Power law distribution of volume fraction is assumed through the plate thickness. The mathematical model includes the quadratic variation of transverse shear stresses within each mathematical layer of the plate. The principle of virtual displacements (PVD) is used to derive Euler–Lagrange differential equations of motions for free vibration problem. The Closed form solution is derived following the Navier’s technique and solving the eigenvalue problem. An original MATLAB computer program is coded for the numerical solution. The results reveal that the effects of side–to–thickness ratio, power–low index and material properties have significant effect on free vibration frequencies of FGM plates