Mathematical development of stress-strain law for rock block contacts

The fracture behavior of rock block contacts has been studied for many years. Unfortunately, up to now, there is not a rigorous formulation or a solid theoretical foundation to support it. A mathematical development to represent the failure mechanism which occurs in the contacts between rock blocks is presented to evaluate the performance of breaking mechanism of such blocks relating it to the morphology of the contact and mechanical parameters of the material. The examined framework includes the evaluation of the surface roughness of first order in the failure mechanism of the granular particles of large size and the development of a theoretical model describing the morphology of the contact between rock blocks.

A finite class of orthogonal functions generated by Routh-Romanovski polynomials

It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh-Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.

On the application of meshfree methods to the nonlinear dynamics of multibody systems

Esta tesis propone una completa formulación termo-mecánica para la simulación no-lineal de mecanismos flexibles basada en métodos libres de malla. El enfoque se basa en tres pilares principales: la formulación de Lagrangiano total para medios continuos, la discretización de Bubnov-Galerkin, y las funciones de forma libres de malla. Los métodos sin malla se caracterizan por la definición de un conjunto de funciones de forma en dominios solapados, junto con una malla de integración de las ecuaciones discretas de balance. Dos tipos de funciones de forma se han seleccionado como representación de las familias interpolantes (Funciones de Base Radial) y aproximantes (Mínimos Cuadrados Móviles). Su formulación se ha adaptado haciendo sus parámetros compatibles, y su ausencia de conectividad predefinida se ha aprovechado para interconectar múltiples dominios de manera automática, permitiendo el uso de mallas de fondo no conformes. Se propone una formulación generalizada de restricciones, juntas y contactos, válida para sólidos rígidos y flexibles, siendo estos últimos discretizados mediante elementos finitos (MEF) o libres de malla. La mayor ventaja de este enfoque reside en que independiza completamente el dominio con respecto de las uniones y acciones externas a cada sólido, permitiendo su definición incluso fuera del contorno. Al mismo tiempo, también se minimiza el número de ecuaciones de restricción necesarias para la definición de uniones realistas. Las diversas validaciones, ejemplos y comparaciones detalladas muestran como el enfoque propuesto es genérico y extensible a un gran número de sistemas. En concreto, las comparaciones con el MEF indican una importante reducción del error para igual número de nodos, tanto en simulaciones mecánicas, como térmicas y termo-mecánicas acopladas. A igualdad de error, la eficiencia numérica de los métodos libres de malla es mayor que la del MEF cuanto más grosera es la discretización. Finalmente, la formulación se aplica a un problema de diseño real sobre el mantenimiento de estructuras masivas en el interior de un reactor de fusión, demostrando su viabilidad en análisis de problemas reales, y a su vez mostrando su potencial para su uso en simulación en tiempo real de sistemas no-lineales. A new complete formulation is proposed for the simulation of nonlinear dynamic of multibody systems with thermo-mechanical behaviour. The approach is founded in three main pillars: total Lagrangian formulation, Bubnov-Galerkin discretization, and meshfree shape functions. Meshfree methods are characterized by the definition of a set of shape functions in overlapping domains, and a background grid for integration of the Galerkin discrete equations. Two different types of shape functions have been chosen as representatives of interpolation (Radial Basis Functions), and approximation (Moving Least Squares) families. Their formulation has been adapted to use compatible parameters, and their lack of predefined connectivity is used to interconnect different domains seamlessly, allowing the use of non-conforming meshes. A generalized formulation for constraints, joints, and contacts is proposed, which is valid for rigid and flexible solids, being the later discretized using either finite elements (FEM) or meshfree methods. The greatest advantage of this approach is that makes the domain completely independent of the external links and actions, allowing to even define them outside of the boundary. At the same time, the number of constraint equations needed for defining realistic joints is minimized. Validation, examples, and benchmarks are provided for the proposed formulation, demonstrating that the approach is generic and extensible to further problems. Comparisons with FEM show a much lower error for the same number of nodes, both for mechanical and thermal analyses. The numerical efficiency is also better when coarse discretizations are used. A final demonstration to a real problem for handling massive structures inside of a fusion reactor is presented. It demonstrates that the application of meshfree methods is feasible and can provide an advantage towards the definition of nonlinear real-time simulation models.

A study of the mechanical behaviour of the human aortic artery by means of non-linear finite element models

En la presente tesis desarrollamos una estrategia para la simulación numérica del comportamiento mecánico de la aorta humana usando modelos de elementos finitos no lineales. Prestamos especial atención a tres aspectos claves relacionados con la biomecánica de los tejidos blandos. Primero, el análisis del comportamiento anisótropo característico de los tejidos blandos debido a las familias de fibras de colágeno. Segundo, el análisis del ablandamiento presentado por los vasos sanguíneos cuando estos soportan cargas fuera del rango de funcionamiento fisiológico. Y finalmente, la inclusión de las tensiones residuales en las simulaciones en concordancia con el experimento de apertura de ángulo. El análisis del daño se aborda mediante dos aproximaciones diferentes. En la primera aproximación se presenta una formulación de daño local con regularización. Esta formulación tiene dos ingredientes principales. Por una parte, usa los principios de la teoría de la fisura difusa para garantizar la objetividad de los resultados con diferentes mallas. Por otra parte, usa el modelo bidimensional de Hodge-Petruska para describir el comportamiento mesoscópico de los fibriles. Partiendo de este modelo mesoscópico, las propiedades macroscópicas de las fibras de colágeno son obtenidas a través de un proceso de homogenización. En la segunda aproximación se presenta un modelo de daño no-local enriquecido con el gradiente de la variable de daño. El modelo se construye a partir del enriquecimiento de la función de energía con un término que contiene el gradiente material de la variable de daño no-local. La inclusión de este término asegura una regularización implícita de la implementación por elementos finitos, dando lugar a resultados de las simulaciones que no dependen de la malla. La aplicabilidad de este último modelo a problemas de biomecánica se estudia por medio de una simulación de un procedimiento quirúrgico típico conocido como angioplastia de balón. In the present thesis we develop a framework for the numerical simulation of the mechanical behaviour of the human aorta using non-linear finite element models. Special attention is paid to three key aspects related to the biomechanics of soft tissues. First, the modelling of the characteristic anisotropic behaviour of the softue due to the collagen fibre families. Secondly, the modelling of damage-related softening that blood vessels exhibit when subjected to loads beyond their physiological range. And finally, the inclusion of the residual stresses in the simulations in accordance with the opening-angle experiment The modelling of damage is addressed with two major and different approaches. In the first approach a continuum local damage formulation with regularisation is presented. This formulation has two principal ingredients. On the one hand, it makes use of the principles of the smeared crack theory to avoid the mesh size dependence of the structural response in softening. On the other hand, it uses a Hodge-Petruska bidimensional model to describe the fibrils as staggered arrays of tropocollagen molecules, and from this mesoscopic model the macroscopic material properties of the collagen fibres are obtained using an homogenisation process. In the second approach a non-local gradient-enhanced damage formulation is introduced. The model is built around the enhancement of the free energy function by means of a term that contains the referential gradient of the non-local damage variable. The inclusion of this term ensures an implicit regularisation of the finite element implementation, yielding mesh-objective results of the simulations. The applicability of the later model to biomechanically-related problems is studied by means of the simulation of a typical surgical procedure, namely, the balloon angioplasty.

Finite Element Model of the Innovation Diffusion: An Application to Photovoltaic Systems

This paper presents a Finite Element Model, which has been used for forecasting the diffusion of innovations in time and space. Unlike conventional models used in diffusion literature, the model considers the spatial heterogeneity. The implementation steps of the model are explained by applying it to the case of diffusion of photovoltaic systems in a local region in southern Germany. The applied model is based on a parabolic partial differential equation that describes the diffusion ratio of photovoltaic systems in a given region over time. The results of the application show that the Finite Element Model constitutes a powerful tool to better understand the diffusion of an innovation as a simultaneous space-time process. For future research, model limitations and possible extensions are also discussed.

Análisis paramétrico determinista y aleatorio de problemas dinámicos en estructuras aeroespaciales

Los fenómenos dinámicos pueden poner en peligro la integridad de estructuras aeroespaciales y los ingenieros han desarrollado diferentes estrategias para analizarlos. Uno de los grandes problemas que se plantean en la ingeniería es cómo atacar un problema dinámico estructural. En la presente tesis se plantean distintos fenómenos dinámicos y se proponen métodos para estimar o simular sus comportamientos mediante un análisis paramétrico determinista y aleatorio del problema. Se han propuesto desde problemas sencillos con pocos grados de libertad que sirven para analizar las diferentes estrategias y herramientas a utilizar, hasta fenómenos muy dinámicos que contienen comportamientos no lineales, daños y fallos. Los primeros ejemplos de investigación planteados cubren una amplia gama de los fenómenos dinámicos, como el análisis de vibraciones de elementos másicos, incluyendo impactos y contactos, y el análisis de una viga con carga armónica aplicada a la que también se le añaden parámetros aleatorios que pueden responder a un desconocimiento o incertidumbre de los mismos. Durante el desarrollo de la tesis se introducen conceptos y se aplican distintos métodos, como el método de elementos finitos (FEM) en el que se analiza su resolución tanto por esquemas implícitos como explícitos, y métodos de análisis paramétricos y estadísticos mediante la técnica de Monte Carlo. Más adelante, una vez ya planteadas las herramientas y estrategias de análisis, se estudian fenómenos más complejos, como el impacto a baja velocidad en materiales compuestos, en el que se busca evaluar la resistencia residual y, por lo tanto, la tolerancia al daño de la estructura. Se trata de un suceso que puede producirse por la caída de herramienta, granizo o restos en la pista de aterrizaje. Otro de los fenómenos analizados también se da en un aeropuerto y se trata de la colisión con un dispositivo frangible, el cual tiene que romperse bajo ciertas cargas y, sin embargo, soportar otras. Finalmente, se aplica toda la metodología planteada en simular y analizar un posible incidente en vuelo, el fenómeno de la pérdida de pala de un turbohélice. Se trata de un suceso muy particular en el que la estructura tiene que soportar unas cargas complejas y excepcionales con las que la aeronave debe ser capaz de completar con éxito el vuelo. El análisis incluye comportamientos no lineales, daños, y varios tipos de fallos, y en el que se trata de identificar los parámetros clave en la secuencia del fallo. El suceso se analiza mediante análisis estructurales deterministas más habituales y también mediante otras técnicas como el método de Monte Carlo con el que se logran estudiar distintas incertidumbres en los parámetros con variables aleatorias. Se estudian, entre otros, el tamaño de pala perdida, la velocidad y el momento en el que se produce la rotura, y la rigidez y resistencia de los apoyos del motor. Se tiene en cuenta incluso el amortiguamiento estructural del sistema. Las distintas estrategias de análisis permiten obtener unos resultados valiosos e interesantes que han sido objeto de distintas publicaciones. ABSTRACT Dynamic phenomena can endanger the integrity of aerospace structures and, consequently, engineers have developed different strategies to analyze them. One of the major engineering problems is how to deal with the structural dynamics. In this thesis, different dynamic phenomena are introduced and several methods are proposed to estimate or simulate their behaviors. The analysis is considered through parametric, deterministic and statistical methods. The suggested issues are from simple problems with few degrees of freedom, in order to develop different strategies and tools to solve them, to very dynamic phenomena containing nonlinear behaviors failures, damages. The first examples cover a wide variety of dynamic phenomena such as vibration analysis of mass elements, including impacts and contacts, and beam analysis with harmonic load applied, in which random parameters are included. These parameters can represent the unawareness or uncertainty of certain variables. During the development of the thesis several concepts are introduced and different methods are applied, such as the finite element method (FEM), which is solved through implicit and explicit schemes, and parametrical and statistical methods using the Monte Carlo analysis technique. Next, once the tools and strategies of analysis are set out more complex phenomena are studied. This is the case of a low-speed impact in composite materials, the residual strength of the structure is evaluated, and therefore, its damage tolerance. This incident may occur from a tool dropped, hail or debris throw on the runway. At an airport may also occur, and it is also analyzed, a collision between an airplane and a frangible device. The devise must brake under these loads, however, it must withstand others. Finally, all the considered methodology is applied to simulate and analyze a flight incident, the blade loss phenomenon of a turboprop. In this particular event the structure must support complex and exceptional loads and the aircraft must be able to successfully complete the flight. Nonlinear behavior, damage, and different types of failures are included in the analysis, in which the key parameters in the failure sequence are identified. The incident is analyzed by deterministic structural analysis and also by other techniques such as Monte Carlo method, in which it is possible to include different parametric uncertainties through random variables. Some of the evaluated parameters are, among others, the blade loss size, propeller rotational frequency, speed and angular position where the blade is lost, and the stiffness and strength of the engine mounts. The study does also research on the structural damping of the system. The different strategies of analysis obtain valuable and interesting results that have been already published.

A time-domain finite element method for seakeeping and wave resistance problems

El objetivo de la tesis es la investigación de algoritmos numéricos para el desarrollo de herramientas numéricas para la simulación de problemas tanto de comportamiento en la mar como de resistencia al avance de buques y estructuras flotantes. La primera herramienta desarrollada resuelve el problema de difracción y radiación de olas. Se basan en el método de los elementos finitos (MEF) para la resolución de la ecuación de Laplace, así como en esquemas basados en MEF, integración a lo largo de líneas de corriente, y en diferencias finitas desarrollados para la condición de superficie libre. Se han desarrollado herramientas numéricas para la resolución de la dinámica de sólido rígido en sistemas multicuerpos con ligaduras. Estas herramientas han sido integradas junto con la herramienta de resolución de olas difractadas y radiadas para la resolución de problemas de interacción de cuerpos con olas. También se han diseñado algoritmos de acoplamientos con otras herramientas numéricas para la resolución de problemas multifísica. En particular, se han realizado acoplamientos con una herramienta numérica basada de cálculo de estructuras con MEF para problemas de interacción fluido-estructura, otra de cálculo de líneas de fondeo, y con una herramienta numérica de cálculo de flujos en tanques internos para problemas acoplados de comportamiento en la mar con “sloshing”. Se han realizado simulaciones numéricas para la validación y verificación de los algoritmos desarrollados, así como para el análisis de diferentes casos de estudio con aplicaciones diversas en los campos de la ingeniería naval, oceánica, y energías renovables marinas. ABSTRACT The objective of this thesis is the research on numerical algorithms to develop numerical tools to simulate seakeeping problems as well as wave resistance problems of ships and floating structures. The first tool developed is a wave diffraction-radiation solver. It is based on the finite element method (FEM) in order to solve the Laplace equation, as well as numerical schemes based on FEM, streamline integration, and finite difference method tailored for solving the free surface boundary condition. It has been developed numerical tools to solve solid body dynamics of multibody systems with body links across them. This tool has been integrated with the wave diffraction-radiation solver to solve wave-body interaction problems. Also it has been tailored coupling algorithms with other numerical tools in order to solve multi-physics problems. In particular, it has been performed coupling with a MEF structural solver to solve fluid-structure interaction problems, with a mooring solver, and with a solver capable of simulating internal flows in tanks to solve couple seakeeping-sloshing problems. Numerical simulations have been carried out to validate and verify the developed algorithms, as well as to analyze case studies in the areas of marine engineering, offshore engineering, and offshore renewable energy.

Resonadores piezoeléctricos de cizalladura: estudio de la formación de biofilms y caracterización de fluidos magneto‐reológicos

La presente tesis revisa y analiza algunos aspectos fundamentales relativos al comportamiento de los sensores basados en resonadores piezoeléctricos TSM (Thickness Shear Mode), así como la aplicación de los mismos al estudio y caracterización de dos medios viscoelásticos de gran interés: los fluidos magnetoreológicos y los biofilms microbianos. El funcionamiento de estos sensores está basado en la medida de sus propiedades resonantes, las cuales varían al entrar en contacto con el material que se quiere analizar. Se ha realizado un análisis multifrecuencial, trabajando en varios modos de resonancia del transductor, en algunas aplicaciones incluso de forma simultánea (excitación pulsada). Se han revisado fenómenos como la presencia de microcontactos en la superficie del sensor y la resonancia de capas viscoelásticas de espesor finito, que pueden afectar a los sensores de cuarzo de manera contraria a lo que predice la teoría convencional (Sauerbrey y Kanazawa), pudiéndonos llevar a incrementos positivos de la frecuencia de resonancia. Además, se ha estudiado el efecto de una deposición no uniforme sobre el resonador piezoeléctrico. Para ello se han medido deposiciones de poliuretano, modelándose la respuesta del resonador con estas deposiciones mediante FEM. El modelo numérico permite estudiar el comportamiento del resonador al modificar distintas variables geométricas (espesor, superficie, no uniformidad y zona de deposición) de la capa depositada. Se ha demostrado que para espesores de entre un cuarto y media longitud de onda aproximadamente, una capa viscoelástica no uniforme sobre la superficie del sensor, amplifica el incremento positivo del desplazamiento de la frecuencia de resonancia en relación con una capa uniforme. Se ha analizado también el patrón geométrico de la sensibilidad del sensor, siendo también no uniforme sobre su superficie. Se han aplicado sensores TSM para estudiar los cambios viscoelásticos que se producen en varios fluidos magneto-reológicos (FMR) al aplicarles distintos esfuerzos de cizalla controlados por un reómetro. Se ha podido ver que existe una relación directa entre diversos parámetros reológicos obtenidos con el reómetro (fuerza normal, G’, G’’, velocidad de deformación, esfuerzo de cizalla…) y los parámetros acústicos, caracterizándose los FMR tanto en ausencia de campo magnético, como con campo magnético aplicado a distintas intensidades. Se han estudiado las ventajas que aporta esta técnica de medida sobre la técnica basada en un reómetro comercial, destacando que se consigue caracterizar con mayor detalle algunos aspectos relevantes del fluido como son la deposición de partículas (estabilidad del fluido), el proceso de ruptura de las estructuras formadas en los FMR tanto en presencia como en ausencia de campo magnético y la rigidez de los microcontactos que aparecen entre partículas y superficies. También se han utilizado sensores de cuarzo para monitorear en tiempo real la formación de biofilms de Staphylococcus epidermidis y Eschericia coli sobre los propios resonadores de cristal de cuarzo sin ningún tipo de recubrimiento, realizándose ensayos con cepas que presentan distinta capacidad de producir biofilm. Se mostró que, una vez que se ha producido una primera adhesión homogénea de las bacterias al sustrato, podemos considerar el biofilm como una capa semi-infinita, de la cual el sensor de cuarzo refleja las propiedades viscoelásticas de la región inmediatamente contigua al resonador, no siendo sensible a lo que sucede en estratos superiores del biofilm. Los experimentos han permitido caracterizar el módulo de rigidez complejo de los biofilms a varias frecuencias, mostrándose que el parámetro característico que indica la adhesión de un biofilm tanto en el caso de S. epidermidis como de E. coli, es el incremento de G’ (relacionado con la elasticidad o rigidez de la capa), el cual viene ligado a un incremento de la frecuencia de resonancia del sensor. ABSTRACT This thesis reviews and analyzes some key aspects of the behavior of sensors based on piezoelectric resonators TSM (Thickness Shear Mode) and their applications to the study and characterization in two viscoelastic media of great interest: magnetorheological fluids and microbial biofilms. The operation of these sensors is based on the analysis of their resonant properties that vary in contact with the material to be analyzed. We have made a multi-frequency analysis, working in several modes of resonance of the transducer, in some applications even simultaneously (by impulse excitation). We reviewed some phenomena as the presence of micro-contacts on the sensor surface and the resonance of viscoelastic layers of finite thickness, which can affect quartz sensors contrary to the conventional theory predictions (Sauerbrey and Kanazawa), leading to positive resonant frequency shifts. In addition, we studied the effect of non-uniform deposition on the piezoelectric resonator. Polyurethane stools have been measured, being the resonator response to these depositions modeled by FEM. The numerical model allows studying the behavior of the resonator when different geometric variables (thickness, surface non-uniformity and deposition zone) of the deposited layer are modified. It has been shown that for thicknesses between a quarter and a half of a wavelength approximately, non-uniform deposits on the sensor surface amplify the positive increase of the resonance frequency displacement compared to a uniform layer. The geometric pattern of the sensor sensitivity was also analyzed, being also non-uniform over its surface. TSM sensors have been applied to study the viscoelastic changes occurring in various magneto-rheological fluids (FMR) when subjected to different controlled shear stresses driven by a rheometer. It has been seen that there is a direct relationship between various rheological parameters obtained with the rheometer (normal force, G', G'', stress, shear rate ...) and the acoustic parameters, being the FMR characterized both in the absence of magnetic field, and when the magnetic field was applied at different intensities. We have studied the advantages of this technique over the characterization methods based on commercial rheometers, noting that TSM sensors are more sensitive to some relevant aspects of the fluid as the deposition of particles (fluid stability), the breaking process of the structures formed in the FMR both in the presence and absence of magnetic field, and the rigidity of the micro-contacts appearing between particles and surfaces. TSM sensors have also been used to monitor in real time the formation of biofilms of Staphylococcus epidermidis and Escherichia coli on the quartz crystal resonators themselves without any coating, performing tests with strains having different ability to produce biofilm. It was shown that, once a first homogeneous adhesion of bacteria was produced on the substrate, the biofilm can be considered as a semi-infinite layer and the quartz sensor reflects only the viscoelastic properties of the region immediately adjacent to the resonator, not being sensitive to what is happening in upper layers of the biofilm. The experiments allow the evaluation of the biofilm complex stiffness module at various frequencies, showing that the characteristic parameter that indicates the adhesion of a biofilm for the case of both S. epidermidis and E. coli, is an increased G' (related to the elasticity or stiffness of the layer), which is linked to an increase in the resonance frequency of the sensor.

Ti/Pd/Ag Contacts to n-Type GaAs for High Current Density Devices

The metallization stack Ti/Pd/Ag on n-type Si has been readily used in solar cells due to its low metal/semiconductor specific contact resistance, very high sheet conductance, bondability, long-term durability, and cost-effectiveness. In this study, the use of Ti/Pd/Ag metallization on n-type GaAs is examined, targeting electronic devices that need to handle high current densities and with grid-like contacts with limited surface coverage (i.e., solar cells, lasers, or light emitting diodes). Ti/Pd/Ag (50 nm/50 nm/1000 nm) metal layers were deposited on n-type GaAs by electron beam evaporation and the contact quality was assessed for different doping levels (from 1.3 × 1018 cm−3 to 1.6 × 1019 cm−3) and annealing temperatures (from 300°C to 750°C). The metal/semiconductor specific contact resistance, metal resistivity, and the morphology of the contacts were studied. The results show that samples doped in the range of 1018 cm−3 had Schottky-like I–V characteristics and only samples doped 1.6 × 1019 cm−3 exhibited ohmic behavior even before annealing. For the ohmic contacts, increasing annealing temperature causes a decrease in the specific contact resistance (ρ c,Ti/Pd/Ag ~ 5 × 10−4 Ω cm2). In regard to the metal resistivity, Ti/Pd/Ag metallization presents a very good metal conductivity for samples treated below 500°C (ρ M,Ti/Pd/Ag ~ 2.3 × 10−6 Ω cm); however, for samples treated at 750°C, metal resistivity is strongly degraded due to morphological degradation and contamination in the silver overlayer. As compared to the classic AuGe/Ni/Au metal system, the Ti/Pd/Ag system shows higher metal/semiconductor specific contact resistance and one order of magnitude lower metal resistivity.

Analysis by finite element calculations of light scattering in laser-textured AZO films for PV thin-film solar cells

In the thin-film photovoltaic industry, to achieve a high light scattering in one or more of the cell interfaces is one of the strategies that allow an enhancement of light absorption inside the cell and, therefore, a better device behavior and efficiency. Although chemical etching is the standard method to texture surfaces for that scattering improvement, laser light has shown as a new way for texturizing different materials, maintaining a good control of the final topography with a unique, clean, and quite precise process. In this work AZO films with different texture parameters are fabricated. The typical parameters used to characterize them, as the root mean square roughness or the haze factor, are discussed and, for deeper understanding of the scattering mechanisms, the light behavior in the films is simulated using a finite element method code. This method gives information about the light intensity in each point of the system, allowing the precise characterization of the scattering behavior near the film surface, and it can be used as well to calculate a simulated haze factor that can be compared with experimental measurements. A discussion of the validation of the numerical code, based in a comprehensive comparison with experimental data is included.

Two-dimensional isostatic meshes in the finite element method

In a Finite Element (FE) analysis of elastic solids several items are usually considered, namely, type and shape of the elements, number of nodes per element, node positions, FE mesh, total number of degrees of freedom (dot) among others. In this paper a method to improve a given FE mesh used for a particular analysis is described. For the improvement criterion different objective functions have been chosen (Total potential energy and Average quadratic error) and the number of nodes and dof's of the new mesh remain constant and equal to the initial FE mesh. In order to find the mesh producing the minimum of the selected objective function the steepest descent gradient technique has been applied as optimization algorithm. However this efficient technique has the drawback that demands a large computation power. Extensive application of this methodology to different 2-D elasticity problems leads to the conclusion that isometric isostatic meshes (ii-meshes) produce better results than the standard reasonably initial regular meshes used in practice. This conclusion seems to be independent on the objective function used for comparison. These ii-meshes are obtained by placing FE nodes along the isostatic lines, i.e. curves tangent at each point to the principal direction lines of the elastic problem to be solved and they should be regularly spaced in order to build regular elements. That means ii-meshes are usually obtained by iteration, i.e. with the initial FE mesh the elastic analysis is carried out. By using the obtained results of this analysis the net of isostatic lines can be drawn and in a first trial an ii-mesh can be built. This first ii-mesh can be improved, if it necessary, by analyzing again the problem and generate after the FE analysis the new and improved ii-mesh. Typically, after two first tentative ii-meshes it is sufficient to produce good FE results from the elastic analysis. Several example of this procedure are presented.