Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions. Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results

Development and applications of the finite point method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications

Development and applications of the Finite Point Method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla. En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodología. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla. A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las características de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos. Finalmente se presentan distintas aplicaciones y extensiones de la metodología desarrollada. Los ejemplos numéricos pretenden demostrar las capacidades del método y también aprovechar las ventajas de la metodología sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras características, el refinamiento automático de la discretización, la presencia de fronteras móviles e interacción fluido-estructura, como así también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodología sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version

An SPH Formulation for Incompressible Newtonian Free Surface Flows

The meshless method Smoothed Particle Hydrodynamics (SPH) is one of the primarily developed numerical methods in the meshless methods family. It has been proven to work well with problems of free surface fluid flows. For the purpose of this thesis, the Smoothed Particle Hydrodynamics method and its current capabilities were studied. Using the obtained knowledge from the literature, a free surface fluid flow application was implemented in the framework of KRATOS multiphysics. Some well known benchmark problems were reproduced using the generated SPH solver. The algorithm's ability to catch the correct free surface evolution and obtain an accurate pressure distribution of free surface problems were analyzed. Furthermore, some programming aspects of coding for meshless methods were discussed from the computational workload point of view. Using the developed application, it was demonstrated that in order to create a generic SPH fluid solver that is able to model any type of fluid, some additional corrections must be applied to the SPH solution. These corrections, although being very convenient to obtain good particle tracking of the fluid motion, affect the pressure distribution of the flow excessively and make the weakly compressible SPH method untrusted for the pressure results of fluid flow problems

On the application of Particle Finite Element Method (PFEM) to problems in civil engineering

Current chapter tries to give an overview of theoretic background necessary to deal with fluid dynamic problems with Finite Element Method (PFEM) approach. Starting form a brief summary of the principle steps of the development of the theory, we analyze the differences between the classical  world of fluids, finding out finally the principle equations that control these problems

Higher-order discontinuous modelling of fracturing in quasi-brittle materials

Quasi-brittle failure is characterised by material degradation, fracturing and potential interaction of fragmented parts. The computational description of this behaviour has presented significant challenges to the mechanics community over the past few decades, driven by the development of technology, the increasing social and economical constraints for safer and more complicated engineering designs and consequently by the increasing requirements for more accurate understanding of macro- and micro-structural processes. Finite element methods have been pushed to their limits in an attempt to resolve strain localisation and ultimately fracturing in a unified and objective manner, while discrete methods have been utilised by artificial connection of discrete bodies which are identified a priori to act as continua. Neither of these attempts comprises a diritta via for modelling the transition from continuum to discontinuum efficiently and this has led to the investigation of alternative techniques. Herein, the numerical modelling of quasi-brittle localisation and fracturing is investigated using the Numerical Manifold Method (NMM) as an alternative unifying framework to industry-established techniques such as the Finite Element Method (FEM) and Discrete Element Method (DEM). One of the particularly interesting aspects of NMM is with respect to its potential for modelling both continuum and discontinuum states and providing an efficient framework for modelling the entire transition between continuum to discontinuum, from a continuum point of view, without remeshing. The attractive nature of this capability advocates potential for modelling mechanics of materials such as concrete, rock and masonry, but also a more general class of quasi-brittle materials. This work investigates and extends NMM primarily with respect to the following characteristics: 1. Discontinuities, such as cracks, are introduced naturally in a discrete manner, but in a continuum setting, without the need for remeshing 2. The approximation is improved globally or locally, for any arbitrary level, without remeshing 3. Integration is undertaken explicitly, for any arbitrary level of local improvement of the approximation Furthermore, NMM is reformulated using a constrained variational approach for generalised three-dimensional problems. Essential boundary conditions are enforced using Lagrange multipliers and projection matrices and potential higher-order boundary issues are investigated. The developments are implemented algorithmically in MATLAB and higher-order enrichment is demonstrated with the use of adaptivity

Development of Micro-Macro Continuum-Discontinuum Coupled Numerical Method

A micro-macro and continuum-discontinuum coupled model and corresponding computer codes are developed in this thesis for rock dynamics study. Firstly, a new micromechanical model for describing the elastic continuum based on the underlying microstructure of material is proposed. The model provides a more general description of material than linear elasticity. Then, a numerical model Distinct Lattice Spring Model (DLSM) is developed based on the RMIB theory. The new proposed model has the advantages of being meshless, and automatic continuum description through underlying discontinuum structure and directly using macroscopic elastic parameters. Following this, the multi-scale DLSM (m-DLSM) is proposed to combine DLSM and NMM. The proposed model uses a tri-layer structure and the macro model can be automatically released into micro model during calculation. Forth ward, the ability of DLSM on modeling dynamic failure is studied. A damage based micro constitutive law is developed. Relationships between the micro constitutive parameters and the macro mechanical parameters of material are provided. The micro parameters can directly be obtained from macro experimental results, i.e., tensile strength and fracture energy, through these equations. Moreover, the ability of DLSM on modeling wave propagation is enhanced and verified. Non-reflection boundary condition and methods to represent discontinuity in DLSM are developed. Finally, the parallelization of DLSM and 2D implicit DLSM are introduced. The main achievements of the whole PhD work and future research works are summarized and prospected in the conclusion part of the thesis

Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy

This paper discusses the finite element modeling of cracking in quasi-brittle materials. The problem is addressed via a mixed strain/displacement finite element formulation and an isotropic damage constitutive model. The proposed mixed formulation is fully general and is applied in 2D and 3D. Also, it is independent of the specific finite element discretization considered; it can be equally used with triangles/tetrahedra, quadrilaterals/hexahedra and prisms. The feasibility and accuracy of the method is assessed through extensive comparison with experimental evidence. The correlation with the experimental tests shows the capacity of the mixed formulation to reproduce the experimental crack path and the force–displacement curves with remarkable accuracy. Both 2D and 3D examples produce results consistent with the documented data. Aspects related to the discrete solution, such as convergence regarding mesh resolution and mesh bias, as well as other related to the physical model, like structural size effect and the influence of Poisson’s ratio, are also investigated. The enhanced accuracy of the computed strain field leads to accurate results in terms of crack paths, failure mechanisms and force displacement curves. Spurious mesh dependency suffered by both continuous and discontinuous irreducible formulations is avoided by the mixed FE, without the need of auxiliary tracking techniques or other computational schemes that alter the continuum mechanical problem

Radial point interpolation method for static and dynamic analysis of three-dimensional solids

Master'sMASTER OF ENGINEERIN