Application of smoothed point interpolation methods to numerical modelling of saturated and unsaturated porous media

This study aims to develop an efficient computational framework for a rigorous coupled flow and deformation analysis of saturated and unsaturated porous media. The governing equations are derived based on equations of equilibrium, and conservation equations of mass and momentum for each phase. For numerical solution of the governing equations, the edge-based smoothed point interpolation method (ESPIM) is employed due to its numerous advantages over the classical techniques. The ESPIM was originally introduced for problems in single phase media. The extension of the technique to multiphase media is not trivial, and therefore as the first development step, ESPIM is extended for the solution of the coupled hydro-mechanical problems in saturated porous media through a novel approach for evaluation of the coupling matrix. Verification of the proposed ESPIM formulation is performed using several benchmark numerical examples. Subsequently, the method of manufactured solutions (MMS) is introduced, for the first time in geomechanics, for a systematic and more rigorous verification of the computational scheme. The proposed numerical framework is then extended to include material nonlinearity. For this purpose, a non-associative Mohr-Coulomb constitutive model is adopted and an algorithm is developed based on the modified Newton-Raphson technique to address the nonlinearities arisen from the elasto-plastic constitutive model. Stress integration is performed using the substepping method. The computational framework is then further extended to include the problems in unsaturated soil mechanics, taking account of coupling among different phases, and the hydraulic hysteresis observed in the behaviour of unsaturated soils. A framework based on the effective stress principle is followed in the formulation and a hysteretic water retention model is taken into account which includes the evolution of water retention curve (WRC) with changes of void ratio. An elasto-plastic constitutive model is employed within the context of bounding surface plasticity theory for predicting the nonlinear behaviour of soil skeleton in saturated and unsaturated porous media. The model is validated by comparing the numerical predictions with experimental or numerical data from the literature for fully and partially saturated soils. The results demonstrate the capability of the proposed numerical framework to predict essential characteristics of variably saturated soils

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

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

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

Aerodynamic damping of an oscillating fan blade: Numerical fluid structure interaction analysis

Thesis (MEng)--Stellenbosch University, 2017.ENGLISH SUMMARY: The thesis’ main objective is to determine the dynamic flow phenomena that dampen a fan blade’s oscillation amplitude using numerical fluid structure interaction (FSI) simulations. The observed flow effects include the formation and shedding of leading edge vortices, downwash and the added mass effect. Leading edge vortices are a major damping contributor and are dependent on the blade’s effective angle of attack. The aim of the thesis is to find a suitable method that is capable of simulating the aerodynamic damping of an axial fan used in an air cooled condenser unit. Therefore, three different numerical models are used to perform the FSI simulation and are compared according to their accuracy, robustness and computational cost. The aerodynamic damping of an oscillating fan blade was experimentally investigated by Basson (2015) and his results are used to validate the three numerical models. The three methods used are a mesh-based FSI simulation, a simplified one dimensional beam model coupled with a heuristic flow model and a meshless FSI simulation. The mesh based and meshless FSI simulations are both suitable for modelling the entire air cooled condenser fan unit, whereas the simplified 1D beam model is incapable of doing so.AFRIKAANS OPSOMMING: Die tesis se hoofdoelwit is om die dinamiese vloei-verskynsels te bepaal wat ’n waaierlem se ossillasie amplitude demp met gebruik van numeriese vloei struktuur interaksie simulasies (FSI). Die waargeneemde vloei effekte sluit in die vorming en vergieting van leirand draaikolke, valstroomvloei en die bykomende massa-effek. Leirand draaikolke lewer ’n groot bydrae tot die demping en is afhanklik van die lem se effektiewe aanvalshoek. Die doel van die tesis is om ’n geskikte metode te vind wat in staat is om die aerodinamiese demping van ’n aksiale waaierlem in ’n lugverkoelde kondensoreenheid te simuleer. Daarom word drie verskillende numeriese modelle gebruik wat die vloei struktuur interaksie simulasie uitvoer en vergelyk volgens hul akkuraatheid, robuustheid en koste van berekening. Die aerodinamiese demping van ’n ossilerende waaierlem is eksperimenteel ondersoek deur Basson (2015) en sy resultate word gebruik om die drie numeriese modelle te bevestig. Die drie metodes is ’n rooster gebaseerde FSI simulasie, ’n vereenvoudigde een-dimensionele balk-model tesame met ’n heuristiese vloeimodel en ’n roosterlose FSI simulasie. Die rooster gebaseerde en roosterlose FSI simulasies is beide geskik vir die modellering van die hele lugverkoelde kondensor waaier eenheid, terwyl die vereenvoudigde 1D balk-model nie in staat is om dit te doen nie

Adaptive meshless point collocation methods: investigation and application to geometrically non-linear solid mechanics

Conventional mesh-based methods for solid mechanics problems suffer from issues resulting from the use of a mesh, therefore, various meshless methods that can be grouped into those based on weak or strong forms of the underlying problem have been proposed to address these problems by using only points for discretisation. Compared to weak form meshless methods, strong form meshless methods have some attractive features because of the absence of any background mesh and avoidance of the need for numerical integration, making the implementation straightforward. The objective of this thesis is to develop a novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems. To address some issues in existing strong form meshless methods, the local maximum entropy point collocation method is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries. r- and h-adaptive strategies are investigated in the proposed method and are further combined into a novel rh-adaptive approach, achieving the prescribed accuracy with the optimised locations and limited number of points. The proposed meshless method with h-adaptivity is then extended to solve geometrically non-linear problems described in a Total Lagrangian formulation, where h-adaptivity is again employed after the initial calculation to improve the accuracy of the solution effciently. This geometrically non-linear method is finally developed to analyse membrane problems, in which the out-of-plane deformation for membranes complicates the governing PDEs and the use of hyperelastic materials makes the computational modelling of membrane problems challenging. The Newton-Raphson arc-length method is adopted here to capture the snap-through behaviour in hyperelastic membrane problems. Several numerical examples are presented for each proposed algorithm to validate the proposed methodology and suggestions are made for future work leading on from the findings of this thesis

Conical spring and localised region methodologies for modelling of soft tissue deformation.

Considerable research efforts have been dedicated to the development of virtual reality simulators that facilitate medical students in learning anatomy and surgery in the virtual environment and to allow surgeons to rehearse the surgical procedures. The level of realism depends upon the simulation accuracy and the computational efficiency of underlying deformable models. Ideally, the deformable models should be able to simulate accurately mechanical behaviours of soft tissues with real-time visual and force feedback. Modelling soft tissue deformation is not an easy task. Due to the complexity of soft tissue properties, many methods have been proposed to model soft tissue properties. One of the most well-known methods is the Finite Element Method (FEM). In this method, the soft tissue is represented by multiple elements that are derived based on complex mathematical formulations. It has been proven that the method is able to simulate soft tissue properties accurately, but it requires high computational cost to produce real-time interaction. In this regard, the Mass Spring Method (MSM) has been proposed as an alternative. The traditional MSM model simulates soft tissue deformation by discretising the soft tissue into several mass points that are connected to each other by linear springs. The major advantage of MSM is it has an excellent computational performance. However, the MSM application is limited to linear deformation, which does not represent the actual behaviour of the soft tissue deformation. In this thesis, an improved MSM model has been proposed to simulate the complex behaviour of soft tissue deformations. The improved MSM model is called conical spring model which considers the general behaviour of soft tissue deformation that is a combination of linear and nonlinear responses. Piecewise approach is used to discretise each response individually, and the conical spring methodology is used to model the deformation behaviours during all the responses. The piecewise approach gives precision in modelling while the conical spring methodology that was founded on stiffness variation, has improved the accuracy of the simulation due to its ability to model any type of linear and nonlinear responses. Moreover, the generated conical spring model is based on the force propagation approach. The computational performance of the model relies on the number of nodes involved in the propagation of the force. Inherently, computational time can be improved by considering the nodes only in a deformation area, and ignoring the other nodes. Soft tissue deformation commonly occurs only within a local region. As the effect of the deformation outside the local region is very little, it can be ignored in real practice. In this thesis, methods to define the local region were proposed. The methods are based on the linear elastic theory. As reported in Chapter 4 of this thesis, the localised region was generated based on displacement value induced when the simulation model was subjected to an external load. The Boussinesq method, which is widely used in the soil mechanics, was used to estimate the induced displacement value. However, the Boussinesq method is limited to the isotropic material. Therefore, as described in Chapter 5, the study has extended the isotropic localised region to anisotropic localised region by introducing an anisotropic factor which was derived based on cross-anisotropic properties. By using the anisotropic factor, the anisotropic localised region is determined from the corresponding isotropic case. Alternatively, in Chapter 6, we have presented a localised region that was generated based on stress value induced during a loading process. It is shown for point load type of contact, in comparison to ABAQUS analysis, stress based localisation has a better accuracy than the displacement based localisation. However, the stress value that is also determined using the Boussinesq method, has no relation to the material properties. Hence, a combination of the Hertzian and the Boussinesq method was used to generate localised regions with respect to the material properties and loading conditions. In the final chapter, contributions of the study were discussed, and some of the future works to expand the research were listed out

On the Numerical Modelling of Machining Processes via the Particle Finite Element Method (PFEM)

Metal cutting or machining is a process in which a thin layer or metal, the chip, is removed by a wedge-shaped tool from a large body. Metal cutting processes are present in big industries (automotive, aerospace, home appliance, etc.) that manufacture big products, but also high tech industries where small piece but high precision is needed. The importance of machining is such that, it is the most common manufacturing processes for producing parts and obtaining specified geometrical dimensions and surface finish, its cost represent 15% of the value of all manufactured products in all industrialized countries. Cutting is a complex physical phenomena in which friction, adiabatic shear bands, excessive heating, large strains and high rate strains are present. Tool geometry, rake angle and cutting speed play an important role in chip morphology, cutting forces, energy consumption and tool wear. The study of metal cutting is difficult from an experimental point of view, because of the high speed at which it takes place under industrial machining conditions (experiments are difficult to carry out), the small scale of the phenomena which are to be observed, the continuous development of tool and workpiece materials and the continuous development of tool geometries, among others reasons. Simulation of machining processes in which the workpiece material is highly deformed on metal cutting is a major challenge of the finite element method (FEM). The principal problem in using a conventional FE model with langrangian mesh is mesh distortion in the high deformation. Traditional Langrangian approaches such as FEM cannot resolve the large deformations very well. Element distortion has been always matter of concern which limited the analysis to incipient chip formation in some studies. Instead, FEM with an Eulerian formulation require the knowledge of the chip geometry in advance, which, undoubtedly, restricts the range of cutting conditions capable of being analyzed. Furthermore serrated and discontinuous chip formation cannot be simulated. The main objective of this work is precisely to contribute to solve some of the problems described above through the extension of the Particle Finite Element Method (PFEM) to thermo-mechanical problems in solid mechanics which involve large strains and rotations, multiple contacts and generation of new surfaces, with the main focus in the numerical simulation of metal cutting process. In this work, we exploit the particle and lagrangian nature of PFEM and the advantages of finite element discretization to simulate the different chip shapes (continuous and serrated) that appear when cutting materials like steel and titanium at different cutting speeds. The new ingredients of PFEM are focused on the insertion and remotion of particles, the use of constrained Delaunay triangulation and a novel transfer operator of the internal variables. The remotion and insertion of particles circumvents the difficulties associated to element distortion, allowing the separation of chip and workpiece without using a physical or geometrical criterion. The constrained Delaunay improves mass conservation and the chip shape through the simulation, and the transfer allows us to minimize the error due to numerical diffusion. The thermo-mechanical problem, formulated in the framework of continuum mechanics, is integrated using an isothermal split in conjunction with implicit, semi-explicit and IMPLEX schemes. The tool has been discretized using a standard three-node triangle finite element. The workpiece has been discretized using a mixed displacement-pressure finite element to deal with the incompressibility constraint imposed by plasticity. The mixed finite element has been stabilized using the Polynomial Pressure Projection (PPP), initially applied in the literature to the Stokes equation in the field of fluid mechanics. The behavior of the tool is described using a Neo-Hookean Hyperelastic constitutive model. The behavior of the workpiece is described using a rate dependent, isotropic, finite strain j2 elastoplasticity with three different yields functions used to describe the strain hardening, the strain rate hardening and the thermal softening (Simo, Johnson Cook, Baker) of different materials under a wide variety of cutting conditions. The friction at the tool chip interface is modeled using the Norton-Hoff friction law. The heat transfer at the tool chip interface includes heat transfer due to conduction and friction. To validate the proposed mixed displacement-pressure formulation, we present three benchmark problems which validate the approach, namely, plain strain Cook´s membrane, the Taylor impact test and a thermo-mechanical traction test. The isothermal-IMPLEX split presented in this work has been validated using a thermo-mechanical traction test. Besides, in order to explore the possibilities of the numerical model as a tool for assisting in the design and analysis of metal cutting processes a set of representative numerical simulations are presented in this work, among them: cutting using a rate independent yield function, cutting using different rake angles, cutting with a deformable tool and a frictionless approach, cutting with a deformable tool including friction and heat transfer, the transition from continuous to serrated chip formation increasing the cutting speed. We have assembled several numerical tec niques which enable the simulation of orthogonal cutting processes. Our simulations demonstrate the ability of the PFEM to predict chip morphologies consistent with experimental observations. Also, our results show that the suitable selection of the global time integration scheme may involve savings in computation time up to 9 times. Furthermore, this work present a sensibility analysis to cutting conditions by means of a Design of Experiments (DoE). The Design of Experiments carried out with PFEM has been compared with DoE carried out with AdvantaEdge, Deform, Abaqus and Experiments. The results obtained with PFEM and other numerical simulations are very similar, while, a comparison of numerical simulations and experiments show some differences in the output variables that depend on the friction phenomena. The results suggest that is necessary to improve the modelization of the friction at the tool-chip interface