Adaptive Meshfree Methods for Partial Differential Equations

There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending on the governing equations, the domains, and the boundary conditions. The MFS is used as the main meshfree method to solve the Laplace equation in this dissertation, and we propose adaptive algorithms in different versions based on the residual type of an error estimator in 2D and 3D domains. Popular techniques for handling parameters and different approaches are considered in each example to obtain satisfactory results. Dirichlet boundary conditions are carefully chosen to validate the efficiency of the adaptive method. The RBF collocation method and the Method of Approximate Particular Solutions (MAPS) are used for solving the Poisson equation. Due to the type of the PDE, different strategies for constructing the adaptive method had to be followed, and proper error estimators are considered for this part. This results in having a new point of view when observing the numerical results. Methodologies of meshfree methods that are employed in this dissertation are introduced, and numerical examples are presented with various boundary conditions to show how the adaptive method performs. We can observe the benefit of using the adaptive method and the improved error estimators provide better results in the experiments

Meshless methods for Maxwell’s equations with applications to magnetotelluric modelling and inversion

The first part of thesis presents new meshless methods for solving time harmonic electromagnetic fields in closed two- or three-dimensional volumes containing heterogeneous materials. This new methods will be used to simulate magnetotelluric experiments, when an Earth conductivity model is given in advanced. Normally, classical approximation methods like finite elements or finite differences are used to solve this task. The algorithms here in this thesis, only need an unstructured point sampling in the modelling domain for the discretization and is able to gain a solution for the partial differential equation without a fixed mesh or grid. This is advantageous when complex model geometries have to be described, because no adapted mesh or grid need to be generated. The meshless methods, described here in this thesis, use a direct discretization technique in combination with a generalized approximation method. This allows to formulate the partial differential equations in terms of linear functionals, which can be approximated and directly form the discretization. For the two-dimensional magnetotelluric problem, a second-order accurate algorithm to solve the partial differential equations was developed and tested with several example calculations. The accuracy of the new meshless methods was compared to analytical solutions, and it was found, that a better accuracy can be achieved with less degrees of freedoms compared to previously published results. For the three-dimensional case, a meshless formulation was given and numerical calculations show the ability of the scheme to handle models with heterogeneous conductivity structures. In the second part of this thesis, the newly developed two-dimensional simulation method will be used in an inversion scheme. Here, the task is to recover the unknown Earth conductivity model with the help of data gained from a magnetotelluric experiment. Due to the previously developed meshless approximation algorithm, some numerical tasks during the inversion can be simplified by reusing the discretization defined on the point sampling from the forward simulation. The newly developed meshless inversion algorithm will be tested with synthetic data to reconstruct known conductivity anomalies. It can be shown, that the inverse algorithm produces correct results, even in the presence of topography

Stencil and kernel optimisation for mesh-free very high-order generalised finite difference method

Generalised Finite Difference Methods and similar mesh-free methods (Point set method, Multipoint method) are based on three main ingredients: a stencil around the reference node, a polynomial reconstruction and a weighted functional to provide the relation sbetween the derivatives at the reference node and the nodes of the stencil.Very few studies were dedicated to the optimal choice of the stencil together with the other parameters that could reduce the global conditioning of the system and bring more stability and better accuracy. We propose a detailed construction of the very high-order polynomial representation and define a functional that assesses the quality of the reconstruction. We propose and implement several techniques of optimisation and demonstrate the advantages in terms of accuracy and stability.J. Figueiredo acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UID/FIS/04650/2019. S. Clain the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional , through COMPETE 2020 – Programa Operational Fatores de Competitividade through FCT – Fundação para a Ciência e a Tecnologia, project N° UIDB/00324/2020

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

A numerical study on the viscous fingering instability of immiscible displacement in Hele-Shaw cells

In this thesis, the viscous fingering instability of radial immiscible displacement is analysed numerically using novel mesh-reduction and interface tracking techniques. Using a reduced Hele-Shaw model for the depth averaged lateral flow, viscous fingering instabilities are explored in flow regimes typical of subsurface carbon sequestration involving supercritical CO2 - brine displacements, i.e. with high capillary numbers, low mobility ratios and inhomogeneous permeability/temperature fields. A high accuracy boundary element method (BEM) is implemented for the solution of homogeneous, finite mobility ratio immiscible displacements. Through efficient, explicit tracking of the sharp fluid-fluid interface, classical fingering processes such as spreading, shielding and splitting are analysed in the late stages of finger growth at low mobility ratios and high capillary numbers. Under these conditions, large differences are found compared with previous high or infinite mobility ratio models and critical events such as plume break-off and coalescence are analysed in much greater detail than has previously been attempted. For the solution of inhomogeneous mobility problems, a novel meshless radial basis function-finite collocation method is developed that utilises a dynamic quadtree dataset and local enforcement of interface matching conditions. When coupled with the BEM, the numerical scheme allows the analysis of variable permeability effects and the transition in (de)stabilising mechanisms that occurs when the capillary number is increased with a fixed, spatially varying permeability. Finally, thermo-viscous fingering is explored in the context of immiscible flows, with a detailed mechanistic study presented to explain, for the first time, the immiscible thermo-viscous fingering process

A numerical study on the viscous fingering instability of immiscible displacement in Hele-Shaw cells

In this thesis, the viscous fingering instability of radial immiscible displacement is analysed numerically using novel mesh-reduction and interface tracking techniques. Using a reduced Hele-Shaw model for the depth averaged lateral flow, viscous fingering instabilities are explored in flow regimes typical of subsurface carbon sequestration involving supercritical CO2 - brine displacements, i.e. with high capillary numbers, low mobility ratios and inhomogeneous permeability/temperature fields. A high accuracy boundary element method (BEM) is implemented for the solution of homogeneous, finite mobility ratio immiscible displacements. Through efficient, explicit tracking of the sharp fluid-fluid interface, classical fingering processes such as spreading, shielding and splitting are analysed in the late stages of finger growth at low mobility ratios and high capillary numbers. Under these conditions, large differences are found compared with previous high or infinite mobility ratio models and critical events such as plume break-off and coalescence are analysed in much greater detail than has previously been attempted. For the solution of inhomogeneous mobility problems, a novel meshless radial basis function-finite collocation method is developed that utilises a dynamic quadtree dataset and local enforcement of interface matching conditions. When coupled with the BEM, the numerical scheme allows the analysis of variable permeability effects and the transition in (de)stabilising mechanisms that occurs when the capillary number is increased with a fixed, spatially varying permeability. Finally, thermo-viscous fingering is explored in the context of immiscible flows, with a detailed mechanistic study presented to explain, for the first time, the immiscible thermo-viscous fingering process

Reduced order modelling in nuclear reaction thermal hydraulics

The context of the present thesis is to assess the potential of Reduced Order Models (ROMs) for nuclear reactor thermal hydraulics applications. ROMs constitute advanced modelling techniques aiming at fast high fidelity simulations. For the purposes of this research, two approaches have been selected and are investigated in depth: the Proper Orthogonal Decomposition (POD) with Galerkin projection (POD-Galerkin) and the hybrid method of Proper Orthogonal Decomposition with Interpolation using Radial Basis Functions, PODI - Galerkin, in the context of parametric model order reduction. Additionally, in terms of the POD method, two sampling techniques are presented and compared: the standard and the nested POD. The aforementioned methods are applied to a parametric case of non-isothermal mixing in a T-junction pipe for laminar and turbulent flow regimes. The flow is governed by the 3D, unsteady Navier - Stokes equations coupled with the energy equation. Furthermore, a ROM for modelling buoyancy driven flows with the Boussinesq approximation is discussed. Two cases are considered: a closed flow, where the method is applied to a benchmark case of a differentially heated square cavity, and an open flow, where a case of a "cold-trap" formation in a U-bend pipe is investigated. The suitability of the above techniques is assessed based on a comparison between the reduced order results and those obtained using high fidelity OpenFOAM solvers.Open Acces