Simple modifications for stabilization of the finite point method

A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill‐conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non‐monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd

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

Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body

Numerično modeliranje dendritskega strjevanja na podlagi formulacije faznega polja in prilagodljivega brezmrežnega rešitvenega postopka

The main aim of the dissertation is to develop a novel numerical approach for an accurate and computationally efficient modelling of dendritic solidification, which is one of the most commonly observed phenomena in the industrial casting of the metallic alloys. The size and the morphology of dendritic structures as well as the distribution of the solute within them critically effect the mechanical and the electro-chemical properties of the solidified material. The numerical modelling of dendritic solidification can be applied for an in-depth understanding and optimisation of the casting process under various solidification conditions and chemical compositions of the alloy under consideration. The dendritic solidification of pure materials and dilute multi-component alloys is considered in the dissertation. The phase field formulation is applied for the modelling of dendritic solidification. The formulation is based on the introduction of the continuous phase field variable that is constant in the bulk of the solid and liquid phases. The phase field variable has a smooth transition from the value denoting the solid phase to the value denoting the liquid phase at the solid-liquid interface over the characteristic interface thickness. A phase field model yields a system of coupled non-linear parabolic partial differential equations that govern the evolution of the phase field and other thermodynamic variables. The meshless radial basis function-generated finite-differences (RBF-FD) method is used for the spatial discretisation of the system of partial differential equations. The forward Euler scheme is applied for the temporal discretisation. Fifth-degree polyharmonic splines are used as the shape functions in the RBF-FD method. A second-order accurate RBF-FD method is achieved by augmenting the shape functions with monomials up to the second degree. The adaptive solution procedure is developed in order to speed-up the calculations. The procedure is based on the quadtree domain decomposition of a rectangular computational domain into rectangular computational sub-domains of different sizes. Each quadtree sub-domain has its own regular or scattered distribution of computational nodes in which the RBF-FD method and the forward Euler scheme apply for the discretisation of the system of partial differential equations. The adaptive solution procedure dynamically ensures the prescribed highest density of the computational nodes at the solid-liquid interface and the lowest-possible density in the bulk of the solid and liquid phases. The adaptive time-stepping is employed to further speed-up the calculations. The stable time step in the forward Euler scheme depends on the density of the computational nodeshence, different time steps can be used in quadtree sub-domains with different node densities. The main originality of the present work is the use of the RBF-FD method for the thorough analysis of the impact of the type of the node distribution and the size of a local sub-domain to the accuracy when the phase field modelling of dendritic solidification for arbitrary preferential growth directions is considered. It is shown how the use of the scattered node distribution reduces the undesirable mesh-induced anisotropy effects, present when the partial differential equations are discretisied on a regular node distribution. The main advantage of the RBF-FD method for the phase field modelling of dendritic growth is the simple discretisation of the partial differential equations on the scattered node distributions. The RBF-FD method is, for the first time, used in combination with the spatial-temporal adaptive solution procedure based on the quadtree domain decomposition. The adaptive solution procedure successfully speeds-up the calculationshowever, the advantages of the use of the scattered node distribution are partly compromised due to the impact of regularity in the quadtree domain decomposition.Glavni cilj disertacije je razvoj novega numeričnega pristopa za natančno in računsko učinkovito modeliranje dendritskega strjevanja. Dendritsko strjevanje je eden najpogosteje opaženih pojavov pri industrijskem ulivanju kovinskih zlitin. Velikost in morfologija dendritskih struktur ter porazdelitev topljencev v njih ključno vplivajo na mehanske in elektro-kemijske lastnosti strjenega materiala. Numerično modeliranje dendritskega strjevanja se lahko uporablja za poglobljeno razumevanje in optimizacijo procesa ulivanja pri različnih pogojih strjevanja in pri različnih kemijskih sestavah obravnavane zlitine. V disertaciji obravnavamo dendritsko strjevanje čistih snovi in razredčenih več-sestavinskih zlitin. Za modeliranje dendritskega strjevanja uporabimo formulacija faznega polja. Formulacija temelji na uvedbi zvezne spremenljivke faznega polja, ki je konstantna v trdni in kapljeviti fazi. Spremenljivka faznega polja ima na medfaznem robu zvezen prehod preko značilne debeline medfaznega roba od vrednosti, ki označuje trdno fazo, do vrednosti, ki označuje kapljevito fazo. Model faznega polja poda sistem sklopljenih nelinearnih paraboličnih parcialnih diferencialnih enačb, ki opisujejo časovni razvoj faznega polja in ostalih termodinamskih spremenljivk. Za krajevno diskretizacijo sistema parcialnih diferencialnih enačb uporabimo brezmrežno metodo z radialnimi baznimi funkcijami generiranih končnih razlik (RBF-KR). Za časovno diskretizacijo uporabimo eksplicitno Eulerjevo shemo. Poliharmonične zlepke petega reda uporabimo kot oblikovne funkcije v metodi RBF-KR. Natančnost drugega reda metode RBF-KR dosežemo z dodajanjem monomov do vključno drugega reda k oblikovnim funkcijam. Za pospešitev izračunov razvijemo prilagodljiv rešitveni postopek. Postopek temelji na razdelitvi pravokotne računske domene na pravokotne računske pod-domene različnih velikosti z uporabo štiriškega drevesa. Vsaka pod-domena na štiriškem drevesu vsebuje svojo lastno regularno ali razmetano porazdelitev računskih točk, v katerih z uporabo metode RBF-KR in eksplicitne Eulerjeve sheme diskretiziramo sistem parcialnih diferencialnih enačb. Prilagodljiv rešitveni postopek dinamično zagotavlja predpisano najvišjo gostoto računskih točk na trdno-kapljevitem medfaznem robu in najmanjšo možno gostoto v notranjosti trdne in kapljevite faze. Za dodatno pohitritev izračunov uporabimo prilagodljivo časovno korakanje. Stabilen časovni korak v eksplicitni Eulerjevi shemi je odvisen od gostote računskih točk, zaradi česar lahko uporabimo različne časovne korake v pod-domenah štiriškega drevesa z različnimi gostotami točk. Glavna novost predstavljenega dela je v uporabi metode RBF-KR za temeljito analizo vpliva tipa porazdelitve računskih točk in velikosti lokalnih pod-domen na natančnost pri modeliranju dendritskega strjevanja pri poljubnih preferenčnih smereh rasti z uporabo metode faznega polja. Pokažemo, kako uporaba razmetanih računskih točk zmanjša neželjen vpliv mrežne anizotropije, ki je prisotna, kadar parcialne diferencialne enačbe diskretiziramo na regularni porazdelitvi računskih točk. Glavna prednost metode RBF-KR za modeliranje dendritskega strjevanja je preprosta diskretizacija parcialnih diferencialnih enačb na razmetanih porazdelitvah računskih točk. Metoda RBF-KR je prvič uporabljena v kombinaciji s krajevno-časovnim prilagodljivim rešitvenim postopkom, ki temelji na razdelitvi računske domene s štiriškim drevesom. Prilagodljiv rešitveni postopek uspešno pohitri izračune, vendar se prednosti uporabe razmetane porazdelitve računskih točk delno zmanjšajo zaradi vpliva regularnosti pri razdelitvi računske domene s štiriškim drevesom

A Singularity-Avoiding Moving Least Squares Scheme for Two Dimensional Unstructured Meshes

Moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data. In particular, they are often employed when solving partial differential equations on unstructured meshes, which are typically needed when the geometry defining the domain is complex. It is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements, however little research has addressed this issue specifically. In this paper, a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions, and the precise conditions under which singularities occur are identified. The theory is then applied in the form of a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner, while attempting to maintain stencil symmetry as closely as possible. Finally, the scheme is used in a convection-diffusion equation solver, and the results of a number of simulations are presented

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semi-implicit (MPS) and meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson’s equation about pressure at each time step is a major task. There are three different approaches to solving Poisson’s equation, i.e. (1) discretizing Laplacian directly by approximating the second-order derivatives, (2) transferring Poisson’s equation into a weak form containing only gradient of pressure and (3) transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation

From Mesh to Meshless : a Generalized Meshless Formulation Based on Riemann Solvers for Computational Fluid Dynamics

Programa Oficial de Doutoramento en Enxeñaría Civil . 5011V01[Abstract] From mesh to meshless: A generalized meshless formulation based on Riemann solvers for Computational Fluid Dynamics This thesis deals with the development of high accuracy meshless methods for the simulation of compressible and incompressible flows. Meshless methods were conceived to overcome the constraints that mesh topology impose on traditional mesh-based numerical methods. Despite the fact that meshless methods have achieved a relative success in some particular applications, the truth is that mesh-based methods are still the preferred choice to compute flows that demand high-accuracy. Instead of assuming that meshless and mesh-based methods are groups of methods that follow independent development paths, in this thesis it is proposed to increase the accuracy of meshless methods by taking guidance of some successful techniques adopted in the mesh-based community. The starting point for the development is inspired by the SPH-ALE scheme proposed by Vila. Especially, the flexibility of the ALE framework and the introduction of Riemann solvers are essential elements adopted. High accuracy is obtained by using the Moving Least Squares (MLS) technique. MLS serves multiple tasks in the implemented scheme: high order reconstruction of Riemann states, more accurate viscous flux evaluation and the replacement of the limited kernel approximation by MLS approximation with polynomial degree consistency by design. The stabilization of the scheme for compressible flows with discontinuities is based on a posteriori stabilization technique (MOOD) that introduces a great improvement compared with the traditional a priori flux limiters. The MLSPH-ALE scheme is the first proposed meshless formulation that uses high order consistent MLS approximation in a versatile ALE framework. In addition, the procedure to obtain the semi-discrete formulation keeps track of a boundary term, which eases the implementation of the boundary conditions. Another important contribution is related with the general concept of the MLSPHALE formulation. The MLSPH-ALE scheme is proved to be a global meshless formulation that under some particular settings provides the same semi-discrete equations that other meshless formulations published. The MLSPH-ALE scheme has been tested for the computation of turbulent flows. The low dissipation inherent to the Riemann solver is compatible with the implicit LES turbulent model. The proposed formulation is able to capture the energy cascade in the subsonic regime where traditional SPH formulations are reported to fail.[Resumen] Desde métodos con malla a métodos sin malla: Una formulación sin malla generalizada basada en solvers de Riemann para Dinámica de Fluidos Computacional Esta tesis aborda el desarrollo de métodos sin malla de alta precisión para la simulación de flujos compresibles e incompresibles. Los métodos sin malla fueron creados para superar las restricciones que la conectividad de la malla impone a los métodos tradicionales. A pesar de haber alcanzado un ´éxito relativo en algunas aplicaciones, la realidad es que los métodos con malla siguen siendo la opción preferida para el cálculo de flujos que demandan alta precisión. En vez de asumir que métodos sin malla y con malla son grupos de métodos que siguen caminos de desarrollo independientes, en esta tesis se propone incrementar la precisión de los métodos sin malla tomando como guía algunas de las técnicas más exitosas empleadas en la comunidad de los métodos con malla. El punto de partida para el desarrollo se inspira en el esquema SPH-ALE propuesto por Vila. De manera especial, la flexibilidad del marco de referencia ALE y la introducción de los solvers de Riemann son elementos esenciales adoptados. La alta precisión se obtiene con la técnica de Mínimos Cuadrados Móviles (MLS). MLS sirve múltiples funciones en la implementación del esquema: alto orden de reconstrucción de los estados de Riemann, evaluaciones más precisas de los flujos viscosos y reemplazo de la aproximación limitada tipo kernel por una aproximación MLS con un grado de consistencia polinómica arbitraria. La estabilización del esquema para flujos compresibles con discontinuidades se basa en una técnica de estabilización a posteriori (MOOD) que introduce una importante mejora con respecto a los tradicionales limitadores de flujo a priori. El esquema MLSPH-ALE es la primera formulación sin malla propuesta que utiliza la aproximación MLS de alto orden en un marco de referencia ALE. Además, el procedimiento dado para obtener la forma semi-discreta realiza el seguimiento de un término en la frontera del dominio que facilita la implementación discreta de las condiciones de contorno. Otra importante contribución está relacionada con el concepto general de la formulación MLSPH-ALE. Se ha demostrado que el esquema MLSPH-ALE es una formulación sin malla global que con ciertas configuraciones particulares es capaz de proporcionar las mismas formas semi-discretas que otras formulaciones publicadas. El método MLSPH-ALE ha sido puesto a prueba frente al cálculo de flujos turbulentos. La baja disipación inherente a los solver de Riemann hace que el esquema sea apto para modelar la turbulencia en un contexto de modelos implícitos LES. La formulación propuesta es capaz de capturar la cascada de energía en el rango de régimen subsónico donde los métodos tradicionales presentan fallos.[Resumo] Desde métodos con malla a métodos sen malla: Unha formulación sen malla xeneralizada baseada en solvers de Riemann para Dinámica de Fluidos Computacional. Esta tese trata sobre o desenvolvemento de métodos sen malla de alta precisión para a simulación de fluxos compresibles e incompresibles. Os métodos sen malla foron creados para superar as restricións que a conectividade da malla impón sobre os métodos tradicionais. A pesar de ter acadado un éxito relativo nalgunhas aplicacións, a realidade é que os métodos con malla seguen sendo a opción preferente para o cálculo de fluxos que demandan alta precisión. No canto de asumir que os métodos sen malla e con malla son grupos que seguen camiños de desenvolvemento independentes, nesta tese proponse incrementar a precisión dos métodos sen malla tomando como guía algunha das técnicas de máis éxito empregadas na comunidade dos métodos con malla. O punto de partida para o desenvolvemento inspírase no esquema SPH-ALE proposto por Vila. A flexibilidade do marco de referencia ALE e a introducción dos solvers de Riemann son os elementos esenciais utilizados nesta tese. A alta precisión acádase coa técnica de Mínimos Cadrados Móbiles (MLS). MLS serve para múltiples tarefas na implementación do esquema: acadar alto orde de reconstrución nos estados de Riemann, avaliacións máis precisas dos fluxos viscosos e troco da aproximación limitada tipo kernel por unha aproximación MLS con grado de consistencia polinómica arbitraria. A estabilización do esquema para fluxos compresibles con descontinuidades baséase nunha técnica de estabilización a posteriori (MOOD) que introduce unha importante mellora con respecto a os tradicionais limitadores de fluxo a priori. O esquema MLSPH-ALE ´e a primeira formulación sen malla proposta que emprega a técnica de aproximación MLS con alta consistencia nun marco de referencia ALE. Ademais, o procedemento seguido para obter a forma semi-discreta realiza o seguimento dun termo na fronteira que facilita a implementación das condicións de contorno. Outra importante contribución relacionase co concepto xeral da formulación MLSPHALE proposta. Demostrase que o esquema MLSPH-ALE é unha formulación sen malla global que con certas configuración particulares rende as mesmas formas semi-discretas que outras formulacións publicadas. O método MLSPH-ALE foi posto a proba fronte o cálculo de fluxos turbulentos. A baixa disipación implícita aportada polo solver de Riemann fai que o esquema sexa apto para acometer o modelado da turbulencia cos modelos implícitos LES. A formulación proposta captura a cascada de enerxía no rango de réxime subsónico, onde os métodos tradicionais SPH presentan deficiencias.This work has been partially supported by the Ministerio de Ciencia, Innovación y Universidades (RTI2018-093366-B-100) of the Spanish Government and by the Consellería de Educación e Ordenación Universitaria of the Xunta de Galicia, cofinanced with FEDER funds and the Universidade da Coruña