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

A moving least square immersed boundary method for SPH with thin-walled structures

This paper presents a novel method for smoothed particle hydrodynamics (SPH) with thin-walled structures. Inspired by the direct forcing immersed boundary method, this method employs a moving least square method to guarantee the smoothness of velocity near the structure surface. It simplifies thin-walled structure simulations by eliminating the need for multiple layers of boundary particles, and improves computational accuracy and stability in three-dimensional scenarios. Supportive three-dimensional numerical results are provided, including the impulsively started plate and the flow past a cylinder. Results of the impulsively started test demonstrate that the proposed method obtains smooth velocity and pressure in the, as well as a good match to the references results of the vortex wake development. In addition, results of the flow past cylinder test show that the proposed method avoids mutual interference on both side of the boundary, remains stable for three-dimensional simulations while accurately calculating the forces acting on structure.Comment: 15 pages,11 figure

MLS-SPH-ALE: A Review of Meshless-FV Methods and a Unifying Formulation for Particle Discretizations

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.[Abstract:] Mesh-based and particle methods were conceived as two different discretization strategies to solve partial differential equations. In the last two decades computational methods have diversified and a myriad of hybrid formulations that combine elements of these two approaches have been developed to solve Computational fluid dynamics problems. In this work we present a review about the meshless-FV family of methods, an analysis is carried out showing that the MLS-SPH-ALE method can be considered as a general formulation from which a set of particle-based methods can be recovered. Moreover, we show the relations between the MLS-SPH-ALE method and the finite volume method. The MLS-SPH-ALE method is a versatile particle-based method that was developed to circumvent the consistency issues of particle methods caused by the use of the kernel approximation. The MLS-SPH-ALE method is developed from the differential equation in ALE form using the partition unity property which is automatically fulfilled by the Moving Least Squares approximation.The authors gratefully acknowledge the support provided by the [Grant PID2021-125447OB-I00] funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”, and the funds by [Grant TED2021–129805B-I00] funded by MCIN/AEI/ 10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. They also acknowledge the funding provided by the Xunta de Galicia (Grant #ED431C 2022/06). J. Fernández-Fidalgo acknowledges the support provided by “Ayudas para la recualificación del sistema universitario español para 2021–2023. Modalidad Margarita Salas RSU.UDC.MS20" by the Ministerio de Universidades of the Spanish Government and European Union through the NextGenerationEU funds.Xunta de Galicia; ED431C 2022/0

High order finite volume schemes on unstructured grids using Moving Least Squares reconstruction. Application to shallow water dynamics

Aceptado para su publicación en International journal for numerical methods in engineering, el 08/06/2005[Abstract] This paper introduces the use of Moving Least Squares (MLS) approximations for the development of high-order finite volume discretizations on unstructured grids. The field variables and their succesive derivatives can be accurately reconstructed using this meshfree technique in a general nodal arrangement. The methodology proposed is used in the construction of low-dissipative highorder high-resolution schemes for the shallow water equations. In particular, second and third-orderreconstruction upwind schemes for unstructured grids based on Roe’s flux difference splitting are developed and applied to inviscid and viscous flows. This class of meshfree reconstruction techniques provide a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes.Ministerio de Ciencia y Tecnología; DPI2001-0556Xunta de Galicia; PGDIT01PXI11802PRXunta de Galicia; PGIDIT03PXIC118002P

A Divergence‐free Mixture Model for Multiphase Fluids

We present a novel divergence free mixture model for multiphase flows and the related fluid-solid coupling. The new mixture model is built upon a volume-weighted mixture velocity so that the divergence free condition is satisfied for miscible and immiscible multiphase fluids. The proposed mixture velocity can be solved efficiently by adapted single phase incompressible solvers, allowing for larger time steps and smaller volume deviations. Besides, the drift velocity formulation is corrected to ensure mass conservation during the simulation. The new approach increases the accuracy of multiphase fluid simulation by several orders. The capability of the new divergence-free mixture model is demonstrated by simulating different multiphase flow phenomena including mixing and unmixing of multiple fluids, fluid-solid coupling involving deformable solids and granular materials

Enhanced SPH modeling of free-surface flows with large deformations

The subject of the present thesis is the development of a numerical solver to study the violent interaction of marine flows with rigid structures. Among the many numerical models available, the Smoothed Particle Hydrodynamics (SPH) has been chosen as it proved appropriate in dealing with violent free-surface flows. Due to its Lagrangian and meshless character it can naturally handle breaking waves and fragmentation that generally are not easily treated by standard methods. On the other hand, some consolidated features of mesh-based methods, such as the solid boundary treatment, still remain unsolved issues in the SPH context. In the present work a great part of the research activity has been devoted to tackle some of the bottlenecks of the method. Firstly, an enhanced SPH model, called delta-SPH, has been proposed. In this model, a proper numerical diffusive term has been added in the continuity equation in order to remove the spurious numerical noise in the pressure field which typically affects the weakly-compressible SPH models. Then, particular attention has been paid to the development of suitable techniques for the enforcement of the boundary conditions. As for the free-surface, a specific algorithm has been designed to detect free-surface particles and to define a related level-set function with two main targets: to allow the imposition of peculiar conditions on the free-surface and to analyse and visualize more easily the simulation outcome (especially in 3D cases). Concerning the solid boundary treatment, much effort has been spent to devise new techniques for handling generic body geometries with an adequate accuracy in both 2D and 3D problems. Two different techniques have been described: in the first one the standard ghost fluid method has been extended in order to treat complex solid geometries. Both free-slip and no-slip boundary conditions have been implemented, the latter being a quite complex matter in the SPH context. The proposed boundary treatment proved to be robust and accurate in evaluating local and global loads, though it is not easy to extend to generic 3D surfaces. The second technique has been adopted for these cases. Such a technique has been developed in the context of Riemann-SPH methods and in the present work is reformulated in the context of the standard SPH scheme. The method proved to be robust in treating complex 3D solid surfaces though less accurate than the former. Finally, an algorithm to correctly initialize the SPH simulation in the case of generic geometries has been described. It forces a resettlement of the fluid particles to achieve a regular and uniform spacing even in complex configurations. This pre-processing procedure avoids the generation of spurious currents due to local defects in the particle distribution at the beginning of the simulation. The delta-SPH model has been validated against several problems concerning fluid-structure interactions. Firstly, the capability of the solver in dealing with water impacts has been tested by simulating a jet impinging on a flat plate and a dam-break flow against a vertical wall. In this cases, the accuracy in the prediction of local loads and of the pressure field have been the main focus. Then, the viscous flow around a cylinder, in both steady and unsteady conditions, has been simulated comparing the results with reference solutions. Finally, the generation and propagation of 2D gravity waves has been simulated. Several regimes of propagation have been tested and the results compared against a potential flow solver. The developed numerical solver has been applied to several cases of free-surface flows striking rigid structures and to the problem of the generation and evolution of ship generated waves. In the former case, the robustness of the solver has been challenged by simulating 2D and 3D water impacts against complex solid surfaces. The numerical outcome have been compared with analytical solutions, experimental data and other numerical results and the limits of the model have been discussed. As for the ship generated waves, the problem has been firstly studied within the 2D+t approximation, focusing on the occurrence and features of the breaking bow waves. Then, a dedicated 3D SPH parallel solver has been developed to tackle the simulation of the entire ship in constant forward motion. This simulation is quite demanding in terms of complexities of the boundary geometry and computational resources required. The wave pattern obtained has been compared against experimental data and results from other numerical methods, showing in both the cases a fair and promising agreement

Incompressible SPH method based on Rankine source solution for violent water wave simulation

With wide applications, the smoothed particle hydrodynamics method (abbreviated as SPH) has become an important numerical tool for solving complex flows, in particular those with a rapidly moving free surface. For such problems, the incompressible Smoothed Particle Hydrodynamics (ISPH) has been shown to yield better and more stable pressure time histories than the traditional SPH by many papers in literature. However, the existing ISPH method directly approximates the second order derivatives of the functions to be solved by using the Poisson equation. The order of accuracy of the method becomes low, especially when particles are distributed in a disorderly manner, which generally happens for modelling violent water waves. This paper introduces a new formulation using the Rankine source solution. In the new approach to the ISPH, the Poisson equation is first transformed into another form that does not include any derivative of the functions to be solved, and as a result, does not need to numerically approximate derivatives. The advantage of the new approach without need of numerical approximation of derivatives is obvious, potentially leading to a more robust numerical method. The newly formulated method is tested by simulating various water waves, and its convergent behaviours are numerically studied in this paper. Its results are compared with experimental data in some cases and reasonably good agreement is achieved. More importantly, numerical results clearly show that the newly developed method does need less number of particles and so less computational costs to achieve the similar level of accuracy, or to produce more accurate results with the same number of particles compared with the traditional SPH and existing ISPH when it is applied to modelling water waves

A moving least square reproducing kernel particle method for unified multiphase continuum simulation

In physically based-based animation, pure particle methods are popular due to their simple data structure, easy implementation, and convenient parallelization. As a pure particle-based method and using Galerkin discretization, the Moving Least Square Reproducing Kernel Method (MLSRK) was developed in engineering computation as a general numerical tool for solving PDEs. The basic idea of Moving Least Square (MLS) has also been used in computer graphics to estimate deformation gradient for deformable solids. Based on these previous studies, we propose a multiphase MLSRK framework that animates complex and coupled fluids and solids in a unified manner. Specifically, we use the Cauchy momentum equation and phase field model to uniformly capture the momentum balance and phase evolution/interaction in a multiphase system, and systematically formulate the MLSRK discretization to support general multiphase constitutive models. A series of animation examples are presented to demonstrate the performance of our new multiphase MLSRK framework, including hyperelastic, elastoplastic, viscous, fracturing and multiphase coupling behaviours etc