An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE

The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding

A Nitsche-based cut finite element method for a fluid--structure interaction problem

We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation which allows us to establish stability and optimal order \emph{a priori} error estimates, see~\cite{MassingLarsonLoggEtAl2013}. We consider here a steady state fluid--structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.Comment: Revised version, 18 pages, 7 figures. Accepted for publication in CAMCo

Multi-Dimensional, Compressible Viscous Flow on a Moving Voronoi Mesh

Numerous formulations of finite volume schemes for the Euler and Navier-Stokes equations exist, but in the majority of cases they have been developed for structured and stationary meshes. In many applications, more flexible mesh geometries that can dynamically adjust to the problem at hand and move with the flow in a (quasi) Lagrangian fashion would, however, be highly desirable, as this can allow a significant reduction of advection errors and an accurate realization of curved and moving boundary conditions. Here we describe a novel formulation of viscous continuum hydrodynamics that solves the equations of motion on a Voronoi mesh created by a set of mesh-generating points. The points can move in an arbitrary manner, but the most natural motion is that given by the fluid velocity itself, such that the mesh dynamically adjusts to the flow. Owing to the mathematical properties of the Voronoi tessellation, pathological mesh-twisting effects are avoided. Our implementation considers the full Navier-Stokes equations and has been realized in the AREPO code both in 2D and 3D. We propose a new approach to compute accurate viscous fluxes for a dynamic Voronoi mesh, and use this to formulate a finite volume solver of the Navier-Stokes equations. Through a number of test problems, including circular Couette flow and flow past a cylindrical obstacle, we show that our new scheme combines good accuracy with geometric flexibility, and hence promises to be competitive with other highly refined Eulerian methods. This will in particular allow astrophysical applications of the AREPO code where physical viscosity is important, such as in the hot plasma in galaxy clusters, or for viscous accretion disk models.Comment: 26 pages, 21 figures. Submitted to MNRA

Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids

A new very high-order technique for solving conservation laws with curved boundary domains is proposed. A Finite Difference scheme on Cartesian grids is coupled with an original ghost cell method that provide accurate approximations for smooth solutions. The technology is based on a specific least square method with restrictions that enables to handle general Robin conditions. Several examples in two-dimensional geometries are presented for the unsteady Convection–Diffusion equation and the Euler equations. A fifth-order WENO scheme is employed with matching fifth-order reconstruction at the boundaries. Arbitrary high-order reconstruction for smooth flows is achievable independently of the underlying differential equation since the method works as a black-box dedicated to boundary condition treatment.This work has been partially supported by the Ministerio de Economı́a y Competitividad (grant #DPI2015- 68431-R) and #RTI2018-093366-B-I00 of the Ministerio de Ciencia, Innovación y Universidades of the Spanish Government and by the Consellerı́a de Educación e Ordenación Universitaria of the Xunta de Galicia (grants #GRC2014/039 and #ED431C 2018/41), cofinanced with FEDER, Spain funds and the Universidade da Coruña, Spain. J. Fernandez-Fidalgo gratefully acknowledges the contributions of the IACOBUS Program, Spain and the INDITEX-UDC, Spain grant that have partially financed the present work. S. Clain acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, Portugal, through COMPETE 2020 – Programa Operational Fatores de Competitividade, and the National Funds through FCT — Fundação para a Ciência e a Tecnologia, Portugal, project No. UID/FIS/04650/2013 and project No. POCI-01-0145-FEDER-02811

High‐order discontinuous Galerkin method for time‐domain electromagnetics on geometry‐independent Cartesian meshes

In this work we present the Cartesian grid discontinuous Galerkin (cgDG) finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high-order discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry-independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artifacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighboring elements. A subdomain matching algorithm and an a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body-fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity

High-order discontinuous Galerkin method for time-domain electromagnetics on geometry-independent Cartesian meshes

This is the peer reviewed version of the following article: Navarro¿García, H., Sevilla, R., Nadal, E., & Ródenas, J. J. (2021). High¿order discontinuous Galerkin method for time¿domain electromagnetics on geometry¿independent Cartesian meshes. International Journal for Numerical Methods in Engineering, 122(24), 7632-7663, which has been published in final form at https://doi.org/10.1002/nme.6846. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[ES] En este trabajo presentamos el método de los elementos finitos con mallados cartesianos y formulación Galerkin discontinua (cgDG), una técnica novedosa que permite la obtención de soluciones numéricas para problemas dominados por términos convectivos. Esta técnica combina la alta precisión y eficiencia de la discretización discontinua de alto orden característica de la formulación Galerkin discontinua con la simplicidad y estructura jerárquica de los mallados cartesianos independientes de la geometría. El correcto tratamiento de los elementos localizados sobre la frontera del dominio de cálculo es crucial a fin de asegurar un buen desempeño del algoritmo. El método tiene en cuenta la definición exacta de la geometría, evitando la aparición de artefactos derivados de una pobre representación de las fronteras. Por otra parte, se ha definido un procedimiento de estabilización que elimina la restricción que impone sobre el paso temporal del integrador explícito la presencia de elementos intersecados con patrones de corte extremos. La estrategia de estabilización elimina los grados de libertad inestables y reasigna los dominios de soporte de sus funciones de forma asociadas a elementos vecinos. En esta publicación presentamos u algoritmo de emparejamiento de subdominios y una estrategia de enriquecimiento a posteriori. La discretización espacial resultante de combinar estas estrategias preserva la estabilidad y precisión de la aproximación con discretizaciones conformes con la geometría. El método se valida a través de un conjunto de ejemplos numéricos de prueba y se aplica de forma satisfactoria a la resolución de problemas de interés en el ámbito de la reflexión de ondas electromagnéticas.[EN] In this work we present the Cartesian grid discontinuous Galerkin (cgDG)finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high-order discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry-independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artefacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighbouring elements. A subdomain matching algorithm and a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body-fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity.Engineering and Physical Sciences Research Council, Grant/Award Number: EP/T009071/1; Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: FPU17/03993; Ministerio de Economia y Competitividad, Grant/Award Number: DPI2017-89816-RNavarro-García, H.; Sevilla, R.; Nadal, E.; Ródenas, JJ. (2021). High-order discontinuous Galerkin method for time-domain electromagnetics on geometry-independent Cartesian meshes. International Journal for Numerical Methods in Engineering. 122(24):7632-7663. https://doi.org/10.1002/nme.6846763276631222

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth