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You searched for subject:(compatible discretization). Showing records 1 – 3 of 3 total matches.

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1. Trask, Nathaniel A. Compatible high-order meshless schemes for viscous flows through l2 optimization.

Degree: PhD, Applied Mathematics, 2015, Brown University

Meshless methods provide an ideal framework for scalably simulating Lagrangian hydrodynamics in domains undergoing large deformation. For these schemes, interfaces can easily be treated without introducing artificial diffusion, and boundary deformation is handled without costly topological updates to the mesh. Unfortunately, abandoning a mesh also means abandoning a natural framework for performing analysis and as a result meshless methods have historically lacked the stability and conservation properties of their finite element/volume counterparts. In this work, we present a number of new schemes that use a combination of ℓ2-optimization and graph theory to achieve highly accurate and robust meshless discretizations. These schemes mimic the algebraic structure of compatible finite-element methods, and as a result inherit many of their favorable properties. We use these methods to develop monolithic schemes for suspension flows driven by electrokinetic effects. While these flows are challenging due to the presence of singular pressure forces and a range of relevant length scales of several orders of magnitude, we demonstrate that these new techniques easily resolve analytic benchmarks without the need for sub-grid scale lubrication models. Advisors/Committee Members: Maxey, Martin (Director), Karniadakis, George (Reader), Shu, Chi-Wang (Reader).

Subjects/Keywords: compatible discretization

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Trask, N. A. (2015). Compatible high-order meshless schemes for viscous flows through l2 optimization. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:674174/

Chicago Manual of Style (16th Edition):

Trask, Nathaniel A. “Compatible high-order meshless schemes for viscous flows through l2 optimization.” 2015. Doctoral Dissertation, Brown University. Accessed January 25, 2020. https://repository.library.brown.edu/studio/item/bdr:674174/.

MLA Handbook (7th Edition):

Trask, Nathaniel A. “Compatible high-order meshless schemes for viscous flows through l2 optimization.” 2015. Web. 25 Jan 2020.

Vancouver:

Trask NA. Compatible high-order meshless schemes for viscous flows through l2 optimization. [Internet] [Doctoral dissertation]. Brown University; 2015. [cited 2020 Jan 25]. Available from: https://repository.library.brown.edu/studio/item/bdr:674174/.

Council of Science Editors:

Trask NA. Compatible high-order meshless schemes for viscous flows through l2 optimization. [Doctoral Dissertation]. Brown University; 2015. Available from: https://repository.library.brown.edu/studio/item/bdr:674174/

2. Bonelle, Jérôme. Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations.

Degree: Docteur es, Mathématiques, 2014, Université Paris-Est

Cette thèse présente une nouvelle classe de schémas de discrétisation spatiale sur maillages polyédriques, nommée Compatible Discrete Operator (CDO) et en étudie l'application aux équations elliptiques et de Stokes. La préservation au niveau discret des caractéristiques essentielles du système continu sert de fil conducteur à la construction des opérateurs. Les opérateurs de de Rham définissent les degrés de liberté en accord avec la nature physique des champs à discrétiser. Les équations sont décomposées de manière à différencier les relations topologiques (lois de conservation) des relations constitutives (lois de fermeture).Les relations topologiques sont associées à des opérateurs différentiels discrets et les relations constitutives à des opérateurs de Hodge discrets. Une particularité de l'approche CDO est l'utilisation explicite d'un second maillage, dit dual, pour bâtir l'opérateur de Hodge discret. Deux familles de schémas CDO sont ainsi considérées : les schémas vertex-based lorsque le potentiel est discrétisé aux sommets du maillage (primal), et les schémas cell-based lorsque le potentiel est discrétisé aux sommets du maillage dual (les sommets duaux étant en bijection avec les cellules primales).Les schémas CDO associés à ces deux familles sont présentés et leur convergence est analysée. Une première analyse s'appuie sur une définition algébrique de l'opérateur de Hodge discret et permet d'identifier trois propriétés clés : symétrie, stabilité et ℙ0-consistance. Une seconde analyse s'appuie sur une définition de l'opérateur de Hodge discret à l'aide d'opérateurs de reconstruction pour lesquels sont identifiées les propriétés à satisfaire. Par ailleurs, les schémas CDO fournissent une vision unifiée d'une large gamme de schémas de la littérature (éléments finis, volumes finis, schémas mimétiques…).Enfin, la validité et l'efficacité de l'approche CDO sont illustrées sur divers cas tests et plusieurs maillages polyédriques

This thesis presents a new class of spatial discretization schemes on polyhedral meshes, called Compatible Discrete Operator (CDO) schemes and their application to elliptic and Stokes equations. In CDO schemes, preserving the structural properties of the continuous equations is the leading principle to design the discrete operators. De Rham maps define the degrees of freedom according to the physical nature of fields to discretize. CDO schemes operate a clear separation between topological relations (balance equations) and constitutive relations (closure laws).Topological relations are related to discrete differential operators, and constitutive relations to discrete Hodge operators. A feature of CDO schemes is the explicit use of a second mesh, called dual mesh, to build the discrete Hodge operator. Two families of CDO schemes are considered: vertex-based schemes where the potential is located at (primal) mesh vertices, and cell-based schemes where the potential is located at dual mesh vertices (dual vertices being in one-to-one correspondence with primal cells).The CDO…

Advisors/Committee Members: Ern, Alexandre (thesis director).

Subjects/Keywords: Opérateur Discret Compatible (CDO); Discrétisation compatible; Discrétisation mimétique; Elliptique; Stokes; Opérateur de Hodge discret; Compatible discretization; Mimetic discretization; Elliptic; Stokes; Discrete Hodge operator; Polyhedral mesh

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Bonelle, J. (2014). Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations. (Doctoral Dissertation). Université Paris-Est. Retrieved from http://www.theses.fr/2014PEST1078

Chicago Manual of Style (16th Edition):

Bonelle, Jérôme. “Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations.” 2014. Doctoral Dissertation, Université Paris-Est. Accessed January 25, 2020. http://www.theses.fr/2014PEST1078.

MLA Handbook (7th Edition):

Bonelle, Jérôme. “Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations.” 2014. Web. 25 Jan 2020.

Vancouver:

Bonelle J. Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations. [Internet] [Doctoral dissertation]. Université Paris-Est; 2014. [cited 2020 Jan 25]. Available from: http://www.theses.fr/2014PEST1078.

Council of Science Editors:

Bonelle J. Opérateurs discrets compatibles pour la discrétisation sur maillages polyédriques des équations elliptiques et de Stokes : Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations. [Doctoral Dissertation]. Université Paris-Est; 2014. Available from: http://www.theses.fr/2014PEST1078


The Ohio State University

3. He, Bo. Compatible discretizations for Maxwell equations.

Degree: PhD, Electrical Engineering, 2006, The Ohio State University

The main focus of this dissertation is the study and development of numerical techniques to solve Maxwell equations on irregular lattices. This is achieved by means of compatible discretizations that rely on some tools of algebraic topology and a discrete analog of differential forms on a lattice. Using discrete Hodge decomposition and Euler’s formula for a network of polyhedra, we show that the number of dynamic degrees of freedom (DoFs) of the electric field equals the number of dynamic DoFs of the magnetic field on an arbitrary lattice (cell complex). This identity reflects an essential property of discrete Maxwell equations (Hamiltonian structure) that any compatible discretization scheme should observe. We unveil a new duality called Galerkin duality, a transformation between two (discrete) systems, primal system and dual system. If the discrete Hodge operators are realized by Galerkin Hodges, we show that the primal system recovers the conventional edge-element FEM and suggests a geometric foundation for it. On the other hand, the dual system suggests a new (dual) type of FEM. We find that inverse Hodge matrices have strong localization properties. Hence we propose two thresholding techniques, viz., algebraic thresholding and topological thresholding, to sparsify inverse Hodge matrices. Based on topological thresholding, we propose a sparse and fully explicit time-domain FEM for Maxwell equations. From a finite-difference viewpoint, topological thresholding provides a general and systematic way to derive stable local finite-difference stencils in irregular grids. We also propose and implement an E-B mixed FEM scheme to discretize first order Maxwell equations in frequency domain directly. This scheme results in sparse matrices. In order to tackle low-frequency instabilities in frequency domain FEM and spurious linear growth of time domain FEM solutions, we propose some gauging techniques to regularize the null space of a curl operator. Advisors/Committee Members: Teixeira, Fernando (Advisor).

Subjects/Keywords: differential forms; chains and cochains; Whitney forms; de Rham diagram; gauging; compatible discretization; Hodge operator; Hodge decomposition; Euler's formula; FDTD; FEM; Galerkin duality; primal and dual; pure Neumann boundary condition; mixed FEM

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

He, B. (2006). Compatible discretizations for Maxwell equations. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299

Chicago Manual of Style (16th Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Doctoral Dissertation, The Ohio State University. Accessed January 25, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

MLA Handbook (7th Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Web. 25 Jan 2020.

Vancouver:

He B. Compatible discretizations for Maxwell equations. [Internet] [Doctoral dissertation]. The Ohio State University; 2006. [cited 2020 Jan 25]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

Council of Science Editors:

He B. Compatible discretizations for Maxwell equations. [Doctoral Dissertation]. The Ohio State University; 2006. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299

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