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Adaptive Discontinuous Galerkin Finite Element Methods For A Diffuse Interface Model Of Biological Growth
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Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth by : Andreas Costa Aristotelous
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth written by Andreas Costa Aristotelous and published by . This book was released on 2011 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This PhD dissertation concentrates on the development and application of adaptive Discontinuous Galerkin Finite Element (DG-FE) methods for the numerical solution of a Cahn-Hilliard-type diffuse interface model for biological growth. Models of this type have become popular for studying cancerous tumor progression in vivo. The work in this dissertation advances the state-of-the-art in the following ways: To our knowledge the work here contains the first primitive-variable, completely discontinuous numerical implementations of a 2D scheme for the Cahn-Hilliard equation as well as a diffuse interface model of cancer growth. We provide numerical evidence that the schemes above are convergent, with the optimal order. The efficiency of the numerical algorithms depends largely on the implementation of fast solvers for the systems of equations resulting from the DG-FE discretizations. We have developed such capabilities based on multigrid and sparse direct solver techniques. We demonstrate proof-of-concept regarding the implementation of a practical spatially adaptive meshing algorithm for the numerical schemes just mentioned and the effective use of a very simple, but powerful, marking strategy based on an inverse estimate. We demonstrate proof-of-concept for a novel simplified diffuse interface model of tumor growth. This model is essentially the Cahn-Hilliard equation with an added source term that is specialized for the context of cancerous tumor progression. We devise and analyze a mixed DG-FE scheme of convex splitting (CS) type for the Cahn-Hilliard equation in any space dimension. Specifically, we prove that our scheme is unconditionally energy stable and unconditionally uniquely solvable. Likewise, we devise and analyze a CS, mixed DG-FE scheme for our diffuse interface cancer model. This scheme is energy stable for any (positive) time step size and for any (positive) space step size that is sufficiently small.
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods by : Haihang You
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods written by Haihang You and published by . This book was released on 2009 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Discontinuous Galerkin Method is one variant of the Finite Element Methods for solving partial differential equations, which was first introduced by Reed and Hill in 1970's [27]. Discontinuous Galerkin Method (DGFEM) differs from the standard Galerkin FEM that continuity constraints are not imposed on the inter-element boundaries. It results in a solution which is composed of totally piecewise discontinuous functions. The absence of continuity constraints on the inter-element boundaries implies that DG method has a great deal of flexibility at the cost of increasing the number of degrees of freedom. This flexibility is the source of many but not all of the advantages of the DGFEM method over the Continuous Galerkin (CGFEM) method that uses spaces of continuous piecewise polynomial functions and other "less standard" methods such as nonconforming methods. As DGFEM method leads to bigger system to solve, theoretical and practical approaches to speed it up are our main focus in this dissertation. This research aims at designing and building an adaptive discontinuous Galerkin finite element method to solve partial differential equations with fast time for desired accuracy on modern architecture.
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations by : Michael Authur Saum
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations written by Michael Authur Saum and published by . This book was released on 2006 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified mathematical and computational framework for implementation of an adaptive discontinuous Galerkin (DG) finite element method (FEM) is developed using the symmetric interior penalty formulation to obtain numerical approximations to solutions of second and fourth order elliptic partial deferential equations. The DG-FEM formulation implemented allows for h-adaptivity and has the capability to work with linear, quadratic, cubic, and quartic polynomials on triangular elements in two dimensions. Two different formulations of DG are implemented based on how fluxes are represented on interior edges and comparisons are made. Explicit representations of two a posteriori error estimators, a residual based type and a "local" based type, are extended to include both Dirichlet and Neumann type boundary conditions on bounded domains. New list-based approaches to data management in an adaptive computational environment are introduced in an effort to utilize computational resources in an efficient and flexible manner.
Book Synopsis Adaptive Discontinuous Petrov-Galerkin Finite-element-methods by : Friederike Hellwig
Download or read book Adaptive Discontinuous Petrov-Galerkin Finite-element-methods written by Friederike Hellwig and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws by : Ralf Hartmann
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws written by Ralf Hartmann and published by . This book was released on 2001 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Geometric Partial Differential Equations - Part I by :
Download or read book Geometric Partial Differential Equations - Part I written by and published by Elsevier. This book was released on 2020-01-14 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering. - About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization - Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading - The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs
Book Synopsis Discontinuous Galerkin Finite Element Methods for (non)conservative Partial Differential Equations by : Sander Rhebergen
Download or read book Discontinuous Galerkin Finite Element Methods for (non)conservative Partial Differential Equations written by Sander Rhebergen and published by . This book was released on 2010 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method by : Shukai Du
Download or read book An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method written by Shukai Du and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph requires basic knowledge of the variational theory of elliptic PDE and the techniques used for the analysis of the Finite Element Method. However, all the tools for the analysis of FEM (scaling arguments, finite dimensional estimates in the reference configuration, Piola transforms) are carefully introduced before being used, so that the reader does not need to go over longforgotten textbooks. Readers include: computational mathematicians, numerical analysts, engineers and scientists interested in new and computationally competitive Discontinuous Galerkin methods. The intended audience includes graduate students in computational mathematics, physics, and engineering, since the prerequisites are quite basic for a second year graduate student who has already taken a non necessarily advanced class in the Finite Element method.
Book Synopsis Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws by : Andreas Eduard Hiltebrand
Download or read book Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws written by Andreas Eduard Hiltebrand and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Overset Mesh Framework for the Hybridizable Discontinuous Galerkin Finite Element Method by : Justin Kauffman
Download or read book An Overset Mesh Framework for the Hybridizable Discontinuous Galerkin Finite Element Method written by Justin Kauffman and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational simulations contain discretizations of both a physical domain and a mathematical model. In this dissertation, an overset mesh framework is used to discretize the physical domain, and the hybridizable discontinuous Galerkin (HDG) finite element method is used to discretize the mathematical model. It is proposed that using an overset mesh framework for the HDG method enables stable solutions to be computed for complex geometries and dynamic meshes. Overset mesh methods are chosen because they are efficient at decomposing geometrically complex domains. The HDG method was chosen because it provides solutions that are arbitrarily high-order accurate, reduces the size of the global discrete problem, and has the ability to solve elliptic, parabolic, and/or hyperbolic problems with a unified form of discretization. An overset mesh method can utilize an inherent property of the HDG method, the decomposition of the solution into global (face) and local (volume) parts. The global solution exists only on the cell boundaries; while, the local solution exists in the interior of each cell and is decoupled between neighboring cells. This decomposition introduces face-volume coupling in the weak form for degrees of freedom on cell boundaries, which is used as the foundation for the overset communication between subdomains.Ultimately, the goal of this work is to simulate full-scale hydrodynamic and fluid-structure interaction (FSI) problems. To achieve these simulations, the necessary building blocks must first be verified and validated in the overset-HDG framework. The building blocks demonstrated in this dissertation are steady convection-diffusion, linear elasticostatics, and pseudo-compressible Navier-Stokes in both Eulerian and arbitrary Lagrangian-Eulerian frames. Computational simulations are performed to demonstrate the applicability and accuracy of the overset-HDG algorithm.
Book Synopsis The Discontinuous Galerkin Finite Element Method by : I. S. Raju
Download or read book The Discontinuous Galerkin Finite Element Method written by I. S. Raju and published by . This book was released on 2022 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Development and Implementation of Discontinuous Galerkin (DG) Finite Element Methods for Topology Optimization by :
Download or read book Development and Implementation of Discontinuous Galerkin (DG) Finite Element Methods for Topology Optimization written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A short design cycle is an important factor in building cost-effective products and staying competitive. In order to meet market demands, structural optimization tools are widely used along with established CAE (Computer Aided Engineering), CAD (Computer Aided Design) and PLM (Product Lifecycle Management) methodologies. Using these tools, the optimal shape of a structure can be predicted at the start of the design process and this can act as an input to the designer in the conceptual design phase and help him design better, stronger and cost-effective structures. The goal of topology optimization is to determine the best distribution of material for a structure such that an objective function such as compliance takes an extremal value, subject to some constraints. After a finite element discretization of the structure, the design variables are normally the "density" of the finite elements. Constraints are placed on these density values and on the volume of the structure to be occupied by the material. In this thesis, we use non-conforming discontinuous Galerkin (DG) methods to overcome some of the numerical problems that occur when the topology optimization problem is solved using conforming finite elements. These numerical problems are the formation of checkerboard type patterns and mesh dependence of the solution. Various non-conforming discontinuous Galerkin formulations are used. The variation of optimal topologies with the penalization parameter present in DG formulations is also studied. The stiffness behavior of checkerboard pattern is also analyzed with both conforming and DG formulations to study the relationship between patch stiffness and occurrence of checkerboards. We find that the non-symmetric interior penalty Galerkin (NIPG) method emerges as a competitive alternative to the standard finite element method and is free from the associated instabilities.
Book Synopsis Discontinuous Galerkin Finite Element Methods Applied to Two-phase, Air-water Flow Problems by : Owen John Eslinger
Download or read book Discontinuous Galerkin Finite Element Methods Applied to Two-phase, Air-water Flow Problems written by Owen John Eslinger and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation by : Lars Pesch
Download or read book Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation written by Lars Pesch and published by . This book was released on 2007 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Space-time Discontinuous Galerkin Finite Element Method for Two-fluid Flows by : Warnerius Egbert Hendrikus Sollie
Download or read book Space-time Discontinuous Galerkin Finite Element Method for Two-fluid Flows written by Warnerius Egbert Hendrikus Sollie and published by . This book was released on 2010 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Discontinuous Galerkin Finite Element Methods for Shallow Water Flow by : Ashley L. Maggi
Download or read book Discontinuous Galerkin Finite Element Methods for Shallow Water Flow written by Ashley L. Maggi and published by . This book was released on 2011 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research has laid the groundwork for a robust computational infrastructure in the context of DG FEM that will significantly cut computational cost by applying fast and efficient state-of-the-art algorithms. The promising results provide motivation for future model development within the framework of a mixed element approach as well as the derivation of higher order numerical integration rules for triangular prism domains.
Book Synopsis Discontinuous Galerkin Methods by : Bernardo Cockburn
Download or read book Discontinuous Galerkin Methods written by Bernardo Cockburn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.
