2,282 research outputs found
Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps
Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property
A partition of unity approach to fluid mechanics and fluid-structure interaction
For problems involving large deformations of thin structures, simulating
fluid-structure interaction (FSI) remains challenging largely due to the need
to balance computational feasibility, efficiency, and solution accuracy.
Overlapping domain techniques have been introduced as a way to combine the
fluid-solid mesh conformity, seen in moving-mesh methods, without the need for
mesh smoothing or re-meshing, which is a core characteristic of fixed mesh
approaches. In this work, we introduce a novel overlapping domain method based
on a partition of unity approach. Unified function spaces are defined as a
weighted sum of fields given on two overlapping meshes. The method is shown to
achieve optimal convergence rates and to be stable for steady-state Stokes,
Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for
FSI in the case of a 2D mock aortic valve simulation. These initial results
point to the potential applicability of the method to a wide range of FSI
applications, enabling boundary layer refinement and large deformations without
the need for re-meshing or user-defined stabilization.Comment: 34 pages, 15 figur
Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows
In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft
Biot model with generalized eigenvalue problems for scalability and robustness to parameters
We consider Biot model with block preconditioners and generalized eigenvalue
problems for scalability and robustness to parameters. A discontinuous Galerkin
discretization is employed with the displacement and Darcy flow flux
discretized as piecewise continuous in elements, and the pore pressure as
piecewise constant in the element with a stabilizing term. Parallel
algorithms are designed to solve the resulting linear system. Specifically, the
GMRES method is employed as the outer iteration algorithm and block-triangular
preconditioners are designed to accelerate the convergence. In the
preconditioners, the elliptic operators are further approximated by using
incomplete Cholesky factorization or two-level additive overlapping Schwartz
method where coarse grids are constructed by generalized eigenvalue problems in
the overlaps (GenEO). Extensive numerical experiments show a scalability and
parametric robustness of the resulting parallel algorithms.Comment: Submitted to the 27th International Conference on Domain
Decomposition Methods (DD27), 8 pages, 1 figur
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods
Large linear systems with sparse, non-symmetric matrices arise in the
modeling of Markov chains or in the discretization of convection-diffusion
problems. Due to their potential to solve sparse linear systems with an effort
that is linear in the number of unknowns, algebraic multigrid (AMG) methods are
of fundamental interest for such systems. For symmetric positive definite
matrices, fundamental theoretical convergence results are established, and
efficient AMG solvers have been developed. In contrast, for non-symmetric
matrices, theoretical convergence results have been provided only recently. A
property that is sufficient for convergence is that the matrix be an M-matrix.
In this paper, we present how the simulation of incompressible fluid flows with
particle methods leads to large linear systems with sparse, non-symmetric
matrices. In each time step, the Poisson equation is approximated by meshfree
finite differences. While traditional least squares approaches do not guarantee
an M-matrix structure, an approach based on linear optimization yields
optimally sparse M-matrices. For both types of discretization approaches, we
investigate the performance of a classical AMG method, as well as an AMLI type
method. While in the considered test problems, the M-matrix structure turns out
not to be necessary for the convergence of AMG, problems can occur when it is
violated. In addition, the matrices obtained by the linear optimization
approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure
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