12 results on '"Heister, Timo"'
Search Results
2. A Flexible, Parallel, Adaptive Geometric Multigrid Method for FEM.
- Author
-
Clevenger, Thomas C., Heister, Timo, Kanschat, Guido, and Kronbichler, Martin
- Subjects
- *
MULTIGRID methods (Numerical analysis) , *PUBLIC libraries , *LEAF anatomy , *MESSAGE passing (Computer science) , *FINITE element method , *ALGORITHMS - Abstract
We present the design and implementation details of a geometric multigrid method on adaptively refined meshes for massively parallel computations. The method uses local smoothing on the refined part of the mesh. Partitioning is achieved by using a space filling curve for the leaf mesh and distributing ancestors in the hierarchy based on the leaves. We present a model of the efficiency of mesh hierarchy distribution and compare its predictions to runtime measurements. The algorithm is implemented as part of the deal.II finite-element library and as such available to the public. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations.
- Author
-
Charnyi, Sergey, Heister, Timo, Olshanskii, Maxim A., and Rebholz, Leo G.
- Subjects
- *
NAVIER-Stokes equations , *CONSERVATION laws (Physics) , *FINITE element method , *ANGULAR momentum (Mechanics) , *NEWTON-Raphson method - Abstract
We study discretizations of the incompressible Navier–Stokes equations, written in the newly developed energy–momentum–angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier–Stokes equations in primitive variables. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
4. The deal.II library, Version 9.0.
- Author
-
Alzetta, Giovanni, Arndt, Daniel, Bangerth, Wolfgang, Boddu, Vishal, Brands, Benjamin, Davydov, Denis, Gassmöller, Rene, Heister, Timo, Heltai, Luca, Kormann, Katharina, Kronbichler, Martin, Maier, Matthias, Pelteret, Jean-Paul, Turcksin, Bruno, and Wells, David
- Subjects
FINITE element method ,FINITE element method software ,FINITE element method data processing ,APPLICATION program interfaces ,LINEAR algebra - Abstract
This paper provides an overview of the new features of the finite element library deal.II version 9.0. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning.
- Author
-
Davydov, Denis, Heister, Timo, Kronbichler, Martin, and Steinmann, Paul
- Subjects
- *
NUMERICAL analysis , *GROUND state energy , *GROUND state (Quantum mechanics) , *DENSITY functional theory , *FINITE element method - Abstract
In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h‐adaptive finite element method is adopted for spatial discretization and implemented with matrix‐free operator evaluation. The ground state energy is found by performing unconstrained minimization with non‐orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward O(N) complexity for the locally adaptive real‐space solution of density functional theory with finite elements. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. The deal.II library, version 8.5.
- Author
-
Arndt, Daniel, Bangerth, Wolfgang, Davydov, Denis, Heister, Timo, Heltai, Luca, Kronbichler, Martin, Maier, Matthias, Pelteret, Jean-Paul, Turcksin, Bruno, and Wells, David
- Subjects
SOFTWARE libraries (Computer programming) ,FINITE element method ,OBJECT-oriented methods (Computer science) ,INFORMATION retrieval ,MANIFOLDS (Mathematics) - Abstract
This paper provides an overview of the new features of the finite element library deal.II version 8.5. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations.
- Author
-
Heister, Timo, Olshanskii, Maxim, and Rebholz, Leo
- Subjects
NAVIER-Stokes equations ,VORTEX motion ,DISCRETIZATION methods ,VELOCITY ,FINITE element method - Abstract
We prove unconditional long-time stability for a particular velocity-vorticity discretization of the 2D Navier-Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity-pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit-explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method's vorticity and velocity are both long-time stable in the $$L^2$$ and $$H^1$$ norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
8. Compressible magma/mantle dynamics: 3-D, adaptive simulations in ASPECT.
- Author
-
Dannberg, Juliane and Heister, Timo
- Subjects
- *
EARTH'S mantle , *MAGMAS , *COMPRESSIBILITY (Fluids) , *THREE-dimensional imaging in geology , *SIMULATION methods & models , *FINITE element method - Abstract
Melt generation and migration are an important link between surface processes and the thermal and chemical evolution of the Earth's interior. However, their vastly different timescales make it difficult to study mantle convection and melt migration in a unified framework, especially for 3-D global models. And although experiments suggest an increase in melt volume of up to 20 per cent from the depth of melt generation to the surface, previous computations have neglected the individual compressibilities of the solid and the fluid phase. Here, we describe our extension of the finite element mantle convection code ASPECT that adds melt generation and migration. We use the original compressible formulation of the McKenzie equations, augmented by an equation for the conservation of energy. Applying adaptive mesh refinement to this type of problems is particularly advantageous, as the resolution can be increased in areas where melt is present and viscosity gradients are high, whereas a lower resolution is sufficient in regions without melt. Together with a high-performance, massively parallel implementation, this allows for high-resolution, 3-D, compressible, global mantle convection simulations coupled with melt migration. We evaluate the functionality and potential of this method using a series of benchmarks and model setups, compare results of the compressible and incompressible formulation, and show the effectiveness of adaptive mesh refinement when applied to melt migration. Our model of magma dynamics provides a framework for modelling processes on different scales and investigating links between processes occurring in the deep mantle and melt generation and migration. This approach could prove particularly useful applied to modelling the generation of komatiites or other melts originating in greater depths. The implementation is available in the Open Source ASPECT repository. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
9. Flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for strongly divergence-free mixed finite element methods for flow problems.
- Author
-
Heister, Timo, Rebholz, Leo G., and Xiao, Mengying
- Subjects
- *
DIRICHLET problem , *DIVERGENCE theorem , *FINITE element method , *NUMERICAL analysis , *INTERPOLATION , *INCOMPRESSIBLE flow - Abstract
We investigate a flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for velocity, u | ∂ Ω = g , for use with finite element methods for incompressible flow problems that strongly enforce mass conservation. Typical enforcement via nodal interpolation is not flux-preserving in general, and it can create divergence error even when divergence-free elements are used. We show with analysis and numerical tests that by slightly (and locally) changing nodal interpolation, the enforcement recovers flux-preservation, is optimally accurate, and delivers divergence-free solutions when used with divergence-free finite elements. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
10. Efficient numerical methods for the large-scale, parallel solution of elastoplastic contact problems.
- Author
-
Frohne, Jörg, Heister, Timo, and Bangerth, Wolfgang
- Subjects
ELASTOPLASTICITY ,CONTACT mechanics ,FINITE element method ,PARALLEL computers ,VARIATIONAL inequalities (Mathematics) - Abstract
Quasi-static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid-preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three-dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
11. Natural vorticity boundary conditions on solid walls.
- Author
-
Olshanskii, Maxim A., Heister, Timo, Rebholz, Leo G., and Galvin, Keith J.
- Subjects
- *
VORTEX motion , *HYDRODYNAMICS , *DIRICHLET principle , *DIRICHLET series , *GALERKIN methods - Abstract
We derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems. The formulation uses a Dirichlet condition for the normal component of vorticity and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e., it is enforced by a right-hand side functional and does not impose a boundary constraint on trial and test spaces. The functional involves the pressure variable, and we discuss several velocity–vorticity formulations where the proposed condition is appropriate. Several numerical experiments are given that illustrate the validity of the new approach. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
12. Multigrid methods for finite element applications with arbitrary-level hanging node configurations
- Author
-
Capodaglio, Giacomo, Bornia, Giorgio, Heister, Timo, Howle, Victoria E., Parameswaran, Siva, and Aulisa, Eugenio
- Subjects
Finite element method ,Local refinement ,Hanging nodes ,Iterative methods ,Multigrid ,Successive subspace correction - Abstract
In this dissertation, multigrid methods for finite element applications with arbitrary-level hanging nodes are considered. When a local midpoint refinement procedure is carried out on the finite element grid, hanging nodes are introduced. The presence of hanging nodes complicates the way the problem has to be addressed for several reasons. For instance, if a continuous finite element solution is sought, extra effort has to be made to enforce continuity. In this work, we propose two different strategies to achieve the desired continuity. Chapter I lays out the first strategy, which relies on the introduction of modified basis functions that are continuous by construction. Finite element spaces are the defined as the spanning sets of these modified basis functions, and the continuity of the finite element solution immediately follows. A detailed computational analysis is presented, where a multigrid algorithm defined on the continuous finite element spaces is used either as a solver, or as a preconditioner for other iterative solvers. Specifically, the conjugate gradient (CG) and the generalized minimal residual (GMRES) will be considered. The numerical results aim to investigate the convergence properties of the multigrid algorithm proposed in this chapter. In Chapter II, a theoretical analysis of multigrid algorithms with successive subspace correction (SSC) smoothers is presented. Here, we obtain convergence estimates under no regularity assumptions on the solution of the underlying partial differential equation (PDE), highlighting a dependence of the convergence bound on the number of smoothing iterations. In this framework, the second strategy to enforce continuity is described. Such a strategy relies on a particular choice of subspaces for the SSC smoother, made according to a multilevel approach that exploits the multigrid hierarchy. Continuity is recovered by decomposing functions on the finite element spaces at finer levels as linear combinations of continuous functions at coarser levels. In this context, the introduction of modified basis functions is not necessary. On the other hand, this second strategy is tied to the multigrid method, since it relies on the multigrid hierarchy and on the SSC smoother. It is important to note that, once continuous finite element spaces are obtained with the approach in Chapter I, a multigrid solver with SSC smoother can be defined also on such spaces. In this case, the choice of subspaces for the space decomposition should be made according to a domain decomposition strategy rather than a multilevel strategy, since continuity is already guaranteed by the modified basis functions, so exploiting the multigrid hierarchy is not necessary. Both the multilevel approach and the domain decomposition approach for the choice of subspaces in the SSC smoother are investigated theoretically in Chapter II. The chapter is concluded with numerical results that compare the convergence performances of the two approaches. In Chapter III, a thorough computational analysis of a multigrid method with SSC smoothers and multilevel strategy for the subspaces is presented. The analysis is motivated by a desire to test the performances of the method for a wider range of settings than those addressed in Chapter II. While a symmetric smoothing procedure is assumed in the theoretical convergence analysis in Chapter II, non-symmetric smoothers may be also used for practical applications, since they require fewer operations. The subspace solver assumed in the theoretical analysis in Chapter II was an unpreconditoned Richardson’s method, however, preconditioners such as Jacobi or Incomplete LU (ILU) factorization are often used in practice. In the computational analysis of Chapter III, we consider symmetric and nonsymmetric smoothing procedures as well as several types of preconditioners for the subspace solver. With the methods studied in this dissertation, the smoothing process is carried out on the entire multigrid space. We refer to this situation as global smoothing. When hanging nodes are present, the usual approach in the literature consists of smoothing only on a subspace of the multigrid space, that does not contain hanging nodes. We refer to this second situation as local smoothing. In Chapter III, we compare the global smoothing convergence results of the proposed method with local smoothing results obtained with existing strategies. Global smoothing provides better convergence properties, especially when the solution of the underlying PDE lacks regularity.
- Published
- 2018
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.