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Timo Heister

Publications and source records attributed to Timo Heister.

15 recordsLinked to original sources
arXiv2023

An adaptive stabilized trace finite element method for surface PDEs

The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two stabilization variants, gradient-jump face and normal-gradient volume, are considered for continuous trace spaces of the first and second degrees, based on the polynomial families $Q_1$ and $Q_2$. We propose a practical error indicator that estimates the `jumps' of finite element solution derivatives across background mesh faces and it avoids integration of any quantities along implicitly defined curvilinear edges of the discrete surface elements. For the $Q_1$ family of piecewise trilinear polynomials on bulk cells, the solve-estimate-mark-refine strategy, combined with the suggested error indicator, achieves optimal convergence rates typical of two-dimensional problems. We also provide a posteriori error estimates, establishing the reliability of the error indicator for the $Q_1$ and $Q_2$ elements and for two types of stabilization. In numerical experiments, we assess the reliability and efficiency of the error indicator. While both stabilizations are found to deliver comparable performance,the lowest degree finite element space appears to be the more robust choice for the adaptive TraceFEM framework.

math.NA↗
arXiv2022

Efficient distributed matrix-free multigrid methods on locally refined meshes for FEM computations

This work studies three multigrid variants for matrix-free finite-element computations on locally refined meshes: geometric local smoothing, geometric global coarsening, and polynomial global coarsening. We have integrated the algorithms into the same framework-the open-source finite-element library deal.II-, which allows us to make fair comparisons regarding their implementation complexity, computational efficiency, and parallel scalability as well as to compare the measurements with theoretically derived performance models. Serial simulations and parallel weak and strong scaling on up to 147,456 CPU cores on 3,072 compute nodes are presented. The results obtained indicate that global coarsening algorithms show a better parallel behavior for comparable smoothers due to the better load balance particularly on the expensive fine levels. In the serial case, the costs of applying hanging-node constraints might be significant, leading to advantages of local smoothing, even though the number of solver iterations needed is slightly higher.

math.NA↗
arXiv2022

Robust preconditioning for a mixed formulation of phase-field fracture problems

In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.

math.NA↗
arXiv2019

The deal.II finite element library: design, features, and insights

deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, $hp$-adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse and worldwide community of developers and users, as well as an educational platform. We therefore also discuss some of the technical and social challenges and lessons learned in running a large community software project over the course of two decades.

cs.MS↗
arXiv2019

Comparison Between Algebraic and Matrix-free Geometric Multigrid for a Stokes Problem on Adaptive Meshes with Variable Viscosity

Problems arising in Earth's mantle convection involve finding the solution to Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 10^9 or more unknowns. One common approach for preconditioning to the velocity block of these systems is to apply an Algebraic Multigrid (AMG) v-cycle (as is done in the ASPECT software, for example), however, with AMG, robustness can be difficult with respect to problem size and number of parallel processes. Additionally, we see an increase in iteration counts with adaptive refinement when using AMG. In contrast, the Geometric Multigrid (GMG) method, by using information about the geometry of the problem, should offer a more robust option. Here we present a matrix-free GMG v-cycle which works on adaptively refined, distributed meshes, and we will compare it against the current AMG preconditioner (Trilinos ML) used in the ASPECT software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 114688 cores and $217$ billion unknowns. All computations are run using the open source, finite element library deal.ii.

math.NA↗
arXiv2019

A Flexible, Parallel, Adaptive Geometric Multigrid method for FEM

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.

math.NA↗
arXiv2018

A new formulation for coupled magma/mantle dynamics

Many open problems in the Earth sciences can only be understood by modelling the porous flow of melt through a viscously deforming solid rock matrix. However, the system of equations describing this process becomes mathematically degenerate in the limit of vanishing melt fraction. Numerical methods that do not consider this degeneracy or avoid it solely by regularising specific material properties generally become computationally expensive as soon as the melt fraction approaches zero in some part of the domain. Here, we present a new formulation of the equations for coupled magma/mantle dynamics that addresses this problem, and allows it to accurately compute large-scale 3-D magma/mantle dynamics simulations with extensive regions of zero melt fraction. We achieve this by rescaling one of the solution variables, the compaction pressure, which ensures that for vanishing melt fraction, the equation causing the degeneracy becomes an identity and the other two equations revert to the Stokes system. This allows us to split the domain into two parts, and to solve the coupled magma/mantle dynamics system only in cells where melt is present. We have implemented this formulation in the open source geodynamic modelling code ASPECT and illustrate the improved performance compared to the previous three-field formulation, showing numerically that the new formulation is optimal in terms of problem size and only minimally sensitive to model parameters. Beyond that, we demonstrate the applicability to realistic problems by showing large-scale 2-D and 3-D models of mid-ocean ridges with complex rheology. Hence, we believe that our new formulation and its implementation in ASPECT will prove a valuable tool for studying the interaction of melt segregating through and interacting with a solid host rock in the Earth and other planetary bodies using high-resolution, three-dimensional simulations.

physics.geo-ph↗
arXiv2018

Parallel solution, adaptivity, computational convergence, and open-source code of 2d and 3d pressurized phase-field fracture problems

We present a scalable, parallel implementation of a solver for the solution of a phase-field model for quasi-static brittle fracture. The code is available as open source. Numerical solutions in 2d and 3d with adaptive mesh refinement show optimal scaling of the linear solver based on algebraic multigrid, and convergence of the phase-field model towards exact values of functionals of interests such as the crack opening displacement or the total crack volume. In contrast to uniform refinement, adaptive mesh refinement allows us to recover optimal convergence rates for the non-smooth solutions encountered in typical test problems. We also present numerical studies of the influence of the finite domain size on functional evaluations used to approximate the infinite domain.

math.NA↗
arXiv2017

Efficient discretizations for the EMAC formulation of the incompressible Navier-Stokes equations

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.

math.NA↗
arXiv2017

High Accuracy Mantle Convection Simulation through Modern Numerical Methods. II: Realistic Models and Problems

Computations have helped elucidate the dynamics of Earth's mantle for several decades already. The numerical methods that underlie these simulations have greatly evolved within this time span, and today include dynamically changing and adaptively refined meshes, sophisticated and efficient solvers, and parallelization to large clusters of computers. At the same time, many of these methods -- discussed in detail in a previous paper in this series -- were developed and tested primarily using model problems that lack many of the complexities that are common to the realistic models our community wants to solve today. With several years of experience solving complex and realistic models, we here revisit some of the algorithm designs of the earlier paper and discuss the incorporation of more complex physics. In particular, we re-consider time stepping and mesh refinement algorithms, evaluate approaches to incorporate compressibility, and discuss dealing with strongly varying material coefficients, latent heat, and how to track chemical compositions and heterogeneities. Taken together and implemented in a high-performance, massively parallel code, the techniques discussed in this paper then allow for high resolution, 3d, compressible, global mantle convection simulations with phase transitions, strongly temperature dependent viscosity and realistic material properties based on mineral physics data.

cs.CE↗
arXiv2016

On conservation laws of Navier-Stokes Galerkin discretizations

We study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

math.NA↗
arXiv2015

Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations

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.

math.AP↗
arXiv2015

Clone and graft: Testing scientific applications as they are built

This article describes our experience developing and maintaining automated tests for scientific applications. The main idea evolves around building on already existing tests by cloning and grafting. The idea is demonstrated on a minimal model problem written in Python.

cs.MS↗
arXiv2011

Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture

We consider a family of finitely presented groups, called Universal Left Invertible Element (or ULIE) groups, that are universal for existence of one--sided invertible elements in a group ring K[G], where K is a field or a division ring. We show that for testing Kaplansky's Direct Finiteness Conjecture, it suffices to test it on ULIE groups, and we show that there is an infinite family of non-amenable ULIE groups. We consider the Invertibles Conjecture and we show that it is equivalent to a question about ULIE groups. We also show that for any group G, direct finiteness of K[ G x H ] for all finite groups H implies stable finiteness of K[G]. Thus, truth of the Direct Finiteness Conjecture implies stable finiteness. By calculating all the ULIE groups over the field K=F_2 of two elements, for ranks (3,n), n<=11 and (5,5), we show that the Direct Finiteness Conjecture and the Invertibles Conjecture (which implies the Zero Divisors Conjecture) hold for these ranks over F_2.

math.RA↗