Space-time Mesh Research Articles

Tessellation and interactive visualization of four-dimensional spacetime geometries

This paper addresses two problems needed to support four-dimensional (3d+t) spacetime numerical simulations. The first contribution is a general algorithm for producing conforming spacetime meshes of moving geometries. Here, the surface points of the geometry are embedded in a four-dimensional space as the geometry moves in time. The geometry is first tessellated at prescribed time steps and then these tessellations are connected in the parameter space of each geometry entity to form tetrahedra. In contrast to previous work, this approach allows the resolution of the geometry to be controlled at each time step. The only restriction on the algorithm is the requirement that no topological changes to the geometry are made (i.e. the hierarchical relations between all geometry entities are maintained) as the geometry moves in time. The validity of the final mesh topology is verified by ensuring the tetrahedralizations represent a closed 3-manifold. For some analytic problems, the 4d volume of the tetrahedralization is also verified. The second problem addressed in this paper is the design of a system to interactively visualize four-dimensional meshes when the 4d view changes, including tetrahedra (embedded in 4d) and pentatopes. Algorithms that either include or exclude a geometry shader are described, and the efficiency of each approach is then compared. Overall, the results suggest that visualizing tetrahedra (either those bounding the domain, or extracted from a pentatopal mesh) using a geometry shader achieves the highest frame rate, realizing interactive frame rates of at least 15 frames per second for meshes with about 50 million tetrahedra.

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

Optimization of Longitudinal Motions of an Elastic Rod Using Periodically Distributed Piezoelectric Forces

The longitudinal vibrations of an elastic rod controlled by a distributed force, which is applied to individual sections of the rod, are studied. It is assumed that the force varies in space in a piecewise constant manner. Such a mechanical system can be implemented using piezoactuators attached along the rod. The dynamics of the system is determined from the solution of the variational problem following the method of integrodifferential relations. The variational problem is solved analytically. To do this, traveling waves of the d’Alembert type are introduced on the space-time mesh, which determine continuous displacements and a dynamic potential. The latter relates the momentum density and stresses. A control problem is posed under the condition of the weighted minimization of the vibrational energy stored by the rod at the terminal time instant, and the mean potential energy generated by the control actions. The extremal motion and the corresponding control law are found explicitly by solving the Euler–Lagrange equations. As an example, the control capabilities for certain configurations of piezoelectric elements are studied.

Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution using higher-order continuous B-spline basis functions in both spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.

The Lagrangian numerical relativity code SPHINCS_BSSN_v1.0

We present version 1.0 of our Lagrangian numerical relativity code SPHINCS_BSSN. This code evolves the full set of Einstein equations, but contrary to other numerical relativity codes, it evolves the matter fluid via Lagrangian particles in the framework of a high-accuracy version of smooth particle hydrodynamics (SPH). The major new elements introduced here are: (i) a new method to map the stress–energy tensor (known at the particles) to the spacetime mesh, based on a local regression estimate; (ii) additional measures that ensure the robust evolution of a neutron star through its collapse to a black hole; and (iii) further refinements in how we place the SPH particles for our initial data. The latter are implemented in our code SPHINCS_ID which now, in addition to LORENE, can also couple to initial data produced by the initial data library FUKA. We discuss several simulations of neutron star mergers performed with SPHINCS_BSSN_v1.0, including irrotational cases with and without prompt collapse and a system where only one of the stars has a large spin (χ = 0.5).

Design and performance of a space–time virtual element method for the heat equation on prismatic meshes

We present a space–time virtual element method for the discretization of the heat equation, which is defined on general prismatic meshes and variable degrees of accuracy. Strategies to handle efficiently the space–time mesh structure are discussed. We perform convergence tests for the h- and hp-versions of the method in case of smooth and singular solutions, and test space–time adaptive mesh refinements driven by a residual-type error indicator.

Spacetime finite element methods for control problems subject to the wave equation

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.

Adaptive space–time finite element methods for parabolic optimal control problems

Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.

On the Implementation of an Adaptive Multirate Framework for Coupled Transport and Flow

This paper presents a multirate in time approach for coupled flow and transport problems combined with goal-oriented error control based on the Dual Weighted Residual (DWR) method. The focus is on the implementation of the multirate concept regarding different time scales for the underlying subproblems. Key ingredients are an arbitrary degree discontinuous Galerkin time discretization, an a posteriori error representation for the transport problem coupled with flow and the concept of space-time slabs based on tensor-product spaces. From the latter, a space-time mesh adaptation with highly economical grids is developed. The performance of the approach is studied carefully by numerical convergence examples as well as an example of physical interest for convection-dominated transport.

A Space-Time Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by non-polynomial complex wave functions that satisfy the Schro\"odinger equation locally on each element of the space-time mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method, and, for the one- and two- dimensional cases, optimal, high-order, h-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.

A stable space-time FE method for the shallow water equations

We consider the finite element (FE) approximation of the shallow water equations (SWE) by considering discretizations in which both space and time are established using an unconditionally stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to break the test space and achieve unconditionally stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We derive a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptivity on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to commonly applied benchmarks for the SWE.

FTK: A Simplicial Spacetime Meshing Framework for Robust and Scalable Feature Tracking.

We present the Feature Tracking Kit (FTK), a framework that simplifies, scales, and delivers various feature-tracking algorithms for scientific data. The key of FTK is our simplicial spacetime meshing scheme that generalizes both regular and unstructured spatial meshes to spacetime while tessellating spacetime mesh elements into simplices. The benefits of using simplicial spacetime meshes include (1) reducing ambiguity cases for feature extraction and tracking, (2) simplifying the handling of degeneracies using symbolic perturbations, and (3) enabling scalable and parallel processing. The use of simplicial spacetime meshing simplifies and improves the implementation of several feature-tracking algorithms for critical points, quantum vortices, and isosurfaces. As a software framework, FTK provides end users with VTK/ParaView filters, Python bindings, a command line interface, and programming interfaces for feature-tracking applications. We demonstrate use cases as well as scalability studies through both synthetic data and scientific applications including tokamak, fluid dynamics, and superconductivity simulations. We also conduct end-to-end performance studies on the Summit supercomputer. FTK is open sourced under the MIT license: https://github.com/hguo/ftk.

Space–time least-squares finite elements for parabolic equations

We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space–time adaptivity. We also present a-priori error analysis on simplicial space–time meshes which are highly structured. Numerical results conclude this work.

Adaptive High-Order Fluid-Structure Interaction Simulations with Reduced Mesh-Motion Errors

This paper demonstrates an adaptive approach for solving fluid–structure interaction problems using high-fidelity numerical methods along with a detailed analysis of mesh-motion errors. A high-order partitioned approach is applied to couple the fluid and the structural subsystems, where the fluid subsystem is discretized using a discontinuous Galerkin finite-element method, while the structures solver uses a continuous Galerkin discretization. An explicit mapping is used as the primary mesh deformation algorithm. High-order time-integration schemes are used by the coupled solver to march forward in time, and the space-time mesh of the fluid subsystem is adapted using output-based methods. The error estimates for the unsteady outputs are evaluated by calculating the uncoupled, unsteady adjoint of the fluid subsystem. Adaptive meshing is used to demonstrate the importance of mesh-motion errors on output convergence, and a comprehensive analysis is conducted to control such errors arising from the mesh deformation algorithm. The benefits of adaptive meshing are demonstrated on a cantilevered Euler–Bernoulli beam placed in a low-Reynolds number flow and on a two-dimensional pitching–plunging airfoil in a high-Reynolds number flow.

Space-time mesh refinement method for simulating transient mixed flows

For design and management of large scale tunnels, it is important to model the transitions between pressurized and free-surface flows inside them. The pressurized and free-surface flow regimes are separated by a discontinuity with different wave speeds on two sides, which makes efficient and accurate simulations still challenging. A robust and conservative space-time adaptive mesh refinement (AMR) strategy is proposed to simulate such transient mixed flow. In the present AMR, the mesh resolution is refined adaptively near the interface between two flow regimes, and the flow states at child meshes are determined by a conservative approach which can avoid numerical oscillations. The numerical results from the AMR strategy are validated with analytical and experimental results, which indicates its good ability to capture the interface between two flow regimes and prevent numerical oscillations. The numerical test in a field scale tunnel shows that AMR can speed up simulations of transient mixed flow by 60 times.

Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the approximation order of the finite element residual. Numerical examples are provided that illustrate the methodology.

Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, which is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residual-type error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.

A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness and to obtain a priori error estimates, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. The analysis results are supported by numerical examples.

Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

Abstract We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.