Space-time Finite Element Research Articles

High-definition simulation of packed-bed liquid chromatography

Numerical simulations of chromatography are conventionally performed using reduced-order models that homogenize aspects of flow and transport in the radial and angular dimensions. This enables much faster simulations at the expense of lumping the effects of inhomogeneities into a column dispersion coefficient, which requires calibration via empirical correlations or experimental results. We present a high-definition model with spatially resolved geometry. A stabilized space–time finite element method is used to solve the model on massively parallel high-performance computers. We simulate packings with up to 10,000 particles. The impact of particle size distribution on velocity and concentration profiles as well as breakthrough curves is studied. Our high-definition simulations provide unique insight into the process. The high-definition data can also be used as a source of ground truth to identify and calibrate appropriate reduced-order models that can then be applied for process design and optimization.

Space–time computational flow analysis: Unconventional methods and first-ever solutions

The Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method was introduced in 1990 as a moving-mesh method for computational analysis of flows with moving boundaries and interfaces (MBI), which is a wide class of problems that includes fluid–particle and fluid–structure interactions and free-surface and multi-fluid flows. The method was inspired by Thomas Hughes’s 1987 work on space–time finite element methods for elastodynamics. The original DSD/SST method is now called “ST-SUPS”, reflecting its stabilization components, which are the Streamline-Upwind/Petrov–Galerkin (SUPG) method, pioneered by Hughes, and the Pressure-Stabilizing/Petrov–Galerkin (PSPG) method, inspired by Hughes’s work on a Stokes-flow Petrov–Galerkin formulation allowing equal-order interpolations for velocity and pressure. Hughes’s work on the residual-based variational multiscale (RBVMS) method inspired the ST-VMS method, which is the VMS version of the DSD/SST. A number of special methods were introduced in connection with the core methods ST-SUPS and ST-VMS. Hughes’s work on isogeometric analysis (IGA) inspired one of those special methods, the “ST-IGA”, with IGA basis functions not only in space but also in time. The core and special ST methods enabled first-ever solutions in some of the most challenging classes of MBI problems, including particle-laden flows, spacecraft parachute fluid–structure interactions, and car and tire aerodynamics. We provide an overview of the ST methods inspired by Hughes’s work and highlight some of the first-ever solutions in these three classes of MBI problems.

A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation

Abstract In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $\dot{W}^\varepsilon $ tending to rough as $\varepsilon \rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $\varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $\varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $\varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency

In this paper, a space-time absolute nodal coordinate formulation cable (SAC) element forming technique based on the Lagrange family of shape functions is proposed. Two distinct SAC elements, each with a distinct spatial shape function, have been generated by this method. Moreover, the external forces such as the bending moment and the air resistance formula have been accounted for. The Lagrange multiplier method, along with the concepts of replacement constraint and supplementary constraint, has been employed to provide a solution for the dynamics of constrained mechanical systems. Additionally, a constraint conversion strategy has been suggested. The solver has been constructed through Hamilton’s law of varying action. The space-time finite element method is used to solve dynamic problems, employing the Newton algorithm and quasi-Newton algorithm. The accuracy and efficiency of the solution has been verified by three simulations and one experiment. The circle-bending static simulation and the double-ended velocity impact dynamic simulation demonstrate the accuracy of the two elements. The correlation between statics and dynamics has been studied for different discretization methods and different solvers’ calculation accuracy and efficiency. Different modeling methods, time steps, order and the application of the quasi-Newton method all have a bearing on the efficiency of the solution. Finally, a comparison with an experiment in the free-pendulum simulation reveals the capability of this model to simulate dynamic problems with air resistance.

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

In this paper, we propose new geometrically unfitted space-time finite element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretization in time, we consider discontinuous Galerkin, as well as related continuous (Petrov–)Galerkin and Galerkin collocation methods. For stabilization with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilization is employed. The article puts an emphasis on the techniques that allow us to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

A Geometric Multigrid Method for Space-Time Finite Element Discretizations of the Navier–Stokes Equations and its Application to 3D Flow Simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier–Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

Space-Time Finite Element Method for Fully Intrinsic Equations of Geometrically Exact Beam

In this paper, a space-time finite element method based on a Galerkin-weighted residual method is proposed to solve the nonlinear fully intrinsic equations of geometrically exact beam which are first-order partial differential equations about time and space. Therefore, it is natural to discretize it in time and space simultaneously. Considering the continuity and intrinsic boundary conditions in the spatial direction and the continuity and periodic boundary conditions in the time direction, the boundary value scheme of space-time finite element for solving the full intrinsic equations is derived. This method has been successfully applied to the static analysis and dynamic response solution of the fully intrinsic equations of nonlinear geometrically exact beam. The numerical results of several examples are compared with the analytical solution, existing algorithms, and literature to illustrate the applicability, accuracy and efficiency of this method.

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.

A cut finite element method for two-phase flows with insoluble surfactants

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier-Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilizes the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.

A new mixed finite element formulation for reorientation in liquid crystalline elastomers

Liquid crystalline elastomers (LCE) belong to the class of soft materials and are capable of large deformations induced by temperature changes and ultraviolet irradiation. These materials are under investigation as actuator materials in light-weight structures. In order to numerically simulate LCE materials by using a space–time finite element method, a continuum model is necessary which combines a thermo-viscoelastic material formulation with a dissipative reorientation process of mesogens. Mesogens are rigid and rod-shaped molecules linked with the polymer chains of the elastomer. The dissipative reorientation of the mesogens can be described by a combination of an independent field of orientation vectors with unit length and micropolar rotational degrees of freedom. The orientation vector field follow from a partial differential equation and possesses initial–boundary conditions. The unit length of the orientation vector field for any time is guaranteed by a local rotation with the micropolar rotational degrees of freedom. The dissipative reorientation process is introduced by a local time evolution equation with respect to the micropolar rotational degrees of freedom. This is analogous to thermo-viscoelasticity and can be variationally-based formulated. In this paper, we present this new variational-based reorientation finite element formulation in a dynamical framework. We arrive at an energy–momentum consistent mixed finite element formulation with a Galerkin-based space–time integration.

Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme

In this paper, continuous space-time finite element methods are developed for approximating a class of quasilinear parabolic problems in space and in time, simultaneously. The basis of the whole approach is on a space-time variational formulation where streamline upwind terms and interface jump terms are further added for stabilizing the discretization in the direction of time. Error estimates are shown and are verified numerically through a series of numerical tests. Emphasis is placed on investigating the asymptotic convergence of the error parts, which are related to the time discretization.

Space-Time Petrov-Discontinuous Galerkin Finite Element Method for Solving Linear Convection-Diffusion Problems

The paper presents the theory of the space-time Petrov-discontinuous Galerkin finite element (PDGFE) method for the discretization of the nonstationary linear convection-diffusion problems. The PDGFE method is modified for the discontinuous Galerkin finite element (DGFE) method in the case of the symmetric interior penalty Galerkin (SIPG) scheme. PDGFE method is applied separately in space using different space gride on different time levels. We prove the properties of the bilinear form aPD, m (u, ν) (V − elliptic and continuity) stability and prove the approximate solution converges with the error of order o (h 2 + τ 3). A numerical experiment is carried out to confirm the theoretical conclusions.

NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems

In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

We present a novel class of high-order space–time finite element schemes for the Poisson–Nernst–Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties.This is accomplished via (1) using finite elements to directly approximate the so-called entropy variableui≔U′(ci)=log(ci) instead of the density variable ci, where U(ci)=(log(ci)−1)ci is the corresponding entropy, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. (2016) under the name of a log-density formulation, guarantees both positivity of densities ci=exp(ui)>0 and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. (2016).

Enabling four-dimensional conformal hybrid meshing with cubic pyramids

The main purpose of this article is to develop a novel refinement strategy for four-dimensional hybrid meshes based on cubic pyramids. This optimal refinement strategy subdivides a given cubic pyramid into a conforming set of congruent cubic pyramids and invariant bipentatopes. The theoretical properties of the refinement strategy are rigorously analyzed and evaluated. In addition, a new class of fully symmetric quadrature rules with positive weights are generated for the cubic pyramid. These rules are capable of exactly integrating polynomials with degrees up to 12. Their effectiveness is successfully demonstrated on polynomial and transcendental functions. Broadly speaking, the refinement strategy and quadrature rules in this paper open new avenues for four-dimensional hybrid meshing, and space-time finite element methods.

Spline-based space-time finite element approach for fluid-structure interaction problems with a focus on fully enclosed domains

Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community.With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI).This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS.The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure.The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.

Effective Viscoplastic-Softening Model Suitable for Brain Impact Modelling

In this paper, we address the numerical aspects and implementation of a nonlinear viscoplastic model of the mechanical behaviour of brain tissue to simulate the dynamic responses related to impact loads which may cause traumatic injury. Among the various viscoelastic models available, we deliberately considered modifying the Norton–Hoff model in order to introduce non-typical viscoplastic softening behaviour that imitates a brain’s response just several milliseconds after a rapid impact. We describe the discretisation and three dimensional implementation of the model, with the aim of obtaining accurate numerical results in a reasonable computational time. Due to the large scale and complexity of the problem, a parallel computation technique, using a space–time finite element method, was used to facilitate the computation boost. It is proven that, after calibrating, the introduced viscoplastic-softening model is better suited for modelling brain tissue behaviour for the specific case of rapid impact loading rather than the commonly used viscoelastic models.